**Meet the editor**

Professor Shuanggen Jin is a Professor at Shanghai Astronomical Observatory of Chinese Academy of Sciences. He received his BSc degree in Geomatics from Wuhan University in 1999 and earned his PhD in Geodesy from the Chinese Academy of Sciences in 2003. His main interests include satellite navigation and positioning, remote sensing and climate change, and aspects of space

and planetary sensing dynamics. He has authored over 80 peer-reviewed journal papers and more than 10 books and chapters. Since 2011 he has been President of IAG Sub-Commission 2.6 , he has been Editor-in-Chief of International Journal of Geosciences since 2010, and is the editor of several international journals. He received the Special Prize of Korea Astronomy and Space Science Institute in 2006, took part in the Chinese Academy of Sciences 100-Talent Program in 2010, was a Fellow of the International Association of Geodesy (IAG) in 2011, and in that year was also involved in the Shanghai Pujiang Talent Program.

Contents

**Preface IX** 

**Part 1 GNSS Signals and System 1** 

Chapter 2 **Baseband Hardware Designs** 

Chapter 4 **Evolution of Integrity Concept** 

Zheng Yao

Fabio Dovis and Tung Hai Ta

Chapter 3 **Unambiguous Processing Techniques of** 

**in Modernised GNSS Receivers 33** 

**– From Galileo to Multisystem 77**

**Part 2 GNSS Navigation and Applications 105** 

Chapter 5 **Estimation of Satellite-User Ranges** 

Bihter Erol and Serdar Erol

Chapter 8 **Achievable Positioning Accuracies in** 

Ahmed El-Mowafy

Chapter 1 **High Sensitivity Techniques for GNSS Signal Acquisition 3** 

Nagaraj C. Shivaramaiah and Andrew G. Dempster

**Binary Offset Carrier Modulated Signals 53**

Mario Calamia, Giovanni Dore and Alessandro Mori

**Through GNSS Code Phase Measurements 107**  Marco Pini, Gianluca Falco and Letizia Lo Presti

Chapter 6 **GNSS in Practical Determination of Regional Heights 127** 

Chapter 7 **Precise Real-Time Positioning Using Network RTK 161** 

**a Network of GNSS Reference Stations 189** 

Paolo Dabove, Mattia De Agostino and Ambrogio Manzino

## Contents

#### **Preface** XI

	- **Part 2 GNSS Navigation and Applications 105**

X Contents


	- **Part 3 GNSS Errors Mitigation and Modelling 357**

## Preface

Global Positioning System (GPS) has been widely used in navigation, positioning, timing, and scientific questions related to precise positioning on Earth's surface as a highly precise, continuous, all-weather and real-time technique, since GPS became fully operational in 1993. In addition, when the GPS signal propagates through the Earth's atmosphere and ionosphere, it is delayed by the atmospheric refractivity. Nowadays, the atmospheric and ionospheric delays can be retrieved from GPS observations, which have facilitated greater advancements in meteorology, climatology, numerical weather models, atmospheric science, and space weather. Furthermore, GPS multipath as one of the main error sources has been recently recognized that GPS reflectometry (GPS-R) from the Earth's surface could be used to sense the Earth's surface environments. Together, with the US's modernized GPS-IIF and planned GPS-III, Russia's restored GLONASS, the coming European Union's GALILEO system, and China's Beidou/COMPASS system, as well as a number of Space Based Augmentation Systems (SBAS), such as Japan's Quasi-Zenith Satellite System (QZSS) and India's Regional Navigation Satellite Systems (IRNSS), more potentials for the next generation multi-frequency and multi-system global navigation satellite systems (GNSS) will be realized. Therefore, it is valuable to provide detailed information on GNSS techniques and applications for readers and users.

This book is devoted to presenting recent results and development in GNSS theory, system, signals, receiver, and applications with a number of chapters. First, the basic framework of GNSS system and signals processing are introduced and illustrated. The core correlator architecture of the next generation GNSS receiver baseband hardware is presented and power consumption estimates are analyzed for the new signals at the core correlator level and at the channel level, respectively. Because the performance of the traditional GNSS is constrained by its inherent capability, an innovative design methodology for future unambiguous processing techniques of Binary offset carrier (BOC) modulated signals is proposed. Some practical design examples with this methodology are tested to show the practicality and to provide reference for further algorithm development. More and more future GNSS systems and the integrity of multi-GNSS system, including GPS, Galileo, GLONASS, and Beidou are very important for future high precision navigation and positioning. Here, the integrity concepts are proposed for the different constellations (GPS/EGNOS and Galileo) and some performances are evaluated.

#### XII Preface

Second, high precise GNSS navigation and positioning are subject to a number of errors sources, such as multipath and atmospheric delays. The challenges and mitigation of GNSS multipath effects are discussed and evaluated. In general, the better multipath mitigation performance can be achieved in moderate-to-high C/N0 scenarios (for example, 30 dB-Hz and onwards). Due to complicated situations and varied environments of GNSS observations, the multipath mitigation remains a challenging topic for future research with the multitude of signal modulations, spreading codes, spectrum placements, and so on. Concerning the atmospheric and ionospheric delays, it is normally mitigated using models or dual-frequency GNSS measurements, including higher order ionospheric propagation effects. In contrast, the delays and corresponding products can be retrieved from ground-based and space borne GNSS radio occultation observations, including high-resolution tropospheric water vapor, temperature and pressure, tropopause parameters, and ionospheric total electron content (TEC) as well, which have been used in meteorology, climatology, atmospheric science, and space weather.

Third, the wide GNSS applications in navigation, positioning, topography, height system, wheeled robots status, and engineering surveying are introduced and demonstrated, including hybrid GNSS positioning, multi-sensor integration, indoor positioning, Network Real Time Kinematic (NRTK), regional height determination, etc. For example, the precise outdoor 3-D localization solution for mobile robots can be determined using a loosely-coupled kalman filter (KF) with a low-cost inertial measurement unit (IMU) and micro electro-mechanical system (MEMS)-based sensors, wheel encoders and GNSS. Also, GNSS can precisely monitor the vibration and characterize the dynamic behavior of large road structures, particularly the bridges. These results are comparable with the displacement transducer and vibration test on a wooden cable-stayed footbridge. In addition, Network RTK methods are presented, as well as their applications, including in engineering surveying, machine automation, and in the airborne mapping and navigation.

This book provides the basic theory, methods, models, applications, and challenges of GNSS navigation and positioning for users and researchers who have GNSS background and experience. Furthermore, it is also useful for the increasing number of the next generation multi-GNSS designers, engineers, and users community. We would like to gratefully thank InTech Publisher, Rijeka, Croatia, for their processes and cordial cooperation with publishing this book.

> **Prof. Shuanggen Jin** Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai, China

**Part 1** 

**GNSS Signals and System** 

**0**

**1**

<sup>1</sup>*Italy* <sup>2</sup>*Vietnam*

**High Sensitivity Techniques**

**for GNSS Signal Acquisition**

The requirements of location based and emergency caller localization services spurred by the E-911 mandate (USA) and the E-112 initiative (EU) have generated the demand for the availability of Global Navigation Satellite Systems (GNSS) in harsh environments like indoors, urban canyons or forests where low power signals dominate. This fact has pushed the

To produce positioning and timing information, a conventional GNSS receiver must go through three main stages: code synchronization; navigation data demodulation; and Position, Velocity and Time (PVT) computation. Code synchronization is in charge of determining the satellites in view, estimating the transmission code epoch and Doppler shift. This stage is usually divided into code acquisition and tracking. The former reduces the code epoch and Doppler shift uncertainties to limited intervals while the latter performs continuous fine delay estimation. In particular, code acquisition can be very critical because it is the first operation performed by the receiver. This is the reason for lots of endeavors having been invested to improve the robustness of the acquisition process toward the HS objective.

Basically, the extension of the coherent integration time is the optimal strategy for improving the acquisition sensitivity in a processing gain sense. However, there are several limitations to the extension of the coherent integration time *Tint*. The presence of data-bit transitions, as the 50bps in the present GPS Coarse-Acquisition (C/A) service, modulating the ranging code is the most impacting. In fact, each transition introduces a sign reversal in successive correlation blocks, such that their coherent accumulation leads to the potential loss of the correlation peak. Therefore, the availability of an external-aiding source is crucial to extend *Tint* to be larger than the data bit duration *Tb* (e.g. for GPS L1 C/A, *Tb* = 20 ms). This approach is referred as the aided (or assisted) signal acquisition, and it is a part of the Assisted GNSS (A-GNSS) positioning method defined by different standardization bodies (3GPP, 2008a;b; OMA, 2007). However, without any external-aiding source, the acquisition stage can use the techniques so-called post-correlation combination to improve its sensitivity. In general, there are 3 post-correlation combination techniques, namely: coherent, non-coherent and differential

**1. Introduction**

development of High Sensitivity (HS) receivers

Fabio Dovis1 and Tung Hai Ta2

<sup>2</sup>*Hanoi University of Science and Technology*

<sup>1</sup>*Politecnico di Torino*

## **High Sensitivity Techniques for GNSS Signal Acquisition**

Fabio Dovis1 and Tung Hai Ta2 <sup>1</sup>*Politecnico di Torino* <sup>2</sup>*Hanoi University of Science and Technology* <sup>1</sup>*Italy* <sup>2</sup>*Vietnam*

## **1. Introduction**

The requirements of location based and emergency caller localization services spurred by the E-911 mandate (USA) and the E-112 initiative (EU) have generated the demand for the availability of Global Navigation Satellite Systems (GNSS) in harsh environments like indoors, urban canyons or forests where low power signals dominate. This fact has pushed the development of High Sensitivity (HS) receivers

To produce positioning and timing information, a conventional GNSS receiver must go through three main stages: code synchronization; navigation data demodulation; and Position, Velocity and Time (PVT) computation. Code synchronization is in charge of determining the satellites in view, estimating the transmission code epoch and Doppler shift. This stage is usually divided into code acquisition and tracking. The former reduces the code epoch and Doppler shift uncertainties to limited intervals while the latter performs continuous fine delay estimation. In particular, code acquisition can be very critical because it is the first operation performed by the receiver. This is the reason for lots of endeavors having been invested to improve the robustness of the acquisition process toward the HS objective.

Basically, the extension of the coherent integration time is the optimal strategy for improving the acquisition sensitivity in a processing gain sense. However, there are several limitations to the extension of the coherent integration time *Tint*. The presence of data-bit transitions, as the 50bps in the present GPS Coarse-Acquisition (C/A) service, modulating the ranging code is the most impacting. In fact, each transition introduces a sign reversal in successive correlation blocks, such that their coherent accumulation leads to the potential loss of the correlation peak. Therefore, the availability of an external-aiding source is crucial to extend *Tint* to be larger than the data bit duration *Tb* (e.g. for GPS L1 C/A, *Tb* = 20 ms). This approach is referred as the aided (or assisted) signal acquisition, and it is a part of the Assisted GNSS (A-GNSS) positioning method defined by different standardization bodies (3GPP, 2008a;b; OMA, 2007).

However, without any external-aiding source, the acquisition stage can use the techniques so-called post-correlation combination to improve its sensitivity. In general, there are 3 post-correlation combination techniques, namely: coherent, non-coherent and differential

for GNSS Signal Acquisition 3

High Sensitivity Techniques for GNSS Signal Acquisition 5

where Π is the rectangular function; *qk* is the PRN code. Because of the properties of the PRN code, *qk* is a periodic sequence with the period *N* chips, *qk* can be rewriten as *qk* = *qmod*(*k*,*N*),

being *Tc*, and *fc* = 1/*Tc* the chip duration (s) and chipping rate (chip per second - cps)

GNSS *fs* is a multiple of *fc* (i.e. *a*/2 is an integer value) and both the values of *fc* and *fs* are normalized by 1.023 MHz; for instance BPSK(5) and BOC(10,5) mean *fc* = 5 × 1.023 MHz and *fs* = 10 × 1.023 MHz. The subcarrier *s*[*n*] can be sine-phased, *s*[*n*] = sgn[sin(2*π fsnTS*)]; or

As introduced in (Kaplan, 2005), the conventional acquisition process (see Fig. 1) strives to determine the presence of a desired signal defined by PRN code (*c*), code delay (*τ*) and Doppler offset (*fD*) in the incoming signal. The uncertainty regions of (*c*, *τ*, *fD*) form a signal

signal, see Fig. 2(a). The acquisition process correlates the incoming signal (*r*[*n*]) with the

*c n* ˆ ˆ [ ] +τ

It is well known that there are several general approaches to code acquisition of a GNSS signals. The basic functional operation is a correlation between a local replica of the code and the incoming signal as depicted in Fig. 1, where a serial approach scheme is reported. Time

1

⋅

1 *<sup>L</sup> L <sup>n</sup>*<sup>=</sup>

*qk*Π(*t* − *kTc*) (2)

*qmod*(*k*,*N*)Π (*nTS* − *kTc*) (3)

*fc*

. Usually in

*qmod*(*k*,*N*)*smod*(*k*,*a*/2)Π (*n* − *kTc*) (4)

*fD*) of which is used to locally generate an equivalent tentative

*<sup>m</sup> Rm S*

+∞ ∑ *k*=−∞

+∞ ∑ *k*=−∞

*c*(*t*) =

*c*[*n*] = *c*(*nTS*) =

∞ ∑ *k*=−∞

tentative signal (*r*ˆ[*n*]) to measure the similarity between the two signals.

*r n***[ ]** ( )

{ } 2 ( ) *IF D S j f f nT*

Fig. 1. Conventional signal acquisition architecture

π+ **ˆ exp**

with *smod*(*k*,*a*/2) ∈ {−1, 1}is the sub-carrier with the frequency *fs* and *<sup>a</sup>* <sup>=</sup> <sup>2</sup> *fs*

cosine-phased, *s*[*n*] = sgn[cos(2*π fsnTS*)] with sgn(*x*) being the signum function of *x*.

*c*[*n*] =


respectively.


**2.2 Conventional acquisition process**

search-space, each cell (*c*ˆ, *τ*ˆ, ˆ

then the digital version of (2) is

combination. In fact, the coherent combination technique is equivalent to the *Tint* extension with the advantage that in this stand-alone scenario *Tint* ≤ *Tb*. The squaring loss (Choi et al., 2002) caused by the non-coherent combination makes this technique less competitive than the others. However, its simplicity and moderate complexity make it suitable for conventional GNSS receivers. Among the three techniques, the differential combination can be considered as a solution trading-off sensitivity and complexity of an acquisition stage (Schmid & Neubauer, 2004; Zarrabizadeh & Sousa, 1997). As an expanded view of the conventional differential combination technique, generalized differential combination is introduced for further sensitivity improvement (Corazza & Pedone, 2007; Shanmugam et al., 2007; Ta et al., 2012).

In addition, modern GNSSes broadcast new civil signals on different frequency bands. Moreover, these new signals are composed of two channels, namely data and pilot (data-less) channels (e.g. Galileo E1 OS, E5, E6; GPS L5, L2C, L1C). These facts yield another approach, usually named *channel combining acquisition* (Gernot et al., 2008; Mattos, 2005; Ta et al., 2010) able to fully exploit the potential of modern navigation signals for sake of sensitivity improvement.

This book chapter strives to identify the issues related to HS signal acquisition and also to introduce in details possible approaches to solve such problems. The remainder of the chapter is organized as follows. Section 2 presents fundamentals of signal acquisition including the common representation of the received signal, the conventional acquisition process. Furthermore, definition of the the performance parameters, in terms of detection probabilities and mean acquisition time are provided. HS acquisition issues and general solutions, namely stand-alone, external-aiding and channel combining approaches, are introduced in Section 3. In Section 4, the stand-alone generalized differential combination technique is presented together with its application to GPS L2C signal in order to show the advantages of such a technique. Section 5 focuses on introducing a test-bed architecture as an example of the external-aiding signal acquisition. The channel combining approach via joint data/pilot signal acquisition strategies for Galileo E1 OS signal is introduced in Section 6. Eventually, some concluding remarks are drawn.

#### **2. Fundamentals of signal acquisition**

#### **2.1 Received signal representation**

The received signal after the Analog to Digital Converter in a Direct Sequence Code Division Multiple Access (DS-CDMA) GNSS system can be represented as

$$r[n] = \sqrt{2C}d[n]c[n+\tau]\cos(2\pi(f\_{IF} + f\_D)nT\_S + \varphi) + n\_W[n] \tag{1}$$

where *C* is the carrier power (W); *d*[*n*] is the navigation data; *c*[*n*] is the spreading code, *fIF*, *fD* denote the Intermediate Frequency (IF) and Doppler shift (Hz) respectively; *TS* = 1/*FS* stands for the sampling period (s) (*FS* is the sampling frequency (Hz)); *ϕ* is the initial carrier phase (rad); *τ* is the initial code delay (samples) ; and *nW* is the Additive White Gaussian Noise (AWGN) with zero mean (*μ* = 0) and variance *σ*<sup>2</sup> *<sup>n</sup>* (*nW* ∼ N (0, *<sup>σ</sup>*<sup>2</sup> *<sup>n</sup>*)).

In fact, most of the current and foreseen signals of GNSSes use either BPSK or BOC modulations (Ta, 2010). For these modulations, *c*[*n*] has the representation as follows:


2 Will-be-set-by-IN-TECH

combination. In fact, the coherent combination technique is equivalent to the *Tint* extension with the advantage that in this stand-alone scenario *Tint* ≤ *Tb*. The squaring loss (Choi et al., 2002) caused by the non-coherent combination makes this technique less competitive than the others. However, its simplicity and moderate complexity make it suitable for conventional GNSS receivers. Among the three techniques, the differential combination can be considered as a solution trading-off sensitivity and complexity of an acquisition stage (Schmid & Neubauer, 2004; Zarrabizadeh & Sousa, 1997). As an expanded view of the conventional differential combination technique, generalized differential combination is introduced for further sensitivity improvement (Corazza & Pedone, 2007; Shanmugam et al.,

In addition, modern GNSSes broadcast new civil signals on different frequency bands. Moreover, these new signals are composed of two channels, namely data and pilot (data-less) channels (e.g. Galileo E1 OS, E5, E6; GPS L5, L2C, L1C). These facts yield another approach, usually named *channel combining acquisition* (Gernot et al., 2008; Mattos, 2005; Ta et al., 2010) able to fully exploit the potential of modern navigation signals for sake of sensitivity

This book chapter strives to identify the issues related to HS signal acquisition and also to introduce in details possible approaches to solve such problems. The remainder of the chapter is organized as follows. Section 2 presents fundamentals of signal acquisition including the common representation of the received signal, the conventional acquisition process. Furthermore, definition of the the performance parameters, in terms of detection probabilities and mean acquisition time are provided. HS acquisition issues and general solutions, namely stand-alone, external-aiding and channel combining approaches, are introduced in Section 3. In Section 4, the stand-alone generalized differential combination technique is presented together with its application to GPS L2C signal in order to show the advantages of such a technique. Section 5 focuses on introducing a test-bed architecture as an example of the external-aiding signal acquisition. The channel combining approach via joint data/pilot signal acquisition strategies for Galileo E1 OS signal is introduced in Section 6. Eventually, some

The received signal after the Analog to Digital Converter in a Direct Sequence Code Division

where *C* is the carrier power (W); *d*[*n*] is the navigation data; *c*[*n*] is the spreading code, *fIF*, *fD* denote the Intermediate Frequency (IF) and Doppler shift (Hz) respectively; *TS* = 1/*FS* stands for the sampling period (s) (*FS* is the sampling frequency (Hz)); *ϕ* is the initial carrier phase (rad); *τ* is the initial code delay (samples) ; and *nW* is the Additive White Gaussian Noise

In fact, most of the current and foreseen signals of GNSSes use either BPSK or BOC

modulations (Ta, 2010). For these modulations, *c*[*n*] has the representation as follows:

2*Cd*[*n*]*c*[*n* + *τ*] cos(2*π*(*fIF* + *fD*)*nTS* + *ϕ*) + *nW*[*n*] (1)

*<sup>n</sup>* (*nW* ∼ N (0, *<sup>σ</sup>*<sup>2</sup>

*<sup>n</sup>*)).

2007; Ta et al., 2012).

improvement.

concluding remarks are drawn.

**2.1 Received signal representation**

**2. Fundamentals of signal acquisition**

*<sup>r</sup>*[*n*] = <sup>√</sup>

(AWGN) with zero mean (*μ* = 0) and variance *σ*<sup>2</sup>

Multiple Access (DS-CDMA) GNSS system can be represented as

$$\mathcal{L}(t) = \sum\_{k=-\infty}^{+\infty} q\_k \Pi(t - kT\_c) \tag{2}$$

where Π is the rectangular function; *qk* is the PRN code. Because of the properties of the PRN code, *qk* is a periodic sequence with the period *N* chips, *qk* can be rewriten as *qk* = *qmod*(*k*,*N*), then the digital version of (2) is

$$\mathfrak{c}[n] = \mathfrak{c}(nT\_{\mathbb{S}}) = \sum\_{k=-\infty}^{+\infty} q\_{mod(k,N)} \Pi \left( nT\_{\mathbb{S}} - kT\_{\mathbb{C}} \right) \tag{3}$$

being *Tc*, and *fc* = 1/*Tc* the chip duration (s) and chipping rate (chip per second - cps) respectively.


$$\mathcal{L}[n] = \sum\_{k=-\infty}^{\infty} q\_{mod(k,N)} s\_{mod(k,a/2)} \Pi \left( n - kT\_c \right) \tag{4}$$

with *smod*(*k*,*a*/2) ∈ {−1, 1}is the sub-carrier with the frequency *fs* and *<sup>a</sup>* <sup>=</sup> <sup>2</sup> *fs fc* . Usually in GNSS *fs* is a multiple of *fc* (i.e. *a*/2 is an integer value) and both the values of *fc* and *fs* are normalized by 1.023 MHz; for instance BPSK(5) and BOC(10,5) mean *fc* = 5 × 1.023 MHz and *fs* = 10 × 1.023 MHz. The subcarrier *s*[*n*] can be sine-phased, *s*[*n*] = sgn[sin(2*π fsnTS*)]; or cosine-phased, *s*[*n*] = sgn[cos(2*π fsnTS*)] with sgn(*x*) being the signum function of *x*.

#### **2.2 Conventional acquisition process**

As introduced in (Kaplan, 2005), the conventional acquisition process (see Fig. 1) strives to determine the presence of a desired signal defined by PRN code (*c*), code delay (*τ*) and Doppler offset (*fD*) in the incoming signal. The uncertainty regions of (*c*, *τ*, *fD*) form a signal search-space, each cell (*c*ˆ, *τ*ˆ, ˆ *fD*) of which is used to locally generate an equivalent tentative signal, see Fig. 2(a). The acquisition process correlates the incoming signal (*r*[*n*]) with the tentative signal (*r*ˆ[*n*]) to measure the similarity between the two signals.

Fig. 1. Conventional signal acquisition architecture

It is well known that there are several general approaches to code acquisition of a GNSS signals. The basic functional operation is a correlation between a local replica of the code and the incoming signal as depicted in Fig. 1, where a serial approach scheme is reported. Time

for GNSS Signal Acquisition 5

High Sensitivity Techniques for GNSS Signal Acquisition 7


(a)

 -

 -

chip (Wilde et al., 2006). As for Doppler shift dimension, � *fD* <sup>=</sup> <sup>2</sup>

0.5 chip

Δ*f d*

**-** 

τˆ, ˆ *f* ( ) *<sup>D</sup>*

<sup>Δ</sup>*fD* <sup>=</sup> <sup>1</sup> 2*T*int

(b)

!--

**Estimated Code Delay**

(c) Sinc function

or � *fD* <sup>=</sup> <sup>1</sup>

complexity and sensitivity.

the magnitude *Sm* = |*Rm*|

**2.3 Acquisition performance parameters**

  -


 

<sup>2</sup>*Tint* as in (Misra & Enge, 2006) are often chosen concerning the trade-off between

<sup>2</sup> of each complex correlator output can be modeled as a random

 -"

(c)

<sup>3</sup>*Tint* as in (Kaplan, 2005)

**Estimated Doppler Shift**

 

 

#-

Fig. 2. (a) Acquisition search-space; (b) Auto-correlation functions of BPSK(1) and BOC(1,1);

signal �*τBPSK* = 0.5 chip. However, for BOC signal, due to the appearance of side-peaks, �*τ* is chosen so that the tracking stage can avoid to lock to the side-peaks. For BOC(1,1), in order to achieve the same average correlation loss as for a BPSK signal, �*τBOC*(1,1) = 0.16

When dealing with real signals, the incoming code is affected by several factors such as propagation distortion and noise, thus resulting in a distorted correlation function. In order to achieve an optimal detection process, the Neyman-Pearson likelihood criterion is used. In fact,

variable with statistical features. Thus, *Sm* is compared with a predetermined threshold (*V*) in order to decide which hypothesis between *H*<sup>0</sup> (*Sm* < *V*) and *H*<sup>1</sup> (*Sm* > *V*) is true, where *H*<sup>0</sup> and *H*<sup>1</sup> respectively represent the absence or presence of the desired peak. Once the decision

#

τ, *f* ( ) *<sup>D</sup>*

> Δτ

 ! - 

(or frequency) parallel acquisition approaches, are often efficiently implemented by using Fast Fourier Transform algorithms (Tsui, 2005).

In general, the complex-valued correlation *R*, which is also referred as Cross Ambiguity Function (CAF), between the incoming and the local generated signals is:

$$\mathcal{R}\_m = \frac{1}{L} \sum\_{n=(m-1)L}^{mL} \{r[n]\hat{c}[n+\hat{\tau}]e^{j(2\pi(f\_{IF}+\hat{f}\_{D\_m}))nT\_S} \}\tag{5}$$
 
$$\stackrel{\Delta}{=} s\_m + w\_m$$

where *m* stands for the index of the coherent integration interval [(*m* − 1)*L*, *mL*], �*L* = *TintFs*� denotes the coherent integration time *Tint* (s) in samples; *sm*, *wm* are the signal and the noise components respectively, and (Holmes, 2007)

$$\begin{cases} s\_{\mathfrak{m}} = \sqrt{2\mathsf{C}}\mathscr{R}[\boldsymbol{\theta}] \text{sinc}(\triangle \overline{f}\_{d\_{\mathfrak{m}}} T\_{\text{int}}) e^{\mathrm{i}(\pi \triangle \overline{f}\_{d\_{\mathfrak{k}}} T\_{\text{int}} + \phi\_{\mathfrak{m}})} \triangleq \mathsf{G}\_{\mathfrak{m}} e^{\mathrm{i}\mathfrak{P}\_{\mathfrak{m}}} \\\ w\_{\mathfrak{m}} = \frac{1}{L} \sum\_{n=(m-1)L}^{mL} n\_{\mathfrak{M}}[n] \widehat{c}[n+\hat{\tau}] e^{\mathrm{i}\left[2\pi(f\_{\text{IF}} + \hat{f}\_{\text{D}\_{\mathfrak{m}}})nT\_{\text{S}}\right]} \end{cases} \tag{6}$$

where *<sup>θ</sup>* <sup>=</sup> *<sup>τ</sup>* <sup>−</sup> *<sup>τ</sup>*� is the difference between actual and estimated code delays and � *<sup>f</sup> dm* <sup>=</sup> *fD* − *f* � *Dm* is the difference between Doppler shifts during the interval *m*, as depicted in Fig. 2(a). (*φ<sup>m</sup>* <sup>=</sup> <sup>2</sup>*π*� *<sup>f</sup> dm*−<sup>1</sup> *Tint* + *<sup>φ</sup>m*−1) is the phase mismatch at the end of the *<sup>m</sup>*-th interval, and R[*θ*] is the cross-correlation function between the incoming signal and the local PRN codes. In an ideal, noiseless case, such cross-corelation would results to be the autocorrelation function of the two PRNs that can be written for a BPSK signal as

$$\mathcal{R}[\theta] = -\frac{1}{L} + \frac{L+1}{L} \Lambda\_0 \left(\frac{\theta}{\lambda}\right) \otimes \sum\_{m=-\infty}^{\infty} \delta[\theta + mL] \tag{7}$$

and for a BOC signal as (Betz, 2001):

$$\mathcal{R}[\theta] = \left[ \Lambda\_0 \left( \frac{\theta}{\frac{\lambda}{a}} \right) + \sum\_{l=1}^{a-1} (-1)^{|l|} \frac{a-|l|}{a} \Lambda\_{\frac{\lambda}{a}} \left( \frac{\theta}{\frac{\lambda}{2}} \right) + \sum\_{l=-(a-1)}^{-1} (-1)^{|l|} \frac{a-|l|}{a} \Lambda\_{\frac{\lambda}{a}} \left( \frac{\theta}{\frac{\lambda}{2}} \right) \right] \tag{8}$$
 
$$\begin{aligned} \circledast &\sum\_{m=-\infty}^{\infty} \delta[\theta + mL] \end{aligned} \tag{9}$$

where *λ* is the samples per chip, and Λ is the triangle function of *x*, centered at *z*, with a base width of *y*

$$\Lambda\_z \left( \frac{\mathbf{x}}{y} \right) = \begin{cases} \left( 1 - \frac{|\mathbf{x}|}{y} \right) z \le |\mathbf{x}| \le z + y - 1 \\\ 0 & \text{elsewhere} \end{cases} \tag{9}$$

From (7) and (8), it can be noted that, when observed over the interval [−*Tc*, *Tc*] around the main peak, the autocorrelation function of BPSK signal has the main peak only, whilst the BOC has (2*a* − 1) peaks. Fig. 2(b) shows the theoretical autocorrelation functions of a BPSK(1) and a BOC(1,1). As seen from Fig. 2(b) and Fig. 2(c), the estimation residuals (*θ*, � *fd*) cause correlation loss on both dimensions. To limit this loss, the cell size (�*τ*, � *fD*) must be chosen carefully taking into account also the pull-in range of the tracking stage. In general, for BPSK 4 Will-be-set-by-IN-TECH

(or frequency) parallel acquisition approaches, are often efficiently implemented by using Fast

In general, the complex-valued correlation *R*, which is also referred as Cross Ambiguity

where *m* stands for the index of the coherent integration interval [(*m* − 1)*L*, *mL*], �*L* = *TintFs*� denotes the coherent integration time *Tint* (s) in samples; *sm*, *wm* are the signal and the noise

*<sup>n</sup>*=(*m*−1)*<sup>L</sup> nW*[*n*]*c*�[*<sup>n</sup>* <sup>+</sup> *<sup>τ</sup>*ˆ]*ej*[2*π*(*fIF*<sup>+</sup> <sup>ˆ</sup>

where *<sup>θ</sup>* <sup>=</sup> *<sup>τ</sup>* <sup>−</sup> *<sup>τ</sup>*� is the difference between actual and estimated code delays and � *<sup>f</sup> dm* <sup>=</sup>

R[*θ*] is the cross-correlation function between the incoming signal and the local PRN codes. In an ideal, noiseless case, such cross-corelation would results to be the autocorrelation function

> � *θ λ* � ⊗

� *θ λ* 2

where *λ* is the samples per chip, and Λ is the triangle function of *x*, centered at *z*, with a base

From (7) and (8), it can be noted that, when observed over the interval [−*Tc*, *Tc*] around the main peak, the autocorrelation function of BPSK signal has the main peak only, whilst the BOC has (2*a* − 1) peaks. Fig. 2(b) shows the theoretical autocorrelation functions of a BPSK(1) and a BOC(1,1). As seen from Fig. 2(b) and Fig. 2(c), the estimation residuals (*θ*, � *fd*) cause correlation loss on both dimensions. To limit this loss, the cell size (�*τ*, � *fD*) must be chosen carefully taking into account also the pull-in range of the tracking stage. In general, for BPSK

<sup>1</sup> <sup>−</sup> <sup>|</sup>*x*<sup>|</sup> *y* � � +

*Dm* is the difference between Doppler shifts during the interval *m*, as depicted in Fig.

{*r*[*n*]*c*�[*<sup>n</sup>* <sup>+</sup> *<sup>τ</sup>*�]*ej*(2*π*(*fIF*+*<sup>f</sup>*

*<sup>j</sup>*(*π*� *<sup>f</sup> dk*

*Tint* + *<sup>φ</sup>m*−1) is the phase mismatch at the end of the *<sup>m</sup>*-th interval, and

∞ ∑ *m*=−∞

> −1 ∑ *l*=−(*a*−1)

*δ*[*θ* + *mL*] (8)

*z* ≤ |*x*| ≤ *z* + *y* − 1

�

*Tint*+*φm*) *Gmej*Φ*<sup>m</sup>*

*Dm* ))*nTS* } (5)

*fDm* )*nTS*] (6)

*δ*[*θ* + *mL*] (7)

� *θ λ* 2 �⎤ ⎦

(−1)|*l*<sup>|</sup> *<sup>a</sup>* − |*l*<sup>|</sup> *a* Λ *l λ a*

0 elsewhere (9)

Function (CAF), between the incoming and the local generated signals is:

*mL* ∑ *n*=(*m*−1)*L*

*sm* <sup>=</sup> <sup>√</sup>2*C*R[*θ*]sinc(� *<sup>f</sup> dm Tint*)*<sup>e</sup>*

*sm* + *wm*

Fourier Transform algorithms (Tsui, 2005).

*Rm* <sup>=</sup> <sup>1</sup> *L*

components respectively, and (Holmes, 2007)

*wm* <sup>=</sup> <sup>1</sup>

*<sup>L</sup>* <sup>∑</sup>*mL*

of the two PRNs that can be written for a BPSK signal as

<sup>R</sup>[*θ*] = <sup>−</sup> <sup>1</sup>

Λ*<sup>z</sup>* � *x y* � = � �

*L* +

(−1)|*l*<sup>|</sup> *<sup>a</sup>* − |*l*<sup>|</sup> *a* Λ *l λ a*

*L* + 1 *<sup>L</sup>* <sup>Λ</sup><sup>0</sup>

⎧ ⎨ ⎩

and for a BOC signal as (Betz, 2001):

∞ ∑ *m*=−∞ � + *a*−1 ∑ *l*=1

*fD* − *f* �

2(a). (*φ<sup>m</sup>* <sup>=</sup> <sup>2</sup>*π*� *<sup>f</sup> dm*−<sup>1</sup>

R[*θ*] =

width of *y*

⎡ ⎣Λ<sup>0</sup> � *θ λ a*

⊗

Fig. 2. (a) Acquisition search-space; (b) Auto-correlation functions of BPSK(1) and BOC(1,1); (c) Sinc function

signal �*τBPSK* = 0.5 chip. However, for BOC signal, due to the appearance of side-peaks, �*τ* is chosen so that the tracking stage can avoid to lock to the side-peaks. For BOC(1,1), in order to achieve the same average correlation loss as for a BPSK signal, �*τBOC*(1,1) = 0.16 chip (Wilde et al., 2006). As for Doppler shift dimension, � *fD* <sup>=</sup> <sup>2</sup> <sup>3</sup>*Tint* as in (Kaplan, 2005) or � *fD* <sup>=</sup> <sup>1</sup> <sup>2</sup>*Tint* as in (Misra & Enge, 2006) are often chosen concerning the trade-off between complexity and sensitivity.

#### **2.3 Acquisition performance parameters**

When dealing with real signals, the incoming code is affected by several factors such as propagation distortion and noise, thus resulting in a distorted correlation function. In order to achieve an optimal detection process, the Neyman-Pearson likelihood criterion is used. In fact, the magnitude *Sm* = |*Rm*| <sup>2</sup> of each complex correlator output can be modeled as a random variable with statistical features. Thus, *Sm* is compared with a predetermined threshold (*V*) in order to decide which hypothesis between *H*<sup>0</sup> (*Sm* < *V*) and *H*<sup>1</sup> (*Sm* > *V*) is true, where *H*<sup>0</sup> and *H*<sup>1</sup> respectively represent the absence or presence of the desired peak. Once the decision

for GNSS Signal Acquisition 7

High Sensitivity Techniques for GNSS Signal Acquisition 9

As described, once *Pf a* and *Pd* are known, the others can be easily computed. These two probabilities are also used to plot the Receiver Operational Characteristic (ROC) curve (see Fig. 3(b)) depicting the behaviors of the *Pf a* versus *Pd* for different values of *V*. This curve is

Theoretical assessment of acquisition performance is not always possible, since it requires also the knowledge of the pdf of the decision variables. For such a reason Monte-Carlo simulation are often employed. In such a case, in order to have suitable confidence in the results, each simulated value of the ROC curve (as in in Fig. 3(b)) has to be the result of the average of million of simulated cases. Therefore, if the sensitivity of a single acquisition scheme in different conditions has to be assessed, it is also useful to consider easy-to-compute

<sup>2</sup> ; *αmean* =

where *Speak* is the maximum of the CAF magnitude and *Sfloor* is the floor of the CAF magnitude (i.e outside the main correlation peak which is 2 chips wide). These metrics highlight the overall trend of post-correlation Signal-to-Noise Ratio (SNR), avoiding time-consuming calculations or simulations. Anyhow, it is important to point out that the comparison of different acquisition schemes based on the peak-to-floor ratios may be not fair

Let us consider a search-space with *Nc* columns and *Nf* rows as in Fig. 2(a), and denote *A* as a successful detection of a serial acquisition engine (Fig. 1) after some miss-detections and false-alarms. The mean duration from the beginning of the process to the instant when *A*

**-** The penalty time *Tf a* and the dwell time *Td*. In fact, *Tf a* is represented through *Td* and the

Therefore, *TA* can be seen as the performance parameter taking into account both the

 *Speak* 2

2 − *Pd* 2*Pd*

+ *Td Pd*

(14)

(15)

*E Sfloor* 2

useful for performance comparison among different acquisition strategies.

parameters, named peak-to-floor ratios, (*αmax*, *αmean*). They are defined as:

if their decision variables show different statistical properties (Ta et al., 2008).

happens is named mean acquisition time, and can be written as (Park et al., 2002)

*TA* = (*NcNf* − 1)(*Td* + *Tf aPf a*)

penalty coefficient *kp*, *Tf a* = *kpTd*. Obviously, *Td* depends on each strategy.

with *Td* and *Tf a* being the dwell time and the penalty time respectively.

**-** The false-alarm (*Pf a*) and detection (*Pd*) probabilities at a single cell.

Equation (15) shows that *TA* depends on the values of :

computational complexity and the sensitivity of a strategy.

 *Speak* 2

max *Sfloor* 

*αmax* =

**2.3.2 Peak-to-floor ratios**

**2.3.3 Mean acquisition time**

**-** The search space size *Nc* × *Nf*

is taken, the parameters ˆ *fD*, *τ*ˆ are taken. Such values must belong to the pull-in range of the tracking stage of the receiver.

#### **2.3.1 Statistical characterization of the detection process**

As previously remarked, the signal acquisition can be seen as a statistical process, and the value taken by the correlator output for each bin of the search space can be modeled as a random variable both when the peak is absent (i.e. *H*0) or present (i.e. *H*1). In each case the random variable is characterized by a probability density function (pdf). Fig. 3(a) shows the signal trial hypothesis test decision when both pdfs are drawn. The threshold

Fig. 3. (a) Possible pdfs of a hypothesis test; (b) Receive Operating Characteristic (ROC) curve *V* is pre-determined based on the requirements of: (i) false-alarm probability (*Pf a*), e.g. *Pf a* = 10−3, or (ii) mean acquisition time (*TA*), e.g. *TA* is minimum.

For a specific value of *V*, there are four possible outcomes as shown in Fig. 3(a). Each outcome is associated with a probability which can be computed by an appropriate integration as (Kaplan, 2005):

• Probability of false-alarm (*Pf a*):

$$P\_{fa} = \int\_{V}^{+\infty} f(s|H\_0)ds\tag{10}$$

• Probability of correct dismissal (*Pcd*):

$$P\_{cd} = 1 - P\_{fa} \tag{11}$$

• Probability of detection (*Pd*):

$$P\_d = \int\_V^{+\infty} f(s|H\_1)ds\tag{12}$$

• Probability of miss-detection (*Pmd*):

$$P\_{md} = 1 - P\_d \tag{13}$$

As described, once *Pf a* and *Pd* are known, the others can be easily computed. These two probabilities are also used to plot the Receiver Operational Characteristic (ROC) curve (see Fig. 3(b)) depicting the behaviors of the *Pf a* versus *Pd* for different values of *V*. This curve is useful for performance comparison among different acquisition strategies.

#### **2.3.2 Peak-to-floor ratios**

6 Will-be-set-by-IN-TECH

As previously remarked, the signal acquisition can be seen as a statistical process, and the value taken by the correlator output for each bin of the search space can be modeled as a random variable both when the peak is absent (i.e. *H*0) or present (i.e. *H*1). In each case the random variable is characterized by a probability density function (pdf). Fig. 3(a) shows the signal trial hypothesis test decision when both pdfs are drawn. The threshold

> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig. 3. (a) Possible pdfs of a hypothesis test; (b) Receive Operating Characteristic (ROC) curve *V* is pre-determined based on the requirements of: (i) false-alarm probability (*Pf a*), e.g.

For a specific value of *V*, there are four possible outcomes as shown in Fig. 3(a). Each outcome is associated with a probability which can be computed by an appropriate integration as

> +∞ *V*

 +∞ *V*

*Pf a* =

*Pd* =

Detection probability Pd

 **-** 

*fD*, *τ*ˆ are taken. Such values must belong to the pull-in range of the

0.02 0.04 0.06 0.08 0.1 <sup>0</sup>

False alarm probability Pfa

*f*(*s*|*H*0)*ds* (10)

*f*(*s*|*H*1)*ds* (12)

*Pcd* = 1 − *Pf a* (11)

*Pmd* = 1 − *Pd* (13)

(b)

is taken, the parameters ˆ

7KUHVKROG*9*

3UREDELOLW\RI'HWHFWLRQ VKDGHGDUHD

(Kaplan, 2005):

• Probability of false-alarm (*Pf a*):

• Probability of detection (*Pd*):

• Probability of correct dismissal (*Pcd*):

• Probability of miss-detection (*Pmd*):

SGIRIQRLVHRQO\LHSGIRID*+* FHOO SGIRIQRLVHZLWKVLJQDOSUHVHQWLHSGIRID*+* FHOO

7KUHVKROG*9* 7KUHVKROG*9*

(a)

3UREDELOLW\RI)DOVH\$ODUP VKDGHGDUHD

tracking stage of the receiver.

**2.3.1 Statistical characterization of the detection process**

3UREDELOLW\RI0LVV'HWHFWLRQ VKDGHGDUHD

3UREDELOLW\RI&RUUHFW'LVPLVVDO VKDGHGDUHD

7KUHVKROG*9*

*Pf a* = 10−3, or (ii) mean acquisition time (*TA*), e.g. *TA* is minimum.

Theoretical assessment of acquisition performance is not always possible, since it requires also the knowledge of the pdf of the decision variables. For such a reason Monte-Carlo simulation are often employed. In such a case, in order to have suitable confidence in the results, each simulated value of the ROC curve (as in in Fig. 3(b)) has to be the result of the average of million of simulated cases. Therefore, if the sensitivity of a single acquisition scheme in different conditions has to be assessed, it is also useful to consider easy-to-compute parameters, named peak-to-floor ratios, (*αmax*, *αmean*). They are defined as:

$$\mathfrak{a}\_{\max} = \frac{\left| \mathbf{S}\_{peak} \right|^2}{\max \left| \mathbf{S}\_{floor} \right|^2}; \ \mathfrak{a}\_{mean} = \frac{\left| \mathbf{S}\_{peak} \right|^2}{E \left[ \left| \mathbf{S}\_{floor} \right|^2 \right]} \tag{14}$$

where *Speak* is the maximum of the CAF magnitude and *Sfloor* is the floor of the CAF magnitude (i.e outside the main correlation peak which is 2 chips wide). These metrics highlight the overall trend of post-correlation Signal-to-Noise Ratio (SNR), avoiding time-consuming calculations or simulations. Anyhow, it is important to point out that the comparison of different acquisition schemes based on the peak-to-floor ratios may be not fair if their decision variables show different statistical properties (Ta et al., 2008).

#### **2.3.3 Mean acquisition time**

Let us consider a search-space with *Nc* columns and *Nf* rows as in Fig. 2(a), and denote *A* as a successful detection of a serial acquisition engine (Fig. 1) after some miss-detections and false-alarms. The mean duration from the beginning of the process to the instant when *A* happens is named mean acquisition time, and can be written as (Park et al., 2002)

$$\overline{T}\_A = (\mathbf{N}\_i \mathbf{N}\_f - 1)(T\_d + T\_{fa} P\_{fa}) \frac{2 - P\_d}{2P\_d} + \frac{T\_d}{P\_d} \tag{15}$$

with *Td* and *Tf a* being the dwell time and the penalty time respectively.

Equation (15) shows that *TA* depends on the values of :


Therefore, *TA* can be seen as the performance parameter taking into account both the computational complexity and the sensitivity of a strategy.

for GNSS Signal Acquisition 9

High Sensitivity Techniques for GNSS Signal Acquisition 11

*M* ∑ *m*=1 *Rm* 

{*r*[*n*]*<sup>c</sup>*[*<sup>n</sup>* <sup>+</sup> *<sup>τ</sup>*]*ej*(2*π*(*fIF*+*<sup>f</sup>*

As seen in (17), the true value of the coherent integration time is no longer *Tint* but increases to *MTint*. Hence, it is fair to state that the coherent combination of {*R*1, ..., *RM*} is equivalent

Unlike the coherent combination, the non-coherent technique combines the squared-envelops of the correlation values {*R*1, ..., *RM*}. The mathematical representation of the decsion variable

> *M* ∑ *m*=1

By using this technique, the main correlation peak also tends to emerge from the noise floor. However, the noise floor is averaged towards a non-zero value. This value is referred as the squaring loss (Choi et al., 2002) and makes the non-coherent combination less effective than

This technique was first introduced in the communication field by (Zarrabizadeh & Sousa, 1997). As far as the satellite navigation field is concerned , (Elders-Boll & Dettmar, 2004; Schmid & Neubauer, 2004) are among the first works using this technique and its variants.

> *M* ∑ *m*=2

As presented in (19), the complex correlator output *Rm* is multiplied by the conjugate of the one obtained at the previous integration interval *Rm*−1. Then the obtained function is accumulated and its envelope becomes the ultimate decision variable. The fact that the signal component remains highly correlated between consecutive correlation intervals, while the noise tends to be de-correlated, results in the improvement of the technique with respect to the non-coherent one. In comparison with the coherent combination, this technique obtains less de-spreading gain, but also requires less computational resources because the search-space size is unchanged (Yu et al., 2007). Therefore, this technique can be seen as a trade-off solution concerning the pros and cons of the coherent and the non-coherent combination techniques.

*RmR*∗ *m*−1 

2


*SN* =

the coherent one. However, the effect is not equivalent to an increasing of *Tint*.

The mathematical representation of the conventional differential combination is

*SD* =  2

 *DM* ))*nTS* }

  2

<sup>2</sup> . (18)

(16)

(17)

(19)

considered. As for the coherent technique, these *M* samples are combined as

*MN* ∑ *n*=0

to increase *Tint* to *MTint*, at the cost of an increased complexity.

However, (16) can be rewritten to

**3.2.2 Non-coherent combination**

**3.2.3 Differential combination**

is then

*SC* = 1 *N* *SC* = 

## **3. High sensitivity acquisition problems**

## **3.1 Acquisition in harsh environments**

The conventional acquisition stage in Fig. 1 is designed to work in open-sky conditions. However, in harsh environments, high sensitivity (HS) acquisition strategies are required. In principle, as a nature of DS-CDMA, the longer the coherent integration time (*Tint*) between the local and the received signals is, the better the de-spreading gain (i.e. signal-to-noise ratio improvement) that can be obtained after the correlation process. However, the presence of unknown data bit transitions limits the value of *Tint* ≤ *Tb* (e.g. *Tint* ≤ 20 ms as for GPS L1 C/A signal) to avoid the correlation loss. This limitation is only neglected if there is an external-aiding source, which provides the data transition information.

The sensitivity improvement obtained by increasing *Tint* is traded-off with an increased computational complexity. As pointed out in Section 2.2, the size of the Doppler step (� *fD*) reduces as *Tint* becomes larger and this fact increases the search-space size. Furthermore, the instability of the receiver clock causes difficulties for the acquisition stage, especially if *Tint* is large, because of the carrier and code Doppler effects. Therefore, one should consider the trade-off between the sensitivity improvement and the complexity increase when changing the value of *Tint*.

Considering the availability of external-aiding sources and the trade-off between the sensitivity and the complexity, the HS strategies can be divided into:


Modern GNSSes broadcast new civil signals on different frequency bands and the new GNSS signals embed the combination of the data channel and a pilot (data-less) channel, per carrier frequency. Examples are E1 OS, E5, E6 signals of Galileo and L5, L2C, L1C signals of GPS. All these facts make possible another approach designed to provide improved acquisition sensitivity:

• Channel combining acquisition approach.

These three approaches are presented in details in the following.

#### **3.2 Stand-alone approach for light harsh enviroments**

Without the availability of external aiding sources, the strategies of this approach use *Tint* ≤ *Tb*. The sensitivity obtained at a specific value of *Tint* is improved by combining the correlator outputs in different ways: coherent, non-coherent and differential combining. These techniques are referred as post-correlation combination techniques.

#### **3.2.1 Coherent combination**

For each cell (*c*ˆ, ˆ *θ*, ˆ *fD*) of the search-space, *M* correlator outputs {*R*1, *R*2, ..., *Rm*, ..., *RM*} obtained by correlating the incoming and the local signals at length *Tint*, see (5), are considered. As for the coherent technique, these *M* samples are combined as

$$S\_{\mathbb{C}} = \left| \sum\_{m=1}^{M} \mathcal{R}\_{m} \right|^{2} \tag{16}$$

However, (16) can be rewritten to

8 Will-be-set-by-IN-TECH

The conventional acquisition stage in Fig. 1 is designed to work in open-sky conditions. However, in harsh environments, high sensitivity (HS) acquisition strategies are required. In principle, as a nature of DS-CDMA, the longer the coherent integration time (*Tint*) between the local and the received signals is, the better the de-spreading gain (i.e. signal-to-noise ratio improvement) that can be obtained after the correlation process. However, the presence of unknown data bit transitions limits the value of *Tint* ≤ *Tb* (e.g. *Tint* ≤ 20 ms as for GPS L1 C/A signal) to avoid the correlation loss. This limitation is only neglected if there is an

The sensitivity improvement obtained by increasing *Tint* is traded-off with an increased computational complexity. As pointed out in Section 2.2, the size of the Doppler step (� *fD*) reduces as *Tint* becomes larger and this fact increases the search-space size. Furthermore, the instability of the receiver clock causes difficulties for the acquisition stage, especially if *Tint* is large, because of the carrier and code Doppler effects. Therefore, one should consider the trade-off between the sensitivity improvement and the complexity increase when changing

Considering the availability of external-aiding sources and the trade-off between the

Modern GNSSes broadcast new civil signals on different frequency bands and the new GNSS signals embed the combination of the data channel and a pilot (data-less) channel, per carrier frequency. Examples are E1 OS, E5, E6 signals of Galileo and L5, L2C, L1C signals of GPS. All these facts make possible another approach designed to provide improved acquisition

Without the availability of external aiding sources, the strategies of this approach use *Tint* ≤ *Tb*. The sensitivity obtained at a specific value of *Tint* is improved by combining the correlator outputs in different ways: coherent, non-coherent and differential combining.

obtained by correlating the incoming and the local signals at length *Tint*, see (5), are

*fD*) of the search-space, *M* correlator outputs {*R*1, *R*2, ..., *Rm*, ..., *RM*}

• Stand-alone approach (to deal with light harsh environments, e.g. light indoor)

• External-aiding approach (to deal with harsh environments, e.g. indoor).

external-aiding source, which provides the data transition information.

sensitivity and the complexity, the HS strategies can be divided into:

These three approaches are presented in details in the following.

These techniques are referred as post-correlation combination techniques.

**3.2 Stand-alone approach for light harsh enviroments**

• Channel combining acquisition approach.

**3.2.1 Coherent combination**

*θ*, ˆ

For each cell (*c*ˆ, ˆ

**3. High sensitivity acquisition problems**

**3.1 Acquisition in harsh environments**

the value of *Tint*.

sensitivity:

$$S\_{\mathbb{C}} = \left| \frac{1}{N} \sum\_{n=0}^{MN} \{r[n] \hat{c}[n+\hat{\tau}]e^{j(2\pi(f\_{\text{IF}} + \hat{f}\_{\text{D}\_M}))nT\_S} \} \right|^2 \tag{17}$$

As seen in (17), the true value of the coherent integration time is no longer *Tint* but increases to *MTint*. Hence, it is fair to state that the coherent combination of {*R*1, ..., *RM*} is equivalent to increase *Tint* to *MTint*, at the cost of an increased complexity.

#### **3.2.2 Non-coherent combination**

Unlike the coherent combination, the non-coherent technique combines the squared-envelops of the correlation values {*R*1, ..., *RM*}. The mathematical representation of the decsion variable is then

$$S\_N = \sum\_{m=1}^{M} |R\_m|^2. \tag{18}$$

By using this technique, the main correlation peak also tends to emerge from the noise floor. However, the noise floor is averaged towards a non-zero value. This value is referred as the squaring loss (Choi et al., 2002) and makes the non-coherent combination less effective than the coherent one. However, the effect is not equivalent to an increasing of *Tint*.

#### **3.2.3 Differential combination**

This technique was first introduced in the communication field by (Zarrabizadeh & Sousa, 1997). As far as the satellite navigation field is concerned , (Elders-Boll & Dettmar, 2004; Schmid & Neubauer, 2004) are among the first works using this technique and its variants. The mathematical representation of the conventional differential combination is

$$S\_D = \left| \sum\_{m=2}^{M} R\_m \mathbf{R}\_{m-1}^\* \right|^2 \tag{19}$$

As presented in (19), the complex correlator output *Rm* is multiplied by the conjugate of the one obtained at the previous integration interval *Rm*−1. Then the obtained function is accumulated and its envelope becomes the ultimate decision variable. The fact that the signal component remains highly correlated between consecutive correlation intervals, while the noise tends to be de-correlated, results in the improvement of the technique with respect to the non-coherent one. In comparison with the coherent combination, this technique obtains less de-spreading gain, but also requires less computational resources because the search-space size is unchanged (Yu et al., 2007). Therefore, this technique can be seen as a trade-off solution concerning the pros and cons of the coherent and the non-coherent combination techniques.

for GNSS Signal Acquisition 11

High Sensitivity Techniques for GNSS Signal Acquisition 13

The mentioned A-GNSS specifications, basically define the procedures for requesting and sending information on user position and assistance data. These are typically of two

• Mobile-based: assistance data are provided to the User Equipment (UE), which measures the pseudo-ranges and provides the position estimation to the proper network service. • Mobile-assisted: the UE measures the pseudoranges and sends them to a location server

In both these two modes, the position estimation may benefit of the knowledge of additional information available to the location server gathered from one or more reference receivers (e.g. differential corrections, precise timing and ephemeris, etc.). In the followings, the challenges of the external aiding approach are discussed. It should be noted that the chosen signal for

The typical effects of both the data wipe-off and non-removed bit transitions are in Fig. 4(a) and Fig. 4(b) respectively. In the first case, the main correlation peak is easily identified whilst in the other one no peak can be distinguished over the floor. Under the AWGN assumption, in

(a) (b)

the correct Doppler and code-phase bins *αmean* is theoretically proportional to post-correlation Signal-to-Noise Ratio (SNR) and it is expected to increase by 3dB when *Tint* doubles. This can be seen in Fig. 5, where we show the effect on *αmean* and *αmax* of a coherent correlation with *C*/*N*<sup>0</sup> = 24 dB-Hz and *Tint* = {100, 500, 1000} ms performed on simulated GPS signals, both with and without data wipe-off. In Fig. 5(a), we observe that at the highest values of *C*/*N*<sup>0</sup> the peak-to-floor ratios change linearly, i.e. *αmean* increases by 3 dB when *Tint* doubles (e.g. from 500 ms to 1000 ms). In this case, *Rpeak* is the correct correlation peak. At the lowest *C*/*N*0, *αmax*

<sup>2</sup> <sup>≈</sup> max <sup>|</sup>*Rfloor*<sup>|</sup>

2

Fig. 4. CAF along code-phase - simulated GPS L1 C/A signal with code-phase of 500 *μ*s, *C*/*N*<sup>0</sup> = 24 dB-Hz and *Tint* = 1 s: (a) with data wipe-off; (b) without data wipe-off

is practically 0 dB and the detected peak is likely a noise peak, thus |*Rpeak*|

which performs the positioning and service-related tasks.

paradigms:

analyses is GPS L1 C/A.

**3.3.1 Navigation data wipe-off**

However, this technique might suffer from the combination loss due to the unknown data transitions. Assuming that the chance of changing data bit sign after each data bit period is 50%, then if full code correlation (i.e. *Tint* = 1 ms) is used, the average degradation due to data overlay is 20 log(18/19) ≈ 0.47 dB. However, in the Galileo case, this loss is scaled to 20 log(1/2) ≈ 6 dB, because the data bit duration is equal to the code length of 4 ms. This fact causes difficulties in applying the differential technique for Galileo E1 OS receivers.

As an expanded view of the conventional differential combination technique, generalized differential combination techniques are introduced to further improve the sensitivity of the acquisition process. These advanced differential techniques will be discussed in details in Section 4.

#### **3.3 External aiding approach for harsh environments**

For this approach, basically, the availability of external aiding sources makes the value of *Tint* able to be larger than *Tb* (i.e. *Tint* > *Tb*). Therefore, in this scenario, increasing *Tint* (or coherent combination of the correlator outputs) is the most suitable solution to give the best sensitivity improvement to the acquisition stage operating in harsh environments. In literature, this approach is also referred as assistance or assisted approach.

As pointed out in (Djuknic & Richton, 2001), the assisted technique enables HS acquisition, since it provides the signal processing chain with preliminary (but approximate) code-phase / Doppler frequency estimates along with fragments of the navigation message. This allows for wiping off data-bit transitions and for extending the coherent integration time. The concept of data-bit assistance has been also introduced by the 3rd Generation Partnership Project (3GPP) in its technical specifications of the Assisted GNSS (A-GNSS) for UMTS (3GPP, 2008a) and GSM/EDGE (3GPP, 2008b) networks.

In general, with all post correlation processing techniques presented in Section 3.2, sensitivity losses are experienced due to


These effects impact the observed Radio Frequency (RF) carrier frequency and can be more relevant with long coherent integrations (Chansarkar & Garin, 2000) as the case of the coherent combination in this external aiding approach.

Finally, a trade-off between sensitivity and complexity is always necessary, particularly for mass-market receivers (e.g. embedded in cellular phones) which require real-time processing but low power consumption. Despite the recent improvements in chip-set sizes and speeds, a real-time indoor-grade high-sensitivity receiver for cellular phones does not exist yet. Reduced sampling rates are mandatory to minimize the computational load of the baseband processing as well as the optimization of the assistance information exchange is fundamental in order to minimize the communication load which is likely to be paid by the user, according to the latest trends, such as the Secure User Plane for Location (SUPL) defined by Open Mobile Alliance (OMA), (Mulassano & Dovis, 2010; OMA, 2007).

The mentioned A-GNSS specifications, basically define the procedures for requesting and sending information on user position and assistance data. These are typically of two paradigms:


In both these two modes, the position estimation may benefit of the knowledge of additional information available to the location server gathered from one or more reference receivers (e.g. differential corrections, precise timing and ephemeris, etc.). In the followings, the challenges of the external aiding approach are discussed. It should be noted that the chosen signal for analyses is GPS L1 C/A.

#### **3.3.1 Navigation data wipe-off**

10 Will-be-set-by-IN-TECH

However, this technique might suffer from the combination loss due to the unknown data transitions. Assuming that the chance of changing data bit sign after each data bit period is 50%, then if full code correlation (i.e. *Tint* = 1 ms) is used, the average degradation due to data overlay is 20 log(18/19) ≈ 0.47 dB. However, in the Galileo case, this loss is scaled to 20 log(1/2) ≈ 6 dB, because the data bit duration is equal to the code length of 4 ms. This fact

As an expanded view of the conventional differential combination technique, generalized differential combination techniques are introduced to further improve the sensitivity of the acquisition process. These advanced differential techniques will be discussed in details in

For this approach, basically, the availability of external aiding sources makes the value of *Tint* able to be larger than *Tb* (i.e. *Tint* > *Tb*). Therefore, in this scenario, increasing *Tint* (or coherent combination of the correlator outputs) is the most suitable solution to give the best sensitivity improvement to the acquisition stage operating in harsh environments. In

As pointed out in (Djuknic & Richton, 2001), the assisted technique enables HS acquisition, since it provides the signal processing chain with preliminary (but approximate) code-phase / Doppler frequency estimates along with fragments of the navigation message. This allows for wiping off data-bit transitions and for extending the coherent integration time. The concept of data-bit assistance has been also introduced by the 3rd Generation Partnership Project (3GPP) in its technical specifications of the Assisted GNSS (A-GNSS) for UMTS (3GPP, 2008a) and

In general, with all post correlation processing techniques presented in Section 3.2, sensitivity

• the residual Doppler error (including the finite search resolution in frequency and the

These effects impact the observed Radio Frequency (RF) carrier frequency and can be more relevant with long coherent integrations (Chansarkar & Garin, 2000) as the case of the coherent

Finally, a trade-off between sensitivity and complexity is always necessary, particularly for mass-market receivers (e.g. embedded in cellular phones) which require real-time processing but low power consumption. Despite the recent improvements in chip-set sizes and speeds, a real-time indoor-grade high-sensitivity receiver for cellular phones does not exist yet. Reduced sampling rates are mandatory to minimize the computational load of the baseband processing as well as the optimization of the assistance information exchange is fundamental in order to minimize the communication load which is likely to be paid by the user, according to the latest trends, such as the Secure User Plane for Location (SUPL) defined by Open Mobile

causes difficulties in applying the differential technique for Galileo E1 OS receivers.

literature, this approach is also referred as assistance or assisted approach.

**3.3 External aiding approach for harsh environments**

GSM/EDGE (3GPP, 2008b) networks.

contribution of the user dynamics)

combination in this external aiding approach.

• the uncertainty on the Local Oscillator (LO) frequency.

Alliance (OMA), (Mulassano & Dovis, 2010; OMA, 2007).

losses are experienced due to

Section 4.

The typical effects of both the data wipe-off and non-removed bit transitions are in Fig. 4(a) and Fig. 4(b) respectively. In the first case, the main correlation peak is easily identified whilst in the other one no peak can be distinguished over the floor. Under the AWGN assumption, in

Fig. 4. CAF along code-phase - simulated GPS L1 C/A signal with code-phase of 500 *μ*s, *C*/*N*<sup>0</sup> = 24 dB-Hz and *Tint* = 1 s: (a) with data wipe-off; (b) without data wipe-off

the correct Doppler and code-phase bins *αmean* is theoretically proportional to post-correlation Signal-to-Noise Ratio (SNR) and it is expected to increase by 3dB when *Tint* doubles. This can be seen in Fig. 5, where we show the effect on *αmean* and *αmax* of a coherent correlation with *C*/*N*<sup>0</sup> = 24 dB-Hz and *Tint* = {100, 500, 1000} ms performed on simulated GPS signals, both with and without data wipe-off. In Fig. 5(a), we observe that at the highest values of *C*/*N*<sup>0</sup> the peak-to-floor ratios change linearly, i.e. *αmean* increases by 3 dB when *Tint* doubles (e.g. from 500 ms to 1000 ms). In this case, *Rpeak* is the correct correlation peak. At the lowest *C*/*N*0, *αmax* is practically 0 dB and the detected peak is likely a noise peak, thus |*Rpeak*| <sup>2</sup> <sup>≈</sup> max <sup>|</sup>*Rfloor*<sup>|</sup> 2

for GNSS Signal Acquisition 13

High Sensitivity Techniques for GNSS Signal Acquisition 15

where *τ*(*t*) is the time-variant propagation delay. With a first-order expansion of the

= [2*π*(*fRF* + *fD*)*t* + *ϕ*0] =

<sup>2</sup>*<sup>π</sup> fRF*

effect, the observed carrier frequency is different from the nominal RF carrier frequency. With a second-order expansion for *τ*(*t*), we could see that also *fD* changes in time and we could take into account a Doppler-rate term *rD*. For a ground GPS receiver in low-dynamics conditions,

The IF down-conversion leaves unmodified the Doppler frequency, as the IF carrier results:

*BPF*{cos[2*π*(*fRF* + *fD*)*t* + *φ*0] · 2 cos[2*π*(*fRF* − *fIF*)*t*]} = cos[2*π*(*fIF* + *fD*)*t* + *φ*<sup>0</sup> + *φRX*] (25)

where BPF {} refers to the front-end filtering operation performed by the down-conversion

The code component is theoretically periodic with fundamental frequency equal to the inverse of the code period. When propagating from the satellite to the receiver, the same time-variant

*<sup>μ</sup>he<sup>j</sup>*2*h<sup>π</sup> <sup>f</sup>*0[*t*−*τ*(*t*)]

*j*2*hπ f*<sup>0</sup> <sup>1</sup><sup>+</sup> *fD fRF t*+*ϑ*<sup>0</sup>

*code*

Due to the Doppler effect, each harmonic is shifted of the same relative frequency offset

. Thus the fundamental frequency of the delayed code is now *f*<sup>0</sup>

*Tcode* <sup>=</sup> *<sup>T</sup>*˜

 <sup>1</sup> <sup>+</sup> *fD*

 <sup>1</sup> <sup>+</sup> *fD fRF*

∞ ∑ *h*=0

= ∞ ∑ *h*=0 *μhe*

*code* is the nominal one. Consequently, the true chip rate is

*Rc* = *R*˜ *<sup>c</sup>*

<sup>1</sup> <sup>+</sup> *fD fRF*

= 2*π fRFt* − 2*π fRFτ*<sup>0</sup> − 2*π fRFa* · *t* = (23)

*t* + *ϕ*<sup>0</sup> 

*dt* is the usual Doppler frequency shift. Due to the Doppler

(24)

(26)

and

(28)

 <sup>1</sup> <sup>+</sup> *fD fRF*

*fRF* (27)

time-variant delay, i.e. *τ*(*t*) = *τ*<sup>0</sup> + *a* · *t* + ..., the carrier phase becomse:

<sup>2</sup>*<sup>π</sup> fRF*[*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*(*t*)] =

the typical intervals are *fD* = −5 kHz ÷5 kHz and *rD* = −1 Hz/s ÷0 Hz/s.

*dτ*(*t*)

stage and *ϕRX* is the related additional phase contribution.

*c*[*t* − *τ*(*t*)] =

delay impacts on all the harmonic components:

Let us denote *ϕ*<sup>0</sup> = −2*π fRFτ*<sup>0</sup> and *fD* = −2*π fRFa* then

where *fD* = −*a fRF* = −*fRF*

 <sup>1</sup> <sup>+</sup> *fD fRF*

where *T*˜

its period duration is:

2*π fRF*[*t* − *τ*(*t*)] = 2*π fRF*(*t* − *τ*<sup>0</sup> − *a* · *t*) =

Fig. 5. *αmean* and *αmax* vs. *C*/*N*<sup>0</sup> and different integration windows: (a) with data wipe-off; (b) without data wipe-off

. *αmean* is constant for low *C*/*N*<sup>0</sup> values because at such a noise level, *Rfloor* is a zero-mean Gaussian random variable and for most of *R*[*k*, *m*] samples:

$$|\mathcal{R}\_{floor}|^2 \le E\{|\mathcal{R}\_{floor}|^2\} + \eta \sqrt{Var\{|\mathcal{R}\_{floor}|^2\}}\tag{20}$$

where *η* is an arbitrary constant. Then:

$$\alpha\_{mean} = \frac{\max\{|R\_{floor}|^2\}}{E\{|R\_{floor}|^2\}} = 1 + \eta \frac{\sqrt{Var\{|R\_{floor}|^2\}}}{E\{|R\_{floor}|^2\}}\tag{21}$$

Since *Rfloor* is complex and Gaussian distributed, then |*Rfloor*| <sup>2</sup> <sup>=</sup> R{*Rfloor*} <sup>+</sup> I{*Rfloor*} is *χ*<sup>2</sup> distributed (2 degrees of freedom) and thus the ratio of mean and variance is constant (Kreiszig, 1999). In Fig. 5(b), it can be seen that without data wipe-off the CAF envelope behaves as if it is made of noise only, even at the highest values of *C*/*N*0.

#### **3.3.2 Doppler effects on carrier and code**

The Doppler effect observed at the receiver location is caused by the time-variant propagation delay of the transmitted signal along its path toward the receiver. This delay changes over time even in case of a low-dynamics user (e.g. pedestrians, etc.), as at least the SV is moving along its own orbit. Even if the rate of change is relatively slow, when long coherent integration windows are used, it can be shown that it impacts on the acquisition sensitivity. Let (22) be the general expression of the received RF signal (noiseless for simplicity):

$$s\_{RX}(t) = \sqrt{2\mathsf{C}}c[t - \tau(t)]\cos\{2\pi f\_{RF}[t - \tau(t)]\}\tag{22}$$

12 Will-be-set-by-IN-TECH

(a) (b)

Fig. 5. *αmean* and *αmax* vs. *C*/*N*<sup>0</sup> and different integration windows: (a) with data wipe-off;

. *αmean* is constant for low *C*/*N*<sup>0</sup> values because at such a noise level, *Rfloor* is a zero-mean

2} *<sup>E</sup>*{|*Rfloor*|2} <sup>=</sup> <sup>1</sup> <sup>+</sup> *<sup>η</sup>*

*χ*<sup>2</sup> distributed (2 degrees of freedom) and thus the ratio of mean and variance is constant (Kreiszig, 1999). In Fig. 5(b), it can be seen that without data wipe-off the CAF envelope

The Doppler effect observed at the receiver location is caused by the time-variant propagation delay of the transmitted signal along its path toward the receiver. This delay changes over time even in case of a low-dynamics user (e.g. pedestrians, etc.), as at least the SV is moving along its own orbit. Even if the rate of change is relatively slow, when long coherent integration windows are used, it can be shown that it impacts on the acquisition sensitivity.

Let (22) be the general expression of the received RF signal (noiseless for simplicity):

<sup>2</sup>} <sup>+</sup> *<sup>η</sup>* 

*Var*{|*Rfloor*|2}

2*Cc*[*t* − *τ*(*t*)] cos{2*π fRF*[*t* − *τ*(*t*)]} (22)

*Var*{|*Rfloor*|2} (20)

*<sup>E</sup>*{|*Rfloor*|2} (21)

<sup>2</sup> <sup>=</sup> R{*Rfloor*} <sup>+</sup> I{*Rfloor*} is

<sup>2</sup> <sup>≤</sup> *<sup>E</sup>*{|*Rfloor*<sup>|</sup>

Gaussian random variable and for most of *R*[*k*, *m*] samples:

*<sup>α</sup>mean* <sup>=</sup> max{|*Rfloor*<sup>|</sup>

Since *Rfloor* is complex and Gaussian distributed, then |*Rfloor*|

*sRX*(*t*) = <sup>√</sup>

behaves as if it is made of noise only, even at the highest values of *C*/*N*0.


where *η* is an arbitrary constant. Then:

**3.3.2 Doppler effects on carrier and code**

(b) without data wipe-off

where *τ*(*t*) is the time-variant propagation delay. With a first-order expansion of the time-variant delay, i.e. *τ*(*t*) = *τ*<sup>0</sup> + *a* · *t* + ..., the carrier phase becomse:

$$\begin{split} 2\pi f\_{RF}[t - \tau(t)] &= 2\pi f\_{RF}(t - \tau\_0 - a \cdot t) = \\ &= 2\pi f\_{RF}t - 2\pi f\_{RF}\tau\_0 - 2\pi f\_{RF}a \cdot t = \\ &= [2\pi (f\_{RF} + f\_D)t + \varphi\_0] = \end{split} \tag{23}$$

Let us denote *ϕ*<sup>0</sup> = −2*π fRFτ*<sup>0</sup> and *fD* = −2*π fRFa* then

$$2\pi f\_{\rm RF}[t-\tau(t)] = \left[2\pi f\_{\rm RF}\left(1+\frac{f\_{\rm D}}{f\_{\rm RF}}\right)t+\varphi\_0\right] \tag{24}$$

where *fD* = −*a fRF* = −*fRF dτ*(*t*) *dt* is the usual Doppler frequency shift. Due to the Doppler effect, the observed carrier frequency is different from the nominal RF carrier frequency. With a second-order expansion for *τ*(*t*), we could see that also *fD* changes in time and we could take into account a Doppler-rate term *rD*. For a ground GPS receiver in low-dynamics conditions, the typical intervals are *fD* = −5 kHz ÷5 kHz and *rD* = −1 Hz/s ÷0 Hz/s.

The IF down-conversion leaves unmodified the Doppler frequency, as the IF carrier results:

$$\text{BPF}\{\cos[2\pi(f\_{\text{RF}} + f\_{\text{D}})t + \phi\_0] \cdot 2\cos[2\pi(f\_{\text{RF}} - f\_{\text{IF}})t] \} = \cos[2\pi(f\_{\text{IF}} + f\_{\text{D}})t + \phi\_0 + \phi\_{\text{RX}}] \tag{25}$$

where BPF {} refers to the front-end filtering operation performed by the down-conversion stage and *ϕRX* is the related additional phase contribution.

The code component is theoretically periodic with fundamental frequency equal to the inverse of the code period. When propagating from the satellite to the receiver, the same time-variant delay impacts on all the harmonic components:

$$\begin{split} \mathbf{c}[t - \tau(t)] &= \sum\_{h=0}^{\infty} \mu\_{h} e^{j2h\pi f\_{0} \left[t - \tau(t)\right]} \\ &= \sum\_{h=0}^{\infty} \mu\_{h} e^{j2h\pi f\_{0} \left(1 + \frac{f\_{D}}{f\_{\text{RF}}}\right) t + \theta\_{0}} \end{split} \tag{26}$$

Due to the Doppler effect, each harmonic is shifted of the same relative frequency offset <sup>1</sup> <sup>+</sup> *fD fRF* . Thus the fundamental frequency of the delayed code is now *f*<sup>0</sup> <sup>1</sup> <sup>+</sup> *fD fRF* and its period duration is:

$$T\_{code} = \frac{\tilde{T}\_{code}}{\left(1 + \frac{f\_D}{f\_{RF}}\right)}\tag{27}$$

where *T*˜ *code* is the nominal one. Consequently, the true chip rate is

$$R\_{\mathcal{C}} = \tilde{R}\_{\mathcal{C}} \left( 1 + \frac{f\_D}{f\_{\mathcal{R}F}} \right) \tag{28}$$

for GNSS Signal Acquisition 15

High Sensitivity Techniques for GNSS Signal Acquisition 17

*fS* <sup>=</sup> *<sup>n</sup> fS*

where *n*/ *fS* is the ideal sampling instant and *fS* is the sampling frequency. The sampled

and it is affected by a time-variant delay with respect to the ideal case. This gives rise to an equivalent Doppler effect, as previously discussed, and hence to an additional correlation loss. Oscillators typically used in GNSS receivers are mostly Crystal Oscillators (XOs) with some degree of frequency stabilization, e.g. Thermally-Compensated Crystal Oscillator (TCXO), with typical accuracy *yLO* <sup>∼</sup> <sup>10</sup>−<sup>6</sup> and Oven-Controlled Crystal Oscillator (OCXO), with typical accuracy *yLO* <sup>∼</sup> <sup>10</sup>−<sup>8</sup> (Vig, 2005). Table 2 shows how a constant offset on the LO frequency may impact both on *αmean*, *αmax* and on the accuracy of the code-delay estimation

*fD*/ *fLO αmax*(*dB*) *αmean* (dB) Code-phase error (chips)

In a new or upgraded GNSS, there are several civil signals broadcast in different frequencies. This fact assures a future for civil GNSS dual-frequency receivers, which are now used only in high-value professional or commercial applications such as survey, machine control and guidance, etc. Beside the predictable advantages, such as ionosphere error elimination and carrier phase measurement improvement, civil dual-frequency receivers also offer sensitivity improvement by making possible combined acquisition strategies. The combined acquisition on different carrier frequencies is guaranteed by the fact that the signal channels belonging to a common GNSS are time synchronized, and the Doppler shifts of these channels are related by the ratio among the carrier frequencies. In literature, (Gernot et al., 2008) uses this approach

New GNSS signals are composed of data and pilot (data-less) channels. These two channels can be multiplexed by Coherent Adaptive Subcarrier Modulation (e.g. Galileo E1 OS), Time Division Multiplexing (GPS L2C) and Quadrature Phase-Shift Keying (Galileo E5; GPS L5,

0 22 31 0 0.5 · <sup>10</sup>−<sup>6</sup> <sup>18</sup> <sup>31</sup> 0.75 1.5 · <sup>10</sup>−<sup>6</sup> <sup>0</sup> <sup>27</sup> 1.5

Table 2. Constant offset on LO frequency. *Tint* = 1 s, *C*/*N*<sup>0</sup> = +∞

• Channel Combining on Different Carrier Frequencies

for combined acquisition of GPS L1 C/A and L2C signals.

• Channel Combining on a Common Frequency:

+ *x*<sup>0</sup> + *y*<sup>0</sup>

*n fS* 

+ *x*<sup>0</sup> + *y*<sup>0</sup>

*n* = 0, 1, 2, ...

*n fS*

(33)

(34)

<sup>+</sup> *xLO <sup>n</sup>*

 *n fS*

sampling clock, the sampling timescale can be defined as:

<sup>=</sup> *<sup>n</sup> fS*

*r*[*n*] = *r*

 *<sup>t</sup>*<sup>=</sup> *<sup>n</sup> fS*

*tS*(*s*)

version of the IF signal is:

in case ofa1s coherent integration.

**3.4 Channel combining approach:**


Table 1. Peak-to-floor ratios and Doppler effect on estimated code phase, *C*/*N*<sup>0</sup> = 24 dB-Hz and *Tint* = 1 s.

During the acquisition phase, if the local code is generated at the nominal chip rate *Rc*, the correlation between local and received codes suffers a loss due to the difference with the true received chip rate *Rc*. Furthermore, such a loss increases with the integration times. A loss of about 8 dB in *αmean* can be estimated at *C*/*N*<sup>0</sup> = 24 dB-Hz (*Tint* = 1 s). Table 1 shows the degradation of the correlation peak and the code-phase estimation error.

#### **3.3.3 Local oscillator stability**

The uncertainty on the nominal value *fLO* of the LO frequency is usually expressed as fractional frequency deviation (Audoin & Guinot, 2001):

$$y\_{LO}(t) = \frac{\triangle f(t)}{f\_{\rm LO}} = \frac{f(t) - f\_{\rm LO}}{f\_{\rm LO}} = \frac{f(t)}{f\_{\rm LO}} - 1\tag{29}$$

where *f*(*t*) is the true instant frequency. *yLO* is affected by environmental conditions (e.g. temperature, pressure), dynamic stress (e.g. acceleration, jerk, etc.), circuital tolerances, etc. The time deviation (i.e. the time difference between the clock with the true oscillator and an ideal clock), is given by:

$$\mathbf{x}\_{LO}(t) = \int\_{-\infty}^{t} \mathbf{y}\_{LO}(u) du\tag{30}$$

With the zero-th order expansion *yLO*(*t*) = *y*<sup>0</sup> + ..., (*y*<sup>0</sup> is a constant frequency offset), the time deviation results:

$$\mathbf{x}\_{LO}(t) = \mathbf{x}\_0 + \mathbf{y}\_0 \cdot t \tag{31}$$

where *x*<sup>0</sup> is an initial synchronization error between real and ideal clocks and *t* is the time elapsed since the initial synchronization epoch. This model can be used to evaluate the effect of the local oscillator accuracy on both the down-conversion and the sampling stages.

During the down-conversion the true mixing signal (used in (25)) is:

$$2\cos[2\pi(f\_{RF} - f\_{IF})(1 + y\_0 t)]\tag{32}$$

The true IF carrier is actually affected by an additional unpredictable shift, that prevents the exact carrier frequency estimation, even with very accurate Doppler aiding information. By means of (31) we can evaluate the impact of the LO on the sampling process. With the true 14 Will-be-set-by-IN-TECH

*fD αmax αmean* Doppler-induced code-phase (kHz) (dB) (dB) estimation error (chips) -5 0.01 14.18 1.683 -2.5 2.66 18.88 0.830 0 12.77 22.48 0 2.5 2.22 18.93 -0.839 5 0.19 14.47 -1.838

Table 1. Peak-to-floor ratios and Doppler effect on estimated code phase, *C*/*N*<sup>0</sup> = 24 dB-Hz

During the acquisition phase, if the local code is generated at the nominal chip rate *Rc*, the correlation between local and received codes suffers a loss due to the difference with the true received chip rate *Rc*. Furthermore, such a loss increases with the integration times. A loss of about 8 dB in *αmean* can be estimated at *C*/*N*<sup>0</sup> = 24 dB-Hz (*Tint* = 1 s). Table 1 shows the

The uncertainty on the nominal value *fLO* of the LO frequency is usually expressed as

where *f*(*t*) is the true instant frequency. *yLO* is affected by environmental conditions (e.g. temperature, pressure), dynamic stress (e.g. acceleration, jerk, etc.), circuital tolerances, etc. The time deviation (i.e. the time difference between the clock with the true oscillator and an

> *t* −∞

With the zero-th order expansion *yLO*(*t*) = *y*<sup>0</sup> + ..., (*y*<sup>0</sup> is a constant frequency offset), the time

where *x*<sup>0</sup> is an initial synchronization error between real and ideal clocks and *t* is the time elapsed since the initial synchronization epoch. This model can be used to evaluate the effect

The true IF carrier is actually affected by an additional unpredictable shift, that prevents the exact carrier frequency estimation, even with very accurate Doppler aiding information. By means of (31) we can evaluate the impact of the LO on the sampling process. With the true

of the local oscillator accuracy on both the down-conversion and the sampling stages.

During the down-conversion the true mixing signal (used in (25)) is:

<sup>=</sup> *<sup>f</sup>*(*t*) <sup>−</sup> *fLO fLO*

<sup>=</sup> *<sup>f</sup>*(*t*) *fLO*

− 1 (29)

*yLO*(*u*)*du* (30)

*xLO*(*t*) = *x*<sup>0</sup> + *y*<sup>0</sup> · *t* (31)

2 cos[2*π*(*fRF* − *fIF*)(1 + *y*0*t*)] (32)

degradation of the correlation peak and the code-phase estimation error.

fractional frequency deviation (Audoin & Guinot, 2001):

*yLO*(*t*) = � *<sup>f</sup>*(*t*)

*fLO*

*xLO*(*t*) =

and *Tint* = 1 s.

**3.3.3 Local oscillator stability**

ideal clock), is given by:

deviation results:

sampling clock, the sampling timescale can be defined as:

$$t\_S(s)\bigg|\_{t=\frac{\eta}{f\_S}} = \frac{n}{f\_S} + x\_{LO}\left(\frac{n}{f\_S}\right) = \frac{n}{f\_S} + x\_0 + y\_0\frac{n}{f\_S}\tag{33}$$

$$n = 0, 1, 2, \dots$$

where *n*/ *fS* is the ideal sampling instant and *fS* is the sampling frequency. The sampled version of the IF signal is:

$$r[n] = r\left(\frac{n}{f\mathfrak{s}} + \mathfrak{x}\_0 + y\_0 \frac{n}{f\mathfrak{s}}\right) \tag{34}$$

and it is affected by a time-variant delay with respect to the ideal case. This gives rise to an equivalent Doppler effect, as previously discussed, and hence to an additional correlation loss. Oscillators typically used in GNSS receivers are mostly Crystal Oscillators (XOs) with some degree of frequency stabilization, e.g. Thermally-Compensated Crystal Oscillator (TCXO), with typical accuracy *yLO* <sup>∼</sup> <sup>10</sup>−<sup>6</sup> and Oven-Controlled Crystal Oscillator (OCXO), with typical accuracy *yLO* <sup>∼</sup> <sup>10</sup>−<sup>8</sup> (Vig, 2005). Table 2 shows how a constant offset on the LO frequency may impact both on *αmean*, *αmax* and on the accuracy of the code-delay estimation in case ofa1s coherent integration.


Table 2. Constant offset on LO frequency. *Tint* = 1 s, *C*/*N*<sup>0</sup> = +∞

#### **3.4 Channel combining approach:**

• Channel Combining on Different Carrier Frequencies

In a new or upgraded GNSS, there are several civil signals broadcast in different frequencies. This fact assures a future for civil GNSS dual-frequency receivers, which are now used only in high-value professional or commercial applications such as survey, machine control and guidance, etc. Beside the predictable advantages, such as ionosphere error elimination and carrier phase measurement improvement, civil dual-frequency receivers also offer sensitivity improvement by making possible combined acquisition strategies. The combined acquisition on different carrier frequencies is guaranteed by the fact that the signal channels belonging to a common GNSS are time synchronized, and the Doppler shifts of these channels are related by the ratio among the carrier frequencies. In literature, (Gernot et al., 2008) uses this approach for combined acquisition of GPS L1 C/A and L2C signals.

• Channel Combining on a Common Frequency:

New GNSS signals are composed of data and pilot (data-less) channels. These two channels can be multiplexed by Coherent Adaptive Subcarrier Modulation (e.g. Galileo E1 OS), Time Division Multiplexing (GPS L2C) and Quadrature Phase-Shift Keying (Galileo E5; GPS L5,

for GNSS Signal Acquisition 17

High Sensitivity Techniques for GNSS Signal Acquisition 19

 

Note that the CDC technique is in fact the GDC taking into account span-1 *A*<sup>1</sup> only, see Fig. 6(b). Basically, the GDC technique can be considered as a coherent integration of the differential combinations at different sample distances. Following the analyzes in (Ta et al., 2012), with small *M* (e.g. *M* ≤ *Tb*/*Tint*), in normal circumstances with normal user dynamic and frequency standards, the average frequency drift is small and tends to zero. Therefore,

*M*−1 ∑ *i*=1 *Ai* 

2

*fdm* in (6) are constant for all *<sup>m</sup>* <sup>∈</sup> [0, *<sup>M</sup>* <sup>−</sup> <sup>1</sup>]. The signal component *<sup>A</sup><sup>S</sup>*

*<sup>G</sup>* <sup>=</sup> *<sup>G</sup>*<sup>1</sup> <sup>=</sup> ... <sup>=</sup> *GM* <sup>=</sup> <sup>√</sup>2*C*R[*τ*]sinc(� *<sup>f</sup> <sup>d</sup>Tint*) (38)

 

<sup>2</sup> <sup>+</sup> <sup>|</sup>(*<sup>M</sup>* <sup>−</sup> <sup>2</sup>)*G*2*ej*4*π*� *<sup>f</sup> <sup>d</sup>Tint* <sup>|</sup>

<sup>2</sup> = (*<sup>M</sup>* <sup>−</sup> <sup>1</sup>)*G*<sup>2</sup> + (*<sup>M</sup>* <sup>−</sup> <sup>2</sup>)*G*<sup>2</sup> <sup>+</sup> ... <sup>+</sup> *<sup>G</sup>*<sup>2</sup> <sup>=</sup> *<sup>M</sup>*(*<sup>M</sup>* <sup>−</sup> <sup>1</sup>)

2

*G*2*ej*2*πi*� *<sup>f</sup> <sup>d</sup>Tint*

*G*2*ej*2*πi*� *<sup>f</sup> <sup>d</sup>Tint* (37)


<sup>2</sup> + ... (41)

<sup>2</sup> *<sup>G</sup>*<sup>2</sup>

(36)

*<sup>i</sup>* of

(39)

*SGDC*

= *M* ∑ *m*=*i*

<sup>2</sup> = 

*SMGDC* =

*M*−1 ∑ *i*=1

Equation (39) shows that the residual carrier phase is still present in the *dGDC*. This fact causes an unpredictable loss, which depends on the specific value of � *f <sup>d</sup>*. To eliminate this loss, Modified Generalized Differential Combination (MGDC) technique (Ta et al., 2012) can be used, see Fig. 6(d). Following this technique, the decision variable of the MGDC technique is

By forming the decision variable in this way, the unpredictable loss caused by the residual carrier phase is canceled completely. However, the non-coherent integrations between all the

Note: for the GDC and MGDC techniques, the number of spans involved can vary from 1 to *M* − 1. By default, all (*M* − 1) possible spans are considered as in (40). If a different number

*M*−1 ∑ *i*=1

Then the decision variable of GDC (Fig. 6(c)) is

an arbitrary span-i (*Ai*) in (35) can be represented as

If the noise is neglected, (40) becomes


<sup>+</sup>|*G*2*ej*2*π*(*M*−1)� *<sup>f</sup> <sup>d</sup>Tint* <sup>|</sup>

*M*−1 ∑ *i*=1

*SMGDC* =

*AS i*|*τ*,� *f <sup>d</sup>*

� *f <sup>d</sup>* = � *f <sup>d</sup>*<sup>1</sup> = ... = � *f dM*

For the GDC technique, substituting (37) into (36), *SGDC* is computed

*SGDC* = |*D*|

*<sup>i</sup>* <sup>|</sup> <sup>=</sup> <sup>|</sup>(*<sup>M</sup>* <sup>−</sup> <sup>1</sup>)*G*2*ej*2*π*� *<sup>f</sup> <sup>d</sup>Tint* <sup>|</sup>

spans make the noise averaging process worse than for GDC.

the values of *Gm*, � ¯

with

L1C). The transmitted power is shared between two channels. Therefore, if the acquisition is performed on both channels, then the better sensitivity improvement can be obtained. In literature, (Mattos, 2005; Ta et al., 2010) use this approach for Galileo E1 OS signal acquisition.

Essentially, for the channel combining acquisition approach (common or different frequencies), in each involved channel, an acquisition strategy belonging to either the stand-alone or the external-aiding approach is performed. Then the acquisition outputs from all the channels are combined in different ways. In Section 6, the joint data/pilot acquisition strategies for Galileo E1 OS signal is introduced as an example for this approach.

#### **4. Stand-alone approach: Generalized differential combination technique**

#### **4.1 Technique description**

As seen in (19), the decision variable of the Conventional Differential Combination (CDC) technique is an accumulation of the products between two consecutive correlator outputs *Rm*, *Rm*−1. In a broader manner, the Generalized Differential Combination (GDC) has been introduced (Corazza & Pedone, 2007; Shanmugam et al., 2007). This technique considers the products of two consecutive correlator outputs as in CDC as well as the products of two correlator outputs at all sample distances or referred as all possible spans, see Fig. 6(a). Let us

Fig. 6. Differential Post Correlation Processing Architecture: (a) Differential operations; (b) Conventional Differential Combination (CDC); (c) Generalized Differential Combination (GDC); (d) Modified Generalized Differential Combination (MGDC)

define a span-i term as:

$$A\_i = \sum\_{m=i+1}^{M} R\_m R\_{m-i}^\* \tag{35}$$

Then the decision variable of GDC (Fig. 6(c)) is

$$S\_{GDC} \stackrel{\Delta}{=} \left| \sum\_{i=1}^{M-1} A\_i \right|^2 \tag{36}$$

Note that the CDC technique is in fact the GDC taking into account span-1 *A*<sup>1</sup> only, see Fig. 6(b). Basically, the GDC technique can be considered as a coherent integration of the differential combinations at different sample distances. Following the analyzes in (Ta et al., 2012), with small *M* (e.g. *M* ≤ *Tb*/*Tint*), in normal circumstances with normal user dynamic and frequency standards, the average frequency drift is small and tends to zero. Therefore, the values of *Gm*, � ¯ *fdm* in (6) are constant for all *<sup>m</sup>* <sup>∈</sup> [0, *<sup>M</sup>* <sup>−</sup> <sup>1</sup>]. The signal component *<sup>A</sup><sup>S</sup> <sup>i</sup>* of an arbitrary span-i (*Ai*) in (35) can be represented as

$$A\_{i\mid \tau, \triangle \overline{f}\_d}^S = \sum\_{m=i}^M G^2 e^{j2\pi i \triangle \overline{f}\_d T\_{int}} \tag{37}$$

with

16 Will-be-set-by-IN-TECH

L1C). The transmitted power is shared between two channels. Therefore, if the acquisition is performed on both channels, then the better sensitivity improvement can be obtained. In literature, (Mattos, 2005; Ta et al., 2010) use this approach for Galileo E1 OS signal acquisition. Essentially, for the channel combining acquisition approach (common or different frequencies), in each involved channel, an acquisition strategy belonging to either the stand-alone or the external-aiding approach is performed. Then the acquisition outputs from all the channels are combined in different ways. In Section 6, the joint data/pilot acquisition

strategies for Galileo E1 OS signal is introduced as an example for this approach.

**4. Stand-alone approach: Generalized differential combination technique**

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As seen in (19), the decision variable of the Conventional Differential Combination (CDC) technique is an accumulation of the products between two consecutive correlator outputs *Rm*, *Rm*−1. In a broader manner, the Generalized Differential Combination (GDC) has been introduced (Corazza & Pedone, 2007; Shanmugam et al., 2007). This technique considers the products of two consecutive correlator outputs as in CDC as well as the products of two correlator outputs at all sample distances or referred as all possible spans, see Fig. 6(a). Let us

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**4.1 Technique description**

5 5 5 50 50

define a span-i term as:

$$\begin{cases} \triangle \overline{f}\_d = \triangle \overline{f}\_{d\_1} = \dots = \triangle \overline{f}\_{d\_M} \\ \text{ } G = G\_1 = \dots = G\_M = \sqrt{2C} \mathcal{R} \{ \pi \} \text{sinc} (\triangle \overline{f}\_d T\_{\text{int}}) \end{cases} \tag{38}$$

For the GDC technique, substituting (37) into (36), *SGDC* is computed

$$\mathcal{S}\_{\rm GDC} = |D|^2 = \left| \sum\_{i=1}^{M-1} G^2 e^{j2\pi i \triangle \overline{f}\_d T\_{\rm int}} \right|^2 \tag{39}$$

Equation (39) shows that the residual carrier phase is still present in the *dGDC*. This fact causes an unpredictable loss, which depends on the specific value of � *f <sup>d</sup>*. To eliminate this loss, Modified Generalized Differential Combination (MGDC) technique (Ta et al., 2012) can be used, see Fig. 6(d). Following this technique, the decision variable of the MGDC technique is

$$S\_{MGDC} = \sum\_{i=1}^{M-1} |A\_i|. \tag{40}$$

If the noise is neglected, (40) becomes

$$S\_{\rm MGDC} = \sum\_{i=1}^{M-1} |A\_i^{\mathbb{S}}| = |(M-1)G^2 e^{i2\pi\triangle\overline{f}\_d T\_{\rm int}}|^2 + |(M-2)G^2 e^{i4\pi\triangle\overline{f}\_d T\_{\rm int}}|^2 + \dots \tag{41}$$

$$+ |G^2 e^{j2\pi(M-1)\triangle\overline{f}\_d T\_{\rm int}}|^2 = (M-1)G^2 + (M-2)G^2 + \dots + G^2 = \frac{M(M-1)}{2}G^2$$

By forming the decision variable in this way, the unpredictable loss caused by the residual carrier phase is canceled completely. However, the non-coherent integrations between all the spans make the noise averaging process worse than for GDC.

Note: for the GDC and MGDC techniques, the number of spans involved can vary from 1 to *M* − 1. By default, all (*M* − 1) possible spans are considered as in (40). If a different number

for GNSS Signal Acquisition 19

High Sensitivity Techniques for GNSS Signal Acquisition 21

MF is loaded with one full modified CM code. The modified CM code is obtained from the original CM code with every alternative sample being zero padded to account for the TM structure. The MF does not produce the correlation results equivalent to the full code period, i.e. *Tint* = 20 ms. Nevertheless, it provides *M* partial correlation results with *Tint* = *l* ms as in Fig. 7. It can be thought of as the partial acquisition process using *M* different local codes of 1-ms length. By setting the local codes in this way, the signal components of all *M* correlator outputs *R*1, ..., *RM* have the same sign. Therefore, the differential combination can be used among these *M* outputs without any loss from the data transition effect. These *M* correlator outputs are then directed to Post Correlation Signal Processing Block, which contains 3 differential combination solutions, namely CDC, GDC and MGDC, as presented in Section 4. The analytical expressions of the performance parameters of these techniques can

Summarizing the techniques introduced in the previous sections, there are five strategies that have to be investigated: non-coherent, CDC, GDC, MGDC and 20-ms coherent combination (full code acquisition). Fig. 8 shows the behavior of the detection probabilities of all the strategies when *Tint* = 1 ms, *Pf a* = 10−<sup>3</sup> and the signal strength (*C*/*N*0) varies. The 20-ms coherent technique, as expected, has the best performance. Among the others, all the differential post correlation processing techniques, i.e. GDC, MGDC, CDC, are better than the non-coherent one. The CDC technique taking into account only Span-1 provides the lowest improvement of 1 dB with respect to the non-coherent. The performance of MGDC with different numbers of spans involved (i.e. span size) is also shown in Fig. 8(a). It can be observed that as the span size increases, the detection capability also improves. For the highest span size (i.e. 19 in the figure), the MGDC can offer an advantage of more than 1 dB over the CDC as well as more than 2 dB over the non-coherent combination. These improvements are preserved even the worst case is considered as can be seen in Fig 8(b). Among the differential techniques, the GDC has the highest performance. If all the spans are considered, the GDC performance approaches that of the coherent one. However, this performance is only guaranteed when the residual carrier phase is known (i.e. the perfect case). In Fig. 8(b), the detection probability of the GDC technique reduces dramatically due to the residual carrier phase. Table 3 compares the simulation results of *TA* for the normal *Tint* ms *TA* (×105) ms Relative Savings 0.5 0.08527 97.15% 1 0.1624 94.5% 2 0.313 89.5% 5 0.769 74.3% 10 1.519 49.3% 20 2.996 0% Table 3. Reduction of Mean Acquisition Time by using MGDC at different partial coherent

integration times with respect to full 20-ms acquisition (*C*/*N*<sup>0</sup> = 23 dB-Hz)

can be achieved by shortening *Tint*.

outdoor operating range of signal power, i.e. above 32 dB-Hz. It can be observed that a significant saving in *TA* of MGDC (with respect to the full CM period correlation acquisition)

be found in (Ta et al., 2012).

**4.2.2 Performance analyses**

Fig. 7. L2C Partial acquisition using matched filter

of spans *i* (1 ≤ *i* ≤ *M* − 1) is used, in the following, the notations for the two techniques will be GDC(*i*) and MGDC(*i*).

#### **4.2 Application of technique to L2C signal**

In this Section, the MGDC technique is used to acquire GPS L2C signal. This signal is chosen because it employs a long PRN code period, which can be used to generate partial correlator outputs with the same sign. Hence, there is no combination loss due to data bit transitions in differential accumulation (see Section 3.2.3).

#### **4.2.1 L2C signal acquisition**

The L2C signal has advantages in interference mitigation due to its advanced PRN code format. This signal is composed of two codes, namely L2 CM and L2 CL. The L2 CM code is 20-ms long containing 10230 chips; while the L2 CL code has a period of 1.5 s with 767250 chips. The CM code is modulo-2 added to data (i.e. it modulates the data) and the resultant sequence of chips is time-multiplexed (TM) with CL code on a chip-by-chip basis. The individual CM and CL codes are clocked at 511.5 kHz while the composite L2C code has a frequency of 1.023 MHz. Code boundaries of CM and CL are aligned and each CL period contains exactly 75 CM periods. This TM L2C sequence modulates the L2 (1227.6 MHz) carrier (GPS-IS, 2006). The original L2C data rate is 25 bps but a half rate convolutional encoder is employed to transmit the data at 50 sps. Consequently, each data symbol matches the CM period of 20 ms.

With these specifications, the common signal representation in (1) is changed to

$$r[n] = \sqrt{2C} \{ d[n] \text{cm}[n+\tau] + c[n+\tau+kP] \} \cos[2\pi(f\_{\text{IF}} + f\_{\text{D}})nT\_{\text{S}} + \varphi] + n\_{W}[n] \tag{42}$$

where *c*m[*n*] and *c*l [*n*] are the received CM and CL codes respectively (samples); *θ* is the received signal delay; *P* refers to the number of samples in a full CM code period (i.e. 20 ms), 0 ≤ *k* ≤ 74 is an integer that gives the CL code delay relative to CM code.

Fig. 7 shows an architecture of the partial acquisition suitable for L2C CM signal. A segmented matched filter (MF) is used as a correlator (Dodds & Moher, 1995; Persson et al., 2001). The MF is loaded with one full modified CM code. The modified CM code is obtained from the original CM code with every alternative sample being zero padded to account for the TM structure. The MF does not produce the correlation results equivalent to the full code period, i.e. *Tint* = 20 ms. Nevertheless, it provides *M* partial correlation results with *Tint* = *l* ms as in Fig. 7. It can be thought of as the partial acquisition process using *M* different local codes of 1-ms length. By setting the local codes in this way, the signal components of all *M* correlator outputs *R*1, ..., *RM* have the same sign. Therefore, the differential combination can be used among these *M* outputs without any loss from the data transition effect. These *M* correlator outputs are then directed to Post Correlation Signal Processing Block, which contains 3 differential combination solutions, namely CDC, GDC and MGDC, as presented in Section 4. The analytical expressions of the performance parameters of these techniques can be found in (Ta et al., 2012).

#### **4.2.2 Performance analyses**

18 Will-be-set-by-IN-TECH

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[*n* + *τ* + *kP*]} cos[2*π*(*fIF* + *fD*)*nTS* + *ϕ*] + *nW*[*n*] (42)

[*n*] are the received CM and CL codes respectively (samples); *θ* is the

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5 5 5 50

of spans *i* (1 ≤ *i* ≤ *M* − 1) is used, in the following, the notations for the two techniques will

In this Section, the MGDC technique is used to acquire GPS L2C signal. This signal is chosen because it employs a long PRN code period, which can be used to generate partial correlator outputs with the same sign. Hence, there is no combination loss due to data bit transitions in

The L2C signal has advantages in interference mitigation due to its advanced PRN code format. This signal is composed of two codes, namely L2 CM and L2 CL. The L2 CM code is 20-ms long containing 10230 chips; while the L2 CL code has a period of 1.5 s with 767250 chips. The CM code is modulo-2 added to data (i.e. it modulates the data) and the resultant sequence of chips is time-multiplexed (TM) with CL code on a chip-by-chip basis. The individual CM and CL codes are clocked at 511.5 kHz while the composite L2C code has a frequency of 1.023 MHz. Code boundaries of CM and CL are aligned and each CL period contains exactly 75 CM periods. This TM L2C sequence modulates the L2 (1227.6 MHz) carrier (GPS-IS, 2006). The original L2C data rate is 25 bps but a half rate convolutional encoder is employed to transmit the data at 50 sps. Consequently, each data symbol matches the CM

received signal delay; *P* refers to the number of samples in a full CM code period (i.e. 20 ms),

Fig. 7 shows an architecture of the partial acquisition suitable for L2C CM signal. A segmented matched filter (MF) is used as a correlator (Dodds & Moher, 1995; Persson et al., 2001). The

With these specifications, the common signal representation in (1) is changed to

0 ≤ *k* ≤ 74 is an integer that gives the CL code delay relative to CM code.

76 76 76

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Fig. 7. L2C Partial acquisition using matched filter

**4.2 Application of technique to L2C signal**

differential accumulation (see Section 3.2.3).

2*C*{*d*[*n*]*c*m[*n* + *τ*] + *c*l

be GDC(*i*) and MGDC(*i*).

**4.2.1 L2C signal acquisition**

period of 20 ms.

*<sup>r</sup>*[*n*] = <sup>√</sup>

where *c*m[*n*] and *c*l

Summarizing the techniques introduced in the previous sections, there are five strategies that have to be investigated: non-coherent, CDC, GDC, MGDC and 20-ms coherent combination (full code acquisition). Fig. 8 shows the behavior of the detection probabilities of all the strategies when *Tint* = 1 ms, *Pf a* = 10−<sup>3</sup> and the signal strength (*C*/*N*0) varies. The 20-ms coherent technique, as expected, has the best performance. Among the others, all the differential post correlation processing techniques, i.e. GDC, MGDC, CDC, are better than the non-coherent one. The CDC technique taking into account only Span-1 provides the lowest improvement of 1 dB with respect to the non-coherent. The performance of MGDC with different numbers of spans involved (i.e. span size) is also shown in Fig. 8(a). It can be observed that as the span size increases, the detection capability also improves. For the highest span size (i.e. 19 in the figure), the MGDC can offer an advantage of more than 1 dB over the CDC as well as more than 2 dB over the non-coherent combination. These improvements are preserved even the worst case is considered as can be seen in Fig 8(b). Among the differential techniques, the GDC has the highest performance. If all the spans are considered, the GDC performance approaches that of the coherent one. However, this performance is only guaranteed when the residual carrier phase is known (i.e. the perfect case). In Fig. 8(b), the detection probability of the GDC technique reduces dramatically due to the residual carrier phase. Table 3 compares the simulation results of *TA* for the normal


Table 3. Reduction of Mean Acquisition Time by using MGDC at different partial coherent integration times with respect to full 20-ms acquisition (*C*/*N*<sup>0</sup> = 23 dB-Hz)

outdoor operating range of signal power, i.e. above 32 dB-Hz. It can be observed that a significant saving in *TA* of MGDC (with respect to the full CM period correlation acquisition) can be achieved by shortening *Tint*.

for GNSS Signal Acquisition 21

High Sensitivity Techniques for GNSS Signal Acquisition 23

In this step, the FFT-based circular correlation stage is used to quickly detect the best PRN selected on the basis of the predicted *C*/*N*<sup>0</sup> and elevation. The value of *Tcoh* is 10ms and; the number of *M* correlator outputs is then non-coherently combined to achieve a sufficient post-correlation SNR; *fD* =100 Hz (trade-off between frequency resolution and search

Code-phase and frequency offsets are caused by: (i) space displacement of test and reference antennas (mostly code-phase offset); (ii) the time offset between the reference receiver and test receiver clocks (code-phase offsets); and (iii) the uncertainty on the test receiver LO frequency (Doppler frequency offset). In this step, these offsets, which are the same for all the PRNs, can be computed by considering the difference of the preliminary estimates (from step 1) with

The offsets obtained with the strong PRN can be used to correct the assistance predictions and finely determine the code-phase/Doppler frequency of other PRNs at the last step (aided long coherent correlation), ensuring the best achievable post-correlation SNR by means of a low-complexity data wipe-off technique. At this step, the frequency bin size of *fD*,3 = 1/*Tcoh* (e.g. *fD*,3 = 1 Hz if *Tcoh* = 1 s) is used, over a frequency search space 100 Hz wide (i.e. the residual uncertainty from step 1), and a code-phase search range 6 chips wide. The knowledge of aiding data would allow for a narrower search space, but the acquisition has to

The code-phase resolution is as low as 1 sample (for both step 1 and 3). The reference signal bandwidth is *B* = 2.046 MHz chosen to match the main lobe (two-sided) of the GPS signal spectrum. Therefore the performed tests have been run with a sampling rate *fS* = 4.092 MHz

Fig. 9. Test bed architecture: reference chain (green) and test chain (blue)

complexity), while the code-phase search space spans over a full code period.

• Step 3 - Aided long coherent correlation with data wipe-off on weaker PRNs

account for possible residual errors between the true and predicted code phases.

• Step 1 - Preliminary fast detection of the strongest PRN

• Step 2 - Determination of the assistance offsets

those provided by the assistance data.

**5.2 Acquisition procedure**

#### **5. External aiding acquisition technique for indoor positioning**

In this section, a test-bed architecture, which is proposed by (Dovis et al., 2010), is introduced as an example of the external-aiding acquisition approach.

#### **5.1 Test-bed architecture**

The test-bed as seen in Fig. 9 includes two chains:

**Test receiver chain:** The main task of this chain is to collect a snapshot of the digitized GPS signal and sends it to a location server through a cellular communication channel. The chain consists of a GPS L1 front-end with the antenna at the test location. The RF front-end is connected to a PC which collects digital sample streams into binary files. The local oscillator is a rubidium (Rb) frequency standard (Datum8040, 1998) running the front-end through a waveform synthesizer (HP, 1990).

**Reference receiver chain:** The main task of this chain is to perform the HS acquisition process taking advantage of the available assistance information. The chain consists of a reference GPS receiver which processes open-sky signals from a fixed (known) location and provides measurements to an assistance server. The latter provides the necessary aiding information to the HS acquisition engine and the GPS Time indication for the synchronization of the sample-stream recorder, performed before starting each signal collection session. The synchronization process introduces an uncertainty on the GPS Time tags, since it is performed by the software running at the PC, which is assumed to be 2s as in this work.

The assistance server is a software tool developed at Telecom Italia Laboratories to support several test activities on Assisted GPS (A-GPS) technologies. It collects data from the reference receiver and generates time-tagged log files with several kind of assistance information to be provided to the HS acquisition engine. Each line of the log file, for each visible SV, contains code-phase, Doppler frequency and Doppler rate estimates.

Fig. 8. Detection probability (*Pd*) of all post-correlation processing techniques at different signal power levels in (a) perfect case: ¯ *fd* <sup>=</sup> 0 Hz; (b) worst case: ¯ *fd* = 12.5 Hz for coherent combination (i.e. full 20-ms acquisition) and ¯ *fd* = 250 Hz for the other techniques.

Fig. 9. Test bed architecture: reference chain (green) and test chain (blue)

#### **5.2 Acquisition procedure**

20 Will-be-set-by-IN-TECH

In this section, a test-bed architecture, which is proposed by (Dovis et al., 2010), is introduced

**Test receiver chain:** The main task of this chain is to collect a snapshot of the digitized GPS signal and sends it to a location server through a cellular communication channel. The chain consists of a GPS L1 front-end with the antenna at the test location. The RF front-end is connected to a PC which collects digital sample streams into binary files. The local oscillator is a rubidium (Rb) frequency standard (Datum8040, 1998) running the front-end through a

**Reference receiver chain:** The main task of this chain is to perform the HS acquisition process taking advantage of the available assistance information. The chain consists of a reference GPS receiver which processes open-sky signals from a fixed (known) location and provides measurements to an assistance server. The latter provides the necessary aiding information to the HS acquisition engine and the GPS Time indication for the synchronization of the sample-stream recorder, performed before starting each signal collection session. The synchronization process introduces an uncertainty on the GPS Time tags, since it is performed

The assistance server is a software tool developed at Telecom Italia Laboratories to support several test activities on Assisted GPS (A-GPS) technologies. It collects data from the reference receiver and generates time-tagged log files with several kind of assistance information to be provided to the HS acquisition engine. Each line of the log file, for each visible SV, contains

by the software running at the PC, which is assumed to be 2s as in this work.

 

Fig. 8. Detection probability (*Pd*) of all post-correlation processing techniques at different

 

*fd* <sup>=</sup> 0 Hz; (b) worst case: ¯

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&('

**5. External aiding acquisition technique for indoor positioning**

as an example of the external-aiding acquisition approach.

The test-bed as seen in Fig. 9 includes two chains:

**5.1 Test-bed architecture**

waveform synthesizer (HP, 1990).

 

\$#\$&#( - - - \$"'-)!!%

 

 

(( \$#&\$ ! (\*-

 

(a)

coherent combination (i.e. full 20-ms acquisition) and ¯

signal power levels in (a) perfect case: ¯

 -+

• Step 1 - Preliminary fast detection of the strongest PRN

In this step, the FFT-based circular correlation stage is used to quickly detect the best PRN selected on the basis of the predicted *C*/*N*<sup>0</sup> and elevation. The value of *Tcoh* is 10ms and; the number of *M* correlator outputs is then non-coherently combined to achieve a sufficient post-correlation SNR; *fD* =100 Hz (trade-off between frequency resolution and search complexity), while the code-phase search space spans over a full code period.

• Step 2 - Determination of the assistance offsets

Code-phase and frequency offsets are caused by: (i) space displacement of test and reference antennas (mostly code-phase offset); (ii) the time offset between the reference receiver and test receiver clocks (code-phase offsets); and (iii) the uncertainty on the test receiver LO frequency (Doppler frequency offset). In this step, these offsets, which are the same for all the PRNs, can be computed by considering the difference of the preliminary estimates (from step 1) with those provided by the assistance data.

• Step 3 - Aided long coherent correlation with data wipe-off on weaker PRNs

The offsets obtained with the strong PRN can be used to correct the assistance predictions and finely determine the code-phase/Doppler frequency of other PRNs at the last step (aided long coherent correlation), ensuring the best achievable post-correlation SNR by means of a low-complexity data wipe-off technique. At this step, the frequency bin size of *fD*,3 = 1/*Tcoh* (e.g. *fD*,3 = 1 Hz if *Tcoh* = 1 s) is used, over a frequency search space 100 Hz wide (i.e. the residual uncertainty from step 1), and a code-phase search range 6 chips wide. The knowledge of aiding data would allow for a narrower search space, but the acquisition has to account for possible residual errors between the true and predicted code phases.

The code-phase resolution is as low as 1 sample (for both step 1 and 3). The reference signal bandwidth is *B* = 2.046 MHz chosen to match the main lobe (two-sided) of the GPS signal spectrum. Therefore the performed tests have been run with a sampling rate *fS* = 4.092 MHz

for GNSS Signal Acquisition 23

High Sensitivity Techniques for GNSS Signal Acquisition 25

the lab window, so to have and indication of the available GPS constellation. The distance between the antenna of the auxiliary receiver and the test indoor antenna is ≤ 10 m. The sky

plot relative to this case-study is depicted in Fig. 10. PRN6 and PRN30 are considered in this section. The assistance log is summarized in Table 4. Then the 3-step procedure in Section 5.2 is applied. Firstly, the strongest signal, which is PRN6 as seen in Table 4, is determined. After that, FFT-based acquisition is activated to search for PRN6 in the signal snapshot collected in indoor environment. Then the following procedure has been used to determine the assistance offsets and the corrected aiding data. For PRN6 the code phase from the assistance log is *τ<sup>a</sup>*

=120 chips. The preliminary fast acquisition on PRN6 estimated a code-phase *<sup>τ</sup><sup>p</sup>*

<sup>6</sup> <sup>−</sup> *<sup>τ</sup><sup>a</sup>*

As the signal snapshot is the same for the two PRNs, there are no time drifts to take into

<sup>30</sup> = *δτ*<sup>6</sup> + *<sup>τ</sup><sup>a</sup>*

The aiding parameters are used for acquiring the weaker satellite, PRN30, in indoor environment. The correlation results are shown in Fig. 11 and in Table 5, it can be noticed that the 3 dB rule still holds. In fact The signal of PRN 30 pass through the roof and the walls of the laboratory. Thus, it was good realizations of typical indoor signals and and it is detected

In this section, the channel combination approach to improve the sensitivity of the acquisition is described. The considered signal is Galileo E1 Open Service signal. The current definition

*δτ*<sup>6</sup> <sup>=</sup> *<sup>τ</sup><sup>p</sup>*

It should be noted that because the full code length is 1023 chips, therefore, *<sup>τ</sup><sup>p</sup>*

The same approach is used to compute the aiding value of Doppler frequency, *f*

**6. Channel combination approach: Joint data/pilot acquisition strategies**

<sup>30</sup> <sup>=</sup> *δτ*<sup>30</sup> <sup>+</sup> *<sup>τ</sup><sup>a</sup>*

kHz. Finally, after step 2, the aiding parameters are listed in Table 5.

6

<sup>6</sup> = 1000.3

<sup>30</sup> can also equal

*<sup>D</sup>*,30 = −0.6025

*p*

<sup>6</sup> = 880.3 chips (47)

<sup>30</sup> = 1060.3 chips (48)

Fig. 10. Skyplot, indoor, Rb

account. Therefore:

by assisted coherent correlation.

to 37.3 chips.

chips (Table 6). The code-phase offset is:

*τp*

designed to meet the Nyquist criterion. Finally the local code rate taking into account the Doppler effect, as presented in (27), is used.

#### **5.3 Data wipe-off mechanism**

In order to increase the coherent integration over the data bit duration (i.e. 20 ms), the acquisition stage performs data wipe-off process. Basically, the conventional data wipe-off process is done as follows

$$R = \frac{1}{N} \sum\_{n=1}^{MN} \hat{d}[n] \cdot \{r[n]\hat{c}[n+\hat{\tau}]e^{j(2\pi(f\_{IF}+\hat{f\_D}))nT\_S}\}\tag{43}$$

with {*d* [*n*]| *n* = 1...*MN*} being the data sequence provided by the assisted data. However, at the acquisition stage, the signal snap-shot and the assisted data are not synchronized. Therefore, in order to determine the correct bit sequence for the signal snap-shot, the acquisition stage needs to test all possible data sequence in a predetermined uncertainty. Then the maximum likelihood estimator is used for decision. Hence, it can be said that the acquisition stage in this scenario searches for the presence of a desired signal on 4-dimensions, namely: PRN, code-phase, frequency and bit-phase (i.e. 4D search-space).

In fact, this mechanism requires an unacceptable computational effort for a single position fix, because for each bit-phase (i.e. a data bit sequence candidate), the whole search-space must be re-computed. As a result, the number of elementary steps (i.e. multiply&add) is

$$(T\_{\rm{coh}} \cdot f\_{\rm{S}}) \times (N\_{\rm{cp}} \cdot N\_{f}) \times N\_{\rm{bit}-seq} = 4.092 \cdot 10^{8} \cdot N\_{\rm{cp}} \cdot N\_{f} \tag{44}$$

with *Ncp*, *Nf* , *Nbit*−*seq* being the numbers of code-phase, Doppler frequency and bit-phase bins respectively; *fS* = 4.092 MHz and *Tcoh* = 1 s.

However, (43) can be rewritten as

$$R = \sum\_{m=1}^{M} \widehat{d}\_{m} R\_{m} \tag{45}$$

where *Rm* is partial correlation value with representation in (5). With this approach, the acquisition stage can compute *R*1, *R*2, ..., *RM* then save these values for testing with all possible values of bit-phase. This approach in fact utilizes the coherent combination presented in (16). For this mechanism, the number of elementary steps is

$$\left[M(f\_S \cdot T\_{\rm coh\_1}) + M \cdot N\_{\rm bit-seq}\right] \cdot N\_{\rm cp}N\_f = 4.192 \cdot 10^6 \cdot N\_{\rm cp} \cdot N\_f \tag{46}$$

with *M* being the number of partial correlations obtained after 1-ms coherent integration time (*Tcoh*<sup>1</sup> ). From (44) and (46), the computational complexity of the partial correlation approach has a reduction of approximately 2 orders of magnitude with respect to the conventional one.

#### **5.4 Performance analyses**

This section demonstrates the application of the test-bed for indoor signal acquisition. The required integration time for indoor signals is longer than for outdoor ones. The sky plot, see Fig. 10, has been generated by means of an auxiliary receiver with the antenna placed out of the lab window, so to have and indication of the available GPS constellation. The distance between the antenna of the auxiliary receiver and the test indoor antenna is ≤ 10 m. The sky

Fig. 10. Skyplot, indoor, Rb

22 Will-be-set-by-IN-TECH

designed to meet the Nyquist criterion. Finally the local code rate taking into account the

In order to increase the coherent integration over the data bit duration (i.e. 20 ms), the acquisition stage performs data wipe-off process. Basically, the conventional data wipe-off

[*n*] · {*r*[*n*]*<sup>c</sup>*[*<sup>n</sup>* <sup>+</sup> *<sup>τ</sup>*]*ej*(2*π*(*fIF*+*<sup>f</sup>*

[*n*]| *n* = 1...*MN*} being the data sequence provided by the assisted data. However, at the acquisition stage, the signal snap-shot and the assisted data are not synchronized. Therefore, in order to determine the correct bit sequence for the signal snap-shot, the acquisition stage needs to test all possible data sequence in a predetermined uncertainty. Then the maximum likelihood estimator is used for decision. Hence, it can be said that the acquisition stage in this scenario searches for the presence of a desired signal on 4-dimensions,

In fact, this mechanism requires an unacceptable computational effort for a single position fix, because for each bit-phase (i.e. a data bit sequence candidate), the whole search-space must

with *Ncp*, *Nf* , *Nbit*−*seq* being the numbers of code-phase, Doppler frequency and bit-phase

*M* ∑ *m*=1 *d* 

where *Rm* is partial correlation value with representation in (5). With this approach, the acquisition stage can compute *R*1, *R*2, ..., *RM* then save these values for testing with all possible values of bit-phase. This approach in fact utilizes the coherent combination presented in (16).

with *M* being the number of partial correlations obtained after 1-ms coherent integration time (*Tcoh*<sup>1</sup> ). From (44) and (46), the computational complexity of the partial correlation approach has a reduction of approximately 2 orders of magnitude with respect to the conventional one.

This section demonstrates the application of the test-bed for indoor signal acquisition. The required integration time for indoor signals is longer than for outdoor ones. The sky plot, see Fig. 10, has been generated by means of an auxiliary receiver with the antenna placed out of

(*Tcoh* · *fS*) <sup>×</sup> (*Ncp* · *Nf*) <sup>×</sup> *Nbit*−*seq* <sup>=</sup> 4.092 · 108 · *Ncp* · *Nf* (44)

[*M*(*fS* · *Tcoh*<sup>1</sup> ) + *<sup>M</sup>* · *Nbit*−*seq*] · *NcpNf* <sup>=</sup> 4.192 · <sup>10</sup><sup>6</sup> · *Ncp* · *Nf* (46)

be re-computed. As a result, the number of elementary steps (i.e. multiply&add) is

*R* =

*<sup>D</sup>*))*nTS* } (43)

*mRm* (45)

Doppler effect, as presented in (27), is used.

*<sup>R</sup>* <sup>=</sup> <sup>1</sup> *N*

bins respectively; *fS* = 4.092 MHz and *Tcoh* = 1 s.

For this mechanism, the number of elementary steps is

However, (43) can be rewritten as

**5.4 Performance analyses**

*MN* ∑ *n*=1 *d*

namely: PRN, code-phase, frequency and bit-phase (i.e. 4D search-space).

**5.3 Data wipe-off mechanism**

process is done as follows

with {*d*

plot relative to this case-study is depicted in Fig. 10. PRN6 and PRN30 are considered in this section. The assistance log is summarized in Table 4. Then the 3-step procedure in Section 5.2 is applied. Firstly, the strongest signal, which is PRN6 as seen in Table 4, is determined. After that, FFT-based acquisition is activated to search for PRN6 in the signal snapshot collected in indoor environment. Then the following procedure has been used to determine the assistance offsets and the corrected aiding data. For PRN6 the code phase from the assistance log is *τ<sup>a</sup>* 6 =120 chips. The preliminary fast acquisition on PRN6 estimated a code-phase *<sup>τ</sup><sup>p</sup>* <sup>6</sup> = 1000.3 chips (Table 6). The code-phase offset is:

$$
\delta \tau\_6 = \tau\_6^p - \tau\_6^a = 880.3 \text{ chips} \tag{47}
$$

As the signal snapshot is the same for the two PRNs, there are no time drifts to take into account. Therefore:

$$
\tau\_{30}^p = \delta \tau\_{30} + \tau\_{30}^a = \delta \tau\_6 + \tau\_{30}^a = 1060.3 \text{ chips} \tag{48}
$$

It should be noted that because the full code length is 1023 chips, therefore, *<sup>τ</sup><sup>p</sup>* <sup>30</sup> can also equal to 37.3 chips.

The same approach is used to compute the aiding value of Doppler frequency, *f p <sup>D</sup>*,30 = −0.6025 kHz. Finally, after step 2, the aiding parameters are listed in Table 5.

The aiding parameters are used for acquiring the weaker satellite, PRN30, in indoor environment. The correlation results are shown in Fig. 11 and in Table 5, it can be noticed that the 3 dB rule still holds. In fact The signal of PRN 30 pass through the roof and the walls of the laboratory. Thus, it was good realizations of typical indoor signals and and it is detected by assisted coherent correlation.

#### **6. Channel combination approach: Joint data/pilot acquisition strategies**

In this section, the channel combination approach to improve the sensitivity of the acquisition is described. The considered signal is Galileo E1 Open Service signal. The current definition

Fig. 12. Joint data/pilot acquisition architectures: (a) Dual Channels - DC; (b) (B×C); (c) Assisted (B-C); (d) Summing Combination - SuC; (e) Comparing Combination - CC

The correlation process performs on both channels to produce *RB*,*<sup>m</sup>* and *RC*,*<sup>m</sup>* (5). After that,

High Sensitivity Techniques for GNSS Signal Acquisition 27

This strategy sums the square envelopes from the two channels, see Fig. 12(a). The decision

<sup>2</sup>) =


*C*,*i*

*M* ∑ *m*=1

(|*SB*,*m*|

<sup>2</sup> <sup>+</sup> <sup>|</sup>*SC*,*m*<sup>|</sup>

<sup>2</sup>) (50)

}|<sup>2</sup> (51)

<sup>2</sup> <sup>+</sup> <sup>|</sup>*RC*,*m*<sup>|</sup>

*M* ∑ *i*=1

This strategy can be seen as another realization of the conventional differential technique presented in Section 3.2.3. The correlator output in a channel is combined with the one from the other channel instead of the delayed copy of itself as in the conventional differential

**6.1 Joint data/pilot acquisition strategies**

*SDCK* =

• Dual Channel (DC):

variable is

• (B×C):

technique.

• Assisted (B-C):

these correlation values are combined as follows:

*M* ∑ *m*=1

In this strategy [see Fig. 12(b)], the decision variable is

(|*RB*, *m*|

*<sup>S</sup>*(*B*×*C*)*<sup>M</sup>* <sup>=</sup>


Table 4. Assistance log, indoor, Rb


Table 5. Aiding data, indoor, Rb

Fig. 11. Long coherent correlation, indoor, Rb, *Tint* = 2000 ms, PRN30

of the this signal (GalileoICD, 2008) includes data (B) and pilot (C) channels which are multiplexed by Coherent Adaptive Sub-carrier Modulation (CASM) (Dafesh et al., 1999). Each channels shares 50 % of the total transmitted power. To represent this signal, the common representation in (1) is changed to

$$r[n] = \frac{1}{\sqrt{2}}\sqrt{2\mathbf{C}}\left(d[n+\tau]b[n+\tau] - c\_{2nd}[n+\tau]c[n+\tau]\right)\cos\left(2\pi(f\_{\text{IF}} + f\_{\text{D}})nT\_{\text{S}} + \phi\right) + m\_{\text{W}}[n] \tag{49}$$

*b*[*n*], *c*[*n*] are respectively the 4-ms primary PRN codes of the data (B) and pilot (C) channels modulated by BOC(1,1) scheme; *d*[*n*] is the navigation data in the B channel; *c*2*nd*[*n*] is the secondary code, which together with *c*[*n*] form a 100-ms tiered code for the C channel (GalileoICD, 2008). Basically, the conventional acquisition stage in Fig. 1 can perform on either B or C channels. This strategy is referred here as Single Channel (SC). However, SC also implies a waste of half of the real capability. Therefore, joint data/pilot acquisition strategies are introduced to utilize the full potential of the E1 OS signal (Mattos, 2005; Ta et al., 2010). In the followings, these strategies are described together with the performance evaluation.

24 Will-be-set-by-IN-TECH

26 Global Navigation Satellite Systems – Signal, Theory and Applications

PRN Elevation (*o*) *C*/*N*<sup>0</sup> (dB-Hz) Code-phase (chips) *fD* (Hz) *rD*(Hz/s) 30 60 - 180 -635.0 -0.4 6 73 32.7 120 1067.5 -0.5

> (*pred*) *<sup>D</sup>* (kHz)

PRN Aiding source *τ*(*pred*) (chips) *f*

Fig. 11. Long coherent correlation, indoor, Rb, *Tint* = 2000 ms, PRN30

of the this signal (GalileoICD, 2008) includes data (B) and pilot (C) channels which are multiplexed by Coherent Adaptive Sub-carrier Modulation (CASM) (Dafesh et al., 1999). Each channels shares 50 % of the total transmitted power. To represent this signal, the common

*b*[*n*], *c*[*n*] are respectively the 4-ms primary PRN codes of the data (B) and pilot (C) channels modulated by BOC(1,1) scheme; *d*[*n*] is the navigation data in the B channel; *c*2*nd*[*n*] is the secondary code, which together with *c*[*n*] form a 100-ms tiered code for the C channel (GalileoICD, 2008). Basically, the conventional acquisition stage in Fig. 1 can perform on either B or C channels. This strategy is referred here as Single Channel (SC). However, SC also implies a waste of half of the real capability. Therefore, joint data/pilot acquisition strategies are introduced to utilize the full potential of the E1 OS signal (Mattos, 2005; Ta et al., 2010). In the followings, these strategies are described together with the performance evaluation.

2*C* (*d*[*n* + *τ*]*b*[*n* + *τ*] − *c*2*nd*[*n* + *τ*]*c*[*n* + *τ*]) cos (2*π*(*fIF* + *fD*)*nTS* + *φ*) + *nW*[*n*]

6 FFT 1000.3 1.1 30 Assistance server 1060.3 or 37.3 -0.6025

Table 4. Assistance log, indoor, Rb

Table 5. Aiding data, indoor, Rb

representation in (1) is changed to

*<sup>r</sup>*[*n*] = <sup>1</sup> √2 √

Fig. 12. Joint data/pilot acquisition architectures: (a) Dual Channels - DC; (b) (B×C); (c) Assisted (B-C); (d) Summing Combination - SuC; (e) Comparing Combination - CC

#### **6.1 Joint data/pilot acquisition strategies**

The correlation process performs on both channels to produce *RB*,*<sup>m</sup>* and *RC*,*<sup>m</sup>* (5). After that, these correlation values are combined as follows:

• Dual Channel (DC):

This strategy sums the square envelopes from the two channels, see Fig. 12(a). The decision variable is

$$S\_{D\mathbb{C}^K} = \sum\_{m=1}^M \left( |R\_{B'}m|^2 + |R\_{\mathbb{C},m}|^2 \right) = \sum\_{m=1}^M \left( |S\_{B,m}|^2 + |S\_{\mathbb{C},m}|^2 \right) \tag{50}$$

• (B×C):

(49)

In this strategy [see Fig. 12(b)], the decision variable is

$$\mathcal{S}\_{(B\times\mathbb{C})^{M}} = \sum\_{i=1}^{M} |\{\mathcal{R}\_{B,i} \cdot \mathcal{R}\_{\mathbb{C},i}^{\*}\}|^{2} \tag{51}$$

This strategy can be seen as another realization of the conventional differential technique presented in Section 3.2.3. The correlator output in a channel is combined with the one from the other channel instead of the delayed copy of itself as in the conventional differential technique.

• Assisted (B-C):

for GNSS Signal Acquisition 27

High Sensitivity Techniques for GNSS Signal Acquisition 29

The analytical expressions of the performance parameters of these strategies are presented in

Fig. 13 clearly shows the improvement of the joint data/pilot strategies over the conventional SC. The benchmark values *Pf a* = 10−<sup>3</sup> and *Pd* = 0.9 for the hypothesis testing in GNSS receivers are used to quantitatively estimate the improvement. When only one full code

*<sup>S</sup>*(*B*−*C*),*m*, *<sup>S</sup>*(*B*+*C*),*<sup>m</sup>*

 

 &

 

 

**2.8 (dB) 1.8 (dB)**

## !" #%


Fig. 13. Detection probability of all the strategies vs. *C*/*N*<sup>0</sup> values when *Pf a* = 10−3: (a)

strategies, the difference in *Pd* is small, but one still can realize that CC is the best one.

period is considered (i.e. *M* = 1), as shown in Fig. 13(a), the joint data/pilot strategies holds the sensitivity enhancement ∼ 3 dB over the conventional SC. Among the joint strategies, the assisted (B-C) outperforms the others, because the assistance data always guarantees the local generated signal matching the most to the received one. As for the other stand-alone joint

When *K* = 50, the assisted (B-C) is far better than the others, because the coherent combination applied in this strategy brings more performance improvement than the other strategies using the non-coherent technique suffering from the squaring loss phenomenon. This loss also reduces the enhancement (from 2.8 dB to 1.8) dB) of the stand-alone joint strategies with respect to the SC, see Fig. 13(b). Among the stand-alone joint strategies, in this scenario, DC takes the position of CC to be the best. While (B×C) degrades significantly, because unlike *K* = 1, for *K* > 1, to secure the accumulation, the absolute values of the differential operation's outputs are used in the non-coherent combination. This fact makes the averaging

Fig. 14 shows the *TA* values of all the strategies. It should be note that *TA* simultaneously consider the influences of both the computational complexity and the sensitivity of a strategy.

 



chosen to be the decision variable

(Ta et al., 2010).

**6.2 Performance analyses**

 

**Joint Strategies**

 

*M* = 1; (b) *M* = 50

not thorough.

 

## !" #%

 

 &

*SCCM* =


*M* ∑ *m*=1

max

The baseband E1 OS signal has the form [*d*(*t*)*b*(*t*) − *c*2*nd* (*t*)*c*(*t*)]. Due to the bi-polar nature of the data and secondary codes, the digital received baseband signal in each code period is always in one of the two representations

$$|b[n] - c[n]| \text{ or } |b[n] + c[n]| \tag{52}$$

This fact paves the way for a new strategy using one of the two equivalent codes (¯ *<sup>b</sup>*[*n*] <sup>−</sup> *<sup>c</sup>*¯[*n*]) or (¯ *b*[*n*] + *c*¯[*n*]) as the local code with the decision depending on the signal representation. Consequently, the two new equivalent channels (B-C) and (B+C) are defined. At a time instance, without the availability of an external-aiding source, because of the unknown navigation data bit, the acquisition stage cannot know the correct representation of the received signal, i.e. (B-C) or (B+C). In addition, the two new equivalent codes are orthogonal and still preserve the properties of the PRN codes (Ta et al., 2010). Therefore, if the chosen equivalent local code is incorrect, the correlation value in the equivalent channel might be null although the tentative parameters (i.e. PRN number, Doppler and code delay) are correct, because of the unknown data bit sign. Hence, the availability of an external-aiding source is crucial.

Without loss of generality, let us assume that the external-aiding source assures the signal structure is (*b*[*n*] − *c*[*n*]), therefore, the (B-C) strategy is applied, see Fig. 12(c). The decision variable of the assisted (B-C) is

$$\mathcal{S}\_{(B-C)^M} \stackrel{\Delta}{=} \left| R\_{(B-C)^M} \right|^2 = \left| \sum\_{m=1}^M R\_{(B-C),m} \right|^2 \tag{53}$$

Note that: for this external-aiding scenario, the coherent combination is used.

However, in one full primary code period, the signal can be only in one of the two representations in (52), it is worth to test both the strategies [i.e. (B-C) and (B+C)] and combine their results. This leads to two new strategies so-called Summing Combination and Comparing Combination.

• Summing Combination (SuC):

In this strategy (see Fig. 12(d)), the (B-C) and (B+C) strategies are simultaneously performed. The square envelope outputs are summed up to form the new decision variable

$$\mathcal{S}\_{\text{SuC}} = \mathcal{S}\_{(B-\mathbb{C})} + \mathcal{S}\_{(B+\mathbb{C})} = |R\_{(B-\mathbb{C})}|^2 + |R\_{(B-\mathbb{C})}|^2 = 2(|R\_B|^2 + |R\_{\mathbb{C}}|^2) \tag{54}$$

In this way, the overall decision variable is no longer affected by the unknown polarity of the data and secondary codes of the received signal. However, multiplying the decision variable by any coefficient does not affect the ultimate performance of a strategy because the signal and the noise powers are increased by the same rate. Therefore, the SuC strategy shares the performance with the DC strategy. For this reason, in the following sections, only the DC strategy is considered.

• Comparing Combination (CC):

This strategy (see Fig. 12(e)) uses a comparator instead of the adder as in the SuC strategy to combine the square envelope outputs of the two equivalent channels. The larger value is chosen to be the decision variable

26 Will-be-set-by-IN-TECH

The baseband E1 OS signal has the form [*d*(*t*)*b*(*t*) − *c*2*nd* (*t*)*c*(*t*)]. Due to the bi-polar nature of the data and secondary codes, the digital received baseband signal in each code period is

This fact paves the way for a new strategy using one of the two equivalent codes

representation. Consequently, the two new equivalent channels (B-C) and (B+C) are defined. At a time instance, without the availability of an external-aiding source, because of the unknown navigation data bit, the acquisition stage cannot know the correct representation of the received signal, i.e. (B-C) or (B+C). In addition, the two new equivalent codes are orthogonal and still preserve the properties of the PRN codes (Ta et al., 2010). Therefore, if the chosen equivalent local code is incorrect, the correlation value in the equivalent channel might be null although the tentative parameters (i.e. PRN number, Doppler and code delay) are correct, because of the unknown data bit sign. Hence, the availability of an external-aiding

Without loss of generality, let us assume that the external-aiding source assures the signal structure is (*b*[*n*] − *c*[*n*]), therefore, the (B-C) strategy is applied, see Fig. 12(c). The decision

> 2 =

However, in one full primary code period, the signal can be only in one of the two representations in (52), it is worth to test both the strategies [i.e. (B-C) and (B+C)] and combine their results. This leads to two new strategies so-called Summing Combination and

In this strategy (see Fig. 12(d)), the (B-C) and (B+C) strategies are simultaneously performed.

In this way, the overall decision variable is no longer affected by the unknown polarity of the data and secondary codes of the received signal. However, multiplying the decision variable by any coefficient does not affect the ultimate performance of a strategy because the signal and the noise powers are increased by the same rate. Therefore, the SuC strategy shares the performance with the DC strategy. For this reason, in the following sections, only the DC

This strategy (see Fig. 12(e)) uses a comparator instead of the adder as in the SuC strategy to combine the square envelope outputs of the two equivalent channels. The larger value is

*M* ∑ *m*=1

<sup>2</sup> <sup>+</sup> <sup>|</sup>*R*(*B*−*C*)<sup>|</sup>

*<sup>R</sup>*(*B*−*C*),*<sup>m</sup>*

 

<sup>2</sup> <sup>=</sup> <sup>2</sup>(|*RB*<sup>|</sup>

<sup>2</sup> <sup>+</sup> <sup>|</sup>*RC*<sup>|</sup>

<sup>2</sup>) (54)

2

(53)

*b*[*n*] + *c*¯[*n*]) as the local code with the decision depending on the signal


always in one of the two representations

(¯

*<sup>b</sup>*[*n*] <sup>−</sup> *<sup>c</sup>*¯[*n*]) or (¯

source is crucial.

variable of the assisted (B-C) is

Comparing Combination.

strategy is considered.

• Comparing Combination (CC):

• Summing Combination (SuC):

*<sup>S</sup>*(*B*−*C*)*<sup>M</sup>*

*SSuC* <sup>=</sup> *<sup>S</sup>*(*B*−*C*) <sup>+</sup> *<sup>S</sup>*(*B*+*C*) <sup>=</sup> <sup>|</sup>*R*(*B*−*C*)<sup>|</sup>

 *<sup>R</sup>*(*B*−*C*)*<sup>M</sup>*

Note that: for this external-aiding scenario, the coherent combination is used.

The square envelope outputs are summed up to form the new decision variable

$$S\_{\mathbb{C}\mathbb{C}^M} = \sum\_{m=1}^M \max\left\{ \mathbb{S}\_{(\mathbb{B}-\mathbb{C}), m'} \mathbb{S}\_{(\mathbb{B}+\mathbb{C}), m} \right\} \tag{55}$$

The analytical expressions of the performance parameters of these strategies are presented in (Ta et al., 2010).

#### **6.2 Performance analyses**

Fig. 13 clearly shows the improvement of the joint data/pilot strategies over the conventional SC. The benchmark values *Pf a* = 10−<sup>3</sup> and *Pd* = 0.9 for the hypothesis testing in GNSS receivers are used to quantitatively estimate the improvement. When only one full code

Fig. 13. Detection probability of all the strategies vs. *C*/*N*<sup>0</sup> values when *Pf a* = 10−3: (a) *M* = 1; (b) *M* = 50

period is considered (i.e. *M* = 1), as shown in Fig. 13(a), the joint data/pilot strategies holds the sensitivity enhancement ∼ 3 dB over the conventional SC. Among the joint strategies, the assisted (B-C) outperforms the others, because the assistance data always guarantees the local generated signal matching the most to the received one. As for the other stand-alone joint strategies, the difference in *Pd* is small, but one still can realize that CC is the best one.

When *K* = 50, the assisted (B-C) is far better than the others, because the coherent combination applied in this strategy brings more performance improvement than the other strategies using the non-coherent technique suffering from the squaring loss phenomenon. This loss also reduces the enhancement (from 2.8 dB to 1.8) dB) of the stand-alone joint strategies with respect to the SC, see Fig. 13(b). Among the stand-alone joint strategies, in this scenario, DC takes the position of CC to be the best. While (B×C) degrades significantly, because unlike *K* = 1, for *K* > 1, to secure the accumulation, the absolute values of the differential operation's outputs are used in the non-coherent combination. This fact makes the averaging not thorough.

Fig. 14 shows the *TA* values of all the strategies. It should be note that *TA* simultaneously consider the influences of both the computational complexity and the sensitivity of a strategy.

for GNSS Signal Acquisition 29

High Sensitivity Techniques for GNSS Signal Acquisition 31

Choi, I. H., Park, S. H., Cho, D. J., Yun, S. J., Kim, Y. B. & Lee, S. J. (2002). A Novel Weak Signal

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GalileoICD (2008). Galileo Open Service, Signal In Space Interface Control Document Draft 1, *Technical report*, European GNSS Supervisory Authority / European Space Agency. Gernot, C., Keefe, K. O. & Lachapelle, G. (2008). Comparison Of L1 C/A L2C Combined Acquisition Techniques, *Proceedings of ENC-GNSS 2008, Toulouse, France*. GPS-IS (2006). Navstar GPS Interface Specification IS-GPS-200 revision D, *Technical report*,

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Fig. 14. Mean acquisition time *TA* vs. *C*/*N*<sup>0</sup> values when *Pf a* = 10−3: (a) *M* = 1; (b) *M* = 50

For all *C*/*N*<sup>0</sup> and *K* values, (B-C) results in the smallest *TA*, because of its high sensitivity (i.e. detection capability) and also moderate complexity (only one correlator required, but assistance is needed). For *K* = 1 and 33 (dB-Hz) ≤ *C*/*N*<sup>0</sup> < 36 (dB-Hz), due to the significant sensitivity improvement of the DC, (B×C), and CC strategies with respect to the SC strategy, their *TA* values are smaller than that of the SC strategy, see Fig. 14(a). However, for *C*/*N*<sup>0</sup> ≥ 35.7 (dB-Hz), the sensitivity improvement of the SC strategy is sufficient to reduce its *TA* to be lower than that of the stand-alone joint strategies. For *K* = 50, due to the sensitivity improvement of all the strategies, the turning point appears earlier at *C*/*N*<sup>0</sup> = 24.3 (dB-Hz). see Fig. 14(b).

#### **7. Conclusions**

This Chapter focused on high sensitivity signal acquisition problems. Throughout the chapter, some challenges to HS signal acquisition such as unknown data transitions, Doppler effects on carrier frequency and PRN code rate, local oscillator instability as well as sensitivity-complexity trade-off were discussed in details in order to define the requirements of HS acquisition strategies suitable for different operational scenarios. Then three HS acquisition approaches, namely stand-alone, external-aiding and channel combining, have been introduced. Finally, the applications of these approaches to specific GNSS signals are demonstrated for readers' better understanding.

#### **8. References**

3GPP (2008a). Radio resource control (RRC) - release 7, *Organizational Partners* .


28 Will-be-set-by-IN-TECH

 

 

 

 

 

"%#\$! \$#

see Fig. 14(b).

**7. Conclusions**

**8. References**

demonstrated for readers' better understanding.

Cambridge University Press.

48: 227–246.

& -

 

 '

 **-**

 

Fig. 14. Mean acquisition time *TA* vs. *C*/*N*<sup>0</sup> values when *Pf a* = 10−3: (a) *M* = 1; (b) *M* = 50

For all *C*/*N*<sup>0</sup> and *K* values, (B-C) results in the smallest *TA*, because of its high sensitivity (i.e. detection capability) and also moderate complexity (only one correlator required, but assistance is needed). For *K* = 1 and 33 (dB-Hz) ≤ *C*/*N*<sup>0</sup> < 36 (dB-Hz), due to the significant sensitivity improvement of the DC, (B×C), and CC strategies with respect to the SC strategy, their *TA* values are smaller than that of the SC strategy, see Fig. 14(a). However, for *C*/*N*<sup>0</sup> ≥ 35.7 (dB-Hz), the sensitivity improvement of the SC strategy is sufficient to reduce its *TA* to be lower than that of the stand-alone joint strategies. For *K* = 50, due to the sensitivity improvement of all the strategies, the turning point appears earlier at *C*/*N*<sup>0</sup> = 24.3 (dB-Hz).

This Chapter focused on high sensitivity signal acquisition problems. Throughout the chapter, some challenges to HS signal acquisition such as unknown data transitions, Doppler effects on carrier frequency and PRN code rate, local oscillator instability as well as sensitivity-complexity trade-off were discussed in details in order to define the requirements of HS acquisition strategies suitable for different operational scenarios. Then three HS acquisition approaches, namely stand-alone, external-aiding and channel combining, have been introduced. Finally, the applications of these approaches to specific GNSS signals are

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*(ION NTM 2000), Anaheim, CA, USA*, pp. 731–737.


 &

"%#\$! \$#

 

**-** 


 '



**1. Introduction**

interesting and useful research topic. The key objectives of this chapter are:

best practices / learnings,

key contributors to this complexity,

on a FPGA based hardware platform, and

The Global Positioning System (GPS) receiver has come a long way from being a specialised tool to a more general purpose everyday use mainstream gadget. This transformation is not only due to the advancements in semiconductor technology and embedded systems but also due to a highly concentrated research effort in the past decade that targeted a high performance, low power and affordable GPS receiver design. Before the ideas for such an efficient GPS receiver design could attain the saturation stage, the GPS modernisation and the development of several satellite navigation systems under the broader "Global Navigation Satellite Systems" (GNSS) umbrella, have brought a new dimension to the problem of efficient GNSS receiver design. The baseband signal processing engine forms an integral part of any

**Baseband Hardware Designs in** 

Nagaraj C. Shivaramaiah and Andrew G. Dempster

**Modernised GNSS Receivers** 

*The University of New South Wales* 

**2**

*Australia* 

GNSS receiver and is a key contributor to the overall cost and power consumption.

This chapter discusses the challenges involved in designing baseband signal processing algorithms for a modernised GNSS receiver. The modernised GNSS receiver in this context includes processing elements not only for the GPS civilian signals GPS L1C/A, GPS L2C, GPS L5 and GPS L1C, but also for the Open Service (OS) signals from other satellite navigation systems that share the same frequency band as that of GPS. The Galileo satellite navigation system is one such example with its E1 and E5 OS signals sharing the GPS L1 and L5 (partial) frequency bands respectively. Though the underlying concept used in all these signals is "spread spectrum", the structure of these signals differ due to different modulation techniques and signal parameters such as chipping rate, spreading code length, signal bandwidth and navigation data rate. These differences make the efficient baseband hardware design an

1. To revisit the existing efficient GPS L1 C/A baseband hardware methodologies and list

2. To analyse the complexity of the modernised GNSS baseband hardware and to identify

3. To explore design alternatives that deal with the key complexity contributors and to

4. To ascertain the practicality of incorporating the design alternatives by implementing them

analyse the implementation feasibility of these design alternatives,


## **Baseband Hardware Designs in Modernised GNSS Receivers**

Nagaraj C. Shivaramaiah and Andrew G. Dempster *The University of New South Wales Australia* 

#### **1. Introduction**

30 Will-be-set-by-IN-TECH

32 Global Navigation Satellite Systems – Signal, Theory and Applications

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Shanmugam, S. K., Nielsen, J. & Lachapelle, G. (2007). Enhanced Differential Detection

Shanmugam, S. K., Watson, R., Nielsen, J. & Lachapelle, G. (2005). Differential Signal

Ta, T. H. (2010). *"Acquisition Architecture for Modern GNSS Signals"*, PhD thesis, Polytechnique

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Tsui, J. B.-Y. (2005). *Fundamentals of Global Positioning System Receivers: a Software Approach*,

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Yu, W., Zheng, B., Watson, R. & Lachapelle, G. (2007). Differential combining for acquiring

Zarrabizadeh, M. H. & Sousa, E. S. (1997). A Differentially Coherent PN Code Acquisition

*USA*, pp. 1998–2009.

*TX, USA*, pp. 2499–2509.

University of Turin, Italy.

*Long Beach, CA, USA*, pp. 212–222.

*ENC-GNSS 2008, Toulouse, France*.

*Electronic Systems* 48, Issue 2.

2nd edn, Wiley-Interscience.

*Sonar and Navigation* 4, Issue 6: 764–779.

*European Navigation Conference ENC-GNSS 2006*.

weak GPS signals, *Signal Processing* 87(5): 824–840.

Single Shot Measurement Positioning, *Proceedings of ION GNSS 2004, Long Beach, CA,*

Scheme for Weak GPS Signal Acquisition, *Proceedings of ION GNSS 2007, Fort Worth,*

Processing Schemes for Enhanced GPS Acquisition, *Proceedings of ION GNSS 2005,*

High-Sensitivity Acquisition Strategies for Indoor Galileo E1 Signal, *Proceedings of*

Strategies for High Sensitivity Galileo E1 Open Service Signal Acquisition, *IET Radar,*

Processing for GPS L2C Signal Acquisition, *IEEE Transactions on Aerospace and*

F. (2006). New Fast Signal Acquisition Unit for GPS/Galileo Receivers, *Proceedings of*

Receiver for CDMA Systems, *IEEE Transactions on Communications* 45(11): 1456–1465.

The Global Positioning System (GPS) receiver has come a long way from being a specialised tool to a more general purpose everyday use mainstream gadget. This transformation is not only due to the advancements in semiconductor technology and embedded systems but also due to a highly concentrated research effort in the past decade that targeted a high performance, low power and affordable GPS receiver design. Before the ideas for such an efficient GPS receiver design could attain the saturation stage, the GPS modernisation and the development of several satellite navigation systems under the broader "Global Navigation Satellite Systems" (GNSS) umbrella, have brought a new dimension to the problem of efficient GNSS receiver design. The baseband signal processing engine forms an integral part of any GNSS receiver and is a key contributor to the overall cost and power consumption.

This chapter discusses the challenges involved in designing baseband signal processing algorithms for a modernised GNSS receiver. The modernised GNSS receiver in this context includes processing elements not only for the GPS civilian signals GPS L1C/A, GPS L2C, GPS L5 and GPS L1C, but also for the Open Service (OS) signals from other satellite navigation systems that share the same frequency band as that of GPS. The Galileo satellite navigation system is one such example with its E1 and E5 OS signals sharing the GPS L1 and L5 (partial) frequency bands respectively. Though the underlying concept used in all these signals is "spread spectrum", the structure of these signals differ due to different modulation techniques and signal parameters such as chipping rate, spreading code length, signal bandwidth and navigation data rate. These differences make the efficient baseband hardware design an interesting and useful research topic.

The key objectives of this chapter are:


Carrier Mixer

*Ncar*

*N1*

*Nif*

IF samples

Carrier Generator

Carrier

boxes are optional )

this part.

2. Correlator Controller

1. Correlation Computation 2. Local Signal Generation

(Braasch & van Dierendonck, 1999).

NCO Code NCO

Code Generator Sub-carrier Generator

*Nref*

**...**

*Nacc -*bit Accumulators

> Timing Control (to all sequential blocks)

Correlation computation

**...**

2R

*Nacc*

CLK

*Nnco1 Nnco2 Nnco3*

1

Decision and Feedback Control

Measurements

Navigation data

Convolution Decoder

2R

*N2*

1

Baseband Hardware Designs in Modernised GNSS Receivers 35

Subcarrier NCO

Local signal generation and control

The controller processes the correlation values produced by the core hardware and makes decisions based on a set of feedback control algorithms. The threshold detection during the acquisition, the carrier and code locked loops during the tracking, the process of dictating the parameters for the local replica carrier and replica code generation, are all included in

The necessity of this second level functionality division is due to the new signals that will be dealt with in future sections. This type of segregation of the core correlator hardware functionality helps accommodate new signals both at the same frequency that may belong to a different constellation and the signals at different frequency bands of the same constellation. The input to the baseband hardware is the sampled and digitized IF signal with *Ni f* -bit quantization at a sampling frequency of *fs* Hz. In order to demodulate the navigation data bits, the baseband module must first remove the carrier and the spreading code from the signal

The IF samples are mixed with the locally generated carrier in the carrier mixer. The local carrier frequency generator aims to match the frequency of the input signal. Both in phase

Fig. 2. A functional diagram of the baseband hardware (thick lines carry *N*• bits, dashed

The core correlator hardware functionality can again be divided into two parts.

Local Reference Mixers

Subcarrier Modulator

Shift Register

*Nref*

1 **...** R

Fig. 1. Typical architecture of a GNSS receiver

5. To provide recommendations and guidelines for the design of a low power, high performance, affordable multi-GNSS baseband hardware.

This chapter substantially draws on one of the authors' conference papers published in ISCAS 2010 (Shivaramaiah & Dempster, 2010).

## **2. GNSS receiver and baseband hardware**

## **2.1 GNSS receiver architecture**

Fig. 1 shows the typical architecture of a GNSS receiver. Each signal from a different frequency band is down-converted and passed through an Analog-to-Digital-Converter (ADC) to obtain the Intermediate Frequency (IF) samples. The baseband signal processing hardware (widely known as the correlator) is usually implemented in hardware. With the help of feedback control algorithms (implemented either as a part of the digital hardware or as a part of the processing in software), the baseband circuit provides accurate estimates of the delay, phase and frequency of the carrier and spreading code in the received signal (tracking). The baseband circuit is also used for the initial coarse estimates of these parameters (acquisition). The processing, usually implemented in software, computes the Position-Velocity-Time (PVT) solution (Braasch & van Dierendonck, 1999; Kaplan & Hegarty, 2006; Parkinson & Spilker, 1995).

### **2.2 Generic baseband architecture for the tracking process in a GNSS receiver**

This section describes a generic architecture for the GNSS baseband that allows the basic functionality of tracking the signal. Though the same architecture can be used for the signal acquisition process, the signal acquisition is not the focus here. The GNSS baseband hardware in its usual definition is comprised of all the signal processing circuits bounded on the input side by the sampled and digitised IF signal, and on the output side by the received signal measurements (carrier phase, code phase, navigation data bits, signal strength, etc.). Fig. 2 shows the functional diagram of generic GNSS baseband hardware for a single signal component. The functionality of each block is described in detail elsewhere in the literature (e.g. Kaplan & Hegarty (2006)) and will be discussed briefly here.

The baseband functionality can be broadly divided into two parts.

1. Core Correlator Hardware

The core hardware is responsible for correlating the input signal with the local replica and producing the correlation values.

Fig. 2. A functional diagram of the baseband hardware (thick lines carry *N*• bits, dashed boxes are optional )

2. Correlator Controller

2 Will-be-set-by-IN-TECH

Digital Baseband (Correlator)

5. To provide recommendations and guidelines for the design of a low power, high

This chapter substantially draws on one of the authors' conference papers published in ISCAS

Fig. 1 shows the typical architecture of a GNSS receiver. Each signal from a different frequency band is down-converted and passed through an Analog-to-Digital-Converter (ADC) to obtain the Intermediate Frequency (IF) samples. The baseband signal processing hardware (widely known as the correlator) is usually implemented in hardware. With the help of feedback control algorithms (implemented either as a part of the digital hardware or as a part of the processing in software), the baseband circuit provides accurate estimates of the delay, phase and frequency of the carrier and spreading code in the received signal (tracking). The baseband circuit is also used for the initial coarse estimates of these parameters (acquisition). The processing, usually implemented in software, computes the Position-Velocity-Time (PVT) solution (Braasch & van Dierendonck, 1999; Kaplan & Hegarty, 2006; Parkinson & Spilker,

**2.2 Generic baseband architecture for the tracking process in a GNSS receiver**

(e.g. Kaplan & Hegarty (2006)) and will be discussed briefly here. The baseband functionality can be broadly divided into two parts.

This section describes a generic architecture for the GNSS baseband that allows the basic functionality of tracking the signal. Though the same architecture can be used for the signal acquisition process, the signal acquisition is not the focus here. The GNSS baseband hardware in its usual definition is comprised of all the signal processing circuits bounded on the input side by the sampled and digitised IF signal, and on the output side by the received signal measurements (carrier phase, code phase, navigation data bits, signal strength, etc.). Fig. 2 shows the functional diagram of generic GNSS baseband hardware for a single signal component. The functionality of each block is described in detail elsewhere in the literature

The core hardware is responsible for correlating the input signal with the local replica and

Processing (Software)

PVT Solution

RF Front end (Downconverter + ADC)

performance, affordable multi-GNSS baseband hardware.

Antenna

Fig. 1. Typical architecture of a GNSS receiver

**2. GNSS receiver and baseband hardware**

2010 (Shivaramaiah & Dempster, 2010).

**2.1 GNSS receiver architecture**

1. Core Correlator Hardware

producing the correlation values.

1995).

The controller processes the correlation values produced by the core hardware and makes decisions based on a set of feedback control algorithms. The threshold detection during the acquisition, the carrier and code locked loops during the tracking, the process of dictating the parameters for the local replica carrier and replica code generation, are all included in this part.

The core correlator hardware functionality can again be divided into two parts.


The necessity of this second level functionality division is due to the new signals that will be dealt with in future sections. This type of segregation of the core correlator hardware functionality helps accommodate new signals both at the same frequency that may belong to a different constellation and the signals at different frequency bands of the same constellation.

The input to the baseband hardware is the sampled and digitized IF signal with *Ni f* -bit quantization at a sampling frequency of *fs* Hz. In order to demodulate the navigation data bits, the baseband module must first remove the carrier and the spreading code from the signal (Braasch & van Dierendonck, 1999).

The IF samples are mixed with the locally generated carrier in the carrier mixer. The local carrier frequency generator aims to match the frequency of the input signal. Both in phase

**2.3.3 Carrier mixer (***N*1**)**

**2.3.5 Local reference mixer (***N*2**)**

output at this stage be denoted by *A*2.

be a multiple of the spreading code period. Let *N*�

*N*� *acc* = log−<sup>1</sup> 2 *A*2 *fs fco McL*

represent the worst-case value at the output of the accumulator. Then

the local carrier frequency. Hence the required accumulator width *Nacc* < *N*�

**2.3.6 Accumulator (***Nacc***)**

**2.3.4 Subcarrier generator & subcarrier modulator (***Nref* **)**

used.

The carrier mixer basically multiplies the input signal with the local carrier bits. Since the resulting values will only have 8 levels {±1, ±2, ±3, ±6}, a 3-bit encoding is sufficient. Observe that with the 3-bit encoding, arithmetic operation cannot be directly performed. Hence if the succeeding stage requires an arithmetic representation then four bits should be

Baseband Hardware Designs in Modernised GNSS Receivers 37

The local code takes on values of either 0 or 1 and hence 1-bit is sufficient for its representation. However, the number of bits required to represent the subcarrier depends on the number of levels in the subcarrier used for the modulation. BOC signals use a 2-level {±1} subcarrier thus requiring only 1-bit for the representation. AltBOC uses 4-levels (dominant component of the subcarrier) which require more bits for the representation and in such situations approximation needs to be used to use smaller bit-width representations. The local spreading code modifies only the sign of the subcarrier at the output of the subcarrier modulation.

Hence, *Nref* will depend on the number of bits used for the subcarrier representation.

This can be easily determined from the number of levels of the two inputs. However, the succeeding stage (the accumulator) is an arithmetic operation and requires binary two's complement representation. This leads to an additional bit at the output. For example, with the 8-level *N*<sup>1</sup> {±1, ±2, ±3, ±6} and the 4-level *Nref* {±1, ±2}, the resultant set will have only 12 levels {±1, ±2, ±3, ±4, ±6, ±12}, but due to the later requirement of signed binary representation the output must be 5-bit wide. Let the sample-maximum (magnitude) of the

The interval between two consecutive accumulator resets is generally determined by the coherent integration duration and the coherent integration duration in turn in most cases will

where *fs* <sup>∈</sup> *<sup>R</sup>*<sup>+</sup> is the sampling frequency in Hz, *fco* <sup>∈</sup> *<sup>R</sup>*<sup>+</sup> is the chipping rate (with any associated Doppler frequency) in Hz, *L* ∈ **N** is the primary code length, *Mc* ∈ **Q** is the number (or fraction) of primary code periods in the coherent integration and *C* is the complex modulation indicator, *C* ∈ {0 = *Normal*, 1 = *Complex*}. (1) clearly satisfies the Design-For-Test (DFT) guidelines, but it is an overkill as all the samples may not end up with a value of *A*2. In reality the sample-maximum is controlled by the input signal strength and

*acc* denote the number of bits required to

*acc*.

(1)

<sup>+</sup> *<sup>C</sup>* <sup>+</sup> <sup>1</sup>

and quadrature phase signals are generated with *Ncar*-bit quantization. The carrier mixer output results in *N*1-bit values.

The local replica code+subcarrier signal, referred to here as the local "reference signal" is *Nref* -bit wide. In the absence of the subcarrier, *Nref* =1 because the spreading code takes only the values of 1 or 0. Since most of the signal tracking algorithms employed in a GNSS receiver use the delay tracking principle, delayed versions of the local reference signal are generated with the help of shift registers. *R* is the number of local reference signal "arms" (sometimes referred to as "taps" or "fingers"), typically three: the Early, the Prompt and the Late).

The local reference mixer generates 2*R* values each *N*2-bits wide as a result of combining in phase and quadrature values with the local reference signal. These individual sample correlation values are accumulated in a *Nacc*-bit accumulator for a predefined "integration duration". The tracking loops act on these accumulator outputs and adjust the local carrier frequency and the code delay so as to maintain lock (to be at the peak of the correlation function). The tracking loops also produce the measurements and also demodulate the navigation data bits present in the signal (Shivaramaiah, 2004).

#### **2.3 Bit-width requirements of the correlator components**

The parameters of interest for the complexity analysis of the core correlator are the number of bits required to represent the intermediate signals, the bit-width of the accumulator and other registers and the minimum frequency of operation required for a particular signal (or any component of a signal thereof). The notations for the number of bits at different stages are shown in Fig. 2, as *N*• along with the thick lines. In the following paragraphs a brief description of each of the underlying modules is given and the number of bits required for the accumulator is derived.

## **2.3.1 ADC/IF (***Ni f* **)**

The signal loss due to the quantisation beyond 2-bits is insignificant as long as the sampling thresholds are sensibly set (Hegarty, 2009). However, 3-bits and more have been used to alleviate the problems with the AGC in the presence of RF interference (Kaplan & Hegarty, 2006). Commercial mass-market receivers normally use 2-bit uniform sign-magnitude quantisation with 4 levels {±1, ±3}(Zarlink, 1999, 2001). Therefore for the examples in this chapter it is safe to assume *Ni f* = 2.

#### **2.3.2 Local carrier generator (***Ncar***)**

The loss due to the local carrier quantisation is studied in Namgoong et al. (2000). Typically, 3-bit uniform NCO phase quantisation and 2-bit amplitude quantisation with 4 levels {±1, ±2} is used. More bits in the phase and amplitude quantisation increases the Spurious-Free-Dynamic-Range (SFDR) and reduces the quantisation noise. However this has a significant impact on the size of succeeding stages.

## **2.3.3 Carrier mixer (***N*1**)**

4 Will-be-set-by-IN-TECH

and quadrature phase signals are generated with *Ncar*-bit quantization. The carrier mixer

The local replica code+subcarrier signal, referred to here as the local "reference signal" is *Nref* -bit wide. In the absence of the subcarrier, *Nref* =1 because the spreading code takes only the values of 1 or 0. Since most of the signal tracking algorithms employed in a GNSS receiver use the delay tracking principle, delayed versions of the local reference signal are generated with the help of shift registers. *R* is the number of local reference signal "arms" (sometimes

The local reference mixer generates 2*R* values each *N*2-bits wide as a result of combining in phase and quadrature values with the local reference signal. These individual sample correlation values are accumulated in a *Nacc*-bit accumulator for a predefined "integration duration". The tracking loops act on these accumulator outputs and adjust the local carrier frequency and the code delay so as to maintain lock (to be at the peak of the correlation function). The tracking loops also produce the measurements and also demodulate the

The parameters of interest for the complexity analysis of the core correlator are the number of bits required to represent the intermediate signals, the bit-width of the accumulator and other registers and the minimum frequency of operation required for a particular signal (or any component of a signal thereof). The notations for the number of bits at different stages are shown in Fig. 2, as *N*• along with the thick lines. In the following paragraphs a brief description of each of the underlying modules is given and the number of bits required for

The signal loss due to the quantisation beyond 2-bits is insignificant as long as the sampling thresholds are sensibly set (Hegarty, 2009). However, 3-bits and more have been used to alleviate the problems with the AGC in the presence of RF interference (Kaplan & Hegarty, 2006). Commercial mass-market receivers normally use 2-bit uniform sign-magnitude quantisation with 4 levels {±1, ±3}(Zarlink, 1999, 2001). Therefore for the examples in this

The loss due to the local carrier quantisation is studied in Namgoong et al. (2000). Typically, 3-bit uniform NCO phase quantisation and 2-bit amplitude quantisation with 4 levels {±1, ±2} is used. More bits in the phase and amplitude quantisation increases the Spurious-Free-Dynamic-Range (SFDR) and reduces the quantisation noise. However this has

referred to as "taps" or "fingers"), typically three: the Early, the Prompt and the Late).

navigation data bits present in the signal (Shivaramaiah, 2004).

**2.3 Bit-width requirements of the correlator components**

output results in *N*1-bit values.

the accumulator is derived.

chapter it is safe to assume *Ni f* = 2.

**2.3.2 Local carrier generator (***Ncar***)**

a significant impact on the size of succeeding stages.

**2.3.1 ADC/IF (***Ni f* **)**

The carrier mixer basically multiplies the input signal with the local carrier bits. Since the resulting values will only have 8 levels {±1, ±2, ±3, ±6}, a 3-bit encoding is sufficient. Observe that with the 3-bit encoding, arithmetic operation cannot be directly performed. Hence if the succeeding stage requires an arithmetic representation then four bits should be used.

## **2.3.4 Subcarrier generator & subcarrier modulator (***Nref* **)**

The local code takes on values of either 0 or 1 and hence 1-bit is sufficient for its representation. However, the number of bits required to represent the subcarrier depends on the number of levels in the subcarrier used for the modulation. BOC signals use a 2-level {±1} subcarrier thus requiring only 1-bit for the representation. AltBOC uses 4-levels (dominant component of the subcarrier) which require more bits for the representation and in such situations approximation needs to be used to use smaller bit-width representations. The local spreading code modifies only the sign of the subcarrier at the output of the subcarrier modulation. Hence, *Nref* will depend on the number of bits used for the subcarrier representation.

## **2.3.5 Local reference mixer (***N*2**)**

This can be easily determined from the number of levels of the two inputs. However, the succeeding stage (the accumulator) is an arithmetic operation and requires binary two's complement representation. This leads to an additional bit at the output. For example, with the 8-level *N*<sup>1</sup> {±1, ±2, ±3, ±6} and the 4-level *Nref* {±1, ±2}, the resultant set will have only 12 levels {±1, ±2, ±3, ±4, ±6, ±12}, but due to the later requirement of signed binary representation the output must be 5-bit wide. Let the sample-maximum (magnitude) of the output at this stage be denoted by *A*2.

## **2.3.6 Accumulator (***Nacc***)**

The interval between two consecutive accumulator resets is generally determined by the coherent integration duration and the coherent integration duration in turn in most cases will be a multiple of the spreading code period. Let *N*� *acc* denote the number of bits required to represent the worst-case value at the output of the accumulator. Then

$$N\_{acc}^{'} = \left\lceil \log\_2^{-1} \left( A\_2 \left\lfloor \frac{f\_s}{f\_{co}} M\_\mathcal{L} L \right\rfloor \right) + \mathcal{C} + 1 \right\rceil \tag{1}$$

where *fs* <sup>∈</sup> *<sup>R</sup>*<sup>+</sup> is the sampling frequency in Hz, *fco* <sup>∈</sup> *<sup>R</sup>*<sup>+</sup> is the chipping rate (with any associated Doppler frequency) in Hz, *L* ∈ **N** is the primary code length, *Mc* ∈ **Q** is the number (or fraction) of primary code periods in the coherent integration and *C* is the complex modulation indicator, *C* ∈ {0 = *Normal*, 1 = *Complex*}. (1) clearly satisfies the Design-For-Test (DFT) guidelines, but it is an overkill as all the samples may not end up with a value of *A*2. In reality the sample-maximum is controlled by the input signal strength and the local carrier frequency. Hence the required accumulator width *Nacc* < *N*� *acc*.

**16x4 LUT**

**x2 (I & Q)**

Correlation 2

Local Ref

signal

Local Code

Data

Sample Correlation

Clk

**Accumulator**

**x6 (E,P,L and I,Q)**

Data

**Logic /** value

**Accumulator**

**x6 (E,P,L and I,Q)**

Add/Sub

Correlation value

Data

Correlation

16

Add/Sub

Sample

4

Clk

Fig. 3. Realisation of the correlation computation blocks (a) Generic implementation (b) an

Longer codes are usually obtained by wide shift registers or a combination of shift registers. Typically the baseband circuit will have the same number of code generators as the number of channels. If the baseband has to implement multiple tracking channels to simultaneously process multiple signals then the additional number of bits in the shift register brings in

Memory codes eliminate the need for a code generator (i.e the shift registers and XOR gates used for the code generation). However the codes for all the pseudo random noise (PRN) sequences must be stored in a circular buffer or ROM. The decision on whether to use a circular buffer or ROM depends on the overall architecture of the receiver. For tracking the signal, it is enough to read the buffer sequentially (like in a FIFO) and no address generation is required. However if there is a requirement to read the local code from a particular delay (which could be the case when the receiver wants to reacquire the signal) then it is better to use the ROM

**LUT**

**x6 (E,P,L and I,Q)**

Baseband Hardware Designs in Modernised GNSS Receivers 39

IF Signal

IF Signal

**(a)**

**(b)**

**Logic / LUT**

**x2 (I & Q)**

Local Carrier

Local Carrier

2

implementation for the GPS L1 C/A signal

**3.1 Longer codes (or longer code period)**

additional hardware which is not insignificant.

**3.1.1 Shift register generated codes**

**3.1.2 Memory codes**

An *R*-arm correlator will have 2*R*(*C* + 1) accumulators (due to the in-phase and quadrature carrier components) and hence accumulator width plays a very important role in correlator complexity. Some correlators use re-sampling prior to the local reference mixer stage (e.g. (Namgoong et al., 2000)), to reduce the number of samples input to the accumulator. However those special techniques are outside the scope of the discussion here.

### **2.4 Efficient realisation of the correlator core for the GPS L1 C/A signal**

As mentioned in the previous section, the input to the correlator is the sampled IF signal. Each sample in the sampled IF signal, when mixed with local carrier and the local reference signal, produces a correlation value ("sample correlation value") which is then fed to the accumulator. Therefore, in the correlator core of Fig. 2, all the blocks do not require sequential logic. The carrier mixer, subcarrier modulation and the local reference mixer are typically implemented as combinational logic. Latching the input signal, carrier NCO, code NCO and the accumulator are implemented as sequential logic. As a result, the combined propagation delay of all the combination logic blocks should be less than the sampling period *tpd* + *tacc su* < 1/ *fs*, where *tpd* is the propagation delay and *tacc su* is the setup time of the accumulator.

The combinational block has to compute the sample correlation value from the three inputs viz. the incoming signal, the local carrier and the local reference signal. The carrier mixer and the local reference mixer can be realised using Look-Up-Tables (LUTs) separately or together. For the single-bit reference signals, the circuit can be further simplified by feeding the local code to select the add or subtract operation of the accumulator. Fig. 3(a) shows a generic way to realise the correlation computation blocks' combinational logic. The number of instantiations of each block is mentioned above the block. Observe that two carrier mixer blocks are required (I and Q), six reference signal mixer blocks are required (early, prompt and late version of reference signals mixed with I and Q carrier mixer outputs) and six accumulator blocks are required for the complete operation.

Fig. 3(b) shows a realisation of the combinational logic using the LUT method for the GPS L1 C/A signal. In Fig. 3, the input signal and the local carrier are assumed to be 2-bit wide and the local reference signal (in this case only the local code) is 1-bit wide. Observe that the sample correlation output is represented using four bits even though there are only eight possible values. This is because the succeeding stage (which is the signed addition, a part of the accumulation process) is an arithmetic operation and hence the sample correlation values need to be represented in 2's complement format. The local code mixer is eliminated by using the local code output as the Add/Sub selection input of the accumulator. The output therefore consists of six correlation values: inphase-early, inphase-prompt, inphase-late, quadrature-early, quadrature-prompt and quadrature-late. These correlation values are fed to the tracking loops for further processing.

#### **3. Impact of the signal structure on the core correlator architecture**

This section analyses the impact of the change in certain parameters of the signal (due to the structure of the new signals) on the architecture of the core correlator.

6 Will-be-set-by-IN-TECH

An *R*-arm correlator will have 2*R*(*C* + 1) accumulators (due to the in-phase and quadrature carrier components) and hence accumulator width plays a very important role in correlator complexity. Some correlators use re-sampling prior to the local reference mixer stage (e.g. (Namgoong et al., 2000)), to reduce the number of samples input to the accumulator. However

As mentioned in the previous section, the input to the correlator is the sampled IF signal. Each sample in the sampled IF signal, when mixed with local carrier and the local reference signal, produces a correlation value ("sample correlation value") which is then fed to the accumulator. Therefore, in the correlator core of Fig. 2, all the blocks do not require sequential logic. The carrier mixer, subcarrier modulation and the local reference mixer are typically implemented as combinational logic. Latching the input signal, carrier NCO, code NCO and the accumulator are implemented as sequential logic. As a result, the combined propagation delay of all the combination logic blocks should be less than the sampling period *tpd* + *tacc*

The combinational block has to compute the sample correlation value from the three inputs viz. the incoming signal, the local carrier and the local reference signal. The carrier mixer and the local reference mixer can be realised using Look-Up-Tables (LUTs) separately or together. For the single-bit reference signals, the circuit can be further simplified by feeding the local code to select the add or subtract operation of the accumulator. Fig. 3(a) shows a generic way to realise the correlation computation blocks' combinational logic. The number of instantiations of each block is mentioned above the block. Observe that two carrier mixer blocks are required (I and Q), six reference signal mixer blocks are required (early, prompt and late version of reference signals mixed with I and Q carrier mixer outputs) and six accumulator

Fig. 3(b) shows a realisation of the combinational logic using the LUT method for the GPS L1 C/A signal. In Fig. 3, the input signal and the local carrier are assumed to be 2-bit wide and the local reference signal (in this case only the local code) is 1-bit wide. Observe that the sample correlation output is represented using four bits even though there are only eight possible values. This is because the succeeding stage (which is the signed addition, a part of the accumulation process) is an arithmetic operation and hence the sample correlation values need to be represented in 2's complement format. The local code mixer is eliminated by using the local code output as the Add/Sub selection input of the accumulator. The output therefore consists of six correlation values: inphase-early, inphase-prompt, inphase-late, quadrature-early, quadrature-prompt and quadrature-late. These correlation values are fed

This section analyses the impact of the change in certain parameters of the signal (due to the

**3. Impact of the signal structure on the core correlator architecture**

structure of the new signals) on the architecture of the core correlator.

*su* is the setup time of the accumulator.

*su* <

those special techniques are outside the scope of the discussion here.

1/ *fs*, where *tpd* is the propagation delay and *tacc*

blocks are required for the complete operation.

to the tracking loops for further processing.

**2.4 Efficient realisation of the correlator core for the GPS L1 C/A signal**

Fig. 3. Realisation of the correlation computation blocks (a) Generic implementation (b) an implementation for the GPS L1 C/A signal

#### **3.1 Longer codes (or longer code period)**

#### **3.1.1 Shift register generated codes**

Longer codes are usually obtained by wide shift registers or a combination of shift registers. Typically the baseband circuit will have the same number of code generators as the number of channels. If the baseband has to implement multiple tracking channels to simultaneously process multiple signals then the additional number of bits in the shift register brings in additional hardware which is not insignificant.

#### **3.1.2 Memory codes**

Memory codes eliminate the need for a code generator (i.e the shift registers and XOR gates used for the code generation). However the codes for all the pseudo random noise (PRN) sequences must be stored in a circular buffer or ROM. The decision on whether to use a circular buffer or ROM depends on the overall architecture of the receiver. For tracking the signal, it is enough to read the buffer sequentially (like in a FIFO) and no address generation is required. However if there is a requirement to read the local code from a particular delay (which could be the case when the receiver wants to reacquire the signal) then it is better to use the ROM

operating frequency requirement of the correlator will go up and also an additional clock

Baseband Hardware Designs in Modernised GNSS Receivers 41

When a signal has more than one component (say pilot and data components), it is wise to compute the correlation values independently for each signal component, thus allowing the subsequent processing blocks to use efficient tracking techniques (Shivaramaiah, 2011). One can optimise the correlation computation blocks by combining the logic for the signal components but that would give a combined correlation value to the tracking loops. This combined correlation value may suffer from loss due to the data and or code bit inversions between the signal components. Therefore it is wise to isolate the different signal components at the correlation computation stage (and combine in the succeeding stages if required).

Receiver bandwidth has a direct impact on the sampling frequency and hence the operating frequency of the circuit. While some baseband blocks can be fed a slower clock than the sampling frequency (but still derived from the sampling frequency), some other blocks have to operate at the sampling frequency itself. Any bandwidth reduction below the minimum required (which is typically the bandwidth occupied by the main lobe(s)) done before the correlation operation stage, will result in rounded auto-correlation peaks, which in turn result in noisier range measurements. Therefore it is a good practice to keep the operating frequency at least equal to the sampling frequency until the carrier mixing stage and at least equal to four times the subcarrier frequency (or the twice the code frequency in the absence of subcarrier)

In the case of AltBOC signals the lines generated within the core correlator portion in Fig. 2 carry complex signals. The local reference mixer LUT must cater for the complex correlation

There are basically two ways to realise this complex reference signal mixer: with the logic or with LUTs. With the logic one would be using adders/subtracters and multipliers of appropriate length to compute the reference signal mixer outputs. With the LUT, there are many ways, each using different sizes of the LUT. In both the cases the resource requirement would significantly increase compared to the GPS L1 C/A correlator (which requires no

Table 1 revisits the centre frequency, typical receiver bandwidth and code lengths of some of the new open service signals. These parameters largely determine the architecture and

**4. Core correlator architectural modifications for the new signals**

complexity of the baseband signal processing stage in a GNSS receiver.

The following are the important points to note from the table.

divider circuit is required.

**3.4 Multiple signal components**

**3.5 Receiver bandwidth and the operating frequency**

from the reference signal mixer stage onwards.

**4.1 New GNSS signals and general requirements**

**3.6 Complex modulation**

reference signal mixer).

operation.

which then demands a separate address generator. The read clock to the FIFO or the ROM is nothing but the output of the code NCO.

Another issue with the memory codes is the way the codes are stored. The codes for all the PRNs cannot be stored in a single memory because it will limit the access of the memory from different channels. Hence the code for each PRN should be stored in a separate memory block. Even in this situation, there is a constraint on the architecture. During the signal acquisition or during the tracking if there is a requirement for more than one GNSS channel to use the same PRN, then the memory block will have to have more than one port which is expensive in terms of the resource and power consumption.

## **3.1.3 Effect on the accumulator bit-width**

Another consequence of longer codes is that the number of bits in the accumulator has to be increased, i.e. the *Nacc* requirement increases (assuming that the accumulator is used to integrate the correlation values for the duration of one code period).

#### **3.2 Subcarrier modulation**

#### **3.2.1 Two-level (1-bit) subcarriers**

With the subcarrier modulation an additional NCO, subcarrier generator and subcarrier modulator may be required depending on the requirement of the tracking loops. If the subcarrier has only two levels then the subcarrier and the replica code bit can be combined with the help of a single XOR gate and the result will also be a 1-bit value. This does not change the other parts of the correlation computation circuit and also the reference signal can still be fed to the Add/Sub input of the accumulator.

#### **3.2.2 Multi-level (> 1-bit) subcarriers**

If the subcarrier has multiple levels (i.e. requiring more than 1-bit) then the process of combining the replica code bit and the subcarrier is not a simple XOR operation, but requires a negation operation which results in the same number of bits as the subcarrier (*Nref*). Secondly, the width of the shift register that generates the early, prompt and late values should be increased to *Nref* . Since the reference signal is not represented by a single bit it cannot be used directly as an input to the accumulator and therefore there needs to be a dedicated reference signal mixer block. Thirdly, the reference signal mixing operation should accommodate this bit-width increase in one of the inputs. As a result, the number of bits required to represent the sample correlation value will increase, which in turn increases the number of bits in the accumulator.

#### **3.3 Modulation type**

The BOC family of signals has a narrow autocorrelation main peak. As a result the spacing between the *R* delayed versions of the reference signals should be reduced in order to achieve better tracking performance (Shivaramaiah & Dempster, 2009). Reduction in the spacing requires the code and the subcarrier NCO to be operating at a higher clock frequency. This constrains the minimum clock frequency requirement of these NCOs. As a result, the overall operating frequency requirement of the correlator will go up and also an additional clock divider circuit is required.

### **3.4 Multiple signal components**

8 Will-be-set-by-IN-TECH

which then demands a separate address generator. The read clock to the FIFO or the ROM is

Another issue with the memory codes is the way the codes are stored. The codes for all the PRNs cannot be stored in a single memory because it will limit the access of the memory from different channels. Hence the code for each PRN should be stored in a separate memory block. Even in this situation, there is a constraint on the architecture. During the signal acquisition or during the tracking if there is a requirement for more than one GNSS channel to use the same PRN, then the memory block will have to have more than one port which is expensive

Another consequence of longer codes is that the number of bits in the accumulator has to be increased, i.e. the *Nacc* requirement increases (assuming that the accumulator is used to

With the subcarrier modulation an additional NCO, subcarrier generator and subcarrier modulator may be required depending on the requirement of the tracking loops. If the subcarrier has only two levels then the subcarrier and the replica code bit can be combined with the help of a single XOR gate and the result will also be a 1-bit value. This does not change the other parts of the correlation computation circuit and also the reference signal can

If the subcarrier has multiple levels (i.e. requiring more than 1-bit) then the process of combining the replica code bit and the subcarrier is not a simple XOR operation, but requires a negation operation which results in the same number of bits as the subcarrier (*Nref*). Secondly, the width of the shift register that generates the early, prompt and late values should be increased to *Nref* . Since the reference signal is not represented by a single bit it cannot be used directly as an input to the accumulator and therefore there needs to be a dedicated reference signal mixer block. Thirdly, the reference signal mixing operation should accommodate this bit-width increase in one of the inputs. As a result, the number of bits required to represent the sample correlation value will increase, which in turn increases the number of bits in the

The BOC family of signals has a narrow autocorrelation main peak. As a result the spacing between the *R* delayed versions of the reference signals should be reduced in order to achieve better tracking performance (Shivaramaiah & Dempster, 2009). Reduction in the spacing requires the code and the subcarrier NCO to be operating at a higher clock frequency. This constrains the minimum clock frequency requirement of these NCOs. As a result, the overall

nothing but the output of the code NCO.

in terms of the resource and power consumption.

still be fed to the Add/Sub input of the accumulator.

integrate the correlation values for the duration of one code period).

**3.1.3 Effect on the accumulator bit-width**

**3.2 Subcarrier modulation**

**3.2.1 Two-level (1-bit) subcarriers**

**3.2.2 Multi-level (> 1-bit) subcarriers**

accumulator.

**3.3 Modulation type**

When a signal has more than one component (say pilot and data components), it is wise to compute the correlation values independently for each signal component, thus allowing the subsequent processing blocks to use efficient tracking techniques (Shivaramaiah, 2011). One can optimise the correlation computation blocks by combining the logic for the signal components but that would give a combined correlation value to the tracking loops. This combined correlation value may suffer from loss due to the data and or code bit inversions between the signal components. Therefore it is wise to isolate the different signal components at the correlation computation stage (and combine in the succeeding stages if required).

#### **3.5 Receiver bandwidth and the operating frequency**

Receiver bandwidth has a direct impact on the sampling frequency and hence the operating frequency of the circuit. While some baseband blocks can be fed a slower clock than the sampling frequency (but still derived from the sampling frequency), some other blocks have to operate at the sampling frequency itself. Any bandwidth reduction below the minimum required (which is typically the bandwidth occupied by the main lobe(s)) done before the correlation operation stage, will result in rounded auto-correlation peaks, which in turn result in noisier range measurements. Therefore it is a good practice to keep the operating frequency at least equal to the sampling frequency until the carrier mixing stage and at least equal to four times the subcarrier frequency (or the twice the code frequency in the absence of subcarrier) from the reference signal mixer stage onwards.

#### **3.6 Complex modulation**

In the case of AltBOC signals the lines generated within the core correlator portion in Fig. 2 carry complex signals. The local reference mixer LUT must cater for the complex correlation operation.

There are basically two ways to realise this complex reference signal mixer: with the logic or with LUTs. With the logic one would be using adders/subtracters and multipliers of appropriate length to compute the reference signal mixer outputs. With the LUT, there are many ways, each using different sizes of the LUT. In both the cases the resource requirement would significantly increase compared to the GPS L1 C/A correlator (which requires no reference signal mixer).

## **4. Core correlator architectural modifications for the new signals**

#### **4.1 New GNSS signals and general requirements**

Table 1 revisits the centre frequency, typical receiver bandwidth and code lengths of some of the new open service signals. These parameters largely determine the architecture and complexity of the baseband signal processing stage in a GNSS receiver.

The following are the important points to note from the table.

number of accumulators remains the same if the CM and CL correlation values are combined (the spreading codes can be combined in time similar to what is done at the transmitter). However, the combination will lead to data bit ambiguity problem (Dempster, 2006). When the components are combined, the increase in the power consumption with respect to the L2C- CM only signal case is negligible. If both the CM and the CL signal components are processed independently then the resource utilisation almost doubles compared to the CM

Baseband Hardware Designs in Modernised GNSS Receivers 43

Because of the use of memory codes, the baseband can eliminate the shift register and store the local spreading code in memory. Therefore there is a small saving in terms of the flip-flops/registers compared to the GPS L1 C/A architecture. However, because of the 8 MHz sampling frequency requirement assumption, one expects to see an increase in the power

Here two sets of memory codes are used each occupying 4092 bits. In addition the number of local reference mixers and accumulators are not only doubled, but also need to operate at higher frequencies due to the higher sampling frequency. For this reason the expected power

For the GPS L5 signal, the code generator shift register requires 13 bits, which is not a significant increase from the 10-bits of GPS L1 C/A. However, the major difference is the higher chipping rate which demands a higher sampling frequency and in turn a higher correlator operating frequency. Due to the longer code length, the accumulators also have to be wide compared to that of the GPS L1 C/A correlator. As a result of the increased operating frequency, the power consumption requirement is expected to drastically increase though the resource utilisation would go up only slightly more than twice that of the GPS L1

For the Galileo E5a and E5b signals, the code generator shift register requires 14-bits. This is only a 1-bit change from the case of GPS L5 correlator and hence all the other circuit parameters (such as bit widths) will be very close to that of the GPS L5 correlator. Hence the expected power consumption for E5a or E5b signal when processed individually, would

consumption is close to twice that of the single signal component (E1b or E1c).

only processing.

**4.3 Galileo E1**

consumption.

**4.4 GPS L5 (pilot and data)**

**4.5.1 Galileo E5a or E5b (pilot and data)**

be close to that of the GPS L5 correlator.

C/A correlator.

**4.5 Galileo E5**

**4.3.1 Single signal component (E1b or E1c)**

**4.3.2 Both the signal components (E1b and E1c)**


\*\* Primary code only, \*\*\* Yet to be available for the Compass signal

Table 1. Some new GNSS signals and their parameters of interest


The operating frequency and the circuit complexity determine the energy efficiency of digital logic and therefore the design of an efficient baseband logic circuit becomes extremely important in the context of baseband hardware targeted to process multi-GNSS signals (Shivaramaiah et al., 2009).

This section discusses the major contributors for the resource utilisation of the correlators for the new signals. The parameters of the correlator that processes GPS L1 C/A signal are used as the reference.

### **4.2 GPS L2C**

### **4.2.1 GPS L2C - CM**

The L2C - CM code generation requires a 27-bit shift register instead of the 10-bit code generator shift register that is used to generate the L1 C/A signal. This in turn increases the code generator read /write and control register bit-widths. The operating frequency remains the same and hence any increase in the power consumption is only due to the increase in the number of shift register bits.

#### **4.2.2 GPS L2C - CM and CL)**

The additions to the L2C - CM only correlator are : another 27-bit shift register, another set of code mixers and accumulators. Since the CM and CL codes are time-multiplexed, the number of accumulators remains the same if the CM and CL correlation values are combined (the spreading codes can be combined in time similar to what is done at the transmitter). However, the combination will lead to data bit ambiguity problem (Dempster, 2006). When the components are combined, the increase in the power consumption with respect to the L2C- CM only signal case is negligible. If both the CM and the CL signal components are processed independently then the resource utilisation almost doubles compared to the CM only processing.

### **4.3 Galileo E1**

10 Will-be-set-by-IN-TECH

GPS L1 C/A 1575.42 (2) BPSK 1023 (N) 1.023

GPS L5 1176.45 (20) BPSK 20460 (N) 10.23

\*\* Primary code only, \*\*\* Yet to be available for the Compass signal

2. increased spreading code lengths and chipping rates demand higher shift register clock

3. use of multi-level subcarriers, as in the case of AltBOC type of modulation, increases the

4. use of memory codes demands additional memory to hold the spreading code for all the

5. increased minimum operating frequency of the baseband hardware mainly due to a) and

The operating frequency and the circuit complexity determine the energy efficiency of digital logic and therefore the design of an efficient baseband logic circuit becomes extremely important in the context of baseband hardware targeted to process multi-GNSS signals

This section discusses the major contributors for the resource utilisation of the correlators for the new signals. The parameters of the correlator that processes GPS L1 C/A signal are used

The L2C - CM code generation requires a 27-bit shift register instead of the 10-bit code generator shift register that is used to generate the L1 C/A signal. This in turn increases the code generator read /write and control register bit-widths. The operating frequency remains the same and hence any increase in the power consumption is only due to the increase in the

The additions to the L2C - CM only correlator are : another 27-bit shift register, another set of code mixers and accumulators. Since the CM and CL codes are time-multiplexed, the

1575.42 (14) MBOC / CBOC 1023 (N), 4096

1191.795 (50) AltBOC 10230 (N), \*\*\* 10.23

GPS L2C 1227.6 (2) BPSK CM-20460 (N),

Modulation type

Code length \* (memory code? Y/N)

CL-767250 (N)

(Y), \*\*

Chipping rate (MHz)

1.023

1.023

Signal name Centre frequency

GPS L1C, Galileo E1, Compass B1

> Galileo E5, Compass B2

frequencies,

satellites, and

(Shivaramaiah et al., 2009).

as the reference.

**4.2.1 GPS L2C - CM**

number of shift register bits.

**4.2.2 GPS L2C - CM and CL)**

**4.2 GPS L2C**

b)

(Typical receiver bandwidth) in MHz

Table 1. Some new GNSS signals and their parameters of interest

number of bits in the local reference signal,

1. increased signal bandwidths demand higher sampling frequencies

#### **4.3.1 Single signal component (E1b or E1c)**

Because of the use of memory codes, the baseband can eliminate the shift register and store the local spreading code in memory. Therefore there is a small saving in terms of the flip-flops/registers compared to the GPS L1 C/A architecture. However, because of the 8 MHz sampling frequency requirement assumption, one expects to see an increase in the power consumption.

#### **4.3.2 Both the signal components (E1b and E1c)**

Here two sets of memory codes are used each occupying 4092 bits. In addition the number of local reference mixers and accumulators are not only doubled, but also need to operate at higher frequencies due to the higher sampling frequency. For this reason the expected power consumption is close to twice that of the single signal component (E1b or E1c).

### **4.4 GPS L5 (pilot and data)**

For the GPS L5 signal, the code generator shift register requires 13 bits, which is not a significant increase from the 10-bits of GPS L1 C/A. However, the major difference is the higher chipping rate which demands a higher sampling frequency and in turn a higher correlator operating frequency. Due to the longer code length, the accumulators also have to be wide compared to that of the GPS L1 C/A correlator. As a result of the increased operating frequency, the power consumption requirement is expected to drastically increase though the resource utilisation would go up only slightly more than twice that of the GPS L1 C/A correlator.

#### **4.5 Galileo E5**

#### **4.5.1 Galileo E5a or E5b (pilot and data)**

For the Galileo E5a and E5b signals, the code generator shift register requires 14-bits. This is only a 1-bit change from the case of GPS L5 correlator and hence all the other circuit parameters (such as bit widths) will be very close to that of the GPS L5 correlator. Hence the expected power consumption for E5a or E5b signal when processed individually, would be close to that of the GPS L5 correlator.

L1/E1 IF Signal

L1 core E1 core

Correlation computation (x Number of channels)

L2C core

Mux

Correlation values

Baseband Hardware Designs in Modernised GNSS Receivers 45

Command

and status

Fig. 4. Components of the baseband module for GPS L1/L2C/L5 and Galileo E1/E5 signals

GPS L1 C/A 4 446 151 - 1.06 Galileo E1b or E1c 8 436 149 4092 1.84

GPS L2C CM only 4 478 210 - 1.13

Galileo E5a or E5b 40 694 203 - 11.80 Galileo E5 100 1010 253 - 39.28 Table 2. Resource utilisation and power consumption estimates of the core correlator (single

Interface Controller

Resource Utilisation Power

Memory (bits)

estimate (mW)

Mux / Demux and Controller

Registers Combina-

tional

8 631 176 8184 2.24

4 737 245 - 1.61

40 701 204 - 11.93

L5 core E5 core

Local signal generation (for all channels)

> Correlator Operating Frequency (MHz)

L2C IF Signal

L5/E5 IF Signal

Signal / Component

Galileo E1 (E1b and E1c)

GPS L2C (CM and CL)

GPS L5 (Pilot and Data)

channel) for different signals

Code memory blocks (all PRNs)

#### **4.5.2 Galileo E5 wideband**

In this case both the E5a and E5b signals are processed together as a single wideband signal (with a bandwidth of at least 51.15 MHz). The local code has to be generated individually for all the four components (E5a-pilot, E5a-data, E5b-pilot and E5b-data) of the signal and the generators require 14-bit shift registers. However, because the four signal components and the complex modulation, the local reference mixer is computationally intensive (more LUTs). In addition, a quadruple number of accumulators are required. As mentioned earlier, independent correlation for all the four signals is performed to allow design freedom for the subsequent stages in combining these four components. As a result of a very high operating frequency and drastic increase in the resource requirements compared to GPS L1 C/A correlator, the power consumption is expected to be very high.

## **4.6 Baseband architecture overview for the GPS and Galileo OS signals**

Fig. 4 shows the baseband architecture assuming that the correlator processes GPA L1 C/A, L2C, L5 and Galileo E1, E5 signals. The computation block is marked separately to the local signal (local carrier and replica reference signal) generation block. This sort of grouping the blocks helps the design because different signals (and different channels in some cases) share common parameters and optimising the hardware becomes easier. In this architecture, it is also assumed that the baseband is commanded (assigned PRNs, Doppler and delay parameters etc) to operate from an external processor. The correlation values of different signals are read and processed in the succeeding decision feedback and control stage.

## **5. Resource requirements for the new signals and recommendations**

## **5.1 Core resource requirements using a straightforward extension of the GPS L1 C/A design**

In order to gauge the resource requirements in terms of the number of registers and combinational logic, the core correlators for the GPS and Galileo open service signals have been implemented on the Altera Cyclone-III family device EP3C120F780C8. The FPGA resource utilisation parameters are listed in Table 2.

The resource and the power consumption values closely match the expected outcomes mentioned in the previous section. While the Galileo E1b or E1c core requires almost the same resources as that of GPS L1 C/A, the power consumption is higher. The power consumption for the single component Galileo E1 is 0.8mW more than that of the GPS L1 C/A. This is due to the presence of the memory block, increased accumulator width and increased operating frequency. The Galileo E1 correlator where both the E1b and E1c signals are processed together has a power estimate of 2.24mW, only about 0.6mW more than the single component. This is because some of the blocks such as the carrier NCO and the carrier mixer are common for both the signal components.

The resource for the GPS L5 signal is increased to 701 registers and 204 combinational units which is due to the increase in the accumulator width and also due to the presence of two signal components. The power consumption estimate of the GPS L5 signal is about 11 times that of the GPS L1 C/A signal and is attributed mainly to the operating frequency. The 12 Will-be-set-by-IN-TECH

In this case both the E5a and E5b signals are processed together as a single wideband signal (with a bandwidth of at least 51.15 MHz). The local code has to be generated individually for all the four components (E5a-pilot, E5a-data, E5b-pilot and E5b-data) of the signal and the generators require 14-bit shift registers. However, because the four signal components and the complex modulation, the local reference mixer is computationally intensive (more LUTs). In addition, a quadruple number of accumulators are required. As mentioned earlier, independent correlation for all the four signals is performed to allow design freedom for the subsequent stages in combining these four components. As a result of a very high operating frequency and drastic increase in the resource requirements compared to GPS L1

Fig. 4 shows the baseband architecture assuming that the correlator processes GPA L1 C/A, L2C, L5 and Galileo E1, E5 signals. The computation block is marked separately to the local signal (local carrier and replica reference signal) generation block. This sort of grouping the blocks helps the design because different signals (and different channels in some cases) share common parameters and optimising the hardware becomes easier. In this architecture, it is also assumed that the baseband is commanded (assigned PRNs, Doppler and delay parameters etc) to operate from an external processor. The correlation values of different

signals are read and processed in the succeeding decision feedback and control stage.

**5.1 Core resource requirements using a straightforward extension of the GPS L1 C/A**

In order to gauge the resource requirements in terms of the number of registers and combinational logic, the core correlators for the GPS and Galileo open service signals have been implemented on the Altera Cyclone-III family device EP3C120F780C8. The FPGA

The resource and the power consumption values closely match the expected outcomes mentioned in the previous section. While the Galileo E1b or E1c core requires almost the same resources as that of GPS L1 C/A, the power consumption is higher. The power consumption for the single component Galileo E1 is 0.8mW more than that of the GPS L1 C/A. This is due to the presence of the memory block, increased accumulator width and increased operating frequency. The Galileo E1 correlator where both the E1b and E1c signals are processed together has a power estimate of 2.24mW, only about 0.6mW more than the single component. This is because some of the blocks such as the carrier NCO and the carrier mixer are common

The resource for the GPS L5 signal is increased to 701 registers and 204 combinational units which is due to the increase in the accumulator width and also due to the presence of two signal components. The power consumption estimate of the GPS L5 signal is about 11 times that of the GPS L1 C/A signal and is attributed mainly to the operating frequency. The

**5. Resource requirements for the new signals and recommendations**

resource utilisation parameters are listed in Table 2.

for both the signal components.

C/A correlator, the power consumption is expected to be very high.

**4.6 Baseband architecture overview for the GPS and Galileo OS signals**

**4.5.2 Galileo E5 wideband**

**design**

Fig. 4. Components of the baseband module for GPS L1/L2C/L5 and Galileo E1/E5 signals


Table 2. Resource utilisation and power consumption estimates of the core correlator (single channel) for different signals

2 4 6 8 10 12 14

L1 L2 E1 L5 E5a E5

Baseband Hardware Designs in Modernised GNSS Receivers 47

Number of channels

Fig. 7 shows the power consumption for different combinations of signals where each signal has been assumed to be using 12 channels. It is interesting to note that a GNSS receiver designed to process all the civilian signals of GPS and Galileo would require slightly short of one watt for the baseband hardware (using the Altera Cyclone-III family device

The challenges that are faced in designing the baseband hardware for a multi-GNSS receiver

The complexity reduction challenges are not of significant concern because of the availability of design tools that help an engineer to handle the kind of complexity present in this situation. However, it is a good practice to have a modular design keeping in mind the scalability of the

In most of the situations, the resource and power consumption are highly interrelated. Exceptions to these situations are generally the changes in the operating frequency. Reduction in the operating frequency will basically reduce only the power consumption though it may indirectly reduce the resource requirement to some extent (such as a simplified clock tree or

architecture to additional signals. The complexity issues are not discussed here.

EP3C120F780C8), which is 38 times that of GPS L1 C/A baseband hardware.

0

Fig. 6. Power consumption of the entire baseband circuit

**5.3 Recommendations for the multi-GNSS baseband design**

can be broadly categorized into three groups

• power consumption reduction challenges, and • resource requirement reduction challenges.

• complexity reduction challenges,

50

100

150

Power consumption estimate (mW)

200

250

300

Fig. 5. Ratio of the power estimate for new signals with respect to GPS L1 C/A

resource and power consumption of the Galileo E5a and E5b signals is close to that of the GPS L5 correlator, as expected. As a result of a very high operating frequency the power consumption of the wideband Galileo E5 correlator shoots up to almost 37 times that of the GPS L1 C/A signal. The power consumption for the E5 signal can be reduced a little bit further by focusing more on how the complex mixers are realised as discussed in Shivaramaiah (2011).

The ratio of the power consumption estimate with respect to the GPS L1 C/A is shown in Fig.5. The power consumption was estimated using the PowerPlay Analyzer tool with real IF signal samples provided as an input1 to the baseband module.

#### **5.2 Complexity comparison results for different baseband configurations**

Fig. 6 shows the power consumption of different signals vs. the number of channels. A "channel" comprises the core correlator, timing control, address and data multiplexer/demultiplexer (for a memory mapped interface to the subsequent stage), and some housekeeping operations. Although the resource consumption is not described in detail here, it should be mentioned that the two major memory spreading code sets in the case of the Galileo signal occupy around 410K bits (E1, 4092 bits, 2 signal components, 50 PRNs) of memory and 10K bits (E5 secondary code, 100 bits, 2 components, 50 satellites) which are totally new additions to the GNSS receiver baseband hardware.

<sup>1</sup> The PowerPlay tool estimates the toggle rate of the internal nets and the output pins based on the input signal and the associated clock-frequency.

14 Will-be-set-by-IN-TECH

L2−CM L2 E1b E1 L5 E5a E5 <sup>0</sup>

**1.07 1.52 1.74 2.11**

signal samples provided as an input1 to the baseband module.

totally new additions to the GNSS receiver baseband hardware.

signal and the associated clock-frequency.

**5.2 Complexity comparison results for different baseband configurations**

Fig. 5. Ratio of the power estimate for new signals with respect to GPS L1 C/A

Signal (Signal Component)

resource and power consumption of the Galileo E5a and E5b signals is close to that of the GPS L5 correlator, as expected. As a result of a very high operating frequency the power consumption of the wideband Galileo E5 correlator shoots up to almost 37 times that of the GPS L1 C/A signal. The power consumption for the E5 signal can be reduced a little bit further by focusing more on how the complex mixers are realised as discussed in Shivaramaiah (2011). The ratio of the power consumption estimate with respect to the GPS L1 C/A is shown in Fig.5. The power consumption was estimated using the PowerPlay Analyzer tool with real IF

Fig. 6 shows the power consumption of different signals vs. the number of channels. A "channel" comprises the core correlator, timing control, address and data multiplexer/demultiplexer (for a memory mapped interface to the subsequent stage), and some housekeeping operations. Although the resource consumption is not described in detail here, it should be mentioned that the two major memory spreading code sets in the case of the Galileo signal occupy around 410K bits (E1, 4092 bits, 2 signal components, 50 PRNs) of memory and 10K bits (E5 secondary code, 100 bits, 2 components, 50 satellites) which are

<sup>1</sup> The PowerPlay tool estimates the toggle rate of the internal nets and the output pins based on the input

**11.25 11.13**

**37.06**

5

10

15

20

Power Consumption (ratio w.r.t. GPS L1 C/A)

25

30

35

40

Fig. 6. Power consumption of the entire baseband circuit

Fig. 7 shows the power consumption for different combinations of signals where each signal has been assumed to be using 12 channels. It is interesting to note that a GNSS receiver designed to process all the civilian signals of GPS and Galileo would require slightly short of one watt for the baseband hardware (using the Altera Cyclone-III family device EP3C120F780C8), which is 38 times that of GPS L1 C/A baseband hardware.

#### **5.3 Recommendations for the multi-GNSS baseband design**

The challenges that are faced in designing the baseband hardware for a multi-GNSS receiver can be broadly categorized into three groups


The complexity reduction challenges are not of significant concern because of the availability of design tools that help an engineer to handle the kind of complexity present in this situation. However, it is a good practice to have a modular design keeping in mind the scalability of the architecture to additional signals. The complexity issues are not discussed here.

In most of the situations, the resource and power consumption are highly interrelated. Exceptions to these situations are generally the changes in the operating frequency. Reduction in the operating frequency will basically reduce only the power consumption though it may indirectly reduce the resource requirement to some extent (such as a simplified clock tree or

**32x5 LUT sIrI**

**32x5 LUT sIrQ**

**32x5 LUT sQrQ**

Baseband Hardware Designs in Modernised GNSS Receivers 49

**32x5 LUT sQrI**

It should be noted that the reference signal mixer example for the complex signal is chosen and dealt in more detail here because it is the correlator block which has a major impact due to the signal structure and is drastically different to the implementation of the reference signal mixer for the GPS L1 C/A signal. It is not difficult to identify other such resource hungry and power hungry blocks and is essentially a part of the baseband hardware design process.

One of the major contributors for the higher power consumption of the correlators that process new signals is the correlator operating frequency. The operating frequency of the correlator is typically the sampling frequency at which the IF signal samples are received. However in most of the situations, once the signal is brought to the baseband after the carrier mix operation (the signal at this point may still contain residual Doppler) the result can be resampled to a lower sampling frequency. The minimum operating frequency for the stages after the carrier mix operation can then be reduced to twice the spreading code chipping rate (Namgoong & Meng, 2001a,b; Namgoong et al., 2000). Reduction below twice the spreading code chipping rate is possible but care should be taken to trade-off wisely the signal loss vs. correlator power consumption advantage. The carrier mixer output should undergo proper filtering before the sampling frequency reduction which will increase the resource requirement (by the amount of resource consumed by the filter). Initial implementation results show that the resource requirements of the filter are not significant and hence it is

The accumulator at the end of the correlator computation chain is a power hungry block. Typically six accumulators are required for the correlator that implements three delayed (early prompt and late) reference signals for the reference signal correlation. The requirement of separate correlation values for the individual signal components increases the requirement of the number of accumulators. For example, the GPS L5 signal requires 12 sets of accumulators

Sample Correlation

<sup>6</sup> <sup>+</sup>

6

+

Carrier Mixer Output

sI

sQ

2 2 rI rQ

Fig. 8. Local reference mixer for the complex modulation signals

**5.4.3 Processing signal components separately vs. processing together**

3

3

Reference Signal

**5.4.2 Operating frequency considerations**

not a significant overhead.

Fig. 7. Power consumption for different multi-signal configurations

reduced fanout requirements due to increased clock period etc.) However, if the reduction in the operating frequency demands a modification in the signal processing chain, then resource requirements may go up. On the other hand, reduction in the resource utilisation will almost always help reduce the power consumption.

The next few paragraphs explore some techniques that enable some progress in overcoming these challenges.

### **5.4 Resource and power consumption reduction opportunities**

## **5.4.1 Design optimisation of the core correlator blocks**

One example of resource reduction is the reference signal mixer for the Galileo E5 signal. The reference signal mixer should be carefully designed to address the complexity vs. propagation delay trade-off.

An architecture for the AltBOC(15,10) modulation (used in Galileo E5 and Compass B3 for example) is shown in Fig. 8. In Fig. 8 it is assumed that the input and the local carrier use two bits and the succeeding stage is not the last arithmetic operation in the chain and hence the carrier mixer output can be encoded with three bits. The local reference signal is assumed to be 2-bit wide which is obtained from a 2-bit subcarrier and 1-bit local code.

The implementation shown in Fig. 8 offers a good trade-off between the complexity and propagation delay requirements compared to both the brute-force logic type of implementation and the brute-force single large-size LUT implementation.

16 Will-be-set-by-IN-TECH

Signal Combination

**L1**

**L1+L2**

**L1+E1**

**E1+E5a**

Fig. 7. Power consumption for different multi-signal configurations

**5.4 Resource and power consumption reduction opportunities**

**5.4.1 Design optimisation of the core correlator blocks**

always help reduce the power consumption.

these challenges.

delay trade-off.

**L1+L2+L5**

0 200 400 600 800 1000 Power consumption estimate (mW)

reduced fanout requirements due to increased clock period etc.) However, if the reduction in the operating frequency demands a modification in the signal processing chain, then resource requirements may go up. On the other hand, reduction in the resource utilisation will almost

The next few paragraphs explore some techniques that enable some progress in overcoming

One example of resource reduction is the reference signal mixer for the Galileo E5 signal. The reference signal mixer should be carefully designed to address the complexity vs. propagation

An architecture for the AltBOC(15,10) modulation (used in Galileo E5 and Compass B3 for example) is shown in Fig. 8. In Fig. 8 it is assumed that the input and the local carrier use two bits and the succeeding stage is not the last arithmetic operation in the chain and hence the carrier mixer output can be encoded with three bits. The local reference signal is assumed to

The implementation shown in Fig. 8 offers a good trade-off between the complexity and propagation delay requirements compared to both the brute-force logic type of

be 2-bit wide which is obtained from a 2-bit subcarrier and 1-bit local code.

implementation and the brute-force single large-size LUT implementation.

**L5+E5a**

**L1+L2+E1+L5 +E5**

**L1+L2+E1+L5 +E5a**

**E1+E5**

**L1+E1+L5+E5**

Fig. 8. Local reference mixer for the complex modulation signals

It should be noted that the reference signal mixer example for the complex signal is chosen and dealt in more detail here because it is the correlator block which has a major impact due to the signal structure and is drastically different to the implementation of the reference signal mixer for the GPS L1 C/A signal. It is not difficult to identify other such resource hungry and power hungry blocks and is essentially a part of the baseband hardware design process.

#### **5.4.2 Operating frequency considerations**

One of the major contributors for the higher power consumption of the correlators that process new signals is the correlator operating frequency. The operating frequency of the correlator is typically the sampling frequency at which the IF signal samples are received. However in most of the situations, once the signal is brought to the baseband after the carrier mix operation (the signal at this point may still contain residual Doppler) the result can be resampled to a lower sampling frequency. The minimum operating frequency for the stages after the carrier mix operation can then be reduced to twice the spreading code chipping rate (Namgoong & Meng, 2001a,b; Namgoong et al., 2000). Reduction below twice the spreading code chipping rate is possible but care should be taken to trade-off wisely the signal loss vs. correlator power consumption advantage. The carrier mixer output should undergo proper filtering before the sampling frequency reduction which will increase the resource requirement (by the amount of resource consumed by the filter). Initial implementation results show that the resource requirements of the filter are not significant and hence it is not a significant overhead.

#### **5.4.3 Processing signal components separately vs. processing together**

The accumulator at the end of the correlator computation chain is a power hungry block. Typically six accumulators are required for the correlator that implements three delayed (early prompt and late) reference signals for the reference signal correlation. The requirement of separate correlation values for the individual signal components increases the requirement of the number of accumulators. For example, the GPS L5 signal requires 12 sets of accumulators

**6. Summary**

correlator.

**7. References**

This chapter analysed the core correlator complexities of modernised GNSS receiver baseband hardware. A core correlator architecture description has been given and the number of bits for the accumulator has been derived. Power consumption estimates were provided for the

Baseband Hardware Designs in Modernised GNSS Receivers 51

It was shown that a GPS and Galileo civil signal receiver baseband would consume approximately 38 times the power of a GPS L1 C/A baseband. The dominant contributor to this increased complexity and power consumption is the Galileo E5 AltBOC signal. In addition, implementation of the core baseband signal processing blocks in FPGA hardware reveals up to eight times the resource requirement compared to the GPS L1 C/A only

It is possible to optimise the hardware targeting the power consumption with the help of resampling and external aiding. However, the performance trade-off should be carefully looked into. Because the enormous resource and power consumption for the Galileo E5 AltBOC correlator is due to the signal structure itself, it is of interest to explore efficient alternatives to the AltBOC signal and one such attempt is made in Shivaramaiah (2011).

Even if a dedicated Application Specific Integrated Circuit (ASIC) replaces the FPGA baseband hardware, as a rule of thumb, and the authors' own experience with multiple generations of GPS L1 C/A correlator ASIC design, there will be a best case reduction of the FPGA power consumption by a factor of 5. In other words, a baseband ASIC will consume about 100 mW for the L1-L5 and about 200-mW for the all civil GPS+Galileo baseband. This power consumption is very high given that it is only for the baseband hardware and not for the entire receiver. Finally, if other global and regional satellite navigation systems (such as GLONASS, Compass, QZSS, IRNSS) are included, then, the "200 times" estimate of Dempster (2007) would not be far away. Hence it can be concluded that development of a baseband hardware for the commercial general purpose multi-GNSS receiver is still a challenging task. A direction towards a promising solution would be to explore the correlator

Braasch, M. & van Dierendonck, A. (1999). GPS receiver architectures and measurements,

Hegarty, C. (2009). Analytical Model for GNSS Receiver Implementation Losses, *U.S. Institute*

Kaplan, E. D. & Hegarty, C. J. (eds) (2006). *Understanding GPS: Principles and Applications*,

Namgoong, W. & Meng, T. (2001a). Minimizing power consumption in direct sequence spread

spectrum correlators by resampling IF samples-Part I: performance analysis, *Circuits and Systems II: Analog and Digital Signal Processing, IEEE Transactions on* 48(5): 450–459.

Dempster, A. G. (2006). Correlators for L2C: Some Considerations, *Inside GNSS* pp. 32–37. Dempster, A. G. (2007). Satellite navigation: New signals, new challenges, *Circuits and Systems,*

*2007. ISCAS 2007. IEEE International Symposium on*, pp. 1725 –1728.

*of Navigation International Technical Meeting, ION GNSS*.

new signals at the core correlator level and at the channel level.

level reconfigurability across the GNSS signals.

*Proceedings of the IEEE*.

Artech House.


Table 3. Resource utilisation and power consumption estimates for the Galileo E5 AltBOC correlator; the reference signal generation is implemented with the help of AltBOC LUT (OSSISICD, 2010)

for each channel and the Galileo E5 requires 24 accumulators per channel. Combining the signal components before the correlation operation is possible but with significant performance degradation. The performance degradation arises mainly due to the data-bit ambiguity. Methods that try to avoid the data-bit ambiguity compromise on the performance parameters of the signal in question. Therefore, again a careful consideration is required to trade-off the performance vs. resource (or power) consumption advantage.

Table 3 shows the power consumption for the Galileo E5 AltBOC correlator if all the four signal components are processed simultaneously. In this case there are only six accumulators required as in the single signal component case. The reference signal in this case is generated according to the AltBOC LUT provided in the Galileo ICD (OSSISICD, 2010). However, the presence of data-bits (assuming that the secondary code phase resolution has already happened) hampers the correlator output and hence the performance. Observe that the power reduction compared to the correlator processing the signal components separately is about 19% which is a significant reduction. In other words it is possible to reduce the correlator power consumption without losing the performance if there is a data aiding mechanism.

#### **5.4.4 Optimising the correlator blocks across signals**

Correlator design optimisation is a separate topic of itself as there are several ways to tackle the resource utilisation issue. Moreover the optimisation is often receiver specific. Three examples are given below where the optimisation is possible in specific correlator blocks.

First, the need for subcarrier NCO can be eliminated (even when the multiplication required is not a power of two) by implementing clock multipliers with simple gates. For example, in the case of Galileo E5, the x1.5 clock can be generated by simple gates that implement <sup>3</sup> 2 multiplier.

Second, the carrier and code NCO for different signals from the same satellite can be combined. This is done by programming and generating the required carrier for one of the signals and deriving the difference in the relative Doppler for the second signal.

Third, the operating frequency for the signals can be adjusted such that the operating frequencies can be derived from a single clock with simple dividers. The advantages of such a clock domain construction are simplification of generation of control and timing signals as well as ease of data transfers across different correlation stages of different signals.

## **6. Summary**

18 Will-be-set-by-IN-TECH

Galileo E5 100 667 519 - 31.98

for each channel and the Galileo E5 requires 24 accumulators per channel. Combining the signal components before the correlation operation is possible but with significant performance degradation. The performance degradation arises mainly due to the data-bit ambiguity. Methods that try to avoid the data-bit ambiguity compromise on the performance parameters of the signal in question. Therefore, again a careful consideration is required to

Table 3 shows the power consumption for the Galileo E5 AltBOC correlator if all the four signal components are processed simultaneously. In this case there are only six accumulators required as in the single signal component case. The reference signal in this case is generated according to the AltBOC LUT provided in the Galileo ICD (OSSISICD, 2010). However, the presence of data-bits (assuming that the secondary code phase resolution has already happened) hampers the correlator output and hence the performance. Observe that the power reduction compared to the correlator processing the signal components separately is about 19% which is a significant reduction. In other words it is possible to reduce the correlator power consumption without losing the performance if there is a data aiding mechanism.

Correlator design optimisation is a separate topic of itself as there are several ways to tackle the resource utilisation issue. Moreover the optimisation is often receiver specific. Three examples are given below where the optimisation is possible in specific correlator blocks.

First, the need for subcarrier NCO can be eliminated (even when the multiplication required is not a power of two) by implementing clock multipliers with simple gates. For example, in the case of Galileo E5, the x1.5 clock can be generated by simple gates that implement <sup>3</sup>

Second, the carrier and code NCO for different signals from the same satellite can be combined. This is done by programming and generating the required carrier for one of the

Third, the operating frequency for the signals can be adjusted such that the operating frequencies can be derived from a single clock with simple dividers. The advantages of such a clock domain construction are simplification of generation of control and timing signals as

signals and deriving the difference in the relative Doppler for the second signal.

well as ease of data transfers across different correlation stages of different signals.

Table 3. Resource utilisation and power consumption estimates for the Galileo E5 AltBOC correlator; the reference signal generation is implemented with the help of AltBOC LUT

Registers Combina-

tional

Resource Utilisation Power

Memory (bits)

estimate (mW)

2

Correlator Operating Frequency (MHz)

trade-off the performance vs. resource (or power) consumption advantage.

**5.4.4 Optimising the correlator blocks across signals**

Signal / Component

(OSSISICD, 2010)

multiplier.

This chapter analysed the core correlator complexities of modernised GNSS receiver baseband hardware. A core correlator architecture description has been given and the number of bits for the accumulator has been derived. Power consumption estimates were provided for the new signals at the core correlator level and at the channel level.

It was shown that a GPS and Galileo civil signal receiver baseband would consume approximately 38 times the power of a GPS L1 C/A baseband. The dominant contributor to this increased complexity and power consumption is the Galileo E5 AltBOC signal. In addition, implementation of the core baseband signal processing blocks in FPGA hardware reveals up to eight times the resource requirement compared to the GPS L1 C/A only correlator.

It is possible to optimise the hardware targeting the power consumption with the help of resampling and external aiding. However, the performance trade-off should be carefully looked into. Because the enormous resource and power consumption for the Galileo E5 AltBOC correlator is due to the signal structure itself, it is of interest to explore efficient alternatives to the AltBOC signal and one such attempt is made in Shivaramaiah (2011).

Even if a dedicated Application Specific Integrated Circuit (ASIC) replaces the FPGA baseband hardware, as a rule of thumb, and the authors' own experience with multiple generations of GPS L1 C/A correlator ASIC design, there will be a best case reduction of the FPGA power consumption by a factor of 5. In other words, a baseband ASIC will consume about 100 mW for the L1-L5 and about 200-mW for the all civil GPS+Galileo baseband. This power consumption is very high given that it is only for the baseband hardware and not for the entire receiver. Finally, if other global and regional satellite navigation systems (such as GLONASS, Compass, QZSS, IRNSS) are included, then, the "200 times" estimate of Dempster (2007) would not be far away. Hence it can be concluded that development of a baseband hardware for the commercial general purpose multi-GNSS receiver is still a challenging task. A direction towards a promising solution would be to explore the correlator level reconfigurability across the GNSS signals.

## **7. References**


**3** 

Zheng Yao

*China* 

*Tsinghua University* 

**Unambiguous Processing Techniques of** 

**Binary Offset Carrier Modulated Signals** 

In recent years, applications of global navigation satellite systems (GNSS) are developing rapidly. The growing public demand for positioning and location services has generated higher requirements for system performance. However, the performance of the traditional GPS is constrained by its inherent capability. In order to cope with both the civil and military expectations in terms of performance, several projects are launched to promote the next generation of GNSS (Hegarty & Chatre, 2008). GPS is undergoing an extensive modernization process (Enge, 2003), while the European satellite system, Galileo, is also under construction. In addition, Russia is restoring their GLONASS (Slater et al., 2004), and

Based on the experience gained during the traditional GPS design and operation, signal structures of these new navigation systems have been well designed (ARINC, 2005, 2006). A large number of modifications have been made intended to address the main weakness of traditional GPS, and to enhance its inherent performance. The accuracy and reliability of those modernized signals and the compatibility between new signals and already-existing

Binary offset carrier (BOC) (Betz, 2001) and multiplexed binary offset carrier (MBOC) modulations (Hein et al., 2006) have been chosen as the chief candidate for several future navigation signals, for example, GPS L1C, GPS M-code, and Galileo open service (OS) signals. BOC modulation is a square-wave modulation scheme. It moves signal energy away from the band center and thus achieves a higher degree of spectral separation between BOC modulated signals and other signals which use traditional binary phase shift keying (BPSK) modulation, such as the GPS C/A code, in order to get a more efficient sharing of the L-band spectrum. Besides, many studies (Avila-Rodriguez, et al., 2007; Betz, 2001; Hein, et al., 2006) show that BOC modulation also provides better inherent resistance to multipath

However, despite these advantages, some problems remain with the use of BOC modulation. According to the theory of matched filtering (Proakis, 2001), when the waveform of the local signal is as same as the received one, the output of the correlator has the highest signal-noise-ratio (SNR). For this reason, in traditional GPS receivers, both of the acquisition and tracking are based upon the auto-correlation function (ACF) of the received

China is in the midst of launching Compass (Gao et al., 2007).

signals have been simultaneously taken into account in the design.

**1. Introduction** 

and narrowband interference.


## **Unambiguous Processing Techniques of Binary Offset Carrier Modulated Signals**

Zheng Yao *Tsinghua University China* 

## **1. Introduction**

20 Will-be-set-by-IN-TECH

52 Global Navigation Satellite Systems – Signal, Theory and Applications

Namgoong, W. & Meng, T. (2001b). Minimizing power consumption in direct sequence

Namgoong, W., Reader, S. & Meng, T. (2000). An all-digital low-power IF GPS synchronizer,

OSSISICD (2010). European gnss (galileo) open service signal in space interface control

Parkinson, B. & Spilker, J. (eds) (1995). *Global Positioning System: Theory and Applications*,

Shivaramaiah, N. C. (2004). *A Fast Acquisition Hardware GPS Correlator*, Master's thesis, Center

Shivaramaiah, N. C. (2011). *Enhanced Receiver Techniques for Galileo E5 AltBOC Signal Processing*,

Shivaramaiah, N. C. & Dempster, A. G. (2009). Design challenges of a Galileo E1 correlator on

Shivaramaiah, N. C. & Dempster, A. G. (2010). On the baseband hardware complexity of

Shivaramaiah, N. C., Dempster, A. G. & Rizos, C. (2009). Application of Prime-factor

Zarlink (1999). *GPS Receiver Hardware Design Application Note AN4855*, 2.0 edn, Zarlink

for Electronics Design and Technology, Indian Institute of Science, Bangalore, India.

PhD thesis, School of Surveying and Spatial Information Systems, University of New

and Mixed-radix FFT Algorithms in Multi-band GNSS Receivers, *Journal of GPS*

*Solid-State Circuits, IEEE Journal of* 35(6): 856–864.

American Institute of Aeronautics and Astronautics.

the Namuru platform, *IGNSS Symp*, Gold Coast, Australia.

modernized GNSS receivers, *IEEE ISCAS*, pp. 3565 –3568.

Zarlink (2001). *GPS 12 channel correlator*, issue 3.2 edn, Zarlink Semiconductor.

South Wales, Sydney, Australia.

*on* 48(5): 460–470.

document.

8: 174–186.

Semiconductor.

spread spectrum correlators by resampling IF samples-Part II: implementation issues, *Circuits and Systems II: Analog and Digital Signal Processing, IEEE Transactions*

> In recent years, applications of global navigation satellite systems (GNSS) are developing rapidly. The growing public demand for positioning and location services has generated higher requirements for system performance. However, the performance of the traditional GPS is constrained by its inherent capability. In order to cope with both the civil and military expectations in terms of performance, several projects are launched to promote the next generation of GNSS (Hegarty & Chatre, 2008). GPS is undergoing an extensive modernization process (Enge, 2003), while the European satellite system, Galileo, is also under construction. In addition, Russia is restoring their GLONASS (Slater et al., 2004), and China is in the midst of launching Compass (Gao et al., 2007).

> Based on the experience gained during the traditional GPS design and operation, signal structures of these new navigation systems have been well designed (ARINC, 2005, 2006). A large number of modifications have been made intended to address the main weakness of traditional GPS, and to enhance its inherent performance. The accuracy and reliability of those modernized signals and the compatibility between new signals and already-existing signals have been simultaneously taken into account in the design.

> Binary offset carrier (BOC) (Betz, 2001) and multiplexed binary offset carrier (MBOC) modulations (Hein et al., 2006) have been chosen as the chief candidate for several future navigation signals, for example, GPS L1C, GPS M-code, and Galileo open service (OS) signals. BOC modulation is a square-wave modulation scheme. It moves signal energy away from the band center and thus achieves a higher degree of spectral separation between BOC modulated signals and other signals which use traditional binary phase shift keying (BPSK) modulation, such as the GPS C/A code, in order to get a more efficient sharing of the L-band spectrum. Besides, many studies (Avila-Rodriguez, et al., 2007; Betz, 2001; Hein, et al., 2006) show that BOC modulation also provides better inherent resistance to multipath and narrowband interference.

> However, despite these advantages, some problems remain with the use of BOC modulation. According to the theory of matched filtering (Proakis, 2001), when the waveform of the local signal is as same as the received one, the output of the correlator has the highest signal-noise-ratio (SNR). For this reason, in traditional GPS receivers, both of the acquisition and tracking are based upon the auto-correlation function (ACF) of the received

Unambiguous Processing Techniques of Binary Offset Carrier Modulated Signals 55

where *ci* is the spreading sequence of binary digits {0,1}, *p*(*t*) is spreading symbol, and *Tc* is the period of the modulated symbol. In conventional GPS, both of C/A code signal and P(Y) code signal use BPSK-R(*n*) modulation whose spreading symbol is the energy normalized

<sup>1</sup> , 0

<sup>⎧</sup> <sup>≤</sup> <sup>&</sup>lt; <sup>⎪</sup> <sup>=</sup> <sup>⎨</sup>

*c*

In principle, the spreading symbol of DSSS signals can be any shape. BOC modulated signal is a variant of basic DSSS signal. The baseband BOC modulated signal can be regarded as the result of multiplying the BPSK-R signal with a sub-carrier which is equal to the sign of a

π

referred to as sine-phased BOC or cosine-phased BOC, respectively. In this Chapter, we focus on sine-phased case. For information on cosine-phased BOC signal unambiguous processing, see (Lohan et al., 2008). Using the terminology from (Betz, 2001), a sine phased BOC modulated signal is denoted as BOC*s*(*m*,*n*), where *m* means the ratio of the square wave frequency *fs* to 1.023 MHz, and *n* represents the ratio of the spreading code rate *fc* to 1.023 MHz. *m* and *n* are constrained to positive integer *m n* ≥ , and the ratio *M* = 2*m*/*n* is

Under the assumption that the spreading sequence has an ideal correlation characteristic,

2

π

π

*s c*

*fT <sup>f</sup> T M*

*s c*

*f fT*

tan , even <sup>2</sup>

2

cos tan , odd <sup>2</sup>

( ) ( )

π

2

<sup>⎧</sup> ⎛ ⎞ <sup>⎪</sup> ⎜ ⎟ <sup>⎪</sup> ⎝ ⎠ <sup>=</sup> <sup>⎨</sup>

*c*

( ) ( )

*<sup>f</sup> fT S f fT <sup>f</sup> T M*

π

2

*c*

It can be seen that due to the effect of subcarrier, BOC modulated signals symmetrically split the main energy component of the signal spectrum and move them away from the band center, so that they have a higher degree of spectral separation with other BPSK-R modulated signals on the same carrier frequency. Moreover, as noted in (Betz, 2001), BOC modulated signals have greater root-mean-square (RMS) bandwidth compared with traditional BPSK signals with the same spreading code frequency. The greater the RMS bandwidth is, the better the inherent ability to mitigate white Gaussian noise and narrowband interference during tracking will be. Consequently, with same *fc*, BOC modulation provides better resistance to thermal noise and narrowband interference than BPSK-R modulation theoretically. However, the ambiguity of the autocorrelation function of sine-BOC modulated signal induces a risk of biased measures in code synchronization.

<sup>⎪</sup> ⎛ ⎞ ⎜ ⎟ <sup>⎪</sup> ⎩ ⎝ ⎠

⎪ ⎩

0, others

*c*

(3)

is the phase of sub-

(5)

)⎤ ⎣ ⎦ (4)

φ

/2, for which the resultant BOC signals are

*t T*

π *f ts* + φ

rectangular pulse with the lasting time *T n <sup>c</sup>* = × 1 / 1.023MHz ( ) :

sine or a cosine waveform:

carrier. Two common values of

BPSK-R ( )

*p t T*

*s ts t* BOC BPSK-R ( ) = ( )sgn sin 2 ⎡ (

are 0 or

referred to as BOC-modulation order, which is constrained to positive integer.

the power spectrum density (PSD) of BOC*<sup>s</sup>* ( *f f s c* , ) can be expressed as (Betz, 2001)

2

π

π

sin

*c*

*c*

where sgn(⋅) is sign function, *fs* is the sub-carrier frequency, and

φ

( )

*s*

BOC 2

signals. Nevertheless, because of the square-wave modulated symbol, a BOC modulated signal has a sawtooth-like, piecewise linear ACF which has multiple non-negligible side peaks along with the main peak. Since there are significant amount of signal energy located at side peaks of BOC ACF, in acquisition stage, under the influence of noise it is quite likely that one of side peak magnitudes exceeds the main peak, and false acquisition will happen. If false acquisition occurs, the code tracking loop will initially lock on the side peak. Similarly, due to the side peaks of ACF, in code tracking loop, the discriminator characteristic curve of a BOC modulated signal has multiple stable false lock points. Once the loop locks on one of the side peaks, it would result in intolerable bias in pseudorange measurements, which is unacceptable for GNSS aiming to provide accurate navigation solution. This problem is reputed as the ambiguity problem for BOC modulated signal acquisition and tracking. And in order to employ BOC modulated signals in the next generation GNSS, solutions have to be found to minimize this bias threat.

In this Chapter, the ambiguity problem of BOC modulated signals as well as its typical solutions is systematically described. An innovative design methodology for future unambiguous processing techniques is also proposed. Some practical design examples on this methodology are also given to show the practicality and to provide reference to further algorithm development.

The rest of the Chapter is organized as follows. In Section 2, the concept and some main characteristics of BOC modulated signals are given, and the ambiguity problem is also described. In Section 3, some existing representative solutions to ambiguity problem are reviewed. Then in Section 4, we present a parameterized chip waveform pattern, and on this basis, give the analytic design framework for side-peak cancellation (SC) based unambiguous BOC signal processing algorithm development. As two application examples of the proposed design framework, the design process of an SC unambiguous acquisition algorithm as well as an SC unambiguous tracking loop is described in Section 5 and Section 6, respectively. Finally, some conclusions are drawn in Section 7.

#### **2. BOC modulated signals**

#### **2.1 Definitions and main characteristics**

In order to take advantage of the frequent phase inversions in the spreading waveform to realize the precise ranging, and to obtain excellent multiple access capability, the majority of GNSS employ direct sequence spread spectrum (DSSS) technique. DSSS can be regarded as an extension of binary phase shift keying (BPSK). The transmitting signal yielded by this technique can be expressed as the product of the un-modulated carrier, data *d*(*t*), as well as the baseband spreading signal *g*(*t*), that is

$$s(t) = A\_s d(t) \lg(t) \cos(2\pi f\_0 t + \theta) \tag{1}$$

where *As* is the amplitude of signal, *f*0 is the carrier frequency in Hz, and θ is the carrier phase in radians. The baseband spreading signal *g*(*t*) can be further represented as

$$\log\left(t\right) = \sum\_{i=-\alpha}^{\alpha} \left(-1\right)^{\varepsilon\_i} p\left(t - iT\_c\right) \tag{2}$$

54 Global Navigation Satellite Systems – Signal, Theory and Applications

signals. Nevertheless, because of the square-wave modulated symbol, a BOC modulated signal has a sawtooth-like, piecewise linear ACF which has multiple non-negligible side peaks along with the main peak. Since there are significant amount of signal energy located at side peaks of BOC ACF, in acquisition stage, under the influence of noise it is quite likely that one of side peak magnitudes exceeds the main peak, and false acquisition will happen. If false acquisition occurs, the code tracking loop will initially lock on the side peak. Similarly, due to the side peaks of ACF, in code tracking loop, the discriminator characteristic curve of a BOC modulated signal has multiple stable false lock points. Once the loop locks on one of the side peaks, it would result in intolerable bias in pseudorange measurements, which is unacceptable for GNSS aiming to provide accurate navigation solution. This problem is reputed as the ambiguity problem for BOC modulated signal acquisition and tracking. And in order to employ BOC modulated signals in the next

In this Chapter, the ambiguity problem of BOC modulated signals as well as its typical solutions is systematically described. An innovative design methodology for future unambiguous processing techniques is also proposed. Some practical design examples on this methodology are also given to show the practicality and to provide reference to further

The rest of the Chapter is organized as follows. In Section 2, the concept and some main characteristics of BOC modulated signals are given, and the ambiguity problem is also described. In Section 3, some existing representative solutions to ambiguity problem are reviewed. Then in Section 4, we present a parameterized chip waveform pattern, and on this basis, give the analytic design framework for side-peak cancellation (SC) based unambiguous BOC signal processing algorithm development. As two application examples of the proposed design framework, the design process of an SC unambiguous acquisition algorithm as well as an SC unambiguous tracking loop is described in Section 5 and Section

In order to take advantage of the frequent phase inversions in the spreading waveform to realize the precise ranging, and to obtain excellent multiple access capability, the majority of GNSS employ direct sequence spread spectrum (DSSS) technique. DSSS can be regarded as an extension of binary phase shift keying (BPSK). The transmitting signal yielded by this technique can be expressed as the product of the un-modulated carrier, data *d*(*t*), as well as

*s t Ad t g t f t* ( ) = *<sup>s</sup>* ( ) ( ) cos 2(

() ( ) ( ) 1 *<sup>i</sup> <sup>c</sup>*

*g t p t iT* ∞ =−∞

where *As* is the amplitude of signal, *f*0 is the carrier frequency in Hz, and

phase in radians. The baseband spreading signal *g*(*t*) can be further represented as

*i*

π <sup>0</sup> +θ

*c*

=− − ∑ (2)

) (1)

θ

is the carrier

generation GNSS, solutions have to be found to minimize this bias threat.

6, respectively. Finally, some conclusions are drawn in Section 7.

algorithm development.

**2. BOC modulated signals** 

**2.1 Definitions and main characteristics** 

the baseband spreading signal *g*(*t*), that is

where *ci* is the spreading sequence of binary digits {0,1}, *p*(*t*) is spreading symbol, and *Tc* is the period of the modulated symbol. In conventional GPS, both of C/A code signal and P(Y) code signal use BPSK-R(*n*) modulation whose spreading symbol is the energy normalized rectangular pulse with the lasting time *T n <sup>c</sup>* = × 1 / 1.023MHz ( ) :

$$p\_{\text{BPSK},\text{R}}\left(t\right) = \begin{cases} \frac{1}{\sqrt{T\_c}}, & 0 \le t < T\_c \\ 0, & \text{others} \end{cases} \tag{3}$$

In principle, the spreading symbol of DSSS signals can be any shape. BOC modulated signal is a variant of basic DSSS signal. The baseband BOC modulated signal can be regarded as the result of multiplying the BPSK-R signal with a sub-carrier which is equal to the sign of a sine or a cosine waveform:

$$s\_{\text{BOC}}(t) = s\_{\text{BPSK}\cdot\text{R}}\left(t\right) \text{sgn}\left[\sin\left(2\pi f\_{\text{s}}t + \phi\right)\right] \tag{4}$$

where sgn(⋅) is sign function, *fs* is the sub-carrier frequency, and φ is the phase of subcarrier. Two common values of φ are 0 or π /2, for which the resultant BOC signals are referred to as sine-phased BOC or cosine-phased BOC, respectively. In this Chapter, we focus on sine-phased case. For information on cosine-phased BOC signal unambiguous processing, see (Lohan et al., 2008). Using the terminology from (Betz, 2001), a sine phased BOC modulated signal is denoted as BOC*s*(*m*,*n*), where *m* means the ratio of the square wave frequency *fs* to 1.023 MHz, and *n* represents the ratio of the spreading code rate *fc* to 1.023 MHz. *m* and *n* are constrained to positive integer *m n* ≥ , and the ratio *M* = 2*m*/*n* is referred to as BOC-modulation order, which is constrained to positive integer.

Under the assumption that the spreading sequence has an ideal correlation characteristic, the power spectrum density (PSD) of BOC*<sup>s</sup>* ( *f f s c* , ) can be expressed as (Betz, 2001)

$$S\_{\rm BOC\_{\sim}}(f) = \begin{cases} T\_c \frac{\sin^2 \left(\pi f T\_c \right)}{\left(\pi f T\_c\right)^2} \tan^2 \left(\frac{\pi f}{2f\_s}\right), & \text{ $M$  even} \\ T\_c \frac{\cos^2 \left(\pi f T\_c \right)}{\left(\pi f T\_c\right)^2} \tan^2 \left(\frac{\pi f}{2f\_s}\right), & \text{ $M$  odd} \end{cases} \tag{5}$$

It can be seen that due to the effect of subcarrier, BOC modulated signals symmetrically split the main energy component of the signal spectrum and move them away from the band center, so that they have a higher degree of spectral separation with other BPSK-R modulated signals on the same carrier frequency. Moreover, as noted in (Betz, 2001), BOC modulated signals have greater root-mean-square (RMS) bandwidth compared with traditional BPSK signals with the same spreading code frequency. The greater the RMS bandwidth is, the better the inherent ability to mitigate white Gaussian noise and narrowband interference during tracking will be. Consequently, with same *fc*, BOC modulation provides better resistance to thermal noise and narrowband interference than BPSK-R modulation theoretically. However, the ambiguity of the autocorrelation function of sine-BOC modulated signal induces a risk of biased measures in code synchronization.

Unambiguous Processing Techniques of Binary Offset Carrier Modulated Signals 57

*M i M*

⎛ ⎞ <sup>−</sup> <sup>=</sup> ⎜ ⎟ ⎝ ⎠

For 2 *M* = , side peaks are 6 dB weaker than the main peak. But for 6 *M* = , the gap between the largest side peak and the main peak is only 2.5 dB. With increase of *M*, the difference between the maximum side peak and the main peak decreases, while the false acquisition

(a)

 (b) Fig. 2. (a) The Statistic in Acquisition Stage and (b) The Discriminator Curve in Tracking

It can be seen from Figure 2(b) that when using a traditional narrow early-minus-late (NEML) tracking loop (Van Dierendonck et al., 1992) with the early-late separation ∆, the discriminator characteristic curve of BOC(*m*,*n*) signal has a smaller linear domain than the one of the BPSK-R(*n*) signal. Besides, the discriminator characteristic curve of a BOC(*m*,*n*) signal has 2*M* - 2 stable false lock points which are due to the side peaks of the autocorrelation function. If a false acquisition occurs, in tracking stage, the code tracking loop will initially lock on a false lock point. Even if there is no false acquisition, the false lock

Figure 3 shows an example of false lock caused by the excessive initial code delay bias in BOC(2*n*,*n*) signal tracking. The *C*/*N*0 in this example is 45 dB-Hz, and the predetection integration time is 1 ms, with the early-late separation 0.1 chips. It can be seen that with the

*i*

ξ

probability increase.

Stage of BOC(2*n*,*n*) Signal

can result from high noise, jitter, or short loss of lock.

2

(7)

#### **2.2 Ambiguous problem**

The difference of the spreading chip waveforms between BPSK-R signals and BOC signals leads their difference in ACF shapes, and thus makes the distinction of acquisition and tracking performance. The ACF of BPSK-R modulated signals is a triangle, but BOC signals have sawtooth-like, piecewise linear ACF. The normalized BOC(*m*,*n*) ACF without filter can be expressed as (Yao, 2009)

$$R\_{\rm RCC}\left(\tau\right) = \begin{bmatrix} \left(-1\right)^{k+1} \left[ \left(-\frac{2k^2 - 2k}{M} + 2k - 1\right) - \left(2M - 2k + 1\right) \left| \tau\right| \right]\_{\prime} & \left| \tau \right| \le T\_{\varepsilon} \\\\ 0, & \text{others} \end{bmatrix} \tag{6}$$

where *k M* = ⋅ ⎡ ⎤ τ ⎢ ⎥ , and ⎡*x*⎤ ⎢ ⎥ means the smallest integer not less than *x* . In Figure 1, the normalized ACF envelopes of BPSK-R(1) signal and BOC(2,1) signal are drawn.

Fig. 1. BPSK-R(1) and BOC(2,1) Normalized ACF Envelopes

From Figure 1 we can see that these two signals have the same spreading chip rate 1.023 MHz, but their ACFs have entirely different shapes. Compared with the triangular ACF of BPSK-R(1) signal, that of BOC(2,1) signal has a sharper main peak, which means better tracking accuracy in thermal noise. However, the ACF of BOC signal has multiple side peaks within 1 τ = ± chips. At the acquisition and tracking stages, these side peaks could be mistaken for the main peak.

When the traditional acquisition and tracking algorithms are employed to process BOC(2*n*,*n*) signal, the shapes of statistic and the discriminator curve are shown in Figure 2(a) and Figure 2(b), respectively.

The traditional acquisition and tracking of DSSS signal have been very well discussed (Ziemer & Peterson, 1985). From Figure 2(a) we can see that since there are significant amount of signal energy located at side peaks of BOC ACF, under the influence of noise it is quite likely that one of side peak magnitudes exceeds the main peak, and false acquisition will happen. For *M*-order BOC signal, the energy ratio between the *i*-th largest side peak and the main peak is

56 Global Navigation Satellite Systems – Signal, Theory and Applications

The difference of the spreading chip waveforms between BPSK-R signals and BOC signals leads their difference in ACF shapes, and thus makes the distinction of acquisition and tracking performance. The ACF of BPSK-R modulated signals is a triangle, but BOC signals have sawtooth-like, piecewise linear ACF. The normalized BOC(*m*,*n*) ACF without filter can

2 2 1 2 <sup>1</sup> <sup>2</sup> <sup>2</sup> <sup>1</sup> ,

<sup>+</sup> <sup>⎧</sup> ⎡ ⎤ ⎛ ⎞ <sup>−</sup> <sup>⎪</sup> <sup>−</sup> ⎢ ⎥ ⎜ ⎟ − +−− −+ ≤ <sup>=</sup> <sup>⎨</sup> ⎣ ⎦ ⎝ ⎠ <sup>⎪</sup>

From Figure 1 we can see that these two signals have the same spreading chip rate 1.023 MHz, but their ACFs have entirely different shapes. Compared with the triangular ACF of BPSK-R(1) signal, that of BOC(2,1) signal has a sharper main peak, which means better tracking accuracy in thermal noise. However, the ACF of BOC signal has multiple side

When the traditional acquisition and tracking algorithms are employed to process BOC(2*n*,*n*) signal, the shapes of statistic and the discriminator curve are shown in Figure

The traditional acquisition and tracking of DSSS signal have been very well discussed (Ziemer & Peterson, 1985). From Figure 2(a) we can see that since there are significant amount of signal energy located at side peaks of BOC ACF, under the influence of noise it is quite likely that one of side peak magnitudes exceeds the main peak, and false acquisition will happen. For *M*-order BOC signal, the energy ratio between the *i*-th largest side peak

= ± chips. At the acquisition and tracking stages, these side peaks could be

0, others

⎢ ⎥ , and ⎡*x*⎤ ⎢ ⎥ means the smallest integer not less than *x* . In Figure 1, the

*k k k Mk T*

τ

*c*

(6)

 τ

( ) ( ) ( ) <sup>2</sup> <sup>1</sup>

normalized ACF envelopes of BPSK-R(1) signal and BOC(2,1) signal are drawn.

**2.2 Ambiguous problem** 

be expressed as (Yao, 2009)

where *k M* = ⋅ ⎡ ⎤

peaks within 1 τ

and the main peak is

mistaken for the main peak.

2(a) and Figure 2(b), respectively.

BOC

τ

τ

*k*

Fig. 1. BPSK-R(1) and BOC(2,1) Normalized ACF Envelopes

*R M*

⎩

$$
\xi\_i = \left(\frac{M-i}{M}\right)^2\tag{7}
$$

For 2 *M* = , side peaks are 6 dB weaker than the main peak. But for 6 *M* = , the gap between the largest side peak and the main peak is only 2.5 dB. With increase of *M*, the difference between the maximum side peak and the main peak decreases, while the false acquisition probability increase.

Fig. 2. (a) The Statistic in Acquisition Stage and (b) The Discriminator Curve in Tracking Stage of BOC(2*n*,*n*) Signal

It can be seen from Figure 2(b) that when using a traditional narrow early-minus-late (NEML) tracking loop (Van Dierendonck et al., 1992) with the early-late separation ∆, the discriminator characteristic curve of BOC(*m*,*n*) signal has a smaller linear domain than the one of the BPSK-R(*n*) signal. Besides, the discriminator characteristic curve of a BOC(*m*,*n*) signal has 2*M* - 2 stable false lock points which are due to the side peaks of the autocorrelation function. If a false acquisition occurs, in tracking stage, the code tracking loop will initially lock on a false lock point. Even if there is no false acquisition, the false lock can result from high noise, jitter, or short loss of lock.

Figure 3 shows an example of false lock caused by the excessive initial code delay bias in BOC(2*n*,*n*) signal tracking. The *C*/*N*0 in this example is 45 dB-Hz, and the predetection integration time is 1 ms, with the early-late separation 0.1 chips. It can be seen that with the

Unambiguous Processing Techniques of Binary Offset Carrier Modulated Signals 59

False lock detection and recovery technique does not remove ambiguity but rather checks false lock. The most representative detection and recovery technique is referred to as bumpjumping technique (Fine & Wilson, 1999). This technique employs the traditional ambiguous code tracking loop and constantly check whether this loop is locked on the main peak of BOC ACF. To do so, bump-jumping technique uses two additional correlators located at the

These two correlators are referred to as very early (VE) and very late (VL) correlators. By measuring and comparing the magnitude of the outputs of these two correlators and the prompt one, bump-jumping technique determines whether the false lock happens. It can be seen from Figure 5 that ignoring the effect of noise, when the code loop locks on the main peak, the magnitude of prompt correlator output is the greatest. And if either VE or VL correlator output is the largest, it means that tracking might be biased, and the loop will

When locked on the main peak, this technique has high tracking accuracy. However, since it is based on the comparing of the main and side peaks magnitudes, the detection may have a high probability of false alarm when the signal-to-noise ratio (SNR) is low. In (Fine & Wilson, 1999), two up/down counter mechanisms are employed to reduce this false alarm probability. After each comparison, if one of the magnitudes of VE and VL correlator outputs exceeds that of the prompt one, the corresponding counter is incremented by one, otherwise the corresponding counter is decremented by one. The counter is not decremented below 0 or incremented above the preset threshold *N*. When the counter reaches the threshold, the loop jumps to the highest peak. By using this counter mechanism, the false alarm probability can be reduced effectively. However, the response time is also increased. Once the false lock happen, this technique needs time to detect and recover from false lock, so it might be intolerable for some critical applications such as aircraft landing.

E L


P

VE VL

E L


VE

P

VL

τ

theoretical location of the two highest side peaks, as shown in Figure 5.

**3.1 False lock detection & recovery** 

"jump" in the appropriate direction.

Fig. 5. Bump-jumping technique

initial code delay bias of 0.14 chips, the loop locks on a false lock point. Although the output of discriminator hovers around zero and DLL maintains lock, the true code phase measurement bias is 0.25 chips, which corresponds to 75 m for BOC(2,1) signal. Such a considerable bias is unacceptable in most of the positioning applications.

Fig. 3. Example of False Lock in Traditional DLL

## **3. Existing unambiguous processing techniques**

During the decade from when BOC modulation was initially proposed to the present time, several solutions have been proposed to solve the ambiguity problem. In summary, the elimination of ambiguity threat can be achieved via two ways: false lock detection and recovery technique, as well as unambiguous processing techniques. More specifically, considering the operation domain, unambiguous processing can be further classified into frequency-domain processing and time-domain processing.

Fig. 4. Existing Ambiguity Elimination Solutions

#### **3.1 False lock detection & recovery**

58 Global Navigation Satellite Systems – Signal, Theory and Applications

initial code delay bias of 0.14 chips, the loop locks on a false lock point. Although the output of discriminator hovers around zero and DLL maintains lock, the true code phase measurement bias is 0.25 chips, which corresponds to 75 m for BOC(2,1) signal. Such a

During the decade from when BOC modulation was initially proposed to the present time, several solutions have been proposed to solve the ambiguity problem. In summary, the elimination of ambiguity threat can be achieved via two ways: false lock detection and recovery technique, as well as unambiguous processing techniques. More specifically, considering the operation domain, unambiguous processing can be further classified into

> Unambiguous Processing

> > Time-domain Processing

Ambiguity Elimination

Frequency -domain Processing

considerable bias is unacceptable in most of the positioning applications.

Fig. 3. Example of False Lock in Traditional DLL

**3. Existing unambiguous processing techniques** 

frequency-domain processing and time-domain processing.

False Lock Detection & Recovery

Fig. 4. Existing Ambiguity Elimination Solutions

False lock detection and recovery technique does not remove ambiguity but rather checks false lock. The most representative detection and recovery technique is referred to as bumpjumping technique (Fine & Wilson, 1999). This technique employs the traditional ambiguous code tracking loop and constantly check whether this loop is locked on the main peak of BOC ACF. To do so, bump-jumping technique uses two additional correlators located at the theoretical location of the two highest side peaks, as shown in Figure 5.

These two correlators are referred to as very early (VE) and very late (VL) correlators. By measuring and comparing the magnitude of the outputs of these two correlators and the prompt one, bump-jumping technique determines whether the false lock happens. It can be seen from Figure 5 that ignoring the effect of noise, when the code loop locks on the main peak, the magnitude of prompt correlator output is the greatest. And if either VE or VL correlator output is the largest, it means that tracking might be biased, and the loop will "jump" in the appropriate direction.

When locked on the main peak, this technique has high tracking accuracy. However, since it is based on the comparing of the main and side peaks magnitudes, the detection may have a high probability of false alarm when the signal-to-noise ratio (SNR) is low. In (Fine & Wilson, 1999), two up/down counter mechanisms are employed to reduce this false alarm probability. After each comparison, if one of the magnitudes of VE and VL correlator outputs exceeds that of the prompt one, the corresponding counter is incremented by one, otherwise the corresponding counter is decremented by one. The counter is not decremented below 0 or incremented above the preset threshold *N*. When the counter reaches the threshold, the loop jumps to the highest peak. By using this counter mechanism, the false alarm probability can be reduced effectively. However, the response time is also increased. Once the false lock happen, this technique needs time to detect and recover from false lock, so it might be intolerable for some critical applications such as aircraft landing.

Fig. 5. Bump-jumping technique

Unambiguous Processing Techniques of Binary Offset Carrier Modulated Signals 61

Although the correlation functions in sideband techniques do not present any side peak, meaning that they are fully unambiguous, this kind of methods suffers from some drawbacks. The principle defect is that this kind of methods completely removes all of the advantages of BOC signal tracking in terms of Gaussian noise and multipath mitigation, since it causes the Gabor bandwidth of the received signal to approach that of the BPSK-R signal. Moreover, two side lobes are combined in non-coherent mode, which introduces correlation losses into the process. It seems that sideband techniques are not appropriate in terms of tracking. However, the correlation functions in this kind of methods have wide main correlation peak, which allows using longer code delay step in acquisition to reduce mean acquisition time. Therefore, sideband techniques can be attractive options in BOC

Time-domain processing techniques are also referred to as side-peaks cancellation (SC) techniques which solve the ambiguity problem by taking advantage of the geometrical property of correlation functions (CF). The basic idea of SC techniques is using synthesized correlation function (SCF) instead of BOC ACF in acquisition and tracking. CF between the received BOC signal and some local auxiliary signals whose chip waveforms may be different from the received one are combined linearly or non-linearly to form the SCF with no side-peak. SC methods are flexible. Due to different auxiliary signal chip waveforms and

The first side-peaks cancellation technique is proposed in (Ward, 2003). This approach removes the ambiguities of the correlation function, but one drawback is that this method destroys the sharp peak of the correlation function. For accurate tracking, preserving a sharp peak of the correlation function is a prerequisite. An innovative unambiguous tracking technique, which is referred to as autocorrelation side-peak cancellation technique (ASPeCT), is described in (Julien et al., 2007). This technique uses ten correlation channels, completely removing the side peaks from the correlation function and keeping the sharp

combination modes, SC methods differentiate from each other greatly.

Upper Sideband Processing

Noncoherently combination

f

ej2 fst

π

BOC signal Local BPSK -R signal

modulated signal acquisition.

**3.3 Time-domain unambiguous processing** 

Fig. 7. Block diagram of BPSK-like method

Lowpass Filter I&D Received

Lower Sideband Processing

(Local signal is multiplied by e-j2 fst) <sup>π</sup>

### **3.2 Frequency-domain unambiguous processing**

Frequency-domain processing techniques are represented by sideband techniques. Sideband technique considers the received BOC signal as the sum of two BPSK signals with carrier frequency symmetrically positioned on each side of the BOC carrier frequency. Thus each side lobe is treated independently as a BPSK signal, which provides an unambiguous correlation function and a wider S-curve steady domain.

The earliest sideband technique was described in (Fishman & Betz, 2000). As shown in Figure 6, the single sideband technique uses only one of the sidebands (either upper or lower) of BOC modulated signal. Both the received signal and the local BOC modulated baseband signal are filtered. Only the upper or lower sidebands of the received and local signals are retained. The shape of the correlation function of these two filtered signals is close to that of two BPSK-R signals. Therefore this correlation function can be used instead of BOC ACF in acquisition and tracking. The double sideband technique uses both the upper and lower sideband of BOC modulated signal. These two sidebands are processed separately before the output of correlators, and then the correlation values are added noncoherently. Compared with single sideband technique, double sideband technique suffers lower non-coherent correlation losses. However, it requires twice the sideband-selection filter number of single sideband technique.

Fig. 6. Block diagram of sideband technique

BPSK-like method (Martin et al., 2003) is another frequency-domain unambiguous processing technique. This method is also based on the consideration of the BOC spectrum as the sum of two BPSK spectrum shifted by ±*fs*. The main difference compared with the method described above is the fact that only one low-pass filter is employed for the received signal. As shown in Figure 7, the filter bandwidth includes the two principal lobes of the spectrum. Another difference is that, the local signal is not the filtered BOC-modulated baseband signal but the BPSK-R signal, shifted by the sub-carrier frequency *fs*. The BPSKlike technique can also be either single or double sideband, according to whether both the sidebands are used and combined non-coherently or only one sideband is used.

The original BPSK-like method can only be used for sine-phased BOC modulations with even BOC order. In (Burian et al., 2006), a modified version of BPSK-like method is proposed to extend BPSK-like method to BOC signals with odd order.

Fig. 7. Block diagram of BPSK-like method

60 Global Navigation Satellite Systems – Signal, Theory and Applications

Frequency-domain processing techniques are represented by sideband techniques. Sideband technique considers the received BOC signal as the sum of two BPSK signals with carrier frequency symmetrically positioned on each side of the BOC carrier frequency. Thus each side lobe is treated independently as a BPSK signal, which provides an unambiguous

The earliest sideband technique was described in (Fishman & Betz, 2000). As shown in Figure 6, the single sideband technique uses only one of the sidebands (either upper or lower) of BOC modulated signal. Both the received signal and the local BOC modulated baseband signal are filtered. Only the upper or lower sidebands of the received and local signals are retained. The shape of the correlation function of these two filtered signals is close to that of two BPSK-R signals. Therefore this correlation function can be used instead of BOC ACF in acquisition and tracking. The double sideband technique uses both the upper and lower sideband of BOC modulated signal. These two sidebands are processed separately before the output of correlators, and then the correlation values are added noncoherently. Compared with single sideband technique, double sideband technique suffers lower non-coherent correlation losses. However, it requires twice the sideband-selection

Upper Sideband Processing

BPSK-like method (Martin et al., 2003) is another frequency-domain unambiguous processing technique. This method is also based on the consideration of the BOC spectrum as the sum of two BPSK spectrum shifted by ±*fs*. The main difference compared with the method described above is the fact that only one low-pass filter is employed for the received signal. As shown in Figure 7, the filter bandwidth includes the two principal lobes of the spectrum. Another difference is that, the local signal is not the filtered BOC-modulated baseband signal but the BPSK-R signal, shifted by the sub-carrier frequency *fs*. The BPSKlike technique can also be either single or double sideband, according to whether both the

The original BPSK-like method can only be used for sine-phased BOC modulations with even BOC order. In (Burian et al., 2006), a modified version of BPSK-like method is

Noncoherently combination

**3.2 Frequency-domain unambiguous processing** 

correlation function and a wider S-curve steady domain.

filter number of single sideband technique.

BOC spectrum

Fig. 6. Block diagram of sideband technique

BOC signal Local BOC signal *f*

Upper Sideband Filter Upper Sideband Filter

I&D Received

Lower Sideband Processing

sidebands are used and combined non-coherently or only one sideband is used.

proposed to extend BPSK-like method to BOC signals with odd order.

Although the correlation functions in sideband techniques do not present any side peak, meaning that they are fully unambiguous, this kind of methods suffers from some drawbacks. The principle defect is that this kind of methods completely removes all of the advantages of BOC signal tracking in terms of Gaussian noise and multipath mitigation, since it causes the Gabor bandwidth of the received signal to approach that of the BPSK-R signal. Moreover, two side lobes are combined in non-coherent mode, which introduces correlation losses into the process. It seems that sideband techniques are not appropriate in terms of tracking. However, the correlation functions in this kind of methods have wide main correlation peak, which allows using longer code delay step in acquisition to reduce mean acquisition time. Therefore, sideband techniques can be attractive options in BOC modulated signal acquisition.

#### **3.3 Time-domain unambiguous processing**

Time-domain processing techniques are also referred to as side-peaks cancellation (SC) techniques which solve the ambiguity problem by taking advantage of the geometrical property of correlation functions (CF). The basic idea of SC techniques is using synthesized correlation function (SCF) instead of BOC ACF in acquisition and tracking. CF between the received BOC signal and some local auxiliary signals whose chip waveforms may be different from the received one are combined linearly or non-linearly to form the SCF with no side-peak. SC methods are flexible. Due to different auxiliary signal chip waveforms and combination modes, SC methods differentiate from each other greatly.

The first side-peaks cancellation technique is proposed in (Ward, 2003). This approach removes the ambiguities of the correlation function, but one drawback is that this method destroys the sharp peak of the correlation function. For accurate tracking, preserving a sharp peak of the correlation function is a prerequisite. An innovative unambiguous tracking technique, which is referred to as autocorrelation side-peak cancellation technique (ASPeCT), is described in (Julien et al., 2007). This technique uses ten correlation channels, completely removing the side peaks from the correlation function and keeping the sharp

Unambiguous Processing Techniques of Binary Offset Carrier Modulated Signals 63

Any SCS waveform can be represented by a coordinate vector. Given a pair of *Tc* and *Ts* ,

( ) ( ) <sup>1</sup> , 1

As one can easily confirm, these *M* functions are orthogonal to each other and any SCS chip waveform *p*SCS (*t*) with the same *Tc* and *M* can be written as a linear combination of

> () () ψ

= () ()

[ ]<sup>T</sup>

1 1 <sup>2</sup> <sup>T</sup> <sup>1</sup>

Given the spreading chip rate *fc* and the vector *d*, the chip waveform *p*(*t*) is determined. Borrowing from the notation of BCS signals (Hegarty et al., 2004), we call the vector *d* shape vector, and use the notation *p*(*t*; *d*, *fc*) to denote a SCS signal whose shape vector is *d* and the chip rate is *fc*. If it is understood from context, we will omit *fc* from the notation.

It can be seen that the chip waveforms employed in most of the modulations in satellite navigation such as BPSK-R, BOC with even order, and BCS are special cases of SCS waveforms. Besides, almost all the auxiliary signal chip waveforms used in SC algorithms also belong to this family. When *d*=1, *p*(*t*;*d*) degenerates to the rectangular pulse, and when

2 1 1, 1, ,1, 1 <sup>×</sup> *d* =− − " <sup>A</sup> , *p*(*t*;*d*) is the chip waveform of a sine phased BOC signal with the

*M M*

 ∫ SCS <sup>0</sup> *Tc*

0

− = <sup>=</sup> ∑ <sup>⋅</sup> 1

*M*

*k*

ψ*<sup>k</sup>* (*t*) , i.e.

<sup>⎧</sup> ≤< + <sup>⎪</sup> <sup>=</sup> <sup>⎨</sup>

0, others

*<sup>k</sup>* (*tk M* ) : 0,1, , 1 = " − } , where

*s s*

*kT t k T*

*k k*

ψ

*p t td* (9)

*k k d M p t t dt* (10)

01 1 *d* = *dd d* ,,, " *<sup>M</sup>*<sup>−</sup> (11)

*d dd* = = (12)

*<sup>k</sup>* (*t*) for *k M* = 0,1, , 1 " − . To meet the energy normalization

ψ

*k c*

⎪ ⎩

SCS

Therefore, each chip symbol *p*SCS (*t*) corresponds to a coordinate vector

ψ

condition of the SCS chip waveform, each vector *d* must satisfy

*t T*

ψ

0 0 Tc Tc/2 Tc 0 Tc

Tc 2 Tc 4

<sup>c</sup> 3T 4

(8)

0

Fig. 8. Examples of SCS Waveforms

we can construct a set of function {

where *<sup>k</sup> d* is the projection of *p*SCS (*t*) onto

**4.2 Vector representation** 

{ψ

*<sup>k</sup>* (*t*)} , that is

in the space spanned by

[ ]<sup>T</sup>

0 Tc/2 Tc

main peak. However, this technique has some limitations, for it is only applicable to sine-BOC(*n*,*n*) signals. Some other side-peaks cancellation methods have been proposed recently (Dovis, et al., 2005; Fante, 2003; Musso et al., 2006; Nunes et al., 2007).

However, the design of SC algorithms is still scarce of uniform theoretical frame and analytical method. There is no easy handling design method for SC algorithms development. The key of SC methods is the selection of local auxiliary signal chip waveforms. Due to lack of mathematical analysis tools, the selection of local signal chip waveforms is mainly based on intuition and trail-and-error. In new SC algorithm design, the shape of auxiliary signal waveforms is limited by the imagination of the designer, thus concentrating on some common shapes such as rectangular pulse, square wave with sine phase or cosine phase, and return to zero (RZ) code wave. When the chip waveform of the received signal is simple, for instance, Manchester code which is used in BOC(*n*,*n*) signals, it is easy to find a corresponding auxiliary chip waveform by using trail-and-error method. However, when the chip waveform of target signal gets more complicated, the design process becomes tough, and a mathematical analysis method is needed.

In the next section, a SC analytic design framework is presented. In this framework, the local auxiliary signal chip waveform can be designed under this framework by means of mathematic analysis so that the waveform shape selection can be more flexible.

## **4. SC analytic design framework**

## **4.1 SCS waveform**

The main difficulty of SC method design is how to select the spreading code chip waveform of local signals. It is desired to define a parameterized local signal model the chip waveform of which has a high degrees of freedom and is easy to generate in receivers to provide more opportunities for waveform optimization. Although there are few investigations about general local signal model for receiver designers since most of the signal receiving techniques are based on matched correlator in GNSS, it is interesting to note that some generalized waveform models are proposed for satellite signal design in order to offer degrees of freedom for shaping the signal spectrum, such as the binary coded symbols (BCS) (Hegarty et al., 2004). The advanced idea can be instructive for SC algorithm design.

For BCS signals, in order to ensure constant modulus, the envelope of *p*(*t*) is restricted to 1. However, when considering auxiliary chip waveform in SC techniques, since local signals do not relate to amplifying and transmitting, they do not need to satisfy the request of constant modulus but their chip waveform should be easy to generate. Therefore, we expand the definition of BCS signal, restricting the chip waveform to being real-valued and having normalized energy. The chip waveform is divided into *M* segments, each with equal length *TTM s c* = / , and in each segment the level remains constant.

Since such waveform looks like steps, for expressional simplicity, we call this kind of chip waveform the step-shape code symbol (SCS) waveform, and call the signal which uses this waveform the SCS signal hereafter. Sticking with the terms used for BOC signals, *M* is referred to as the order of SCS signal. Some examples of SCS waveforms are shown in Figure 8.

Fig. 8. Examples of SCS Waveforms

#### **4.2 Vector representation**

62 Global Navigation Satellite Systems – Signal, Theory and Applications

main peak. However, this technique has some limitations, for it is only applicable to sine-BOC(*n*,*n*) signals. Some other side-peaks cancellation methods have been proposed recently

However, the design of SC algorithms is still scarce of uniform theoretical frame and analytical method. There is no easy handling design method for SC algorithms development. The key of SC methods is the selection of local auxiliary signal chip waveforms. Due to lack of mathematical analysis tools, the selection of local signal chip waveforms is mainly based on intuition and trail-and-error. In new SC algorithm design, the shape of auxiliary signal waveforms is limited by the imagination of the designer, thus concentrating on some common shapes such as rectangular pulse, square wave with sine phase or cosine phase, and return to zero (RZ) code wave. When the chip waveform of the received signal is simple, for instance, Manchester code which is used in BOC(*n*,*n*) signals, it is easy to find a corresponding auxiliary chip waveform by using trail-and-error method. However, when the chip waveform of target signal gets more complicated, the design

In the next section, a SC analytic design framework is presented. In this framework, the local auxiliary signal chip waveform can be designed under this framework by means of

The main difficulty of SC method design is how to select the spreading code chip waveform of local signals. It is desired to define a parameterized local signal model the chip waveform of which has a high degrees of freedom and is easy to generate in receivers to provide more opportunities for waveform optimization. Although there are few investigations about general local signal model for receiver designers since most of the signal receiving techniques are based on matched correlator in GNSS, it is interesting to note that some generalized waveform models are proposed for satellite signal design in order to offer degrees of freedom for shaping the signal spectrum, such as the binary coded symbols (BCS) (Hegarty et al., 2004). The advanced idea can be instructive for SC algorithm

For BCS signals, in order to ensure constant modulus, the envelope of *p*(*t*) is restricted to 1. However, when considering auxiliary chip waveform in SC techniques, since local signals do not relate to amplifying and transmitting, they do not need to satisfy the request of constant modulus but their chip waveform should be easy to generate. Therefore, we expand the definition of BCS signal, restricting the chip waveform to being real-valued and having normalized energy. The chip waveform is divided into *M* segments, each with equal

Since such waveform looks like steps, for expressional simplicity, we call this kind of chip waveform the step-shape code symbol (SCS) waveform, and call the signal which uses this waveform the SCS signal hereafter. Sticking with the terms used for BOC signals, *M* is referred to as the order of SCS signal. Some examples of SCS waveforms are shown in

length *TTM s c* = / , and in each segment the level remains constant.

(Dovis, et al., 2005; Fante, 2003; Musso et al., 2006; Nunes et al., 2007).

process becomes tough, and a mathematical analysis method is needed.

**4. SC analytic design framework** 

**4.1 SCS waveform** 

design.

Figure 8.

mathematic analysis so that the waveform shape selection can be more flexible.

Any SCS waveform can be represented by a coordinate vector. Given a pair of *Tc* and *Ts* , we can construct a set of function {ψ*<sup>k</sup>* (*tk M* ) : 0,1, , 1 = " − } , where

$$\varphi\_k(t) = \begin{cases} \frac{1}{\sqrt{T\_c}}, & kT\_s \le t < (k+1)T\_s\\ 0, & \text{others} \end{cases} \tag{8}$$

As one can easily confirm, these *M* functions are orthogonal to each other and any SCS chip waveform *p*SCS (*t*) with the same *Tc* and *M* can be written as a linear combination of {ψ*<sup>k</sup>* (*t*)} , that is

$$p\_{\text{SCS}}\left(t\right) = \sum\_{k=0}^{M-1} \varphi\_k\left(t\right) \cdot d\_k \tag{9}$$

where *<sup>k</sup> d* is the projection of *p*SCS (*t*) onto ψ*<sup>k</sup>* (*t*) , i.e.

$$d\_k = M \int\_0^{r\_\epsilon} p\_{\text{SCS}}\left(t\right) \nu\_k\left(t\right) dt\tag{10}$$

Therefore, each chip symbol *p*SCS (*t*) corresponds to a coordinate vector

$$\mathbf{d} = \begin{bmatrix} d\_{0'}d\_{1'} \cdots d\_{M-1} \end{bmatrix}^{\mathrm{T}} \tag{11}$$

in the space spanned by ψ *<sup>k</sup>* (*t*) for *k M* = 0,1, , 1 " − . To meet the energy normalization condition of the SCS chip waveform, each vector *d* must satisfy

$$\frac{1}{M} \left\| d \right\|^2 = \frac{1}{M} d^\top d = 1 \tag{12}$$

Given the spreading chip rate *fc* and the vector *d*, the chip waveform *p*(*t*) is determined. Borrowing from the notation of BCS signals (Hegarty et al., 2004), we call the vector *d* shape vector, and use the notation *p*(*t*; *d*, *fc*) to denote a SCS signal whose shape vector is *d* and the chip rate is *fc*. If it is understood from context, we will omit *fc* from the notation.

It can be seen that the chip waveforms employed in most of the modulations in satellite navigation such as BPSK-R, BOC with even order, and BCS are special cases of SCS waveforms. Besides, almost all the auxiliary signal chip waveforms used in SC algorithms also belong to this family. When *d*=1, *p*(*t*;*d*) degenerates to the rectangular pulse, and when [ ]<sup>T</sup> 2 1 1, 1, ,1, 1 <sup>×</sup> *d* =− − " <sup>A</sup> , *p*(*t*;*d*) is the chip waveform of a sine phased BOC signal with the

Unambiguous Processing Techniques of Binary Offset Carrier Modulated Signals 65

By using (15) one can derive other expression form of an *M*-order BOC signal ACF, which is

<sup>⎧</sup> ⎡ ⎤ −− ≤ <sup>&</sup>lt; <sup>⎪</sup> ⎢ ⎥ ⎣ ⎦ <sup>⎪</sup> <sup>=</sup> <sup>⎨</sup> − + ⎡ ⎤ ≤ < <sup>⎪</sup> ⎣ ⎦

<sup>1</sup> 21 21 <sup>1</sup> B OC

0, others

where 0,1, , 1 *k M* = − " . And the CCF between an *M*-order BOC signal and a SCS signal is

( )( ) ( ) ( )

+

*k i*

<sup>1</sup> 1 ,0 1

<sup>1</sup> 1 ,1 0

*i*

<sup>⎧</sup> − ≤≤ − <sup>⎪</sup>

⎪⎪ <sup>=</sup> <sup>⎨</sup> − − ≤< <sup>⎪</sup>

τ

0 (1*-M*)*Ts … T* -2*Ts -Ts <sup>s</sup>* 2*Ts* (*M*-1)*Ts*

From (15) and (17), it can be seen that the CCF of two SCS signals mostly depends on the ACCF of their chip waveforms when these two signals have the same spreading sequences. In SC algorithm design, since the chip waveform shape of the received signal is known, CCF entirely depends on the shape of local signal spreading chip waveform. Note that each SCS

*r*1

*r*2

τ;*d<sup>L</sup>* ) .

*…*

function is piecewise linear between *<sup>s</sup> kT* and ( 1) *<sup>s</sup> k T* + , and *R kT r* B/L ( *s k* ) = , for

*r*0

*Rxx'* ( ) τ

*c c c*

− + − − +

*c c c c*

*R r rr d* (19)

*d kM*

*k M*

( )( ) ( )

−+ − −

<sup>⎧</sup> − + ≤ < <sup>⎪</sup> <sup>⎪</sup> <sup>=</sup> <sup>⎨</sup> − + ≤< <sup>⎪</sup>

,

<sup>1</sup> B/L <sup>1</sup>

0, others

( )

− −

*M k <sup>i</sup>*

= − − <sup>−</sup>

*i M k i k <sup>k</sup>*

∑

=

*i*

∑

⎪

0,

*M*

*M*

( )

*<sup>r</sup> d Mk*

⎪ ≥ ⎩

*r-*1

*r*-2

− +

*M kT kT k T T M k kk M M kT MT k MT k M T <sup>L</sup> T M kM kM kM <sup>M</sup>*

1 ,

1 ,

*c*

*c*

τ

τ

+

1

; ,

*r rr*

( ) ( ) ( ) ( )

+ −− − − + +

<sup>2</sup> 1 2 21 2 1 2 1

*k Mk M kkM kT k T T M MM k k Mk k k k MT k M T TM M M R* (18)

( ) ( ) ( )( ) ( ) ( )

+ − − −− − − +

τ

τ

τ

1

;*d<sup>L</sup>* ) . Note that within (-*Tc*,*Tc*) the correlation

*rM*-1

τ

*c c*

τ

*c c*

(20)

( )

⎪ ⎩

τ

*c*

*c*

τ

where *<sup>L</sup> d* is the shape vector of SCS signal, and

Fig. 9 shows a schematic diagram of *R*B/L (

*r*1*-M*

Fig. 9. Schematic diagram of correlation function *R*B/L (

**4.4 SC algorithm design process under SC framework** 

*kMM* ∈− + − [ 1, 1] and *k* ∈ ].

⎪⎩

τ

( )

τ

order 2 *M* = A . Note that in an odd order SCS signal, the chip waveform is time-varying. For example, for sin-BOC signal with 3 *M* = , the shape vector of the spreading chip is (1,-1,1)T in the time interval *t nT n T* ∈⎡ + 2 ,2 1 *c c* ( ) ) ⎣ , while it is (-1,1,-1)T in *t n T nT* ∈⎡ − (2 1 ,2 ) *c c* ) ⎣ . We assume that *M* is even hereafter.

#### **4.3 CCF of SCS signals**

All the SC techniques are based on the shapes of CCF between the received signal and the local signal. Here we consider the CCF of two SCS signals which have the same chip rate *<sup>c</sup> f* , spreading sequence {*ci*} , and the order *M* , while the chip waveform are difference.

By using (2) and (9), a SCS baseband signal can be expressed as

$$\log \left( t; \mathbf{d} \right) = \sum\_{n = -\nu}^{\ast \nu} \sum\_{k=0}^{M-1} \left( -1 \right)^{c\_s} d\_k \nu\_k \left( t - nMT\_s \right) \tag{13}$$

The CCF of two SCS signals is

$$\begin{split} \mathcal{R}\_{\mathcal{K}}\left(\tau; \mathbf{d}, \mathbf{d}'\right) &= \frac{1}{T} \int\_{0}^{T} g\left(t\right) g'\left(t + \tau\right) dt \\ &= \frac{1}{T} \sum\_{n} \sum\_{m} \sum\_{k=0}^{M-1} \sum\_{q=0}^{M-1} \left(-1\right)^{c\_{n} + c\_{m}} d\_{k} d\_{q}' \int\_{0}^{T} \boldsymbol{\nu}\_{k} \left(t - nMT\_{s}\right) \boldsymbol{\nu}\_{q} \left(t - mMT\_{s} + \tau\right) dt \end{split} \tag{14}$$

where *T NT* = *<sup>c</sup>* is the period of the spreading sequence. The integral in (14) is nonzero only when ψ *k s* (*t nMT* − ) and ψ *q s* (*t mMT* − +τ ) have the overlapping parts. The delay τ can be expressed as the summation of three parts *c s* τ = *aT bT* + + ε , where *a* is an integer, *b M* = − 0,1, , 1 " ,and ε ∈[0,*Ts* ) . And after some algebraic simplifications, (14) can be rewritten as (Yao & Lu, 2011)

$$\begin{split} R\_{\boldsymbol{\chi}\boldsymbol{\xi}'} \left( \boldsymbol{\tau} \right) &= R\_{\boldsymbol{\chi}\boldsymbol{\xi}'} \left( a \boldsymbol{T}\_{\boldsymbol{\varepsilon}} + b \boldsymbol{T}\_{\boldsymbol{s}} + \boldsymbol{\varepsilon} \right) \\ &= R\_{\boldsymbol{\varepsilon}} \left( a \right) \left[ r\_{b} \left( 1 - \frac{\boldsymbol{\varepsilon}}{\boldsymbol{T}\_{\boldsymbol{s}}} \right) + r\_{b \circ 1} \left( \frac{\boldsymbol{\varepsilon}}{\boldsymbol{T}\_{\boldsymbol{s}}} \right) \right] + R\_{\boldsymbol{\varepsilon}} \left( a + \mathbf{1} \right) \left[ r\_{b \circ M} \left( 1 - \frac{\boldsymbol{\varepsilon}}{\boldsymbol{T}\_{\boldsymbol{s}}} \right) + r\_{b \circ M + 1} \left( \frac{\boldsymbol{\varepsilon}}{\boldsymbol{T}\_{\boldsymbol{s}}} \right) \right] \end{split} \tag{15}$$

where

$$R\_c(a) \triangleq \frac{1}{N} \sum\_{n=0}^{N-1} \left(-1\right)^{c\_n + c\_{n+a}} \tag{16}$$

and the aperiodic cross-correlation function (ACCF) of *d* and *d*′ is

$$r\_b(\mathbf{d}, \mathbf{d}') = \begin{cases} \frac{1}{M} \sum\_{i=0}^{M-1-b} d\_i d'\_{b+i}, & 0 \le b \le M-1 \\\frac{1}{M} \sum\_{i=0}^{M-1-b} d\_{i-b} d'\_i, & 1 - M \le b < 0 \\\ & \mathbf{0}, \mathbf{} \end{cases} \tag{17}$$

By using (15) one can derive other expression form of an *M*-order BOC signal ACF, which is

$$R\_{\rm BOC}\left(\tau\right) = \begin{cases} \left(-1\right)^{k+1} \left[ \frac{r(2M-2k-1)}{T\_c} - \frac{2(M-1)k - 2k^2 + M}{M} \right], & \frac{kT\_c}{M} \le \tau < \frac{(k+1)T\_c}{M} \\\\ \left(-1\right)^{k+1} \left[ \frac{r(2k-1)}{T\_c} + \frac{(M-k)(2k-1) - k}{M} \right], & \frac{(k-M)T\_c}{M} \le \tau < \frac{(k-M+1)T\_c}{M} \\\\ 0, & \text{others} \end{cases} \tag{18}$$

where 0,1, , 1 *k M* = − " . And the CCF between an *M*-order BOC signal and a SCS signal is

$$R\_{\mathbf{B}/\mathcal{L}}\left(\mathbf{r};\mathbf{d}\_{\mathcal{L}}\right) = \begin{cases} \left(\frac{rM - kT\_{\varepsilon}}{T\_{\varepsilon}}\right) \left(r\_{k+1} - r\_{k}\right) + r\_{k\prime} & \frac{kT\_{\varepsilon}}{M} \le \tau < \frac{(k+1)T\_{\varepsilon}}{M} \\\\ \left(\frac{rM - kT\_{\varepsilon} \circ MT\_{\varepsilon}}{T\_{\varepsilon}}\right) \left(r\_{k-M+1} - r\_{k-M}\right) + r\_{k-M\prime} & \frac{(k-M)T\_{\varepsilon}}{M} \le \tau < \frac{(k-M+1)T\_{\varepsilon}}{M} \\ 0, & \text{others} \end{cases} \tag{19}$$

where *<sup>L</sup> d* is the shape vector of SCS signal, and

64 Global Navigation Satellite Systems – Signal, Theory and Applications

order 2 *M* = A . Note that in an odd order SCS signal, the chip waveform is time-varying. For example, for sin-BOC signal with 3 *M* = , the shape vector of the spreading chip is (1,-1,1)T in the time interval *t nT n T* ∈⎡ + 2 ,2 1 *c c* ( ) ) ⎣ , while it is (-1,1,-1)T in *t n T nT* ∈⎡ − (2 1 ,2 ) *c c* ) ⎣ . We

All the SC techniques are based on the shapes of CCF between the received signal and the local signal. Here we consider the CCF of two SCS signals which have the same chip rate *<sup>c</sup> f* ,

( ) () ( ) <sup>1</sup>

*k k s*

() ( )( )

∈[0,*Ts* ) . And after some algebraic simplifications, (14) can be

ψψ

= *aT bT* + +

*s s s s*

+

+ − − +

*T T T T*

⎡ ⎛ ⎞ ⎛⎞⎤ ⎡ ⎛ ⎞ ⎛⎞⎤

*kq k s q s*

) have the overlapping parts. The delay

*d d t nMT t mMT dt*

ε

ε

∑ <sup>−</sup> (16)

 τ

, where *a* is an integer,

 ε (14)

can be

(15)

τ

*t nMT*

*<sup>d</sup>* =− − ∑ ∑ (13)

ψ

= − − −+ ′

τ

( ) 1 1 1 1 ( ) 1

= −+ + + −+ <sup>⎢</sup> ⎜ ⎟ ⎜⎟⎥ ⎢ ⎜ ⎟ ⎜⎟⎥ ⎢⎣ ⎝ ⎠ ⎝⎠⎥ ⎢ ⎦ ⎣ ⎝ ⎠ ⎝⎠⎥⎦

<sup>−</sup> <sup>+</sup>

<sup>1</sup> ,0 1

*dd b M*

*b M*

*d d* ∑ (17)

*c b b c b M b M*

*Rar r Ra r r*

( ) ( ) 1 0 <sup>1</sup> <sup>1</sup> *n na <sup>N</sup> c c*

*N*

− −

*M b*

∑

*i M b*

= − −

=

*b ib i i*

*M*

*M*

⎪

⎪ ⎪

⎩

0,

*r dd M b*

*n*

=

<sup>1</sup> , ,1 0

⎪ ≥

<sup>⎪</sup> ′ ′ <sup>=</sup> <sup>⎨</sup> − ≤<

−

*i bi*

+

<sup>⎧</sup> ′ <sup>≤</sup> ≤ − <sup>⎪</sup>

 ε

where *T NT* = *<sup>c</sup>* is the period of the spreading sequence. The integral in (14) is nonzero only

spreading sequence {*ci*} , and the order *M* , while the chip waveform are difference.

0 ; 1 *<sup>n</sup> <sup>M</sup> <sup>c</sup>*

<sup>0</sup> 0 0

*M M c c <sup>T</sup>*

τ

*n k g t d*

1 1

 τ

− − <sup>+</sup>

∑∑∑ ∑ ∫

<sup>1</sup> <sup>1</sup> *n m*

= =

 ε

*c*

and the aperiodic cross-correlation function (ACCF) of *d* and *d*′ is

( )

*R a*

ε

*n mk q*

*q s* (*t mMT* − +

expressed as the summation of three parts *c s*

+∞ − =−∞ =

By using (2) and (9), a SCS baseband signal can be expressed as

( ) () ( )

*R g t g t dt T*

∫

*T*

′ ′ = +

1

*T*

ψ

ε

( ) ( )

= ++

*gg gg c s*

*R R aT bT*

0

assume that *M* is even hereafter.

The CCF of two SCS signals is

τ

*k s* (*t nMT* − ) and

rewritten as (Yao & Lu, 2011)

τ

′ ′

*gg*

*b M* = − 0,1, , 1 " ,and

when ψ

where

′

; ,

*d d*

**4.3 CCF of SCS signals** 

$$r\_k = \begin{cases} \frac{1}{M} \sum\_{i=0}^{M-1-k} \left(-1\right)^i d\_{k+i} & 0 \le k \le M-1\\ \frac{1}{M} \sum\_{i=0}^{M-1-k} \left(-1\right)^{i-k} d\_{i'} & 1-M \le k < 0\\ 0 & \left|k\right| \ge M \end{cases} \tag{20}$$

Fig. 9 shows a schematic diagram of *R*B/L (τ ;*d<sup>L</sup>* ) . Note that within (-*Tc*,*Tc*) the correlation function is piecewise linear between *<sup>s</sup> kT* and ( 1) *<sup>s</sup> k T* + , and *R kT r* B/L ( *s k* ) = , for *kMM* ∈− + − [ 1, 1] and *k* ∈ ].

Fig. 9. Schematic diagram of correlation function *R*B/L (τ;*d<sup>L</sup>* ) .

#### **4.4 SC algorithm design process under SC framework**

From (15) and (17), it can be seen that the CCF of two SCS signals mostly depends on the ACCF of their chip waveforms when these two signals have the same spreading sequences. In SC algorithm design, since the chip waveform shape of the received signal is known, CCF entirely depends on the shape of local signal spreading chip waveform. Note that each SCS

Unambiguous Processing Techniques of Binary Offset Carrier Modulated Signals 67

min , , , ( 1 2 ) . . , 1, ,

S

*st i n*

and the final step is to find *n* optimal shape vectors ( ) opt *d*<sup>1</sup> , ( ) opt *d*<sup>2</sup> ,…, ( ) opt *d<sup>n</sup>* from UVS, which

Note that usually at different processing stages, the optimization objects are difference. For example, in acquisition, the optimization objects may be the maximum SNR or the widest SCF main peak, while at tracking stage it may be the ability of multipath rejection, the greatest slope or the widest linear range of the discriminator curve, or even some compromises between them. In next two sections, we will give two examples of SC algorithm design under the steps described above. The design process of an SC unambiguous acquisition

Under the analytic design framework described above, an unambiguous acquisition technique named General Removing Ambiguity via Side-peak Suppression (GRASS) technique is developed. This technique is suitable for generic sin-BOC(*kn*,*n*) signals and it is convenient to implement. The detailed performance analysis of this technique can be found in (Yao et al., 2010a). This section puts its emphasis on the design process of this technique.

Theoretically, when the number of local auxiliary signals is unlimited, SCF can be shaped into any desired forms. However, from the view of engineering, the more local signals are used, the more correlators are needed in a receiver which is directly related to the complexity and power consumption. Moreover, the noncoherent combination of too much correlator results may aggravate SNR deterioration. Therefore, in our design, the number of

Since the signal acquisition is a process of searching pronounced energy peak in a 2 dimentional space, the requirement to the shape of SCF in acquisition is relatively generous compared to code tracking. A SCF having main peak without positive side peak is enough. Therefore in GRASS technique only one local auxiliary SCS signal with a matched BOC signal is employed to suppress the side peaks of BOC ACF in noncoherent mode. The SCF

( ) ( ) ( ) 2 2 *RR R* B B/L Δ

where *R*B is the ACF of BOC signal, *R*B/L is the CCF between the received BOC signal and

the SCF used in (Julien et al., 2007) in form. However, as shown later, GRASS technique is

The objective is to keep the main peak of BOC ACF envelop while remove all the positive side peaks (the negative side peaks do not interfere with the statistical test since only

 = Δ− Δ τα

 τ(23)

is the weight coefficient. It can be seen that (23) is similar with

τ

not only suitable for BOC(*n*,*n*) signals but also for other BOC ( *kn n*, ) signals.

*i i*

⎪ ∈ = ⎩

*f*

*d*

algorithm as well as an SC unambiguous tracking loop is described respectively.

⎧⎪ ⎨

correspond to the optimal local chip waveforms.

local auxiliary signals should be as few as possible.

α

**5. GRASS technique** 

**5.1 Step 1 - SCF selection** 

used is as follow:

the local SCS signal, and

**5.2 Step 2 - CCF shape constraint** 

*n*

(22)

" ;

"

*dd d*

chip waveform corresponds to an unique point in *M*-dimensional space whose coordinate is (*dd d* 01 1 ,,, " *<sup>M</sup>*<sup>−</sup> ) , so by changing the value of *<sup>k</sup> d* , one can adjust the shape of CCF. That is, CCF is a function of *d* .

After building the relation between the shape of CCF and the value of a vector, the search for good chip waveform can be equivalent to an optimization problem which can be formulated as

$$\begin{cases} \min f\left(\mathbf{d}\_1, \mathbf{d}\_2, \cdots, \mathbf{d}\_n\right); \\ \text{s.t.} \quad \mathcal{g}\_i\left(\mathbf{d}\_1, \mathbf{d}\_2, \cdots, \mathbf{d}\_n\right) \ge 0, i = 1, \cdots, m \end{cases} \tag{21}$$

where *f* is the objective function, *<sup>i</sup> g* is the constraint function, *m* is the number of constraints, and 1 , , " *<sup>n</sup> d d* are shape vectors of local signals. Then the development of SC algorithm becomes the solving of an optimization problem with a set of inequalities constraints and can be achieved through four steps (Yao & Lu, 2011), as shown in Figure 10.

Fig. 10. SC Algorithm Design Process under SC Framework

The first step is SCF selection. In SC algorithms, the SCF *R* is used instead of ACF in acquisition and tracking. The choice of SCF determines the combination mode of CCFs, and provides for the number of local auxiliary signals. The more local signals are used to form SCF, the more flexible the shape control is, but the more correlators are needed as well.

The second step is CCF shape constraint. Once the SCF is determined, the shape of each CCF employed in SCF can be restricted. Since CCF is piecewise linear between its values at integer multiples of *T M <sup>c</sup>* / , and *R kT M r gg*′( *c k* / ) = , the constraint of the *i*'th CCF shape can be translated into the restriction on values of those ( )*<sup>i</sup> <sup>k</sup> r* .

The third step is referred to as unambiguous vector subset (UVS) establishment. Using the corresponding relationship between ( )*<sup>i</sup> <sup>k</sup> r* and the shape vector *di* of local signal, one can further translate the restriction on the value of ( )*<sup>i</sup> <sup>k</sup> r* into the limitation of *di*. In most cases, the values of each ( )*<sup>i</sup> <sup>k</sup> r* which makes the SCF unambiguous are not unique, thus the feasible solution of *di* is not a fixed point. The feasible region of *di* is denoted as *<sup>M</sup>* S*<sup>i</sup>* ⊂ \ and is called UVS of *di*.

The final step is local waveform optimization. As the UVS has been established, with an optimization object, (21) can be rewritten as

$$\begin{cases} \min f\left(\mathbf{d}\_1, \mathbf{d}\_2, \dots, \mathbf{d}\_n\right); \\ \text{s.t.} \quad \mathbf{d}\_i \in \mathcal{S}\_{i'} \; i = 1, \dots, n \end{cases} \tag{22}$$

and the final step is to find *n* optimal shape vectors ( ) opt *d*<sup>1</sup> , ( ) opt *d*<sup>2</sup> ,…, ( ) opt *d<sup>n</sup>* from UVS, which correspond to the optimal local chip waveforms.

Note that usually at different processing stages, the optimization objects are difference. For example, in acquisition, the optimization objects may be the maximum SNR or the widest SCF main peak, while at tracking stage it may be the ability of multipath rejection, the greatest slope or the widest linear range of the discriminator curve, or even some compromises between them. In next two sections, we will give two examples of SC algorithm design under the steps described above. The design process of an SC unambiguous acquisition algorithm as well as an SC unambiguous tracking loop is described respectively.

## **5. GRASS technique**

66 Global Navigation Satellite Systems – Signal, Theory and Applications

chip waveform corresponds to an unique point in *M*-dimensional space whose coordinate is (*dd d* 01 1 ,,, " *<sup>M</sup>*<sup>−</sup> ) , so by changing the value of *<sup>k</sup> d* , one can adjust the shape of CCF. That is,

After building the relation between the shape of CCF and the value of a vector, the search for good chip waveform can be equivalent to an optimization problem which can be

> . . , , , 0, 1, , *n*

*st g i m*

where *f* is the objective function, *<sup>i</sup> g* is the constraint function, *m* is the number of constraints, and 1 , , " *<sup>n</sup> d d* are shape vectors of local signals. Then the development of SC algorithm becomes the solving of an optimization problem with a set of inequalities constraints and can be achieved through four steps (Yao & Lu, 2011), as shown in Figure 10.

Step 1: SCF Selection

Step 2: CCF Shape Constraint

Step 3: UVS establishment

Step 4: Local Waveform Optimization

The first step is SCF selection. In SC algorithms, the SCF *R* is used instead of ACF in acquisition and tracking. The choice of SCF determines the combination mode of CCFs, and provides for the number of local auxiliary signals. The more local signals are used to form SCF, the more flexible the shape control is, but the more correlators are needed as well.

The second step is CCF shape constraint. Once the SCF is determined, the shape of each CCF employed in SCF can be restricted. Since CCF is piecewise linear between its values at integer multiples of *T M <sup>c</sup>* / , and *R kT M r gg*′( *c k* / ) = , the constraint of the *i*'th CCF shape can

The third step is referred to as unambiguous vector subset (UVS) establishment. Using the

solution of *di* is not a fixed point. The feasible region of *di* is denoted as *<sup>M</sup>* S*<sup>i</sup>* ⊂ \ and is called

The final step is local waveform optimization. As the UVS has been established, with an

*<sup>k</sup> r* .

*<sup>k</sup> r* which makes the SCF unambiguous are not unique, thus the feasible

*<sup>k</sup> r* and the shape vector *di* of local signal, one can

*<sup>k</sup> r* into the limitation of *di*. In most cases,

" "

*dd d* (21)

( ) ( ) 1 2 1 2

*dd d* ;

*i n*

⎪ ≥ = ⎩ "

min , , ,

*f*

⎧⎪ ⎨

Fig. 10. SC Algorithm Design Process under SC Framework

be translated into the restriction on values of those ( )*<sup>i</sup>*

further translate the restriction on the value of ( )*<sup>i</sup>*

corresponding relationship between ( )*<sup>i</sup>*

optimization object, (21) can be rewritten as

the values of each ( )*<sup>i</sup>*

UVS of *di*.

CCF is a function of *d* .

formulated as

Under the analytic design framework described above, an unambiguous acquisition technique named General Removing Ambiguity via Side-peak Suppression (GRASS) technique is developed. This technique is suitable for generic sin-BOC(*kn*,*n*) signals and it is convenient to implement. The detailed performance analysis of this technique can be found in (Yao et al., 2010a). This section puts its emphasis on the design process of this technique.

#### **5.1 Step 1 - SCF selection**

Theoretically, when the number of local auxiliary signals is unlimited, SCF can be shaped into any desired forms. However, from the view of engineering, the more local signals are used, the more correlators are needed in a receiver which is directly related to the complexity and power consumption. Moreover, the noncoherent combination of too much correlator results may aggravate SNR deterioration. Therefore, in our design, the number of local auxiliary signals should be as few as possible.

Since the signal acquisition is a process of searching pronounced energy peak in a 2 dimentional space, the requirement to the shape of SCF in acquisition is relatively generous compared to code tracking. A SCF having main peak without positive side peak is enough. Therefore in GRASS technique only one local auxiliary SCS signal with a matched BOC signal is employed to suppress the side peaks of BOC ACF in noncoherent mode. The SCF used is as follow:

$$
\tilde{R}\left(\Delta\tau\right) = R\_{\mathbb{B}}^2 \left(\Delta\tau\right) - \alpha R\_{\mathbb{B}/\mathbb{L}}^2 \left(\Delta\tau\right) \tag{23}
$$

where *R*B is the ACF of BOC signal, *R*B/L is the CCF between the received BOC signal and the local SCS signal, and α is the weight coefficient. It can be seen that (23) is similar with the SCF used in (Julien et al., 2007) in form. However, as shown later, GRASS technique is not only suitable for BOC(*n*,*n*) signals but also for other BOC ( *kn n*, ) signals.

#### **5.2 Step 2 - CCF shape constraint**

The objective is to keep the main peak of BOC ACF envelop while remove all the positive side peaks (the negative side peaks do not interfere with the statistical test since only positive values could pass the threshold). In consideration of the shape of BOC ACF, it is desirable that the envelop of *R*B/L be zigzag and symmetric with respect to 0 τ = . Moreover, ( ) <sup>2</sup> B/L *R* 0 should be zero in order to ensure that the magnitude of main peak is unaffected after the subtracting.

As explained in the previous section, the above constraints of the CCF shape can be translated into the restriction on *<sup>k</sup> r* via (19). The constraint ( ) <sup>2</sup> B/L *R* 0 0 = is equivalent to

$$r\_0 = 0\tag{24}$$

Unambiguous Processing Techniques of Binary Offset Carrier Modulated Signals 69

( )

*k Mk*

*dd k*

: 10

− − <sup>⎧</sup> <sup>=</sup> <sup>⎫</sup> ⎪⎪ ⎪⎪ <sup>=</sup> <sup>⎨</sup> ∈ −> <sup>⎬</sup> <sup>⎪</sup> <sup>⎪</sup> ⎪⎩ ⎭ <sup>=</sup> = − <sup>⎪</sup>

*k Bc* ( ) / *<sup>M</sup> <sup>k</sup> r R kT M*

0 max *<sup>k</sup> <sup>k</sup> M k M r*

≠

α

So far the effect of thermal noise has not been considered. In fact, the coefficient

SNR in the SCF is. Therefore, from the viewpoint of sensitivity, it is desired that

max

≠

opt <sup>0</sup> arg min arg min max

1

− − − − − + −

*M M M*

1

<sup>⎪</sup> <sup>=</sup> = = − <sup>⎩</sup> "

Figure 11 (a)-(c) depict the optimum local SCS waveforms for BOC(*n*,*n*), BOC(2*n*,*n*), and

can be found that the SC method proposed in (Julien, et al., 2007) is equivalent to this case. The local symbol is simply a rectangular pulse. For BOC(2*n*,*n*), ( )

opt <sup>3</sup> *d* = −− 5,1, 1, 1,1,5 and min

0 1 2 3

*d d*

⎨

= 5 . For BOC(3*n*,*n*), ( )<sup>T</sup> <sup>1</sup>

<sup>⎧</sup> = = <sup>⎪</sup>

*i Mi M*

BOC(3*n*,*n*) signals respectively. For BOC(*n*,*n*) signals, 2 *M* = *.* So ( )<sup>T</sup>

*dd i* <sup>−</sup>

α

noise components in *R*B/L . Under a given pre-correlation SNR, the larger

α *d M d*

"

S *d* R (30)

*M*

must satisfies

<sup>−</sup> ≥ = (31)

, all the undesired positive side peaks of <sup>2</sup> *R*B can

<sup>−</sup> <sup>≥</sup> . (32)

<sup>−</sup> <sup>=</sup> *d* (33)

*M k M r*

opt 0 1 <sup>1</sup> *d* = *dd d* , ," *<sup>M</sup>*<sup>−</sup> (35)

<sup>−</sup> = = *d d <sup>d</sup>* S S (34)

α

α

(36)

α

<sup>T</sup> <sup>9</sup> 1 1 opt 5 33 *d* = 1, , ,1 and

= 1 . It

opt *d* = 1,1 and min

= 9 . It can be seen that the

amplifies

is, the lower

αbe as

, 0,1, , 1

*M*

αis

( ) <sup>0</sup>

α, that is

*L L <sup>k</sup> <sup>k</sup>*

*<sup>k</sup> k L M k M r*

∈ ∈ ≠

[ ]<sup>T</sup>

( ) ( ) <sup>1</sup>

*M*

α

<sup>1</sup> 2 3 <sup>2</sup> , 1,2, , 1 *<sup>i</sup>*

1 2

α

− = =

*M i i M k i L i i*

∑ ∑

α

With an appropriate weight coefficient

**5.4 Step 4 - Local waveform optimization** 

small as possible. So that with a given *dL* the optimum

and in UVS, the optimum *dL* is the one minimizing

It can be proved that the explicit expression of (34) is

for *k M* = − 1,2, , 1 " , or

where

and min α

min α

= − 2 3 *M* .

be canceled by subtraction. From (18), the coefficient

α

and the axial symmetry of *R*B/L means

$$\|r\_i\| = \|r\_{-i}\|\tag{25}$$

The requirement of zigzag shape can be realized through making adjacent *<sup>k</sup> r* have opposite sign, that is

$$\begin{cases} r\_i r\_{i+1} < 0, & i > 0 \\ r\_i r\_{i-1} < 0, & i < 0 \end{cases} \tag{26}$$

Actually, under the restrictions of (24) and (26), (25) can be simplified to

$$
\mathbf{r}\_i = -\mathbf{r}\_{-i} \tag{27}
$$

because by (20) and (24) we have

$$\begin{split} \left| r\_{M/2} + r\_{-M/2} \right| &= \frac{1}{M} \left| \sum\_{i=0}^{M/2-1} \left( -\mathbf{1} \right)^{i} d\_{i+M/2} + \sum\_{l=0}^{M/2-1} \left( -\mathbf{1} \right)^{i-M/2} d\_{i} \right| \\ &= \frac{1}{M} \left| \left( \sum\_{i=0}^{M/2-1} + \sum\_{l=M/2}^{M-1} \right) \left( -\mathbf{1} \right)^{i-M/2} d\_{i} \right| \\ &= \frac{1}{M} \left| \sum\_{i=0}^{M-1} \left( -\mathbf{1} \right)^{i} d\_{i} \right| = \left| r\_{0} \right| = 0 \end{split} \tag{28}$$

so that *M M* /2 /2 *r r* = − <sup>−</sup> . Then using (26), we obtain (27).

#### **5.3 Step 3 - UVS establishment**

Substituting (20) into (24), (26) and (27), after some straightforward algebraic simplification, we have the set of inequalities constraints on the elements of *<sup>L</sup> d* :

$$\begin{cases} \sum\_{i=0}^{k} (-1)^{i} d\_{i} > 0, & k = 0, 1, \dots, \frac{M}{2} - 1 \\ d\_{k} = d\_{M-1-k}, & k = 0, 1, \dots, \frac{M}{2} - 1 \\ \sum\_{i=0}^{M-1} d\_{i}^{2} = M \end{cases} \tag{29}$$

Note that the last term in (29) is the energy normalization constraint of SCS waveform. So that the UVS can be represented as

$$\mathcal{S} = \left\{ \mathbf{d}\_{\perp} \in \mathbb{R}^{M} : \sum\_{i=0}^{M-1} \left( \mathbf{-1} \right)^{i} d\_{i} > 0 \right\} \tag{30}$$
 
$$d\_{k} = d\_{M-1-k}, k = 0, 1, \dots, \frac{M}{2} - 1$$

With an appropriate weight coefficient α , all the undesired positive side peaks of <sup>2</sup> *R*B can be canceled by subtraction. From (18), the coefficient αmust satisfies

$$\left| \alpha \left| r\_k \right| \ge \left| R\_g \left( kT\_c \ne M \right) \right| = \frac{M - k}{M} \tag{31}$$

for *k M* = − 1,2, , 1 " , or

68 Global Navigation Satellite Systems – Signal, Theory and Applications

positive values could pass the threshold). In consideration of the shape of BOC ACF, it is

B/L *R* 0 should be zero in order to ensure that the magnitude of main peak is unaffected

As explained in the previous section, the above constraints of the CCF shape can be

The requirement of zigzag shape can be realized through making adjacent *<sup>k</sup> r* have opposite

0, 0 0, 0

() ()

− − <sup>−</sup>

*M M i i <sup>M</sup>*

0 0 /2 1 <sup>1</sup> /2

= = <sup>−</sup> <sup>−</sup> <sup>−</sup>

∑ ∑

<sup>1</sup> 1 1

/2 1 /2 1 /2

( )

"

2

*M*

"

, 0,1, , 1

*i*

*d*

0

1 1

( )

= − ==

1

− =

∑

0

*i*

0 /2

*i iM <sup>M</sup> <sup>i</sup>*

∑ ∑

= =

<sup>1</sup> 1 0

Substituting (20) into (24), (26) and (27), after some straightforward algebraic simplification,

<sup>⎧</sup> <sup>−</sup> >= − <sup>⎪</sup>

*d k*

*<sup>k</sup> <sup>i</sup> <sup>M</sup>*

⎨ = = −

Note that the last term in (29) is the energy normalization constraint of SCS waveform. So

1 2

1 0, 0,1, , 1

*i*

*d r*

<sup>1</sup> <sup>1</sup>

⎛ ⎞ = +− ⎜ ⎟ ⎝ ⎠

*M M i M i i i <sup>M</sup> <sup>M</sup> i M*

*r r d d*

+= − +−

*rr i rr i* + − <sup>⎧</sup> <sup>&</sup>lt; <sup>&</sup>gt; <sup>⎨</sup> <sup>&</sup>lt; <sup>&</sup>lt; <sup>⎩</sup>

*i i i i*

Actually, under the restrictions of (24) and (26), (25) can be simplified to

/2 /2 /2

− +

*M*

*M*

*M*

we have the set of inequalities constraints on the elements of *<sup>L</sup> d* :

∑

− =

*M*

1 2 0

0

∑

<sup>⎪</sup> <sup>=</sup> ⎪⎩

*i i*

⎪⎪

*i*

⎪

( ) =

*k Mk*

*d M*

− −

*dd k*

*i*

so that *M M* /2 /2 *r r* = − <sup>−</sup> . Then using (26), we obtain (27).

τ

B/L *R* 0 0 = is equivalent to

(26)

(28)

(29)

<sup>0</sup>*r* = 0 (24)

*i i r r* = <sup>−</sup> (25)

*i i r r* = − <sup>−</sup> (27)

= . Moreover,

desirable that the envelop of *R*B/L be zigzag and symmetric with respect to 0

translated into the restriction on *<sup>k</sup> r* via (19). The constraint ( ) <sup>2</sup>

( ) <sup>2</sup>

sign, that is

after the subtracting.

and the axial symmetry of *R*B/L means

because by (20) and (24) we have

**5.3 Step 3 - UVS establishment** 

that the UVS can be represented as

$$\alpha \ge \max\_{k \ne 0} \frac{M - k}{M \| r\_k \|} \,. \tag{32}$$

#### **5.4 Step 4 - Local waveform optimization**

So far the effect of thermal noise has not been considered. In fact, the coefficient α amplifies noise components in *R*B/L . Under a given pre-correlation SNR, the larger α is, the lower SNR in the SCF is. Therefore, from the viewpoint of sensitivity, it is desired that α be as small as possible. So that with a given *dL* the optimum αis

$$\alpha = \max\_{k \neq 0} \frac{M - k}{M \left| r\_k \left( d\_\perp \right) \right|} \tag{33}$$

and in UVS, the optimum *dL* is the one minimizing α, that is

$$\mathbf{d}\_{\text{opt}} = \arg\min\_{\mathbf{d}\_{\mathbb{L}} \in \mathcal{S}} \boldsymbol{\alpha} = \arg\min\_{\mathbf{d}\_{\mathbb{L}} \in \mathcal{S}} \max\_{k \neq 0} \frac{M - k}{M|r\_k|} \tag{34}$$

It can be proved that the explicit expression of (34) is

$$\mathbf{d}\_{\rm opt} = \begin{bmatrix} d\_{0\ \prime} d\_{1\ \prime} \cdots d\_{M-1} \end{bmatrix}^{\rm T} \tag{35}$$

where

$$\begin{cases} d\_0 = d\_{M-1} = \frac{M-1}{\sqrt{2M-3}}\\ d\_i = d\_{M-i+1} = \frac{\left(-1\right)^{i-1}}{\sqrt{2M-3}}, \left(i = 1, 2, \dots, \frac{M}{2} - 1\right) \end{cases} \tag{36}$$

and min α= − 2 3 *M* .

Figure 11 (a)-(c) depict the optimum local SCS waveforms for BOC(*n*,*n*), BOC(2*n*,*n*), and BOC(3*n*,*n*) signals respectively. For BOC(*n*,*n*) signals, 2 *M* = *.* So ( )<sup>T</sup> opt *d* = 1,1 and min α = 1 . It can be found that the SC method proposed in (Julien, et al., 2007) is equivalent to this case. The local symbol is simply a rectangular pulse. For BOC(2*n*,*n*), ( ) <sup>T</sup> <sup>9</sup> 1 1 opt 5 33 *d* = 1, , ,1 and min α = 5 . For BOC(3*n*,*n*), ( )<sup>T</sup> <sup>1</sup> opt <sup>3</sup> *d* = −− 5,1, 1, 1,1,5 and min α= 9 . It can be seen that the

Unambiguous Processing Techniques of Binary Offset Carrier Modulated Signals 71

From the figures it can be seen that with any *M*, no major positive side peak exists in SCF. Although there are some pits on the SCF, their magnitudes are all below zero, so they bring

Although all examples shown above are for sin-BOC signals, it is easy to demonstrate that GRASS technique can also be applied to cos-BOC signals by simply replacing basis function

( ) { } ( ) ( ) <sup>1</sup> sgn sin 2 , <sup>1</sup>

As another example of the application of SC design framework, in this section we show the design process of an unambiguous code tracking technique named pseudo-correlationfunction-based unambiguous delay lock loop (PUDLL) (Yao et al., 2010b) which is applicable to any BOC(*kn*,*n*) signals. At tracking stage, because the discriminator characteristic curve is based on the first derivative of SCF, the requirement to the shape of

It is desired that the SCF used in code tracking has an ideal triangular main peak with no side peak. Since the CCF between the received BOC signal and the local SCS signal is piecewise linear, utilizing this characteristic, and using the absolute-magnitude operation to change the direction of lines on one side of the zero crossing point, following by the linear combination, it is possible to obtain the SCF without any side peak. Therefore, in this

*RR R RR* (

*R R* 1 2 (τ) = (−

> τ) = (−

of the local SCS chip waveform. Without loss of generality, assume *R R* 1 2 (

τ

) = + −+ 1 2 12 (

 τ

where *R*1 and *R*2 are CCFs between BOC signal and two local SCS signals *g t* 1 1 ( ) ;*d* and *g t* 2 2 ( ;*d* ) , respectively, in which 1 *d* and 2 *d* are the shape vectors of 1 *g* and 2 *g* respectively.

 ττ) ( ) ( ) ( ) (38)

τ

τ

τ= is

 ) and *R R* 1 2 (τ) = − −(

) . (39)

τ

τ) = − −(

*i i r r* ′ = − (40)

) lies in the polarity

τ

) , so that

τ

A sufficient condition for (38) being symmetric with respect to 0

<sup>⎧</sup> ⎡ ⎤ − ≤< + <sup>⎪</sup> ⎣ ⎦ ′ <sup>=</sup> <sup>⎨</sup>

π

0, others

*ss s s*

(37)

*f t kT kT t k T*

no threat to the acquisition.

*k c*

ψ

SCF is more stringent than in acquisition.

**6.1 Step 1 – SCF selection** 

example, the SCF is chosen as

**6.2 Step 2 – CCF shape constraint** 

at each endpoint of CCF segment

In fact, the only difference between *R R* 1 2 (

where *r R iT M i c* = <sup>1</sup> ( / ) and *r R iT M i c* ′ = <sup>2</sup> ( / ) .

*t T*

⎪ ⎩

(8) with

**6. PUDLL** 

complexity of the chip shape increases as the BOC-modulation order is raised. For 4 *M* = or higher, the optimum local SCS waveform is hard to obtain by geometrical intuition. Figure 12, 13, and 14 show the envelop of *R*<sup>B</sup> , *R*B/L and the SCF for 2 *M* = , 4, 6, respectively.

Fig. 11. Optimum Local SCS Waveforms for (a) *M*=2, (b) *M*=4, (c) *M*=6

Fig. 12. The Envelops of *R*<sup>B</sup> , *R*B/L and the SCF for *M*=2

Fig. 13. The Envelops of *R*<sup>B</sup> , *R*B/L and the SCF for *M*=4

Fig. 14. The Envelops of *R*<sup>B</sup> , *R*B/L and the SCF for *M*=6

From the figures it can be seen that with any *M*, no major positive side peak exists in SCF. Although there are some pits on the SCF, their magnitudes are all below zero, so they bring no threat to the acquisition.

Although all examples shown above are for sin-BOC signals, it is easy to demonstrate that GRASS technique can also be applied to cos-BOC signals by simply replacing basis function (8) with

$$\nu\_k'(t) = \begin{cases} \frac{1}{\sqrt{T\_c}} \text{sgn} \left\{ \sin \left[ 2\pi f\_s (t - kT\_s) \right] \right\}, & kT\_s \le t < (k+1)T\_s\\ & 0, & \text{others} \end{cases} \tag{37}$$

#### **6. PUDLL**

70 Global Navigation Satellite Systems – Signal, Theory and Applications

complexity of the chip shape increases as the BOC-modulation order is raised. For 4 *M* = or higher, the optimum local SCS waveform is hard to obtain by geometrical intuition. Figure 12, 13, and 14 show the envelop of *R*<sup>B</sup> , *R*B/L and the SCF for 2 *M* = , 4, 6, respectively.

000 Tc Tc Tc




Code Delay (Chips)

Code Delay (Chips)

Code Delay (Chips)

5


9 5

1 <sup>3</sup>

(a) (b) (c)

(a) *M=2*

Fig. 12. The Envelops of *R*<sup>B</sup> , *R*B/L and the SCF for *M*=2

(b) *M=4*

Fig. 13. The Envelops of *R*<sup>B</sup> , *R*B/L and the SCF for *M*=4

(c ) *M=6*

Fig. 14. The Envelops of *R*<sup>B</sup> , *R*B/L and the SCF for *M*=6




0

Correlation Function

0.5

1

0

0.5

Correlation Function

1

Correlation Function

Fig. 11. Optimum Local SCS Waveforms for (a) *M*=2, (b) *M*=4, (c) *M*=6

As another example of the application of SC design framework, in this section we show the design process of an unambiguous code tracking technique named pseudo-correlationfunction-based unambiguous delay lock loop (PUDLL) (Yao et al., 2010b) which is applicable to any BOC(*kn*,*n*) signals. At tracking stage, because the discriminator characteristic curve is based on the first derivative of SCF, the requirement to the shape of SCF is more stringent than in acquisition.

#### **6.1 Step 1 – SCF selection**

It is desired that the SCF used in code tracking has an ideal triangular main peak with no side peak. Since the CCF between the received BOC signal and the local SCS signal is piecewise linear, utilizing this characteristic, and using the absolute-magnitude operation to change the direction of lines on one side of the zero crossing point, following by the linear combination, it is possible to obtain the SCF without any side peak. Therefore, in this example, the SCF is chosen as

$$\tilde{R}\left(\tau\right) = \left|R\_1\left(\tau\right)\right| + \left|R\_2\left(\tau\right)\right| - \left|R\_1\left(\tau\right) + R\_2\left(\tau\right)\right|\tag{38}$$

where *R*1 and *R*2 are CCFs between BOC signal and two local SCS signals *g t* 1 1 ( ) ;*d* and *g t* 2 2 ( ;*d* ) , respectively, in which 1 *d* and 2 *d* are the shape vectors of 1 *g* and 2 *g* respectively.

#### **6.2 Step 2 – CCF shape constraint**

A sufficient condition for (38) being symmetric with respect to 0 τ= is

$$\left| R\_1 \left( \tau \right) \right| = \left| R\_2 \left( -\tau \right) \right|. \tag{39}$$

In fact, the only difference between *R R* 1 2 (τ ) = (−τ ) and *R R* 1 2 (τ ) = − −( τ ) lies in the polarity of the local SCS chip waveform. Without loss of generality, assume *R R* 1 2 (τ ) = − −( τ ) , so that at each endpoint of CCF segment

$$
\mathbf{r}'\_i = -\mathbf{r}\_i \tag{40}
$$

where *r R iT M i c* = <sup>1</sup> ( / ) and *r R iT M i c* ′ = <sup>2</sup> ( / ) .

Unambiguous Processing Techniques of Binary Offset Carrier Modulated Signals 73

In the case 4 *M* = (Yao, 2008), UVS can be expressed as

1

1

= *d d <sup>M</sup>*−1 0 / , the shape vector in S′ can be expressed as

⎪ ⎩

*h*

κ

( )

τ κ

*R*

Figure 15 (a) and (b) show some SCFs with different

power (EMLP) loop which uses SCF instead of BOC(*n*,*n*) ACF.

of the subset of UVS

defining

κ

expression of *R*�

can make a triangular SCF for all *M* even.

the base line half width of which is

and the height of the peak is

**6.4 Step 4 – Local waveform optimization** 

{ } 22 2

4 0 3 3 2

<sup>⎧</sup> <sup>⎧</sup> ⎫ ⎧ ≤≤≤ ⎫⎫ <sup>⎪</sup> <sup>⎪</sup> ⎪ ⎪ ⎪⎪ <sup>⎪</sup> ⎪ ⎪⎪ > ≥ <sup>≤</sup> ⎪⎪ = ∈ <sup>⎨</sup> ⎨ ⎬⎨ ⎬⎬ = = −≤ − ≤ <sup>⎪</sup> ⎪ ⎪⎪ ⎪⎪ <sup>⎪</sup> ⎪ ⎪⎪ −+−≠ ⎪⎪ ⎩ ⎭ ⎩ ⎭⎩ <sup>⎭</sup>

For larger *M*, as the degrees of freedom of 1 *d* increase, the explicit expressions for the constraints on 1 *d* become complex and hard to derive. The full UVS can be obtained through the use of numerical method. However, it is easy to verify that any element in one

0 1

Under the energy normalization restriction (12), 1 *d* in (50) has one degree of freedom. So by

( 2 2 )

Utilizing (51), (19), and (38), without considering front-end filtering, we can obtain the

<sup>1</sup> 1 1 ,0, ,0, *M M* κ

> ( ) ( ) ( )

κ

( ) ( )

( ) ( )

<sup>−</sup> <sup>=</sup>

*M*

respectively. Figure 16 depicts the discriminator characteristic curve of the early-minus-late

2 1 1

1 2 *Tc <sup>w</sup> M*

<sup>⎧</sup> <sup>&</sup>lt; <sup>⎪</sup> <sup>=</sup> <sup>⎨</sup>

, ;

κ

κτ

*M M i d d*

*d d d d dd dddd*

0 0

1 2 312 0

0 0, 1,2, , 2

*di M* <sup>−</sup> <sup>⎧</sup> <sup>⎧</sup> > ≥ ⎫⎫ <sup>⎪</sup> <sup>⎪</sup> ′ = ∈ <sup>⎨</sup> ⎨ ⎬⎬ = = − ⎩ ⎭ <sup>⎪</sup> ⎩ ⎭ � <sup>⎪</sup>

0

1 12 01 S = *d* ∈ += >≥ R : 2, 0 *dd dd* (48)

301

*ddd*

0

0123

S *d* R (50)

T

+ + *d* = � (51)

( )

� (52)

κ

κ

<sup>−</sup> <sup>=</sup> <sup>−</sup> (53)

(54)

for BOC(*n*,*n*) and BOC(2*n*,*n*) signals,

 κ κ

2 ( )

τ

*c c*

0, others

2 4 21 1 <sup>2</sup> <sup>1</sup>

− +− − <sup>−</sup> <sup>+</sup>

( )

( ) <sup>2</sup>

+

κ

κ

κ

κ

κ

*c M T T M T <sup>M</sup>*

 κ *dddd*

S *d* R ∪ (49)

From (38), it can be proved that *R* is also piecewise linear. In order to shape the SCF into an ideal triangle, one have to make all of the ending points of lines in *R* be zero except the central one.

To ensure the triangular shape, SCF must satisfy the following request:

$$\begin{cases} \tilde{\mathcal{R}}(0) \neq 0\\ \tilde{\mathcal{R}}(kT\_c \,/ \, M) = 0, \, (k \neq 0) \end{cases} \tag{41}$$

where the first term is equivalent to

$$r\_0 = 0\tag{42}$$

and using (38) and (40), the second term can be simplified as

$$r\_k r\_{-k} \le 0 \tag{43}$$

for 0 *k* ≠ .

The constraints (42) and (43) are necessary but not sufficient, since the absolute-magnitude operation introduces additional endpoints at the zero crossing points in *R*1 and *R*<sup>2</sup> . If *R*<sup>1</sup> has a zero crossing point 0 τ within the interval ⎡*kT M k T M c c* /, 1 / ( + ⎤ ) ⎣ ⎦ , ( *k* > 0) , easily proved, it must be

$$
\tau\_0 = \frac{kT\_c}{\mathcal{M}} + \frac{\left|r\_k\right|}{\left|r\_k\right| + \left|r\_{k+1}\right|} \frac{T\_c}{\mathcal{M}} \tag{44}
$$

From (43) we know that *R*2 must have a zero crossing point within the same interval, which is

$$
\tau\_o' = \frac{kT\_c}{\mathcal{M}} + \frac{\left|r\_{-k}\right|}{\left|r\_{-k}\right| + \left|r\_{-k-1}\right|} \frac{T\_c}{\mathcal{M}} \tag{45}
$$

In order to eliminate the inclined lines on both sides of the zero crossing point, we must have 0 0 τ =τ′ , which can be simplified as

$$r\_k r\_{-k-1} = r\_{-k} r\_{k+1} \tag{46}$$

for 0 *k* > .

#### **6.3 Step 3 – UVS establishment**

The necessary and sufficient conditions for SCF being triangular are (40), (42), (43), and (46). From (40), we obtain that 1 *d* and 2 *d* are mirror images of each other, i.e.

$$d\_k' = d\_{M-k-1} \tag{47}$$

where *<sup>k</sup> d* and *<sup>k</sup> d*′ are the entries of 1 *d* and 2 *d* , respectively. When 2 *M* = , by using the relationship (20), the UVS can be represented as

$$\mathcal{S} = \left\{ d\_1 \in \mathbb{R}^2 : d\_1^2 + d\_2^2 = \mathcal{D}, d\_0 > d\_1 \ge 0 \right\} \tag{48}$$

In the case 4 *M* = (Yao, 2008), UVS can be expressed as

72 Global Navigation Satellite Systems – Signal, Theory and Applications

From (38), it can be proved that *R* is also piecewise linear. In order to shape the SCF into an ideal triangle, one have to make all of the ending points of lines in *R* be zero except the

( )( )

The constraints (42) and (43) are necessary but not sufficient, since the absolute-magnitude operation introduces additional endpoints at the zero crossing points in *R*1 and *R*<sup>2</sup> . If *R*<sup>1</sup>

1

1

+

*c c k k k kT r T Mr rM*

*c c k k k kT r T Mr r M*

− −−

From (43) we know that *R*2 must have a zero crossing point within the same interval, which

In order to eliminate the inclined lines on both sides of the zero crossing point, we must

The necessary and sufficient conditions for SCF being triangular are (40), (42), (43), and (46).

where *<sup>k</sup> d* and *<sup>k</sup> d*′ are the entries of 1 *d* and 2 *d* , respectively. When 2 *M* = , by using the

*R kT M k*

⎪ = ≠ ⎩

/ 0, 0 *<sup>c</sup>*

(41)

<sup>0</sup>*r* = 0 (42)

0 *k k r r*<sup>−</sup> ≤ (43)

<sup>+</sup> (44)

<sup>+</sup> (45)

*k k kk* 1 1 *rr r r* <sup>−</sup> − −+ = (46)

*k Mk* <sup>1</sup> *d d* <sup>−</sup> <sup>−</sup> ′ = (47)

within the interval ⎡*kT M k T M c c* /, 1 / ( + ⎤ ) ⎣ ⎦ , ( *k* > 0) , easily

To ensure the triangular shape, SCF must satisfy the following request:

*R*

and using (38) and (40), the second term can be simplified as

τ

′ , which can be simplified as

relationship (20), the UVS can be represented as

**6.3 Step 3 – UVS establishment** 

0

= +

<sup>−</sup>

′ = +

τ

0

From (40), we obtain that 1 *d* and 2 *d* are mirror images of each other, i.e.

τ

( )

⎧⎪ ≠ ⎨

0 0

central one.

for 0 *k* ≠ .

is

have 0 0 τ =τ

for 0 *k* > .

where the first term is equivalent to

has a zero crossing point 0

proved, it must be

$$\mathcal{S} = \left\{ \mathbf{d}\_1 \in \mathbb{R}^4 \, \middle| \, \begin{aligned} &d\_0 > d\_3 \ge 0\\ &d\_1 = d\_2 = 0\\ &d\_1 = d\_2 = 0 \end{aligned} \right\} \cup \left\{ \begin{aligned} &0 \le d\_3 \le d\_0 \le d\_1\\ &d\_3 \le d\_2\\ &-d\_3 \le d\_1 - d\_2 \le d\_0\\ &d\_0 - d\_1 + d\_2 - d\_3 \ne 0 \end{aligned} \right\} \tag{49}$$

For larger *M*, as the degrees of freedom of 1 *d* increase, the explicit expressions for the constraints on 1 *d* become complex and hard to derive. The full UVS can be obtained through the use of numerical method. However, it is easy to verify that any element in one of the subset of UVS

$$\mathcal{S}' = \left\{ \mathbf{d}\_1 \in \mathbb{R}^M \, \middle| \begin{cases} d\_0 > d\_{M-1} \ge 0 \\ d\_i = 0, i = 1, 2, \dots, M - 2 \end{cases} \right\} \tag{50}$$

can make a triangular SCF for all *M* even.

#### **6.4 Step 4 – Local waveform optimization**

Under the energy normalization restriction (12), 1 *d* in (50) has one degree of freedom. So by defining κ= *d d <sup>M</sup>*−1 0 / , the shape vector in S′ can be expressed as

$$\mathbf{d}\_1 = \left(\sqrt{\frac{M}{1 \star \mathbf{x}^2}}, \mathbf{0}, \dots, \mathbf{0}, \kappa \sqrt{\frac{M}{1 \star \mathbf{x}^2}}\right)^T \tag{51}$$

Utilizing (51), (19), and (38), without considering front-end filtering, we can obtain the expression of *R*�

$$\tilde{R}\left(\tau;\kappa\right) = \begin{cases} \frac{M\left(2\kappa - 4\right) \left|\tau\right| \circ 2\left(1 - \kappa\right)\tau\_{\epsilon}}{\sqrt{M\left(1 + \kappa^{2}\right)^{T}}}, & \left|\tau\right| < \frac{(1 - \kappa)\tau\_{\epsilon}}{M\left(2 - \kappa\right)}\\ & 0, & \text{others} \end{cases} \tag{52}$$

the base line half width of which is

$$kw(\kappa) = \frac{(1-\kappa)T\_{\varsigma}}{M(2-\kappa)}\tag{53}$$

and the height of the peak is

$$\ln\left(\kappa\right) = \frac{2\left(1-\kappa\right)}{\sqrt{M\left(1+\kappa^2\right)}}\tag{54}$$

Figure 15 (a) and (b) show some SCFs with different κ for BOC(*n*,*n*) and BOC(2*n*,*n*) signals, respectively. Figure 16 depicts the discriminator characteristic curve of the early-minus-late power (EMLP) loop which uses SCF instead of BOC(*n*,*n*) ACF.

Unambiguous Processing Techniques of Binary Offset Carrier Modulated Signals 75

its solutions is systematically described. An innovative design methodology for future unambiguous processing techniques is also proposed. Under the proposed design framework, the development of SC algorithm becomes the solving of an optimization problem with a set of inequalities constraints and can be achieved through four steps.

As two practical examples, the design process of an SC unambiguous acquisition algorithm as well as an SC unambiguous tracking loop is described respectively to demonstrate the practicality of the proposed framework and to provide reference to further SC algorithm development. Although the optimization objects are difference in these two algorithms, the methods of analysis and the design steps under the analytic design framework are unified. It is proved that both of these two algorithms can completely eliminate the ambiguity problem in acquisition and tracking. Moreover, these two algorithms have outstanding

Future works will focus on the development of new algorithms under the proposed framework. Considering the complexities, both of the example algorithms in this Charpter employ only two local signals. In fact, by using more local signals, degrees of freedom in design can be further increased and the shape control will be more flexible. Besides, this analytic design framework can be used not only on unambiguous algorithm development

ARINC. (2005). NAVSTAR GPS space segment/navigation L5 User interfaces. In *IS-GPS-705*. ARINC. (2006). Navstar GPS space segment/user L1C interfaces. In *IS-GPS-800*. El Segundo,

Avila-Rodriguez, J.-A., Hein, G. W., Wallner, S., Issler, J.-L., Ries, L., Lestarquit, L., Latour, A. d.,

Burian, A., Lohan, E. S., & Renfors, M. (2006). BPSK-like methods for hybrid-search

Dovis, F., Mulassano, P., & Presti, L. L. (2005). A novel algorithm for the code tracking of

Enge, P. (2003). GPS modernization: capabilities of the new civil signals. *Proceedings of*

Fante, R. L. (2003). Unambiguous tracker for GPS binary-offset-carrier signals, *Proceedings of* 

Fine, P., & Wilson, W. (1999). Tracking algorithm for GPS offset carrier signals. *Proceedings of* 

Fishman, P., & Betz, J. W. (2000). Predicting performance of direct acquisition for the M-code signal. *Proceedings of ION NTM 2000*, pp. 574-582, Anaheim, CA, US, 2000. Gao, G. X., Chen, A., Lo, S., Lorenzo, D. d., & Enge, P. (2007). GNSS over China - the

*Australian International Aerospace Congress*, pp. 1-22. Brisbane, 2003.

Compass MEO satellite codes. *Inside GNSS,* 2007, 2(5), pp. 36-43.

*Symposium*, pp. 141-145, Albuquerque, NM, US, 2003.

*ION NTM 1999*, pp. 671-676, San Diego, CA, US, 1999.

Godet, J., Bastide, F., Pratt, T., & Owen, J. (2007). The MBOC modulation: the final touch to the Galileo frequency and signal plan, *Proceedings of ION GNSS 20th International Technical Meeting of the Satellite Division*, pp. 1515-1529, Fort Worth, TX, US, 2007. Betz, J. W. (2001). Binary offset carrier modulations for radionavigation, *Navigation: J. Inst.* 

acquisition of Galileo signals, *Proceedings of IEEE ICC 2006*, pp. 5211-5216, Istanbul,

BOC(*n*,*n*) modulated signals, *Proceedings of ION GNSS 2005*, pp. 152-155, Long

*the 59th Annual Meeting of The Institute of Navigation and CIGTF 22nd Guidance Test* 

compatibility. Both of them are suitable for generic even-order sine-BOC signal.

but also on finding good local waveform to resist the effect of multipath.

**8. References** 

CA, US.

*Navig.*, 48(4), 227-246.

Turkey, 2006.

Beach, CA, 2005.

Fig. 15. SCF for (a) BOC(*n*,*n*) and (b) BOC(2*n*,*n*) Signals, with Different κ

Fig. 16. The Discriminator Characteristic Curve for BOC(*n*,*n*) Signals, with Different κ

It can be noted that by using SCF, this technique completely removes the false lock points. And it can also be seen that both shapes of SCF and discriminator characteristic curve are functions of κ . Consequently, changing the value of κ can adjust the linear range of the discriminator and the slope of the curve, thus changes the multipath and thermal noise mitigation performances of the tracking loop (Yao, Cui, et al., 2010). With different optimization objective, the optimum κand the optimum chip waveform are not the same.

#### **7. Conclusion**

In this Chapter, the ambiguity problem of BOC modulated signals which have been chosen as the chief candidate for several navigation signals in the next generation GNSS as well as its solutions is systematically described. An innovative design methodology for future unambiguous processing techniques is also proposed. Under the proposed design framework, the development of SC algorithm becomes the solving of an optimization problem with a set of inequalities constraints and can be achieved through four steps.

As two practical examples, the design process of an SC unambiguous acquisition algorithm as well as an SC unambiguous tracking loop is described respectively to demonstrate the practicality of the proposed framework and to provide reference to further SC algorithm development. Although the optimization objects are difference in these two algorithms, the methods of analysis and the design steps under the analytic design framework are unified. It is proved that both of these two algorithms can completely eliminate the ambiguity problem in acquisition and tracking. Moreover, these two algorithms have outstanding compatibility. Both of them are suitable for generic even-order sine-BOC signal.

Future works will focus on the development of new algorithms under the proposed framework. Considering the complexities, both of the example algorithms in this Charpter employ only two local signals. In fact, by using more local signals, degrees of freedom in design can be further increased and the shape control will be more flexible. Besides, this analytic design framework can be used not only on unambiguous algorithm development but also on finding good local waveform to resist the effect of multipath.

## **8. References**

74 Global Navigation Satellite Systems – Signal, Theory and Applications


It can be noted that by using SCF, this technique completely removes the false lock points. And it can also be seen that both shapes of SCF and discriminator characteristic curve are

discriminator and the slope of the curve, thus changes the multipath and thermal noise mitigation performances of the tracking loop (Yao, Cui, et al., 2010). With different

In this Chapter, the ambiguity problem of BOC modulated signals which have been chosen as the chief candidate for several navigation signals in the next generation GNSS as well as

Fig. 16. The Discriminator Characteristic Curve for BOC(*n*,*n*) Signals, with Different

. Consequently, changing the value of

κ

Code Delay (Chips)

κ

and the optimum chip waveform are not the same.

κ

κ=0.1 κ=0.3

κ

can adjust the linear range of the

Fig. 15. SCF for (a) BOC(*n*,*n*) and (b) BOC(2*n*,*n*) Signals, with Different





0

Discriminator Output

functions of

**7. Conclusion** 

κ

optimization objective, the optimum

0.5

1

1.5

2

ARINC. (2005). NAVSTAR GPS space segment/navigation L5 User interfaces. In *IS-GPS-705*.


**4** 

*Italy* 

*University of Florence* 

**Evolution of Integrity Concept – From Galileo to Multisystem** 

Mario Calamia, Giovanni Dore and Alessandro Mori

The Galileo navigation system introduced the integrity concept, intended as a continuous control of the information broadcasted by satellites. Although the RAIM technique represents the first example of integrity monitoring, it is able to detect only local errors made at the receiver level. The integrity monitoring applied by EGNOS could instead be seen as the forerunner of the Galileo system. Even if there are many differences in the definition of integrity for the two systems, the aim is the same for both: to protect the user against the failure of the system, warning him in the shortest time and with the greatest

The integrity of a navigation system can be defined as follows: "integrity relates to the trust that can be placed in the correctness of information supplied by a navigation system. Specifically, a navigation system is required to deliver an alarm when the error in the derived user position solution exceeds an allowable level (alarm limit). This warning must be issued to the user within a given period of time (time-to-alarm) and with a given

In the near future a central role will be played by the integrity receiver's capability. This service can be considered essential in the safety critical application domain, particularly in aviation. For these applications, the system's capability of protecting the user against system

Integrity includes the system's ability to supply, at the right time, reliable warnings to the user (alarm). The main problem with this service is how to determine what can be considered safe. This depends on the requirements of the different fields of application. The following parameters are traditionally used to define the safety of the service for a specific

• Alarm Limit (AL): the maximum error allowed in the position domain before an alarm

• Time To Alarm (TTA): the time that elapses between an error's overcoming of the AL

• Integrity Risk (IR): the probability that the alarm will not be delivered within the TTA. Allowable values of AL, TTA and IR depend on the specific application of the navigation system. The Galileo system provides a high level of integrity of the navigation signal. The

**1. Introduction** 

precision possible.

application:

is generated.

probability (integrity risk)" (Oehler et al., 2004).

and the reception of the alarm by the user's receiver.

failure is of primary importance.


## **Evolution of Integrity Concept – From Galileo to Multisystem**

Mario Calamia, Giovanni Dore and Alessandro Mori *University of Florence Italy* 

## **1. Introduction**

76 Global Navigation Satellite Systems – Signal, Theory and Applications

Hegarty, C. J., Betz, J. W., & Saidi, A. (2004). Binary coded symbol modulations for GNSS. *Proceedings of ION 60th Annual Meeting*, pp. 56-64, Dayton, OH, US, 2004. Hegarty, C. J., & Chatre, E. (2008). Evolution of the global navigation satellite system

Hein, G. W., Avila-Rodriguez, J.-A., Wallner, S., Pratt, A. R., Owen, J., Issler, J.-L., Betz, J. W.,

Julien, O., Macabiau, C., Cannon, M. E., & Lachapelle, G. (2007). ASPeCT: unambiguous

*Transactions on Aerospace and Electronic Systems,*Vol. 43, No.1, pp. 150-162. Lohan, E. S., Burian, A., & Renfors, M. (2008). Low-complexity unambiguous acquisition

Martin, N., Leblond, V., Guillotel, G., & Heiries, V. (2003). BOC(x,y) signal acquisition

Musso, M., Cattoni, A. F., & Regazzoni, C. S. (2006). A new fine tracking algorithm for

Nunes, F., Sousa, F., & Leitao, J. (2007). Gating functions for multipath mitigation in GNSS

Slater, J. A., Weber, R., & Fragner, E. (2004). The IGS GLONASS pilot project – transitioning

VanDierendonck, A. J., Fenton, P., & Ford, T. (1992). Theory and performance of narrow correlator spacing in GPS receiver, *Navigation: J. Inst. Navig.,* Vol. 39, pp. 115 - 124. Ward, P. W. (2003). A design technique to remove the correlation ambiguity in binary offset

Yao, Z. (2008). A new unambiguous tracking technique for sine-BOC(2*n*,*n*) signals. *Proceedings of ION GNSS 2008,* pp. 1490-1496, Savannah, GA, US, 2008. Yao, Z. (2009). *Code Synchronization and Carrier Tracking Algorithms for New Generation of* 

Yao, Z., Cui, X., Lu, M., & Feng, Z. (2010b). Pseudo-correlation-function-based unambiguous

Yao, Z., & Lu, M. (2011). Side-peaks cancellation analytic design framework with

Yao, Z., Lu, M., & Feng, Z. (2010a). Unambiguous sine-phased binary offset carrier

Ziemer, R. E., & Peterson, R. L. (1985). *Digitial Communications and Spread Spectrum Systems*.

Proakis, J. G. (2001). *Digital Communications*, Boston: McGraw-Hill Companies, Inc.

Hegarty, C. J., Lenahan, L. S., Rushanan, J. J., Kraay, A. L., & Stansell, T. A. (2006). MBOC: the new optimized spreading modulation recommended for GALILEO L1 OS and GPS L1C. *Proceedings of IEEE/ION PLANS 2006*, pp. 883-892, San Diego, CA,

sine-BOC(n,n) acquisition/tracking technique for navigation applications, *IEEE* 

methods for BOC-modulated CDMA signals, *Int. J. Commun. Syst. Network,* 26

techniques and performances. *Proceedings of ION GPS 2003*, pp. 188-198, Portland,

binary offset carrier modulated signals. *Proceedings of ION GNSS 2006*, pp. 834-840,

BOC signals, *IEEE Transactions on Aerospace and Electronic Systems,* Vol. 43, No. 3

an experiment into an operational GNSS service. *Proceedings of ION GNSS 2004*, pp.

carrier (BOC) spread spectrum signals. *Proceedings of ION AM 2003*, pp. 146-155,

tracking technique for sine-BOC signals. *IEEE Transactions on Aerospace and* 

applications in BOC signals unambiguous processing, *Proceedings of ION ITM 2011*,

modulated signal acquisition technique, *IEEE Transactions on Wireless* 

(GNSS). *Proceedings of IEEE,* 96(12), pp. 1902-1917.

US, 2006.

(2008), pp. 503–522.

Fort Worth, TX, US, 2006.

1749 – 1757, Long Beach, CA, US, 2004.

*GNSS.* PhD Thesis, Tsinghua University, Beijing.

*Electronic Systems,* Vol. 46, No. 4, pp. 1782-1796.

pp. 775-785, San Diego, CA, US, 2011.

*Communications,* Vol. 9, No. 2, pp. 577-580.

New York: Macmillan Publishing Company.

Albuquerque, NM, US, 2003.

(2007), pp. 941-964.

OR, US, 2003.

The Galileo navigation system introduced the integrity concept, intended as a continuous control of the information broadcasted by satellites. Although the RAIM technique represents the first example of integrity monitoring, it is able to detect only local errors made at the receiver level. The integrity monitoring applied by EGNOS could instead be seen as the forerunner of the Galileo system. Even if there are many differences in the definition of integrity for the two systems, the aim is the same for both: to protect the user against the failure of the system, warning him in the shortest time and with the greatest precision possible.

The integrity of a navigation system can be defined as follows: "integrity relates to the trust that can be placed in the correctness of information supplied by a navigation system. Specifically, a navigation system is required to deliver an alarm when the error in the derived user position solution exceeds an allowable level (alarm limit). This warning must be issued to the user within a given period of time (time-to-alarm) and with a given probability (integrity risk)" (Oehler et al., 2004).

In the near future a central role will be played by the integrity receiver's capability. This service can be considered essential in the safety critical application domain, particularly in aviation. For these applications, the system's capability of protecting the user against system failure is of primary importance.

Integrity includes the system's ability to supply, at the right time, reliable warnings to the user (alarm). The main problem with this service is how to determine what can be considered safe. This depends on the requirements of the different fields of application. The following parameters are traditionally used to define the safety of the service for a specific application:


Allowable values of AL, TTA and IR depend on the specific application of the navigation system. The Galileo system provides a high level of integrity of the navigation signal. The

Evolution of Integrity Concept – From Galileo to Multisystem 79

• Integrity Flags (IF): this is a warning relative to a satellite that is transmitting a signal with an excessive error. IF is based on the short term observation of the clock's

• Signal-in-Space Monitoring Accuracy (SISMA): this is an estimation of the accuracy of

Threshold

0-10 -5 0 5 10 15 20 25 30 35 40

• SWs set as NOT OK or NOT MONITORED are discarded from the position computation. • The user receiver computes the Protection Level using SISA and SISMA parameters.

The Galileo Integrity system is based on the concept of Protection Level. Its main purpose is to calculate the error's bound in the position estimate, in order to be able to control this error

The user receiver judges the accuracy of the computed position solution, typically in term of Horizontal Protection Level (HPL) and Vertical Protection Level (VPL), by means of an estimate of the system errors, an estimate of the local errors and the knowledge of the number and geometry of the SWs used for the positioning algorithm. The computed Protection Level is then compared with a specific Alert Limit, in order to determine the

The original definition of integrity belongs to the position domain, but it can be translated into the Signal-in-Space domain. As a matter of fact, the position error can be replaced by

Using the parameters described above, the user could check the integrity, as follows:

**3**

variations, the ephemerides and the RF signals.

**1**

**4**

**2**

• In a faulty free condition, SISA overbounds the SISE distribution.

the Signal-in-Space Error (SISE).

0.14

0.12

0.1

0.08

0.06

0.04

0.02

• PL is compared with the specific AL.

**2.1 Faulty free protection level** 

with a sufficient level of confidence.

availability of the navigation service.

the SISE and the Protection level by the SISA.

Fig. 1. Integrity events

global integrity concept is the answer to the needs of different types of users who are all looking for different services in terms of signal and performance.

A new concept of Integrity will be introduced in the following paragraphs. In particular, starting from the Galileo Integrity concepts, we will illustrate a few solutions to the integrity problem and describe a new one, in which data of different constellations (GPS/EGNOS and Galileo) are combined in order to improve the accuracy and the availability of the navigation data.

## **2. Galileo integrity**

The integrity concept developed in Galileo has the aim of ensuring the correct computation of the user's position and provide a valid alarm to the user if the error in the position solution has exceeded a fixed threshold - the Alert Limit - relative to the specific application (Martini, 2006). The user can be in one of the following conditions (Table 1):


Table 1. Examples of integrity

In order to estimate all the errors that might occur in different situations, we have adopted a Gaussian model (J. Rife et al., 2004), whose standard deviation derives from the standard deviation of the error distribution and from the accuracy of the system. Moreover, each Gaussian distribution might have a bias, representing the presence of a faulty condition. The following Figure (Figure 1) shows the system's estimate of the error distribution, illustrating the situations displayed in Table 1. The first two cases concern a faulty free condition: the error is modelled with a zero-mean Gaussian distribution. In this case, the system only has an estimation of the error. This estimation could be considered as a sample of the abovementioned Gaussian distribution, and this sample could be above (1) or below (2) the specific threshold. In case 1, the system is working in nominal condition, whereas case 2 concerns a False Alarm condition. The failure is modelled as the presence of a bias in the error distribution. This bias could be higher than the threshold (case 3), and in that case the system would certainly detect it.

Otherwise, the mentioned bias could be higher than the threshold, but the sample of the distribution could be below this limit (case 4). This case is referred as Missed Detection condition (Martini, 2006).

The Galileo system provides three elements to preserve user integrity:

• Signal-in-Space Accuracy (SISA): this is the expectation of the errors relative to the SW's clock and ephemerides, based on long term observations.


Fig. 1. Integrity events

78 Global Navigation Satellite Systems – Signal, Theory and Applications

global integrity concept is the answer to the needs of different types of users who are all

A new concept of Integrity will be introduced in the following paragraphs. In particular, starting from the Galileo Integrity concepts, we will illustrate a few solutions to the integrity problem and describe a new one, in which data of different constellations (GPS/EGNOS and Galileo) are combined in order to improve the accuracy and the availability of the

The integrity concept developed in Galileo has the aim of ensuring the correct computation of the user's position and provide a valid alarm to the user if the error in the position solution has exceeded a fixed threshold - the Alert Limit - relative to the specific application

> **System Alert for Satellite**

2 Fault-Free Nominal YES NOT-OK False Alert 3 Faulty Non-nominal YES NOT-OK True Alert 4b Faulty Non-nominal YES NOT-Monitored True Alert 4c Faulty Non-nominal NO OK Error below

In order to estimate all the errors that might occur in different situations, we have adopted a Gaussian model (J. Rife et al., 2004), whose standard deviation derives from the standard deviation of the error distribution and from the accuracy of the system. Moreover, each Gaussian distribution might have a bias, representing the presence of a faulty condition. The following Figure (Figure 1) shows the system's estimate of the error distribution, illustrating the situations displayed in Table 1. The first two cases concern a faulty free condition: the error is modelled with a zero-mean Gaussian distribution. In this case, the system only has an estimation of the error. This estimation could be considered as a sample of the abovementioned Gaussian distribution, and this sample could be above (1) or below (2) the specific threshold. In case 1, the system is working in nominal condition, whereas case 2 concerns a False Alarm condition. The failure is modelled as the presence of a bias in the error distribution. This bias could be higher than the threshold (case 3), and in that case the

Otherwise, the mentioned bias could be higher than the threshold, but the sample of the distribution could be below this limit (case 4). This case is referred as Missed Detection

• Signal-in-Space Accuracy (SISA): this is the expectation of the errors relative to the SW's

The Galileo system provides three elements to preserve user integrity:

clock and ephemerides, based on long term observations.

**Satellite User** 

**Msg. Comment** 

Threshold

(Martini, 2006). The user can be in one of the following conditions (Table 1):

1 Fault-Free Nominal NO OK

looking for different services in terms of signal and performance.

navigation data.

**2. Galileo integrity** 

**Case System Case System State** 

Table 1. Examples of integrity

system would certainly detect it.

condition (Martini, 2006).

Using the parameters described above, the user could check the integrity, as follows:


## **2.1 Faulty free protection level**

The Galileo Integrity system is based on the concept of Protection Level. Its main purpose is to calculate the error's bound in the position estimate, in order to be able to control this error with a sufficient level of confidence.

The user receiver judges the accuracy of the computed position solution, typically in term of Horizontal Protection Level (HPL) and Vertical Protection Level (VPL), by means of an estimate of the system errors, an estimate of the local errors and the knowledge of the number and geometry of the SWs used for the positioning algorithm. The computed Protection Level is then compared with a specific Alert Limit, in order to determine the availability of the navigation service.

The original definition of integrity belongs to the position domain, but it can be translated into the Signal-in-Space domain. As a matter of fact, the position error can be replaced by the SISE and the Protection level by the SISA.

Evolution of Integrity Concept – From Galileo to Multisystem 81

The assumptions made for the derivation of the user integrity equation are summarized as

• In a Faulty Free mode, the true SISE for a satellite is zero mean Gaussian distributed

• For each instance in time, one satellite of those flagged as OK is considered to be faulty but not detected (Faulty Mode). The distribution for the SISE of a faulty satellite is

Once the distribution of the error in the reference frame is known (Gaussian overbounding distribution with SISA and SISMA respectively), the derivation of the associate integrity risk

Therefore, the error distribution for the vertical (one dimensional Gaussian distribution) and horizontal (Chi Squared distribution with two degree of freedom) cases needs to be derived, and the corresponding integrity risk can be easily computed by analyzing the integral for both distributions with respect to the given alert limit. Finally, the integrity risk at the alert limits HAL (Horizontal) and VAL (vertical) are computed by adding the vertical and

> 2 2

*FF*

1 , , , ,

*VAL VAL P erf erf*

2 *FM HAL cdf* ξ

*j u V FM u V FM*

2

<sup>+</sup> ⎜ ⎟ ⎜ ⎟ − + ⎜ ⎟⎜ ⎟ ⎜ ⎟ − + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠

μ

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞ + − ⎛ ⎞

, ,

*u V u V*

 μ (2)

 σ

2

−

*HAL*

ξ

2 2 2

σ

<sup>1</sup> 1 1

⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠

The Integrity Risk computed by the user represents the probability of exceeding the specified alert limits, since the system works according to the hypothesis described above. The Integrity Risk guaranteed by Galileo is partially allocated to user computation and partially to the system itself. This means that a proper design and implementation of the system must guarantee that the system have a sufficiently low probability of being in a condition in which the performance relevant assumption is no longer valid. Only this will ensure that the true overall integrity risk is below the required limit, in accordance with the specified level of service when this service is declared available by the integrity system.

In order to better understand the PHMI formula (Eq. 2) and all the elements contributing to its design, it is necessary to show the main passages leading to the construction of that equation. These passages could be collected into an algorithm leading to the HPCA

*u*

δ

Gaussian with an expectation value TH and a standard deviation SISMA.

, ,

=+ =

*VAL erf <sup>e</sup>*

⎛ ⎞ =− + + ⎜ ⎟ ⎝ ⎠

,

*fail sat*

2

σ

1

**2.5 HMI probability computation algorithm (HPCA)** 

*P*

+ −

*j*

=

*N*

∑

*N*

∑

=

, ,

*u V FF*

, 2, ,

algorithm (HMI Probability Computation Algorithm) (Luongo et al., 2004).

χ

2

*fail sat H*

1

where N is the number of satellites used for the positioning algorithm.

*j*

follows:

is straightforward.

with standard deviation SISA.

horizontal contributions (Oehler et al., 2004).

*P VAL HAL P P*

1

*HMI IntRisk V IntRisk H*

(, )

• In general, a faulty satellite will be flagged as Don't Use.

#### **2.2 Faulty case**

In case of a system failure, the range measurement will be affected by a bias that gets added to the other errors. The aim of the system is to detect this bias. For this reason the Galileo system consists of a Ground Segment (GSS) that is able to monitor range measurements. If the bias exceeds an established integrity threshold, the user will become aware of this via an alarm.

The error detected by the ground segment can be modelled using a zero-mean Gaussian distribution with variance <sup>2</sup> σ . Since the false alarm probability can be considered as the area limited by this function between threshold and infinite, we can calculate this threshold as follows:

$$\text{TH} = k\_{fa} \cdot \sqrt{\sigma\_{SISA}^2 + \sigma\_{u,L}^2} \tag{1}$$

where *fa k* derives from the false alarm probability.

The alarm is notified by setting the Integrity Flag relative to the satellite with failure in the information delivered to the users. This satellite must not then be considered by the user in the xPL computation and in the positioning algorithm. The combination between the IF and the PL can ensure the integrity of the information received in the position domain. Moreover, the implementation of a RAIM algorithm comes to the aid of the integrity monitoring, in order to face up to errors caused by local effects (i.e., multipath, interference, jamming and ionospheric effects).

### **2.3 Evolution of integrity concept**

The evolution of the Galileo integrity concept concerns only the verification of system integrity. In particular, based on the above-mentioned definitions, the checking methodology has been modified: the vertical and the horizontal protection levels have been combined in a unique concept, and the user has to compute a probability, named Hazardous Misleading Information Probability (PHMI), which will be compared to the threshold. Once the distribution of the error in the desired reference frame is known (Gaussian overbounding distributions with SISA and SISMA), it will be simple to derive the associated integrity risk both in the faulty and the faulty free conditions appointed to the user equations. Therefore, the error distributions for the vertical (one dimensional Gaussian distribution) and horizontal (Chi Squared distribution with two degrees of freedom) cases need to be derived, and the corresponding integrity risk can be easily computed by analyzing the integral for both distributions with the given alert limits. The integrity risk at the alert limits VAL and HAL are finally computed by adding the vertical and horizontal contributions (Dore & Calamia, 2009).

#### **2.4 Galileo integrity risk**

Based on the aforementioned quantities (SISE, SISA, SISMA, IF and TH), the user receiver can derive the integrity risk for the user position solution. This integrity risk is always computed for a given alert limit. Whenever the derived IR at the AL is larger than the allowed IR, the user equipment will raise an alert (Oehler et al., 2004).

80 Global Navigation Satellite Systems – Signal, Theory and Applications

In case of a system failure, the range measurement will be affected by a bias that gets added to the other errors. The aim of the system is to detect this bias. For this reason the Galileo system consists of a Ground Segment (GSS) that is able to monitor range measurements. If the bias exceeds an established integrity threshold, the user will become aware of this via an

The error detected by the ground segment can be modelled using a zero-mean Gaussian

area limited by this function between threshold and infinite, we can calculate this threshold

2 2 *TH k* =⋅ + *fa SISA u L* σ

The alarm is notified by setting the Integrity Flag relative to the satellite with failure in the information delivered to the users. This satellite must not then be considered by the user in the xPL computation and in the positioning algorithm. The combination between the IF and the PL can ensure the integrity of the information received in the position domain. Moreover, the implementation of a RAIM algorithm comes to the aid of the integrity monitoring, in order to face up to errors caused by local effects (i.e., multipath, interference,

The evolution of the Galileo integrity concept concerns only the verification of system integrity. In particular, based on the above-mentioned definitions, the checking methodology has been modified: the vertical and the horizontal protection levels have been combined in a unique concept, and the user has to compute a probability, named Hazardous Misleading Information Probability (PHMI), which will be compared to the threshold. Once the distribution of the error in the desired reference frame is known (Gaussian overbounding distributions with SISA and SISMA), it will be simple to derive the associated integrity risk both in the faulty and the faulty free conditions appointed to the user equations. Therefore, the error distributions for the vertical (one dimensional Gaussian distribution) and horizontal (Chi Squared distribution with two degrees of freedom) cases need to be derived, and the corresponding integrity risk can be easily computed by analyzing the integral for both distributions with the given alert limits. The integrity risk at the alert limits VAL and HAL are finally computed by adding the vertical and horizontal

Based on the aforementioned quantities (SISE, SISA, SISMA, IF and TH), the user receiver can derive the integrity risk for the user position solution. This integrity risk is always computed for a given alert limit. Whenever the derived IR at the AL is larger than the

allowed IR, the user equipment will raise an alert (Oehler et al., 2004).

 σ

. Since the false alarm probability can be considered as the

, (1)

**2.2 Faulty case** 

distribution with variance <sup>2</sup>

jamming and ionospheric effects).

**2.3 Evolution of integrity concept** 

contributions (Dore & Calamia, 2009).

**2.4 Galileo integrity risk** 

σ

where *fa k* derives from the false alarm probability.

alarm.

as follows:

The assumptions made for the derivation of the user integrity equation are summarized as follows:


Once the distribution of the error in the reference frame is known (Gaussian overbounding distribution with SISA and SISMA respectively), the derivation of the associate integrity risk is straightforward.

Therefore, the error distribution for the vertical (one dimensional Gaussian distribution) and horizontal (Chi Squared distribution with two degree of freedom) cases needs to be derived, and the corresponding integrity risk can be easily computed by analyzing the integral for both distributions with respect to the given alert limit. Finally, the integrity risk at the alert limits HAL (Horizontal) and VAL (vertical) are computed by adding the vertical and horizontal contributions (Oehler et al., 2004).

$$\begin{split} P\_{\text{HM}}(\text{VAL},\text{HAL}) &= P\_{\text{Int}Risk,V} + P\_{\text{Int}Risk,H} = \\ &= 1 - \text{erf}\left(\frac{\text{VAL}}{\sqrt{2}\sigma\_{u,V,FF}}\right) + e^{-\frac{\text{HAL}^2}{2\varepsilon\_{\text{NF}}^2}} \\ &+ \frac{1}{2} \sum\_{j=1}^{N} P\_{\text{fail},\text{sat}\_j} \left( \left(1 - \text{erf}\left(\frac{\text{VAL} + \mu\_{u,V}}{\sqrt{2}\sigma\_{u,V,FM}}\right)\right) + \left(1 - \text{erf}\left(\frac{\text{VAL} - \mu\_{u,V}}{\sqrt{2}\sigma\_{u,V,FM}}\right)\right) \right) + \\ &+ \sum\_{j=1}^{N} P\_{\text{fail},\text{sat}} \left(1 - \chi^2\_{2,\delta\_{u,V},H} \text{erf}\left(\frac{\text{HAL}^2}{\varepsilon\_{\text{FM}}^2}\right)\right) \end{split} \tag{2}$$

where N is the number of satellites used for the positioning algorithm.

The Integrity Risk computed by the user represents the probability of exceeding the specified alert limits, since the system works according to the hypothesis described above. The Integrity Risk guaranteed by Galileo is partially allocated to user computation and partially to the system itself. This means that a proper design and implementation of the system must guarantee that the system have a sufficiently low probability of being in a condition in which the performance relevant assumption is no longer valid. Only this will ensure that the true overall integrity risk is below the required limit, in accordance with the specified level of service when this service is declared available by the integrity system.

#### **2.5 HMI probability computation algorithm (HPCA)**

In order to better understand the PHMI formula (Eq. 2) and all the elements contributing to its design, it is necessary to show the main passages leading to the construction of that equation. These passages could be collected into an algorithm leading to the HPCA algorithm (HMI Probability Computation Algorithm) (Luongo et al., 2004).

Evolution of Integrity Concept – From Galileo to Multisystem 83

 *u L*, [*i*] is the predicted standard deviation of the local errors (troposphere, noise, multipath) on the ith signal. The values of the standard deviation of local errors can be read in the UERE table (Galileo satellites). Typical values are those reported in the following Table (F. Luongo et al., 2004). The computation is performed using the following interpolation function:

, ( ) *ELi*

value of the UERE for the ith satellite, while a and b can be computed by the following

<sup>1</sup> ( ) *<sup>T</sup> T T <sup>i</sup> ab A A A*

The "Position Solution Matrix" module computes the position solution matrix. The main

( ) <sup>1</sup> *T T K G WG G W* <sup>−</sup>

where K is the Position Solution Matrix, G is the Observation Matrix and W is the Weighting

The "Fault Free Position Error" module computes the characteristic of the position error in fault free mode (fault free geometry: all the useable satellites). In particular, it evaluates the standard deviation of the distribution that overbounds the vertical position error and the variance of the distribution that overbounds the horizontal position error. The principal

2

 σ

<sup>2</sup> 22 22 2 4 2 2 *XX YY XX YY H XY*

 σσ

*<sup>H</sup>* is the variance of the model used to overbound the horizontal position error (along the

 + − ⎛ ⎞ =+ + ⎜ ⎟

⎝ ⎠

*<sup>V</sup>* is the standard deviation of the model (zero-mean normal CDF) used to

σ

σσ

*m n*, components are obtained using this general expression:

**ID 01 02 03 04 05 06 07 08**  Elevation [rad] 0.1745 0.2618 0.3491 0.5236 0.6981 0.8727 1.0472 1.5708 σi [m] 1.0300 0.7800 0.6700 0.6000 0.5800 0.5700 0.5600 0.5500

*u L* σ

Where A is a matrix that depends on the elevation angles,

The predicted standard deviation of total pseudorange error (

Matrix obtained by inverting the Covariance Matrix.

ξ

overbound the vertical position error in fault free mode.

semi-mayor axis of the error ellipse) in fault free mode.

internal variable of the HMI Probability Computation Algorithm (HPCA).

10

σ

*i a be*− ⋅ =+⋅ (5)

σ

<sup>−</sup> = ⋅ ⋅⋅ (6)

σ

= (7)

*V ZZ* = (8)

 σ

*<sup>i</sup>* is the standard deviation

*u RX* , ) is considered an

(9)

σ

equation:

Table 2. UERE table

formula is reported:

where

2 ξ

The σ σ

formulas are reported as follows:

The objective of HPCA is to compute the predicted HMI probability in any integrity exposure time interval (150 s) for a given GMS integrity by monitoring state and user geometry.

This algorithm includes the following modules:


Figure 2 shows the block diagram of the HPCA Algorithm.

Fig. 2. HPCA algorithm

The "UERE Computation module" computes the predicted standard deviation of total pseudorange error on each signal from visible satellites. The principal formula is as follows:

$$
\sigma\_{\boldsymbol{u}, \boldsymbol{RX}} \begin{bmatrix} \boldsymbol{i} \end{bmatrix} = \sqrt{\sigma\_{\boldsymbol{S} \mid \boldsymbol{SA}}^2 \begin{bmatrix} \boldsymbol{i} \end{bmatrix} + \sigma\_{\boldsymbol{u}, \boldsymbol{L}}^2 \begin{bmatrix} \boldsymbol{i} \end{bmatrix}} \tag{3}
$$

It takes into account the signal in space as well as the local errors.

[ ] *SISA* σ *i* is the SISA value for the ith satellite used at user level; it is equal to the SISA value broadcasted in the navigation message increased by a factor of 1.1.

$$\left[\sigma\_{SISA}^2\left[i\right]\right] = \left(1.1 \cdot SISA\left[i\right]\right)^2\tag{4}$$

σ *u L*, [*i*] is the predicted standard deviation of the local errors (troposphere, noise, multipath) on the ith signal. The values of the standard deviation of local errors can be read in the UERE table (Galileo satellites). Typical values are those reported in the following Table (F. Luongo et al., 2004). The computation is performed using the following interpolation function:

$$
\sigma\_{u,L}(\mathbf{i}) = a + b \cdot e^{-10 \cdot El\_i} \tag{5}
$$

Where A is a matrix that depends on the elevation angles, σ *<sup>i</sup>* is the standard deviation value of the UERE for the ith satellite, while a and b can be computed by the following equation:

$$\begin{array}{c|c|c|c|c|c|c} \text{ } & ab \text{ } & ^\text{I} = (A^T \cdot A)^{-1} \cdot A^T \cdot \sigma\_i & & & & & & \\ \hline \\ \hline \text{ID} & 0\mathbf{1} & 0\mathbf{2} & 0\mathbf{3} & \mathbf{04} & \mathbf{05} & \mathbf{06} & \mathbf{07} & \mathbf{08} \\ \hline \text{Elevation [rad]} & 0.1745 & 0.2618 & 0.3491 & 0.5236 & 0.6981 & 0.8727 & 1.0472 & 1.5708 \\ \hline \sigma\_i \text{ [m]} & 1.0300 & 0.7800 & 0.6700 & 0.6000 & 0.5800 & 0.5700 & 0.5600 & 0.5500 \\ \end{array}$$

Table 2. UERE table

82 Global Navigation Satellite Systems – Signal, Theory and Applications

The objective of HPCA is to compute the predicted HMI probability in any integrity exposure time interval (150 s) for a given GMS integrity by monitoring state and user

> Obtain Geometry and Integrity Data

UERE Computation

 Position Solution Matrix Computation

HMI Probability Computation

The "UERE Computation module" computes the predicted standard deviation of total pseudorange error on each signal from visible satellites. The principal formula is as follows:

*i* is the SISA value for the ith satellite used at user level; it is equal to the SISA value

[ ] ( ) [ ] <sup>2</sup> <sup>2</sup> 1.1 *SISA*

[] [] [] 2 2 *u RX SISA u L* , ,

σσ

*i ii* = + (3)

*i SISA i* = ⋅ (4)

 Set HPCA\_index  Faulty Position Error Computation

 FaultFree Position Error Computation

σ

It takes into account the signal in space as well as the local errors.

broadcasted in the navigation message increased by a factor of 1.1.

σ

geometry.

This algorithm includes the following modules:

• Position Solution Matrix Computation Module • Fault Free Position Error Computation Module • Faulty Position Error Computation Module • HMI Probability Computation Module

Figure 2 shows the block diagram of the HPCA Algorithm.

• UERE Computation Module

Fig. 2. HPCA algorithm

[ ] *SISA* σ

The predicted standard deviation of total pseudorange error (σ *u RX* , ) is considered an internal variable of the HMI Probability Computation Algorithm (HPCA).

The "Position Solution Matrix" module computes the position solution matrix. The main formula is reported:

$$\mathbf{K} = \left(\mathbf{G}^T \mathcal{W} \mathbf{G}\right)^{-1} \mathbf{G}^T \mathcal{W} \tag{7}$$

where K is the Position Solution Matrix, G is the Observation Matrix and W is the Weighting Matrix obtained by inverting the Covariance Matrix.

The "Fault Free Position Error" module computes the characteristic of the position error in fault free mode (fault free geometry: all the useable satellites). In particular, it evaluates the standard deviation of the distribution that overbounds the vertical position error and the variance of the distribution that overbounds the horizontal position error. The principal formulas are reported as follows:

$$
\sigma\_V = \sqrt{\sigma\_{ZZ}^2} \tag{8}
$$

$$
\xi\_H^2 = \frac{\sigma\_{XX}^2 + \sigma\_{YY}^2}{2} + \sqrt{\left(\frac{\sigma\_{XX}^2 - \sigma\_{YY}^2}{2}\right)^2 + \sigma\_{XY}^4} \tag{9}
$$

where σ *<sup>V</sup>* is the standard deviation of the model (zero-mean normal CDF) used to overbound the vertical position error in fault free mode.

2 ξ *<sup>H</sup>* is the variance of the model used to overbound the horizontal position error (along the semi-mayor axis of the error ellipse) in fault free mode.

The σ*m n*, components are obtained using this general expression:

Evolution of Integrity Concept – From Galileo to Multisystem 85

without additional ILS systems. For CAT III approaches (zero visibility), even the SBAS will

EGNOS provides a European-wide, standardized and quality-assured augmentation service suitable for different fields of applications. Integrity is a key quality and safety parameter, and it alerts users when the system exceeds tolerance limits. EGNOS broadcasts wide-area differential corrections to improve accuracy, and alerts users within six seconds if something

The receiver combines satellite/user geometry information, with EGNOS-corrected pseudoranges, and internal estimates of the tropospheric delay to compute the user position. Ideally, the user would like to have the difference between the computed position and the true position - the true position error (PE) - to be less than the AL. However, since the true position is not known, the PE cannot be determined, and an alternative approach is

In fact, the receiver continuously estimates a predicted position error, known as the protection level (PL), for each position solution. The PL can be estimated using the UDRE and GIVE parameters and other local error-bound estimates. It is scaled for compatibility with the probability of non-integrity detection so that the PL should always be larger than

Integrity assessments are based on PL and AL. A new PL is estimated for each computed position solution, then it is compared with the required AL, and an integrity alert is triggered if PL>AL. There is an underlying assumption, that PL>PE, when assessing integrity, and this corresponds to the "safe" zone to the left of the leading diagonal in Figure 3. In the nominal operation case, PL<AL and the system is available. If PL>AL for a particular operation, the EGNOS integrity cannot support the operation, and the system is

There is also an "unsafe" zone to the right of the leading diagonal where PL<PE and the integrity assessment provide misleading information (Figure 3). The case at the bottom left corner of the diagram (PL<PE<AL) is also "safe," theoretically, because the AL has not been exceeded, but it should be noted that EGNOS also protects against these out-of-tolerance

Different parameters, used in the XPL computation, must be elaborated by the ground

• the variance σ2UDRE,i of a zero-mean normal distribution that describes the user differential range error (UDRE) for each ranging source after the application of fast and

of an ionospheric model based on the broadcast grid ionospheric vertical error (GIVE).

*local i*, of a zero-mean normal distribution that relates the pseudo range

*tropo i*, of a zero-mean normal distribution that defines the residual pseudo

σ*GIVE i*, )

long-term corrections and excluding atmospheric effects and receiver errors; • the variance σ2UIRE,i of a zero-mean normal distribution that describes the L1 residual user ionospheric range error (UIRE) for each ranging source after ionospheric corrections have been applied. This variance is determined from the variance ( <sup>2</sup>

not suffice, and ILS is still required.

goes wrong (integrity).

required.

PE.

unavailable.

situations (ESA, 2005).

• the variance <sup>2</sup>

• the variance <sup>2</sup>

segment (Roturier et al., 2001):

σ

σ

error due to local receiver noise and multipath;

range error of a tropospheric correction model.

$$
\sigma\_{m,n} = \sum\_{i=1}^{N} \mathcal{K}\left[m, i\right] \cdot \mathcal{K}\left[n, i\right] \cdot \sigma\_{u,RX}^{2}\left[i\right] \tag{10}
$$

where "i" indicates the ith satellite, "m" and "n" the reference axis: X, Y or Z.

The "Faulty Position Error" module computes the characteristic of the position error in faulty mode (faulty geometry: one single failure satellite). In particular, the standard deviation of the distribution that overbounds the vertical position error and the variance of the distribution that overbounds the horizontal position error are computed. The principal formulas are reported here:

$$
\sigma\_V = \sqrt{\sigma\_{ZZ\\_F}^2} \tag{11}
$$

$$\xi\_H^2 = \frac{\sigma\_{XX\\_F}^2 + \sigma\_{YY\\_F}^2}{2} + \sqrt{\left(\frac{\sigma\_{XX\\_F}^2 - \sigma\_{YY\\_F}^2}{2}\right)^2 + \sigma\_{XY\\_F}^4} \tag{12}$$

where σ *<sup>V</sup>* is the standard deviation of the model (zero-mean normal CDF) used to overbound the vertical position error in faulty mode.

2 ξ *<sup>H</sup>* is the variance of the model used to overbound the horizontal position error (along the semi-mayor axis of the error ellipse) in faulty mode.

The σ*mn F* , \_ components are obtained using the following general expression:

$$\boldsymbol{\sigma}\_{m,n\\_F} = \sum\_{i=1}^{N} \mathbb{K}[m,i] \cdot \mathbb{K}[n,i] \cdot \boldsymbol{\sigma}\_{u,\mathcal{R}X}^2 \left[i\right] + \mathbb{K}[m,i\_0] \cdot \mathbb{K}[n,i\_0] \cdot \left(\sigma\_{\text{SISMA}}^2[i\_0] - \sigma\_{\text{SISA}}^2[i\_0]\right) \tag{13}$$

where "i" indicates the ith satellite, "m" and "n" the reference axis: X, Y or Z.

The "HMI Probability" module computes the probability of HMI. The principal formulas are reported as follows:

$$P\_{\rm HMI} = P\_{\rm HMI, Fault-Free} + p\_{\rm fail} P\_{\rm HMI, Faulty} \tag{14}$$

$$P\_{\text{HMI},\text{Fault}-\text{Free}} = P\_{\text{HMI},\text{Fault}-\text{Free},V} + P\_{\text{HMI},\text{Fault}-\text{Free},H} \tag{15}$$

$$P\_{\text{HMI},\text{Fault}} = P\_{\text{HMI},\text{Fault},V} + P\_{\text{HMI},\text{Fault},\text{H}} \tag{16}$$

#### **3. EGNOS integrity**

The GPS system is neither accurate nor reliable enough to be accepted as the only instrument of navigation for critical applications. One of the reasons is that no reliable and quick (within seconds) information can reach the user if any problems with the system occur. As a consequence, the GPS system cannot be used for landing approaches, for instance. Airplanes still have to use ILS-systems (Instrument Landing Systems) in case of poor visibility. But the installation and the maintenance of ILS-systems in every airport is expensive. With the SBAS systems, CAT I approaches (limited visibility) will be possible 84 Global Navigation Satellite Systems – Signal, Theory and Applications

, ,

, ,

*m n u RX*

The "Faulty Position Error" module computes the characteristic of the position error in faulty mode (faulty geometry: one single failure satellite). In particular, the standard deviation of the distribution that overbounds the vertical position error and the variance of the distribution that overbounds the horizontal position error are computed. The principal

2

\_ 2 2

*<sup>V</sup>* is the standard deviation of the model (zero-mean normal CDF) used to

⎝ ⎠

[ ] [ ] [] [ ] [ ] [] [] ( ) 2 2 <sup>2</sup>

= ⋅⋅ + ⋅ ⋅ − ∑ (13)

*Kmi Kni i Kmi Kni i i*

σσ

 σ

<sup>2</sup> 22 22 2 4 \_\_ \_\_

+ − ⎛ ⎞ =+ + ⎜ ⎟

*<sup>H</sup>* is the variance of the model used to overbound the horizontal position error (along the

, \_ , 0 0 0 0

The "HMI Probability" module computes the probability of HMI. The principal formulas

The GPS system is neither accurate nor reliable enough to be accepted as the only instrument of navigation for critical applications. One of the reasons is that no reliable and quick (within seconds) information can reach the user if any problems with the system occur. As a consequence, the GPS system cannot be used for landing approaches, for instance. Airplanes still have to use ILS-systems (Instrument Landing Systems) in case of poor visibility. But the installation and the maintenance of ILS-systems in every airport is expensive. With the SBAS systems, CAT I approaches (limited visibility) will be possible

*mn F u RX SISMA SISA*

*mn F* , \_ components are obtained using the following general expression:

,, , ,

where "i" indicates the ith satellite, "m" and "n" the reference axis: X, Y or Z.

*XX F YY F XX F YY F H XY F*

1

where "i" indicates the ith satellite, "m" and "n" the reference axis: X, Y or Z.

σ

*i*

=

σ

σσ

ξ

1

σσ

*i*

are reported as follows:

**3. EGNOS integrity** 

=

*N*

overbound the vertical position error in faulty mode.

semi-mayor axis of the error ellipse) in faulty mode.

formulas are reported here:

where

2 ξ

The σ σ

*N*

[ ] [ ] [] <sup>2</sup>

*Kmi Kni i*

 σ

= ⋅⋅ ∑ (10)

*V ZZ F* = \_ (11)

(12)

 σ

 σ

*P P pP HMI HMI Fault Free* = , , <sup>−</sup> + *fail HMI Faulty* (14)

*PP P HMI Fault* , ,, ,, *<sup>y</sup>* = *HMI Faulty V HMI Fault* + *<sup>y</sup> <sup>H</sup>* (16)

*PP P HMI Fault Free HMI Fault Free V HMI Fault Free H* , ,, ,, −− − = + (15)

σ

without additional ILS systems. For CAT III approaches (zero visibility), even the SBAS will not suffice, and ILS is still required.

EGNOS provides a European-wide, standardized and quality-assured augmentation service suitable for different fields of applications. Integrity is a key quality and safety parameter, and it alerts users when the system exceeds tolerance limits. EGNOS broadcasts wide-area differential corrections to improve accuracy, and alerts users within six seconds if something goes wrong (integrity).

The receiver combines satellite/user geometry information, with EGNOS-corrected pseudoranges, and internal estimates of the tropospheric delay to compute the user position. Ideally, the user would like to have the difference between the computed position and the true position - the true position error (PE) - to be less than the AL. However, since the true position is not known, the PE cannot be determined, and an alternative approach is required.

In fact, the receiver continuously estimates a predicted position error, known as the protection level (PL), for each position solution. The PL can be estimated using the UDRE and GIVE parameters and other local error-bound estimates. It is scaled for compatibility with the probability of non-integrity detection so that the PL should always be larger than PE.

Integrity assessments are based on PL and AL. A new PL is estimated for each computed position solution, then it is compared with the required AL, and an integrity alert is triggered if PL>AL. There is an underlying assumption, that PL>PE, when assessing integrity, and this corresponds to the "safe" zone to the left of the leading diagonal in Figure 3. In the nominal operation case, PL<AL and the system is available. If PL>AL for a particular operation, the EGNOS integrity cannot support the operation, and the system is unavailable.

There is also an "unsafe" zone to the right of the leading diagonal where PL<PE and the integrity assessment provide misleading information (Figure 3). The case at the bottom left corner of the diagram (PL<PE<AL) is also "safe," theoretically, because the AL has not been exceeded, but it should be noted that EGNOS also protects against these out-of-tolerance situations (ESA, 2005).

Different parameters, used in the XPL computation, must be elaborated by the ground segment (Roturier et al., 2001):


Evolution of Integrity Concept – From Galileo to Multisystem 87

Integrity assessments are based on XPL and XAL: a new XPL is estimated for each computed position solution. It is compared with the required XAL, and an integrity alert is

The integrity of a navigation system can be checked by using external systems such as SBAS to monitor the correctness of the signals used to calculate position. One of the main drawbacks to this approach is the inherent delay that is introduced when an error is detected, due to the time taken to uplink the information on errors. This section will focus on internal monitoring, and in particular on RAIM. RAIM stands for Receiver Autonomous Integrity Monitoring and is used to denote a monitoring algorithm that uses nothing but the measurements of one particular navigation subsystem, usually a GPS receiver. Conventional RAIM algorithms are designed to protect users from a single satellite failure at a time. However, recent developments have shown that RAIM has the potential to provide integrity even in case of multiple failures for challenging flight categories such as LPV-200 and APV-

Measurement information is used to compute a position. A test statistic is derived from this position computation. It gets passed to an error detector that will warn the user whenever something is wrong. The error detection procedure will have to obey navigation requirements, and it is important to determine the detection power (or 'error detectability'). It depends on the measurement quality and the configuration, and it is this detection power computation that monitors the system's integrity, determining whether the system has the ability to provide timely warnings when the system is in error. If this is not the case, it will inform the user that it might be unsafe to use the system. It should be noted that position computation algorithms always assume that noise on the measurements has a zero mean. An error or bias, as it is commonly called, is therefore defined as the non-zero mean of

The slope, which relates the induced position error to the test statistic, can be calculated directly from geometry and is different for each satellite. The satellite with the largest slope is the most difficult to detect. It produces the largest position error for a given test statistic

Max Slope

Test Statistic

Position Error

triggered if XPL > XAL.

**4. RAIM integrity** 

II (Ciollaro, 2009).

measurement noise.

**4.1 Satellite slope** 

(Figure 4) (Ciollaro, 2009).

Fig. 4. Satellite slope

Position Error (PE)

Fig. 3. EGNOS protection level

Assuming that the error in the range domain can be overbounded by a zero-mean Gaussian probability density function, the variance of that distribution is:

$$
\sigma^2 \sigma^2 = \sigma^2 \sigma^2 \, \_{i,flt} + \sigma^2 \, \_{LiRE,i} + \sigma^2 \, \_{air,i} + \sigma^2 \, \_{tropo,i} \tag{17}
$$

where the variance <sup>2</sup> σ *<sup>i</sup>*, *flt* may be easily derived from <sup>2</sup> σ *UDRE i*, . From Eq. (17) and for a given user/satellite geometry, it is quite simple to derive the vertical (horizontal) protection level VPL (HPL) equation by:


Compared to the first step, the variance in the position domain residual error is a linear combination of σ<sup>2</sup> i and is also representative of a zero-mean normal law.

$$
\sigma^2\_{Vposition} = \sum\_{i=1}^{N} s\_{\,^1V,i}^2 \sigma\_{\,^i}^2 \tag{18}
$$

where *SV,i* are geometrical parameters.

The second step is achieved by multiplying the position domain variance by a factor k that propagates this variance to a level compatible with the integrity requirement.

$$\text{VPL}\_{\text{EGNOS}} = \text{K}\_{V} \sqrt{\sum\_{i=0}^{N} s\_{\text{-},i}^{2} \sigma\_{\text{-}i}^{2}} \tag{19}$$

Integrity assessments are based on XPL and XAL: a new XPL is estimated for each computed position solution. It is compared with the required XAL, and an integrity alert is triggered if XPL > XAL.

## **4. RAIM integrity**

86 Global Navigation Satellite Systems – Signal, Theory and Applications

 System Unavailable AL<PE<PL

 Misleading Information PL<PE<AL

Position Error (PE)

Assuming that the error in the range domain can be overbounded by a zero-mean Gaussian

user/satellite geometry, it is quite simple to derive the vertical (horizontal) protection level

1. passing from the pseudo range variance domain to the position variance domain,

2. scaling the position domain variance to the integrity requirement. VPL (HPL) is scaled for compatibility with the probability of non integrity detection so that the VPL (HPL)

Compared to the first step, the variance in the position domain residual error is a linear

2 2 2

The second step is achieved by multiplying the position domain variance by a factor k that

*s*

*VPL K s*

*N Vposition V i i i*

=

, 1

σ

2 2 , 0

σ

*N EGNOS V V i i i*

=

 σ

 σ*i i* =+ ++ , ,, , *flt UiRE i air i tropo i* (17)

*UDRE i*, . From Eq. (17) and for a given

<sup>=</sup> ∑ (18)

<sup>=</sup> ∑ (19)

σ

22 2 2 2

 σ

*<sup>i</sup>*, *flt* may be easily derived from <sup>2</sup>

noting that the integrity definitions are all in the position domain;

combination of σ2i and is also representative of a zero-mean normal law.

σ

propagates this variance to a level compatible with the integrity requirement.

Alert Limit (AL)

Hazardously Misleading Information PL<AL<PE

 System Unavailable & Misleading Information AL<PL<PE

 System Unavailable PE<AL<PL

Alert Limit (AL)

Protection Level (PL)

Fig. 3. EGNOS protection level

σ

should always be larger than PE.

where *SV,i* are geometrical parameters.

where the variance <sup>2</sup>

VPL (HPL) equation by:

 Nominal Operations PE<PL<AL

probability density function, the variance of that distribution is:

σσ

The integrity of a navigation system can be checked by using external systems such as SBAS to monitor the correctness of the signals used to calculate position. One of the main drawbacks to this approach is the inherent delay that is introduced when an error is detected, due to the time taken to uplink the information on errors. This section will focus on internal monitoring, and in particular on RAIM. RAIM stands for Receiver Autonomous Integrity Monitoring and is used to denote a monitoring algorithm that uses nothing but the measurements of one particular navigation subsystem, usually a GPS receiver. Conventional RAIM algorithms are designed to protect users from a single satellite failure at a time. However, recent developments have shown that RAIM has the potential to provide integrity even in case of multiple failures for challenging flight categories such as LPV-200 and APV-II (Ciollaro, 2009).

Measurement information is used to compute a position. A test statistic is derived from this position computation. It gets passed to an error detector that will warn the user whenever something is wrong. The error detection procedure will have to obey navigation requirements, and it is important to determine the detection power (or 'error detectability'). It depends on the measurement quality and the configuration, and it is this detection power computation that monitors the system's integrity, determining whether the system has the ability to provide timely warnings when the system is in error. If this is not the case, it will inform the user that it might be unsafe to use the system. It should be noted that position computation algorithms always assume that noise on the measurements has a zero mean. An error or bias, as it is commonly called, is therefore defined as the non-zero mean of measurement noise.

## **4.1 Satellite slope**

The slope, which relates the induced position error to the test statistic, can be calculated directly from geometry and is different for each satellite. The satellite with the largest slope is the most difficult to detect. It produces the largest position error for a given test statistic (Figure 4) (Ciollaro, 2009).

Test Statistic

Fig. 4. Satellite slope

Evolution of Integrity Concept – From Galileo to Multisystem 89

specific operation. For example, for LPV-200, the whole Integrity Risk is allocated to the

It is not possible to obtain a direct measurement of the position error. Therefore, the overall consistency of the solution has to be investigated (Walter & Enge, 1995). As long as there are more than four measurements, the system is overdetermined and cannot be solved accurately. This is why a least squares solution is performed in the first place. Since all of the conditions cannot be met realistically and exactly, there is always an error residual to the fit. Therefore, we need to be able to estimate the fit and assume that, if there is a good fit, the

An estimate of the ranging errors from the least squares fit and the basic measurement

( )() *wls wls*

From these error estimates it is possible to define a scalar measure, defined as the Weighted

[( ) ] [( ) ] *T T WSSE W I P wls wls* =

The square root of WSSE plays the role of the basic observable, because it yields a linear relationship between a satellite bias error and the associated induced test statistic. The test

Typically, a certain threshold, which depends on the required probability of false alarm, is selected. If the statistic exceeds that threshold, then the position fix is assumed to be unsafe. On the other hand, if the statistic is below the threshold, then the position fix is assumed to

The statistic-vertical error plane is thus broken up into four regions consisting of: normal operation points, missed detections, successful detections and false alarms. Ideally, there would never be any missed detections or false alarms. In reality, a certain number of missed detections and false alarms are allowed, based on the Pmd and Pfa requirements, respectively.

With the advent of Galileo, users will be provided with multiple signals coming from different satellite systems. This will improve position accuracy, because the number of satellites in view per user will be almost doubled. Moreover, the higher measurements

 ε

= *y* −⋅ = −⋅ ⋅= − ⋅ *Gx I GK y I P y* (26)

<sup>1</sup> ( ) *T T P G K GG W G G W* <sup>−</sup> = ⋅= ⋅ ⋅ ⋅ ⋅ (27)

⋅⋅ = −⋅ ⋅⋅−⋅ *y W IP y* (28)

( ) *WSSE <sup>T</sup>* = *y* ⋅ ⋅− ⋅ *WIP y* (29)

vertical domain, since this is the most demanding requirement.

**4.3 RAIM tst statistic** 

equation is given by:

which is equivalent to:

**5. Multisystem integrity** 

where:

be valid.

position error is most likely small.

Sum of the Squared Errors (WSSE):

ε

ε

statistic can be defined in both the horizontal and vertical planes.

The slope is a geometric parameter that can be directly computed from the specific satelliteuser geometry, based on the following equations, in the horizontal and vertical planes respectively:

$$Hsslope\_i = \frac{\sqrt{K\_{1i}^2 + K\_{2i}^2} \sigma\_i}{\sqrt{1 - P\_{ii}}} \tag{20}$$

$$Vslope\_i = \frac{|K\_{3i}|\sigma\_i}{\sqrt{1 - P\_{ii}}} \tag{21}$$

where <sup>1</sup> ( ) *T T K G WG G W* <sup>−</sup> = ⋅⋅ ⋅ ⋅ is the weighted pseudo-inverse of the design matrix, where W the inverse of the covariance matrix, while *P GK* = ⋅ . The geometric contribution to the slope is given by the K and P matrices.

#### **4.2 RAIM protection levels**

The Protection Levels in the vertical and horizontal planes can be described by the following equations (Walter & Enge, 1995), for the vertical and horizontal cases, respectively:

$$\text{VPL}\_{\text{FD}} = \max\{V\_{\text{slque}}\} T(N\_\prime P\_{\text{ft}}) + k(P\_{\text{ml}}) \sigma\_V \tag{22}$$

$$HPL\_{FD} = \max\{H\_{slope}\} T(N\_{\prime}P\_{ft}) + k(P\_{md})\sigma\_{H} \tag{23}$$

where:


$$1 - P\_{fa} = \frac{1}{\Gamma(a)} \int\_0^{r^2} e^{-s} s^{a-1} ds \tag{24}$$

where a is the number of degrees of freedom divided by two, or, in terms of the number of measurements N and unknowns M:

$$a = \frac{N - M}{2} \tag{25}$$


It should be noted that, when using RAIM, it is common to allocate the whole Integrity Risk, and so the whole Pmd is confined to only one plane (vertical or horizontal) according to the specific operation. For example, for LPV-200, the whole Integrity Risk is allocated to the vertical domain, since this is the most demanding requirement.

#### **4.3 RAIM tst statistic**

88 Global Navigation Satellite Systems – Signal, Theory and Applications

The slope is a geometric parameter that can be directly computed from the specific satelliteuser geometry, based on the following equations, in the horizontal and vertical planes

*i*

*K K Hslope <sup>P</sup>*

*i*

*<sup>K</sup> Vslope <sup>P</sup>*

where <sup>1</sup> ( ) *T T K G WG G W* <sup>−</sup> = ⋅⋅ ⋅ ⋅ is the weighted pseudo-inverse of the design matrix, where W the inverse of the covariance matrix, while *P GK* = ⋅ . The geometric contribution to the

The Protection Levels in the vertical and horizontal planes can be described by the following

max{ } ( , ) ( ) *HPL H T N P k P FD* = *slope fa md H* +

• T(N,Pfa) is the test statistic threshold, and it is a function of the number of satellites (N) and the desired probability of false alarm (Pfa). Given the probability of false alarms, the

2

0

( ) *<sup>T</sup> s a P es d fa <sup>s</sup>*

where a is the number of degrees of freedom divided by two, or, in terms of the number of

2 *N M*

*<sup>H</sup>* are the standard deviations of the error in the position domain in the

• k(PMD) is the number of standard deviations corresponding to the specified Probability of Missed Detection. The smaller the PMD value, the higher the number of standard deviations should be considered, since longer tails for the Gaussian distribution should

It should be noted that, when using RAIM, it is common to allocate the whole Integrity Risk, and so the whole Pmd is confined to only one plane (vertical or horizontal) according to the

Γ*a* 1

equations (Walter & Enge, 1995), for the vertical and horizontal cases, respectively:

• Vslope and Hslope are the satellite error slope in the vertical and horizontal planes

threshold can be found by inverting the incomplete gamma function:

<sup>1</sup> <sup>1</sup>

max{ } ( , ) ( ) *VPL V T N P k P FD* = *slope fa md V* +

2 2 1 2 1 *i ii*

+

3 1 *i i*

*ii*

*ii*

σ

σ

<sup>=</sup> − (20)

<sup>=</sup> − (21)

σ

σ

− − − = ∫ (24)

*<sup>a</sup>* <sup>−</sup> <sup>=</sup> (25)

(22)

(23)

respectively:

where:

•

σ*<sup>V</sup>* and

slope is given by the K and P matrices.

measurements N and unknowns M:

be taken into account.

vertical and horizontal planes.

σ

**4.2 RAIM protection levels** 

It is not possible to obtain a direct measurement of the position error. Therefore, the overall consistency of the solution has to be investigated (Walter & Enge, 1995). As long as there are more than four measurements, the system is overdetermined and cannot be solved accurately. This is why a least squares solution is performed in the first place. Since all of the conditions cannot be met realistically and exactly, there is always an error residual to the fit. Therefore, we need to be able to estimate the fit and assume that, if there is a good fit, the position error is most likely small.

An estimate of the ranging errors from the least squares fit and the basic measurement equation is given by:

$$
\boldsymbol{\varepsilon}\_{\text{wls}} = \boldsymbol{y} - \boldsymbol{G} \cdot \boldsymbol{\chi}\_{\text{wls}} = (\boldsymbol{I} - \boldsymbol{G} \cdot \boldsymbol{K}) \cdot \boldsymbol{y} = (\boldsymbol{I} - \boldsymbol{P}) \cdot \boldsymbol{y} \tag{26}
$$

where:

$$P = \mathbf{G} \cdot \mathbf{K} = \mathbf{G} \{ \mathbf{G}^T \cdot \mathbf{W} \cdot \mathbf{G} \}^{-1} \cdot \mathbf{G}^T \cdot \mathbf{W} \tag{27}$$

From these error estimates it is possible to define a scalar measure, defined as the Weighted Sum of the Squared Errors (WSSE):

$$\text{WSSE} = \boldsymbol{\varepsilon}\_{\text{wls}}^T \cdot \boldsymbol{\mathcal{W}} \cdot \boldsymbol{\varepsilon}\_{\text{wls}} = [(I - P) \cdot \boldsymbol{y}]^T \cdot \boldsymbol{\mathcal{W}} \cdot [(I - P) \cdot \boldsymbol{y}] \tag{28}$$

which is equivalent to:

$$\text{WSSE} = y^T \cdot \mathcal{W} \cdot (I - P) \cdot y \tag{29}$$

The square root of WSSE plays the role of the basic observable, because it yields a linear relationship between a satellite bias error and the associated induced test statistic. The test statistic can be defined in both the horizontal and vertical planes.

Typically, a certain threshold, which depends on the required probability of false alarm, is selected. If the statistic exceeds that threshold, then the position fix is assumed to be unsafe. On the other hand, if the statistic is below the threshold, then the position fix is assumed to be valid.

The statistic-vertical error plane is thus broken up into four regions consisting of: normal operation points, missed detections, successful detections and false alarms. Ideally, there would never be any missed detections or false alarms. In reality, a certain number of missed detections and false alarms are allowed, based on the Pmd and Pfa requirements, respectively.

#### **5. Multisystem integrity**

With the advent of Galileo, users will be provided with multiple signals coming from different satellite systems. This will improve position accuracy, because the number of satellites in view per user will be almost doubled. Moreover, the higher measurements

Evolution of Integrity Concept – From Galileo to Multisystem 91

*SISA GPS SISA GPS UDRE* , ,

*SISMA GPS SISMA GPS UDRE* , ,

Taking into account the different integrity allocation between the Galileo concept, which implies the use of four failure mechanisms, and the EGNOS concept, based on a failure assumption, the contribution of the GPS satellite to IR computation is reduced only to the

> σ

> > σσ

Moreover, in order to estimate the standard deviation of the error, the following equation

2 2 22 , , ( )

 σ

 EGNOS (Europe)

 EGNOS Integrity Message

EGNOS/Galileo Converter

EGNOS Converted Integrity Message

Galileo Integrity Algorithm: **HPCA**

 Galileo Integrity Message

In this section, we describe a new proposed multisystem integrity algorithm. The algorithm merges integrity data originated by the Galileo and EGNOS systems and employs a Receiver

**5.3 Multisystem integrity (MSI) algorithm implementation** 

 σ

> σ

= *f* (30)

= *f* (31)

*SISA GPS UDRE* , = (32)

Integration

*u L GPS UIRE Air Tro* = ++ *po* (33)

 IR (Integrity Risk)

σ

Then it is possible to assume , 0 *SISMA GPS f* = and , 1 *SISA GPS f* = , that is:

σ

 GPS (USA)

 Galileo (Europe)

Fig. 5. Galileo-based integrity algorithm

Consistency Check (FD)

σ

σ

faulty free mode.

can be used:

 GPS Navigation Message

> Galileo Navigation Message

RAIM

redundancy will help guarantee a safer position and the detection of errors. This will also result in an improved availability as well as meet the requirements for more demanding flight categories. Therefore, it is necessary to introduce a base-line for a combined system, defining new parameters, a new integrity algorithm and possible ways to combine the two independent systems.

With the term "Multisystem", we intend the improvement of the accuracy and availability of the navigation solution using the combined Galileo and GPS signals. In this context, it is essential for the user to be able to take advantage of the integrity information coming from both Galileo and GPS satellite constellations, in order to prevent users from making errors that might represent an excessive risk. The multisystem integrity algorithm has to establish a link between the two generations of GNSS, defining the relation for integrating different integrity monitoring schemes (Pecchioni et al. 2007) (Ciollaro, 2009).

## **5.1 Definition of a new integrity algorithm (EGNOS + Galileo)**

Two different approaches have been studied to define the new integrity algorithm. They represent two opposite ways of solving the problem of how to combine different integrity concepts: the first one has been called "One-System-Based Integrity," and the second one "Parallel Integrity" (Dore & Calamia, 2009).

The first approach is based on the use of only one algorithm for both systems with an a priori definition of integrity inputs. The integrity analysis can be made either by converting the EGNOS integrity message into an equivalent Galileo integrity message or vice versa, by using the inverse transformation from a Galileo to an EGNOS-like message. In the first case, known as Galileo-Based-Integrity-Algorithm (GBIA), the Galileo Integrity is used as a baseline; in the second case, called EGNOS-Based-Integrity-Algorithm (EBIA), the One-System Integrity is the EGNOS algorithm.

The second approach is based on the use of independent (parallel) algorithms, one for each System, and on an a posteriori integration of the integrity results. The integrity analysis can be performed by monitoring the values assumed by both the Integrity Risk and the Protection Level. If the IR is used as monitored variable, the scheme will be called IR-PIA; otherwise, if the monitored variable is the PL, the method will be called PL-PIA. It is worth noting that the computational load for the IR/PL conversion is expected to be higher than the PL/IR conversion, because an iterative method must be applied (Ciollaro, 2009).

## **5.2 Galileo based integrity algorithm**

The approach chosen for this study is GBIA. The integrity data in fact arrives from the two systems, Galileo and EGNOS, and is implemented inside the Integrity Risk equation of Galileo, in order to estimate the HMI Probability.

Figure 5 shows the block diagram of a GBIA system. The fundamental block of this diagram is the EGNOS/Galileo converter, which has the aim of converting the EGNOS Integrity message into a message that can be used by the Galileo Integrity Algorithm.

The main functions implemented by the EGNOS-Galileo converter are the following (Ciollaro, 2009):

90 Global Navigation Satellite Systems – Signal, Theory and Applications

redundancy will help guarantee a safer position and the detection of errors. This will also result in an improved availability as well as meet the requirements for more demanding flight categories. Therefore, it is necessary to introduce a base-line for a combined system, defining new parameters, a new integrity algorithm and possible ways to combine the two

With the term "Multisystem", we intend the improvement of the accuracy and availability of the navigation solution using the combined Galileo and GPS signals. In this context, it is essential for the user to be able to take advantage of the integrity information coming from both Galileo and GPS satellite constellations, in order to prevent users from making errors that might represent an excessive risk. The multisystem integrity algorithm has to establish a link between the two generations of GNSS, defining the relation for integrating different

Two different approaches have been studied to define the new integrity algorithm. They represent two opposite ways of solving the problem of how to combine different integrity concepts: the first one has been called "One-System-Based Integrity," and the second one

The first approach is based on the use of only one algorithm for both systems with an a priori definition of integrity inputs. The integrity analysis can be made either by converting the EGNOS integrity message into an equivalent Galileo integrity message or vice versa, by using the inverse transformation from a Galileo to an EGNOS-like message. In the first case, known as Galileo-Based-Integrity-Algorithm (GBIA), the Galileo Integrity is used as a baseline; in the second case, called EGNOS-Based-Integrity-Algorithm (EBIA), the One-

The second approach is based on the use of independent (parallel) algorithms, one for each System, and on an a posteriori integration of the integrity results. The integrity analysis can be performed by monitoring the values assumed by both the Integrity Risk and the Protection Level. If the IR is used as monitored variable, the scheme will be called IR-PIA; otherwise, if the monitored variable is the PL, the method will be called PL-PIA. It is worth noting that the computational load for the IR/PL conversion is expected to be higher than

The approach chosen for this study is GBIA. The integrity data in fact arrives from the two systems, Galileo and EGNOS, and is implemented inside the Integrity Risk equation of

Figure 5 shows the block diagram of a GBIA system. The fundamental block of this diagram is the EGNOS/Galileo converter, which has the aim of converting the EGNOS Integrity

The main functions implemented by the EGNOS-Galileo converter are the following

the PL/IR conversion, because an iterative method must be applied (Ciollaro, 2009).

message into a message that can be used by the Galileo Integrity Algorithm.

integrity monitoring schemes (Pecchioni et al. 2007) (Ciollaro, 2009).

**5.1 Definition of a new integrity algorithm (EGNOS + Galileo)** 

"Parallel Integrity" (Dore & Calamia, 2009).

System Integrity is the EGNOS algorithm.

**5.2 Galileo based integrity algorithm** 

(Ciollaro, 2009):

Galileo, in order to estimate the HMI Probability.

independent systems.

$$
\sigma\_{\text{SISA,GPS}} = f\_{\text{SISA,GPS}} \sigma\_{\text{LDRE}} \tag{50}
$$

$$
\sigma\_{\text{SISMA,GPS}} = f\_{\text{SISMA,GPS}} \sigma\_{\text{LDRE}} \tag{31}
$$

Taking into account the different integrity allocation between the Galileo concept, which implies the use of four failure mechanisms, and the EGNOS concept, based on a failure assumption, the contribution of the GPS satellite to IR computation is reduced only to the faulty free mode.

Then it is possible to assume , 0 *SISMA GPS f* = and , 1 *SISA GPS f* = , that is:

$$
\sigma\_{SISA,GPS} = \sigma\_{\text{UDRE}} \tag{32}
$$

Moreover, in order to estimate the standard deviation of the error, the following equation can be used:

Fig. 5. Galileo-based integrity algorithm

#### **5.3 Multisystem integrity (MSI) algorithm implementation**

In this section, we describe a new proposed multisystem integrity algorithm. The algorithm merges integrity data originated by the Galileo and EGNOS systems and employs a Receiver

Evolution of Integrity Concept – From Galileo to Multisystem 93

where a and b are parameters that depend directly on Eli (Luongo, 2004). This estimation

The IR equation has been implemented by means of a numerical code in a computer. FF and FM, in Eq. 2, suggest the faulty and faulty free modes. In fact, the Galileo system assumes two separate scenarios: one in which the satellites are all set as use, and the other in which one of the satellites set as use is supposed not to be functioning. When you are in the faulty mode, in the case of Galileo satellites, the SISMA element comes out; in an EGNOS case, only the faulty free mode is instead expected and, because we could not find an equivalent for the Galileo SISMA in its navigation message, we are going to consider the following

• Faulty free: for the Integrity Risk computation we consider all satellites in view, GPS

• Faulty mode: the involved satellites are only those belonging to the Galileo

The information available a priori for the new algorithm consists of two text files containing position (X,Y and Z components) and velocity (X,Y and Z components) of the SV belonging

Pseudoranges are obtained by the true satellite-user distance, adding a zero-mean Gaussian

Regarding the SISA and SISMA evaluation, we have considered actual values, adding a

0.87 0, 0.7 0,

received. We must also describe the behaviour of the positioning algorithm in the combined constellation case. Generally speaking, if we define Xk, Yk and Zk as the coordinates of the Kth satellite and X, Y and Z as the coordinates of the user position, we are able to compute the

> () () *kk k*( ) *c u d ct*

 =+ + δ ε

( ) ( )

σ

σ

*SISA SISMA*

= , in order to simulate a sort of degradation on the signal

<sup>222</sup> ( )( )( ) *kk k k d XX YY ZZ* = − +−+− (37)

ρ

(36)

ρ

(38)

) as follows

to the two constellations considered, and obtained through a constellation simulator.

noise with variance depending on SISA and the elevation angles of the satellites.

*SISA N SISMA N*

distance between the satellite and the user ( *<sup>k</sup> d* ) and the pseudoranges ( *<sup>k</sup>*

ρ

: residual error on k-th satellite. *b* : clock's offset.

= + = +

constellation; hence the index of the sum concerns only those satellites.

, ,

*uLi* σ

was performed for both cases, the Galileo and GPS satellites.

situation:

Gaussian noise:

(Misra & Enge, 2001):

and

where: *k* ρ ε

and Galileo are set as OK.

In this case, 0.01 σ

 *SISA SISMA* = σ

**5.3.2 Inputs of the implemented algorithm** 

10

*Eli*

*a be*<sup>−</sup> =+⋅ (35)

Autonomous Integrity (RAIM) technique (Weighted RAIM). One of the potential uses of this algorithm consists in the combination of the IR algorithm with the RAIM technique. RAIM is able to detect failures that have not been detected by the IR algorithm. In case of multiple failures, when the WRAIM technique fails, the IR algorithm triggers an alarm.

In this Section we describe the characteristics of this innovative algorithm, pointing out the reason for using the IR equation for the combined constellation Galileo/EGNOS and the reason for taking advantage of a RAIM technique. The EGNOS integrity equation provides a way to measure the integrity based on the incoming signal variances and the satellite geometry. The same is also true for the Galileo IR equation, obviously bearing in mind which data is the Galileo integrity data.

This algorithm is supposed to enable the user to take advantage of the data transmitted by the Galileo and EGNOS systems: the user receiver must consider a single and large constellation in order to strengthen the positioning algorithm and improve the accuracy. This idea is simply in need of the definition of a new integrity concept, which would be able to combine the techniques mentioned above.

## **5.3.1 IR equation**

First of all, we have to explain why the protection level concept turns into the integrity risk concept in a Galileo environment. In an EGNOS domain, IR is the probability that the horizontal (vertical) PL exceeds the horizontal (vertical) AL without the user receiving any alarm whatsoever. This definition requires a clear distinction between the horizontal and vertical cases. Therefore, it is necessary to split IR into two a priori fixed quantities.

On the contrary, as far as a Galileo integrity equation is concerned, the users do not have to evaluate the horizontal and vertical protection levels, but the global IR directly, without making any strict allocations. In fact, different applications need distinct integrity requirements for the horizontal and vertical situations: for example, for a ship the vertical component of the error is not that important for a ship, but it is instead essential for a plane. This last observation leads us to choose the Galileo integrity equation to perform the multisystem integrity algorithm.

The first step in the definition of a new integrity algorithm concerning a combined constellation (Galileo+GPS), is the characterization of the equivalent elements belonging to the two navigation systems. In order to perform the test on the position solution, we opted for the relationship between <sup>2</sup> σ *SISA* of Galileo and <sup>2</sup> σ *UDRE* of EGNOS. First of all, these are quantities defined in the same domain SIS. Secondly, they are related to the same typology of error (clock and ephemeris).

The local contribution to the variance of the error in the SIS depends on the elevation angles of the satellite belonging to the two constellations considered. As mentioned before, in order to consider a single constellation composed by both GPS and Galileo SW, we have considered the variance of the error in the SIS as follows:

$$
\sigma\_i^2 = \sigma\_{\text{SISA}/\text{UIDRE},i}^2 + \sigma\_{u,L,i}^2 \tag{34}
$$

where the first term, in the case of an EGNOS satellite, derives from Eq. 32; the second term instead represents the local error contribution and can be estimated via the following equation:

$$
\sigma\_{u,L,i} = a + b \cdot e^{-10El\_i} \tag{35}
$$

where a and b are parameters that depend directly on Eli (Luongo, 2004). This estimation was performed for both cases, the Galileo and GPS satellites.

The IR equation has been implemented by means of a numerical code in a computer. FF and FM, in Eq. 2, suggest the faulty and faulty free modes. In fact, the Galileo system assumes two separate scenarios: one in which the satellites are all set as use, and the other in which one of the satellites set as use is supposed not to be functioning. When you are in the faulty mode, in the case of Galileo satellites, the SISMA element comes out; in an EGNOS case, only the faulty free mode is instead expected and, because we could not find an equivalent for the Galileo SISMA in its navigation message, we are going to consider the following situation:


#### **5.3.2 Inputs of the implemented algorithm**

The information available a priori for the new algorithm consists of two text files containing position (X,Y and Z components) and velocity (X,Y and Z components) of the SV belonging to the two constellations considered, and obtained through a constellation simulator.

Pseudoranges are obtained by the true satellite-user distance, adding a zero-mean Gaussian noise with variance depending on SISA and the elevation angles of the satellites.

Regarding the SISA and SISMA evaluation, we have considered actual values, adding a Gaussian noise:

$$\begin{aligned} \text{SISA} &= 0.87 + N(0, \sigma\_{\text{SISA}})\\ \text{SISMA} &= 0.7 + N(0, \sigma\_{\text{SISMA}}) \end{aligned} \tag{36}$$

In this case, 0.01 σ *SISA SISMA* = σ = , in order to simulate a sort of degradation on the signal received. We must also describe the behaviour of the positioning algorithm in the combined constellation case. Generally speaking, if we define Xk, Yk and Zk as the coordinates of the Kth satellite and X, Y and Z as the coordinates of the user position, we are able to compute the distance between the satellite and the user ( *<sup>k</sup> d* ) and the pseudoranges ( *<sup>k</sup>* ρ ) as follows (Misra & Enge, 2001):

$$d^k = \sqrt{(X^k - X)^2 + (Y^k - Y)^2 + (Z^k - Z)^2} \tag{37}$$

and

92 Global Navigation Satellite Systems – Signal, Theory and Applications

Autonomous Integrity (RAIM) technique (Weighted RAIM). One of the potential uses of this algorithm consists in the combination of the IR algorithm with the RAIM technique. RAIM is able to detect failures that have not been detected by the IR algorithm. In case of multiple

In this Section we describe the characteristics of this innovative algorithm, pointing out the reason for using the IR equation for the combined constellation Galileo/EGNOS and the reason for taking advantage of a RAIM technique. The EGNOS integrity equation provides a way to measure the integrity based on the incoming signal variances and the satellite geometry. The same is also true for the Galileo IR equation, obviously bearing in mind

This algorithm is supposed to enable the user to take advantage of the data transmitted by the Galileo and EGNOS systems: the user receiver must consider a single and large constellation in order to strengthen the positioning algorithm and improve the accuracy. This idea is simply in need of the definition of a new integrity concept, which would be able

First of all, we have to explain why the protection level concept turns into the integrity risk concept in a Galileo environment. In an EGNOS domain, IR is the probability that the horizontal (vertical) PL exceeds the horizontal (vertical) AL without the user receiving any alarm whatsoever. This definition requires a clear distinction between the horizontal and

On the contrary, as far as a Galileo integrity equation is concerned, the users do not have to evaluate the horizontal and vertical protection levels, but the global IR directly, without making any strict allocations. In fact, different applications need distinct integrity requirements for the horizontal and vertical situations: for example, for a ship the vertical component of the error is not that important for a ship, but it is instead essential for a plane. This last observation leads us to choose the Galileo integrity equation to perform the

The first step in the definition of a new integrity algorithm concerning a combined constellation (Galileo+GPS), is the characterization of the equivalent elements belonging to the two navigation systems. In order to perform the test on the position solution, we opted

quantities defined in the same domain SIS. Secondly, they are related to the same typology

The local contribution to the variance of the error in the SIS depends on the elevation angles of the satellite belonging to the two constellations considered. As mentioned before, in order to consider a single constellation composed by both GPS and Galileo SW, we have

2 2 2

where the first term, in the case of an EGNOS satellite, derives from Eq. 32; the second term instead represents the local error contribution and can be estimated via the following equation:

σσ

σ

 σ

*i SISA UDRE i u L i* = / , ,, + (34)

*UDRE* of EGNOS. First of all, these are

*SISA* of Galileo and <sup>2</sup>

vertical cases. Therefore, it is necessary to split IR into two a priori fixed quantities.

failures, when the WRAIM technique fails, the IR algorithm triggers an alarm.

which data is the Galileo integrity data.

to combine the techniques mentioned above.

**5.3.1 IR equation** 

multisystem integrity algorithm.

for the relationship between <sup>2</sup>

of error (clock and ephemeris).

σ

considered the variance of the error in the SIS as follows:

$$
\rho\_c^{(k)} = d^{(k)} + c\delta t\_u + \tilde{\varepsilon}\_\rho^{(k)} \tag{38}
$$

where:

*k* ρε: residual error on k-th satellite. *b* : clock's offset. Applying a linearization to the (38), we get the expression of the pseudo range model:

$$
\underline{\sf A}\underline{\rho} = \sf G A \underline{X} + \underline{\varepsilon}\_{\rho} \tag{39}
$$

Evolution of Integrity Concept – From Galileo to Multisystem 95

We chose the step function because it is able to characterize a lasting failure on a satellite. In fact, when we are looking at aeronautical applications, any failures lasting more than six seconds (TTA) are relevant. SISA results from the predictions on a satellite clock and ephemeris errors, and these error estimations are based on long term observations: SISA increases mark out long term failures. SISA derives from a large data batch, so the anomalous behaviour of just one sample is not relevant. On the other hand, pseudorange variations point out instantaneous failures. In case of failures, the new algorithm is able to

• long term bias and short term bias due to local errors (multipath, receiver noise) and errors caused by the SV, the SV payload and the navigation message (i.e. ephemeris

In a "no failure" condition we are able to judge the behaviour of the new algorithm compared to the single constellation case, and we can also evaluate the performances

Figure 6 illustrates the RAIM statistic in normal operations (Vertical case), without failure, and the correct functioning of this part of the algorithm. In this case the RAIM algorithm has been simulated independently from the IR algorithm, in order to estimate how it behaves

We tested the IR algorithm in the same way, for the two constellations and in absence of

Figures 6 and 7 show that the RAIM statistic presents some samples that exceed the threshold. In particular, these samples do not exceed the VPL (Vertical Protection Level), therefore they are in the False Alarm zone. This tells us that the RAIM statistic presents a low probability of triggering an alarm, whenever it is not necessary (the main reason for this behaviour of the WRAIM could be seen in the largest sensibility to the outliers of this integrity algorithm). Instead, the IR algorithm has a lower false alarm probability than the previous case, consequently to the fact that the threshold is never exceeded, and the system

We simulated the local error by adding a bias (fixed value) to the pseudoranges. Our intent was to emulate the contribution of some types of errors (i.e. multipath) that are not present in the SIS transmitted (local errors) and consequently are not detectable by the ground

The pseudoranges are calculated by using the true distance between the satellites and the

the noise, in order to simulate the malfunctioning in the biased case, by a certain epoch we added a fixed value to the range measurement. Since the IR algorithm is not able to detect these kinds of errors, we present the results of the WRAIM part of the proposed algorithm

σ

*<sup>i</sup>* (Eq. 34). In addition to

• long term bias due to errors from the clock and ephemeris data (IR equation);

offered by the code in term of probability of false alarm and missed detection.

does not trigger any alarms when the SIS is not affected by any bias.

segment of EGNOS or Galileo, but only by a RAIM technique.

receiver, adding a Gaussian noise that depends on the variance <sup>2</sup>

• Bias on SISA and SISMA.

data, clock) (RAIM algorithm).

protect users from:

**5.3.3.1 No failure mode** 

with many samples in an epoch.

**5.3.3.2 Error on pseudoranges** 

for this first model of failure.

failures (Figure 7).

The matrix G is named Design Matrix, and it consists of the linear coefficients obtained by the partial derivatives of the observation's equations with respect to the estimated coordinates. This matrix characterizes the user-satellite geometry. The number of the columns of G agree with the number of unknowns to be determined (X,Y,Z and b), while the rows equal the number of the available observations (number of satellites in view for both navigation systems). The union of the Galileo and the GPS constellations causes a change in the G matrix. The number of unknowns in fact become five, in order to compute the clock's offset for both systems. In order to estimate the user position's ( *X*iΔ ) we have to apply the weighted least mean square method to the pseudo range model, organizing the weight matrix (W) with the information contained in the navigation message sent by EGNOS or by the Galileo satellites (considering only SWs in view, or those with an elevation angle greater than 10°):

$$
\Delta \tilde{X} = \left(\mathbf{G}^T \cdot \mathcal{W} \cdot \mathbf{G}\right)^{-1} \cdot \mathbf{G}^T \cdot \mathcal{W} \cdot \Delta \rho \tag{40}
$$

where G and W are two matrices of dimension N×5 and N×N respectively, with N representing the number of the satellites used in the positioning algorithm.

#### **5.3.3 Outputs of the implemented algorithm**

In this Section we will describe the characteristics of the implemented multisystem integrity algorithm. We will discuss the results of a few simulation tests organized by different typologies (with or without failure) and different durations, in order to test the validity of the proposed algorithm and confirm the expected results.

A peculiarity of this algorithm is the allocation of the Integrity Risk, valid for the computation of the *PHMI* , and the *PFA* (False Alarm Probability), required to estimate the RAIM statistics. The false alarm probability of RAIM and the Integrity Risk of Galileo are related to the time required for a specific flight operation. For example, in the case of safety of life applications, this time is equal to 150 seconds. Our study refers to these applications.

The proposed algorithm elaborates the position computation, the RAIM statistics and the IR equation in every second. It is therefore useful to refer to the probability mentioned above as to one second. In order to perform this conversion, we use the binomial distribution, obtaining the value of *PHMI* and *PFA* , both initially set1 at <sup>7</sup> 0.5 10<sup>−</sup> <sup>×</sup> , referred to as one epoch (second).

The failures have been reproduced in two different ways:

• Introduction of a step function, at a given test epoch, on the pseudo range of a satellite in view.

<sup>1</sup> Equally split between the two integrity requirements from the initial value of <sup>7</sup> 1 10 / 150*s* <sup>−</sup> × as defined by the ICAO for the avionic integrity requirements.

• Bias on SISA and SISMA.

94 Global Navigation Satellite Systems – Signal, Theory and Applications

ρΔε

The matrix G is named Design Matrix, and it consists of the linear coefficients obtained by the partial derivatives of the observation's equations with respect to the estimated coordinates. This matrix characterizes the user-satellite geometry. The number of the columns of G agree with the number of unknowns to be determined (X,Y,Z and b), while the rows equal the number of the available observations (number of satellites in view for both navigation systems). The union of the Galileo and the GPS constellations causes a change in the G matrix. The number of unknowns in fact become five, in order to compute the clock's

weighted least mean square method to the pseudo range model, organizing the weight matrix (W) with the information contained in the navigation message sent by EGNOS or by the Galileo satellites (considering only SWs in view, or those with an elevation angle greater

( ) <sup>~</sup> <sup>1</sup> *T T*

representing the number of the satellites used in the positioning algorithm.

*X G WG G W*

where G and W are two matrices of dimension N×5 and N×N respectively, with N

In this Section we will describe the characteristics of the implemented multisystem integrity algorithm. We will discuss the results of a few simulation tests organized by different typologies (with or without failure) and different durations, in order to test the validity of

A peculiarity of this algorithm is the allocation of the Integrity Risk, valid for the computation of the *PHMI* , and the *PFA* (False Alarm Probability), required to estimate the RAIM statistics. The false alarm probability of RAIM and the Integrity Risk of Galileo are related to the time required for a specific flight operation. For example, in the case of safety of life applications, this time is equal to 150 seconds. Our study refers to these applications. The proposed algorithm elaborates the position computation, the RAIM statistics and the IR equation in every second. It is therefore useful to refer to the probability mentioned above as to one second. In order to perform this conversion, we use the binomial distribution, obtaining the value of *PHMI* and *PFA* , both initially set1 at <sup>7</sup> 0.5 10<sup>−</sup> <sup>×</sup> , referred to as one

• Introduction of a step function, at a given test epoch, on the pseudo range of a satellite

1 Equally split between the two integrity requirements from the initial value of <sup>7</sup> 1 10 / 150*s* <sup>−</sup> × as defined

ρ

= + (39)

Δ

Δρ

= ⋅⋅ ⋅ ⋅⋅ (40)

<sup>−</sup>

) we have to apply the

Applying a linearization to the (38), we get the expression of the pseudo range model:

Δ*G X*

offset for both systems. In order to estimate the user position's ( *X*i

Δ

**5.3.3 Outputs of the implemented algorithm** 

the proposed algorithm and confirm the expected results.

The failures have been reproduced in two different ways:

by the ICAO for the avionic integrity requirements.

than 10°):

epoch (second).

in view.

We chose the step function because it is able to characterize a lasting failure on a satellite. In fact, when we are looking at aeronautical applications, any failures lasting more than six seconds (TTA) are relevant. SISA results from the predictions on a satellite clock and ephemeris errors, and these error estimations are based on long term observations: SISA increases mark out long term failures. SISA derives from a large data batch, so the anomalous behaviour of just one sample is not relevant. On the other hand, pseudorange variations point out instantaneous failures. In case of failures, the new algorithm is able to protect users from:


#### **5.3.3.1 No failure mode**

In a "no failure" condition we are able to judge the behaviour of the new algorithm compared to the single constellation case, and we can also evaluate the performances offered by the code in term of probability of false alarm and missed detection.

Figure 6 illustrates the RAIM statistic in normal operations (Vertical case), without failure, and the correct functioning of this part of the algorithm. In this case the RAIM algorithm has been simulated independently from the IR algorithm, in order to estimate how it behaves with many samples in an epoch.

We tested the IR algorithm in the same way, for the two constellations and in absence of failures (Figure 7).

Figures 6 and 7 show that the RAIM statistic presents some samples that exceed the threshold. In particular, these samples do not exceed the VPL (Vertical Protection Level), therefore they are in the False Alarm zone. This tells us that the RAIM statistic presents a low probability of triggering an alarm, whenever it is not necessary (the main reason for this behaviour of the WRAIM could be seen in the largest sensibility to the outliers of this integrity algorithm). Instead, the IR algorithm has a lower false alarm probability than the previous case, consequently to the fact that the threshold is never exceeded, and the system does not trigger any alarms when the SIS is not affected by any bias.

#### **5.3.3.2 Error on pseudoranges**

We simulated the local error by adding a bias (fixed value) to the pseudoranges. Our intent was to emulate the contribution of some types of errors (i.e. multipath) that are not present in the SIS transmitted (local errors) and consequently are not detectable by the ground segment of EGNOS or Galileo, but only by a RAIM technique.

The pseudoranges are calculated by using the true distance between the satellites and the receiver, adding a Gaussian noise that depends on the variance <sup>2</sup> σ *<sup>i</sup>* (Eq. 34). In addition to the noise, in order to simulate the malfunctioning in the biased case, by a certain epoch we added a fixed value to the range measurement. Since the IR algorithm is not able to detect these kinds of errors, we present the results of the WRAIM part of the proposed algorithm for this first model of failure.

Evolution of Integrity Concept – From Galileo to Multisystem 97

Figures 9 and 10 show how the RAIM statistic behaves in the presence of a bias of 20 meters inserted from the 30-th epoch in a satellite belonging to the Galileo constellation, and a bias of 10 meters, from the 50-th epoch, in a GPS satellite. In both cases the RAIM statistic (green and blue curves) exceeds the Test Statistic Threshold by a probability of 100%. The Figures show the instantaneous behaviour of the RAIM. This way of representing the RAIM process

Fig. 8. RAIM and bias on pseudorange – satellite Galileo PRN 4 – RAIM algorithm

**Weighted RAIM**

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> <sup>60</sup> <sup>70</sup> <sup>80</sup> <sup>90</sup> <sup>100</sup> <sup>0</sup>

Test Statistic Threshold

**Time [s]**

Fig. 9. RAIM and bias on pseudorange – satellite Galileo PRN 4 – Multisystem Integrity

2

Algorithm

4

6

8

10

**RAIM Statistic**

12

14

16

18

Fig. 6. WRAIM in faulty free condition

Fig. 7. IR algorithm in faulty free condition

96 Global Navigation Satellite Systems – Signal, Theory and Applications

Fig. 6. WRAIM in faulty free condition

Fig. 7. IR algorithm in faulty free condition

Figures 9 and 10 show how the RAIM statistic behaves in the presence of a bias of 20 meters inserted from the 30-th epoch in a satellite belonging to the Galileo constellation, and a bias of 10 meters, from the 50-th epoch, in a GPS satellite. In both cases the RAIM statistic (green and blue curves) exceeds the Test Statistic Threshold by a probability of 100%. The Figures show the instantaneous behaviour of the RAIM. This way of representing the RAIM process

Fig. 8. RAIM and bias on pseudorange – satellite Galileo PRN 4 – RAIM algorithm

Fig. 9. RAIM and bias on pseudorange – satellite Galileo PRN 4 – Multisystem Integrity Algorithm

Evolution of Integrity Concept – From Galileo to Multisystem 99

is not typical; in fact, in the example depicted in Figure 6, different samples of one epoch2 are considered in order to obtain an estimate of the proper functioning of the algorithm. Figure 8 shows this kind of analysis made for the biased case (Galileo satellite, Vertical case). The RAIM technique clearly detects the error in the pseudorange; in fact, the ellipse of point leaves the normal operation region exceeding the TST. The same result can be achieved also

We can obtain different results by adding a bias on all pseudo ranges relative to all satellite in view. Through this kind of simulation we can reach the results shown in Figure 11.

In Figure 11, the RAIM statistic remains around zero value: the receiver assigns the 10 metres bias entirely to the two temporal unknowns, the GPS time clock's offset and the

In conclusion, we figured out that the RAIM statistic is not able to detect the failures on more than one satellite at the same time. This is a limit for this algorithm, which leads us to conclude that the RAIM algorithm does not work properly when used as single integrity

We simulated the error on the signal in space by adding a bias on the standard deviation of the noise considered in the SISA and SISMA computation of two random satellites belonging to one of the two constellations considered. In this failure mode, SISA and SISMA values have been implemented as in Eq. 36, assuming the following value for the respective

> 10 7

<sup>=</sup> (41)

=

Figure 12 shows the behaviour of the implemented algorithm, in particular the IR equation, when the bias on SISA and SISMA is considered in two Galileo satellites. In this case the

Comparing this with a single constellation case (only a Galileo satellite), Figure 13 shows the behaviour of the Integrity Risk algorithm in the Galileo case, considering the same size of bias for the same satellite. What is clear from this comparison is the decrease of alarms (~10%, in the second case) triggered by the system achieved by using the combined constellations. This means that in a dual constellation the combined system provides a safe

As described in the previous Section, since these disturbances are not related to a variation

The trend of the statistics is similar to that in Figure 11, in which the curve never exceeds the

In conclusion, Figure 14 shows the behaviour of the described algorithm when the bias is applied to the SISA value belonging to two GPS satellites; in this case the biased SISMA is

<sup>2</sup> the satellite configuration remains the same during the simulation; however, the noise added to the

), we are not able to detect those errors through the RAIM statistic.

*SISA SISMA*

σ

σ

in the GPS biased case.

Galileo time clock's offset.

standard deviation:

position for the user.

in the pseudoranges (

pseudoranges varies.

**5.3.3.3 Error on the SISA/SISMA value** 

algorithm triggers alarms with a probability of 2%.

Δρ

Threshold; for the sake of brevity we didn't report this picture.

system.

Fig. 10. RAIM and bias on pseudorange – satellite GPS PRN 8

Fig. 11. RAIM and bias on all pseudoranges

98 Global Navigation Satellite Systems – Signal, Theory and Applications

**Weighted RAIM**

<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>2</sup>

Test Statistic Threshold

**Time [s]**

**Weighted RAIM**

Test Statistic Threshold

<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>1</sup>

**Time [s]**

Fig. 10. RAIM and bias on pseudorange – satellite GPS PRN 8

3

2

Fig. 11. RAIM and bias on all pseudoranges

3

4

**RAIM Statistic**

5

6

7

4

5

6

**RAIM Statistic**

7

8

9

10

11

is not typical; in fact, in the example depicted in Figure 6, different samples of one epoch2 are considered in order to obtain an estimate of the proper functioning of the algorithm. Figure 8 shows this kind of analysis made for the biased case (Galileo satellite, Vertical case). The RAIM technique clearly detects the error in the pseudorange; in fact, the ellipse of point leaves the normal operation region exceeding the TST. The same result can be achieved also in the GPS biased case.

We can obtain different results by adding a bias on all pseudo ranges relative to all satellite in view. Through this kind of simulation we can reach the results shown in Figure 11.

In Figure 11, the RAIM statistic remains around zero value: the receiver assigns the 10 metres bias entirely to the two temporal unknowns, the GPS time clock's offset and the Galileo time clock's offset.

In conclusion, we figured out that the RAIM statistic is not able to detect the failures on more than one satellite at the same time. This is a limit for this algorithm, which leads us to conclude that the RAIM algorithm does not work properly when used as single integrity system.

#### **5.3.3.3 Error on the SISA/SISMA value**

We simulated the error on the signal in space by adding a bias on the standard deviation of the noise considered in the SISA and SISMA computation of two random satellites belonging to one of the two constellations considered. In this failure mode, SISA and SISMA values have been implemented as in Eq. 36, assuming the following value for the respective standard deviation:

$$
\begin{aligned}
\sigma\_{SISA} &= 10 \\
\sigma\_{SISMA} &= 7
\end{aligned}
\tag{41}
$$

Figure 12 shows the behaviour of the implemented algorithm, in particular the IR equation, when the bias on SISA and SISMA is considered in two Galileo satellites. In this case the algorithm triggers alarms with a probability of 2%.

Comparing this with a single constellation case (only a Galileo satellite), Figure 13 shows the behaviour of the Integrity Risk algorithm in the Galileo case, considering the same size of bias for the same satellite. What is clear from this comparison is the decrease of alarms (~10%, in the second case) triggered by the system achieved by using the combined constellations. This means that in a dual constellation the combined system provides a safe position for the user.

As described in the previous Section, since these disturbances are not related to a variation in the pseudoranges ( Δρ ), we are not able to detect those errors through the RAIM statistic. The trend of the statistics is similar to that in Figure 11, in which the curve never exceeds the Threshold; for the sake of brevity we didn't report this picture.

In conclusion, Figure 14 shows the behaviour of the described algorithm when the bias is applied to the SISA value belonging to two GPS satellites; in this case the biased SISMA is

 <sup>2</sup> the satellite configuration remains the same during the simulation; however, the noise added to the pseudoranges varies.

Evolution of Integrity Concept – From Galileo to Multisystem 101

not present because this quantity is not broadcasted by the GPS satellite and, as we stated previously, the integrity data delivered by these satellites does not alter the Faulty section of the *PHMI* equation. For this reason, in this context, the above-mentioned probability does

<sup>0</sup> <sup>200</sup> <sup>400</sup> <sup>600</sup> <sup>800</sup> <sup>1000</sup> <sup>1200</sup> <sup>1400</sup> <sup>1600</sup> <sup>1800</sup> <sup>2000</sup> -2

IR Threshold

Alarms=0

**Time [s]**

An interesting development of the described study is the analysis of the new RAIM technique, the Advanced RAIM. ARAIM, proposed in 2010 by the GEAS (GNSS Evolutionary Architecture Study), could be considered as an evolution of the classical RAIM. This new solution takes advantage of the availability of different new navigation systems (i.e. Galileo)

ARAIM is an extension of the single-frequency RAIM. Both are based on an airborne comparison of each satellite measurement to the consensus of the other available satellite measurements (GEAS, 2010). However, the differences between the two techniques are also important. ARAIM should be pursued for the worldwide vertical guidance of civil aircraft based on two or more GNSS constellations radiating at two ARNS/RNSS frequencies (L1 and L5). The main characteristic of the Advanced RAIM would support vertical guidance to decision heights of 200 feet (LPV-200), whereas single-frequency RAIM only supports LNAV guidance. As such, ARAIM must protect vertical errors at levels of 35 meters, while RAIM only needs to detect lateral errors of 200 meters or so. In addition, LPV-200 corresponds to a severe major hazard level (10-7), and LNAV is only major (10-5).

Fig. 14. IR algorithm combined constellation and bias on SISA, satellites GPS PRN3 and

not reach high values. Indeed, the statistics never exceed the Threshold.

x 10-10 **Integrity Risk**

0

PRN10

**6. RAIM evolution: ARAIM** 

in order to improve receiver performances.

2

4

6

8

10

**HMI Probability**

12

14

16

18

Fig. 12. IR algorithm combined constellation and bias on SISA and SISMA, satellites Galileo PRN15 and PRN22

Fig. 13. Galileo IR algorithm and bias on SISA and SISMA, satellites Galileo PRN15 and PRN22

100 Global Navigation Satellite Systems – Signal, Theory and Applications

Alarms=38

<sup>10</sup> x 10-9 **Integrity Risk**

0 200 400 600 800 1000 1200 1400 1600 1800 2000

**Time [s]**

Fig. 12. IR algorithm combined constellation and bias on SISA and SISMA, satellites Galileo

**Integrity Risk**

<sup>0</sup> <sup>200</sup> <sup>400</sup> <sup>600</sup> <sup>800</sup> <sup>1000</sup> <sup>1200</sup> <sup>1400</sup> <sup>1600</sup> <sup>1800</sup> <sup>2000</sup> <sup>0</sup>

IR Threshold

Alarms=230

**Time [s]**

Fig. 13. Galileo IR algorithm and bias on SISA and SISMA, satellites Galileo PRN15 and

IR Threshold

0

5

PRN22

10

**HMI Probability**

PRN15 and PRN22

x 10-9

1

2

3

4

5

**HMI Probability**

6

7

8

9

not present because this quantity is not broadcasted by the GPS satellite and, as we stated previously, the integrity data delivered by these satellites does not alter the Faulty section of the *PHMI* equation. For this reason, in this context, the above-mentioned probability does not reach high values. Indeed, the statistics never exceed the Threshold.

Fig. 14. IR algorithm combined constellation and bias on SISA, satellites GPS PRN3 and PRN10

## **6. RAIM evolution: ARAIM**

An interesting development of the described study is the analysis of the new RAIM technique, the Advanced RAIM. ARAIM, proposed in 2010 by the GEAS (GNSS Evolutionary Architecture Study), could be considered as an evolution of the classical RAIM. This new solution takes advantage of the availability of different new navigation systems (i.e. Galileo) in order to improve receiver performances.

ARAIM is an extension of the single-frequency RAIM. Both are based on an airborne comparison of each satellite measurement to the consensus of the other available satellite measurements (GEAS, 2010). However, the differences between the two techniques are also important. ARAIM should be pursued for the worldwide vertical guidance of civil aircraft based on two or more GNSS constellations radiating at two ARNS/RNSS frequencies (L1 and L5). The main characteristic of the Advanced RAIM would support vertical guidance to decision heights of 200 feet (LPV-200), whereas single-frequency RAIM only supports LNAV guidance. As such, ARAIM must protect vertical errors at levels of 35 meters, while RAIM only needs to detect lateral errors of 200 meters or so. In addition, LPV-200 corresponds to a severe major hazard level (10-7), and LNAV is only major (10-5).

Evolution of Integrity Concept – From Galileo to Multisystem 103

used for the error distributions and the parameters to be considered in the integrity equation. Indeed, although more complete, the new integrity concept introduced by Galileo

A possible development of the proposed algorithm could be the definition of SISMA analogous for the GPS satellites in order to contribute to the IR equation under faulty

The present study is only a preliminary analysis. In order to better evaluate the performances of the proposed algorithm, we need to use realistic data (i.e. pseudoranges measurements obtained through a real GNSS receiver) as inputs of the implemented code.

Some of the concepts illustrated in this Chapter have been developed by the authors and

• GIRASOLE (Galileo Safety of Life Receivers Development), March 2005 – September 2006, financed by the GSA (European GNSS Agency) under the contract

• SWAN (Sistemi software per Applicazioni di Navigazione), December 2007 – June 2009, financed by the Italian Space Agency (ASI) under the contract DC-IPC-2006-160; • PEGASUS (Platform of Enhanced Gnss receiver for Application in Sol User Segment), December 2010 – December 2011, financed by the Italian Space Agency (ASI) under the

G. Dore, M. Calamia, "Evolution of Integrity Concept : from Galileo to Multisystem", ENC-

V. Oehler, F. Luongo, J. P. Boyero, R. Stalford, H. L. Trautenberg, J. Hahn, F. Amarillo, M.

T. Walter and P. Enge, "Weighted RAIM for Precision Approach", Stanford University,

C. Pecchioni, "L'integrity nei sistemi combinati di navigazione satellitare: confronti,

P. Misra and P. Enge, "Global Positioning System, Signal, Measurements and Performance",

M. Ciollaro, "GNSS Multisystem Integrity for Precision Approaches in Civil Aviation",

F. Luongo, V. Oehler and R. Stalford, "HPCA Input/Output Test Data", Galileo Industries,

algoritmi e verifiche", Università degli Studi di Firenze, Sept. 2006.

"Galileo Integrity User Equations – Working Paper", GAL-TNO-GLI-SYST-I/0630.

Università degli Studi di Napoli "Federico II", Feb. 2009.

Crisci, B. Schalarmann, J. F. Flamand, "The Galileo Integrity Concept", ION GNSS 17th International Technical Meeting of the Satellite Division, 21-24 Sept. 2004. C. Pecchioni, M. Ciollaro, M. Calamia, "Combined Galileo and EGNOS Integrity Signal: a

multisystem integrity algorithm", 2nd Workshop on GNSS Signals & Signal

their collaborators at the University of Florence, also within the following projects:

is more complex and less intuitive than SBAS and RAIM protection level concept.

conditions.

**8. Acknowledgements** 

GJU/05/2415/CTR/GALILEOSOL;

GNSS 2009, Naples (Italy), May 2009.

contract. ASI I/024/10/0.

Processing, Apr. 2007.

Ganga-Jamuna Press, 2001.

**9. References** 

1995.

Apr. 2004.

The proper functioning of this system does require some assistance from the ground; for example, the ISM (Integrity Support Message), which is a message developed using reference receivers on the ground, is communicated to the aircraft. The ISM message conveys the safety assertions associated with each of the underlying satellite systems to the sovereign responsible for a given airspace. These messages would contain performance estimates for each satellite to be used for navigation. ARAIM therefore uses a multiplicity of satellites in a dual-constellation environment to take responsibility for all faults that arise between dispatch and the completion of approach.

As described in the previous Section, one of the potential uses of the Multisystem Integrity algorithm is represented by the combination of the IR algorithm with the RAIM technique. ARAIM is still in a feasibility status, and a comparison—in order to test and verify the requirements and highlight the differences between the two approaches—between its results and Multi System Integrity cannot yet be performed. Indeed, the two systems have the same aim: to improve the reliability of the position solution provided by the system in particular conditions (LPV-200 for ARAIM), taking advantage of different navigation systems.

## **7. Conclusions**

The totality of the tests made on the implemented code has been planned with the aim of characterizing the performances of the algorithm respectively in faulty free and in faulty mode. The use of the Galileo and EGNOS system as a single and augmented constellation allows us to develop the positioning algorithm and improve the position accuracy. Furthermore, the combination of the two SVs systems enables us to obtain some benefits from the RAIM point of view.

Our proposed solution starts from the integrity equation defined for the Galileo system and adapts it to the combined Galileo + EGNOS system, or rather, it combines the integrity data supplied separately by the two navigation systems, with the aim of computing the Hazardous Misleading Information Probability. We focused our attention on the IR equation: the implemented code reproduces the IR equation as it is presented in literature, that is, with the SISA values relative to Galileo and GPS satellites, and SISMA relative only to the Galileo ones, in faulty free and faulty mode, respectively. The results obtained testing the algorithm in the presence of failure have provided positive indications on the implemented IR equation: in these cases, the HMI probability increases with the value of the bias.

Alhough the IR protects the user against extended failure, whose effects revert on the SISE estimation, the RAIM technique could instead highlight instantaneous errors on the distances measured by a Galileo or a GPS satellite. RAIM and IR compensate each other, or rather, the RAIM indicates failure unperceived by the IR and vice versa; therefore the combination of the RAIM technique with the integrity equation has proved to be a good idea. This technique is based on a very different concept than protection levels and leads to different results. However, the Galileo integrity concept is more complete than the GPS/SBAS and RAIM integrity concepts and offers more protection from failures. However, this concept needs to be investigated further, in particular regarding the assumptions to be used for the error distributions and the parameters to be considered in the integrity equation. Indeed, although more complete, the new integrity concept introduced by Galileo is more complex and less intuitive than SBAS and RAIM protection level concept.

A possible development of the proposed algorithm could be the definition of SISMA analogous for the GPS satellites in order to contribute to the IR equation under faulty conditions.

The present study is only a preliminary analysis. In order to better evaluate the performances of the proposed algorithm, we need to use realistic data (i.e. pseudoranges measurements obtained through a real GNSS receiver) as inputs of the implemented code.

## **8. Acknowledgements**

102 Global Navigation Satellite Systems – Signal, Theory and Applications

The proper functioning of this system does require some assistance from the ground; for example, the ISM (Integrity Support Message), which is a message developed using reference receivers on the ground, is communicated to the aircraft. The ISM message conveys the safety assertions associated with each of the underlying satellite systems to the sovereign responsible for a given airspace. These messages would contain performance estimates for each satellite to be used for navigation. ARAIM therefore uses a multiplicity of satellites in a dual-constellation environment to take responsibility for all faults that arise

As described in the previous Section, one of the potential uses of the Multisystem Integrity algorithm is represented by the combination of the IR algorithm with the RAIM technique. ARAIM is still in a feasibility status, and a comparison—in order to test and verify the requirements and highlight the differences between the two approaches—between its results and Multi System Integrity cannot yet be performed. Indeed, the two systems have the same aim: to improve the reliability of the position solution provided by the system in particular conditions (LPV-200 for ARAIM), taking advantage of different navigation

The totality of the tests made on the implemented code has been planned with the aim of characterizing the performances of the algorithm respectively in faulty free and in faulty mode. The use of the Galileo and EGNOS system as a single and augmented constellation allows us to develop the positioning algorithm and improve the position accuracy. Furthermore, the combination of the two SVs systems enables us to obtain some benefits

Our proposed solution starts from the integrity equation defined for the Galileo system and adapts it to the combined Galileo + EGNOS system, or rather, it combines the integrity data supplied separately by the two navigation systems, with the aim of computing the Hazardous Misleading Information Probability. We focused our attention on the IR equation: the implemented code reproduces the IR equation as it is presented in literature, that is, with the SISA values relative to Galileo and GPS satellites, and SISMA relative only to the Galileo ones, in faulty free and faulty mode, respectively. The results obtained testing the algorithm in the presence of failure have provided positive indications on the implemented IR equation: in these cases, the HMI probability increases with the value of the

Alhough the IR protects the user against extended failure, whose effects revert on the SISE estimation, the RAIM technique could instead highlight instantaneous errors on the distances measured by a Galileo or a GPS satellite. RAIM and IR compensate each other, or rather, the RAIM indicates failure unperceived by the IR and vice versa; therefore the combination of the RAIM technique with the integrity equation has proved to be a good idea. This technique is based on a very different concept than protection levels and leads to different results. However, the Galileo integrity concept is more complete than the GPS/SBAS and RAIM integrity concepts and offers more protection from failures. However, this concept needs to be investigated further, in particular regarding the assumptions to be

between dispatch and the completion of approach.

systems.

bias.

**7. Conclusions** 

from the RAIM point of view.

Some of the concepts illustrated in this Chapter have been developed by the authors and their collaborators at the University of Florence, also within the following projects:


## **9. References**


**Part 2** 

**GNSS Navigation and Applications** 

