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## **Meet the editor**

Professor Wuqiang Yang (FIET, FInstMC, FIEEE, CEng) received all his degrees (BEng, MSc, PhD) from Tsinghua University. After 3 years as a Lecturer at Tsinghua University, he joined UMIST in 1991 and currently is a Professor at University of Manchester. His research is focused on electrical capacitance tomography. He has published 300 papers, and holds 10 patents. He is an edi-

torial board member of 5 journals, a guest editor of several journal special issues, and reviews papers for 40 professional journals, including 6 IEEE Transactions. He has been invited by many universities, research institutions and international conferences worldwide to give lectures, seminars, or keynotes. He is a visiting professor at 6 universities. Since 2002, his biography has been in Who's Who in the World. He is currently an IEEE Instrumentation and Measurement Society Distinguished Lecturer.

Contents

**Preface VII**

Junli Liang and Ding Liu

**with Sensor Arrays 21** 

Chapter 3 **Predictive Angle Tracking Algorithm** 

Chapter 1 **Passive Source Localization of Sensor Arrays 1** 

Juan J. Villacorta and Mariano Raboso

**Based on Extended Kalman Filter 37** 

Chapter 4 **Shape Recognition and Position Measurement** 

Chapter 6 **Gas Sensor Array with Broad Applicability 81**  Andrzej Szczurek and Monika Maciejewska

Kozo Ohtani and Mitsuru Baba

Chapter 5 **Detection of Alkylating Agents** 

Chapter 7 **Neuro-Fuzzy Classifiers/Quantifiers for E-Nose Applications 109** 

Ravi Kumar

Chapter 2 **Experimental Calibration for Electronic Beamforming** 

Lara del Val, María I. Jiménez, Alberto Izquierdo,

Sheng-Yun Hou, Shun-Hsyung Chang and Hsien-Sen Hung

**of an Object Using an Ultrasonic Sensor Array 53** 

**Using Optical, Electrical and Mechanical Means 67**  Yoav Eichen, Yulia Gerchikov, Elena Borzin, Yair Gannot, Ariel Shemesh, Shai Meltzman, Carmit Hertzog-Ronen, Shay Tal, Sara Stolyarova, Yael Nemirovsky and Nir Tessler

## Contents

#### **Preface XI**


Preface

medicine.

efficient management.

It is my great pleasure to be an editor for InTech Open Access publisher, one of the most successful Open Access book publishers in the fields of science, technology and

Sensor arrays are used to overcome the limits of simple and/or individual conventional sensors. Obviously, it is more complicated to deal with some issues related to sensor arrays, e.g. signal processing, than the conventional sensors. Some of the issues are addressed in this book, with emphasis on signal processing, calibration and some advanced applications, e.g. how to place sensors as an array for accurate measurement, how to calibrate a sensor array by experiment, how to use a sensor array to track non-stationary targets efficiently and effectively, how to use an ultrasonic sensor array for shape recognition and position measurement, how to use sensor arrays to detect chemical agents, and applications of gas sensor arrays, including e-nose. This book should be useful for those who would like to learn about recent developments in sensor arrays, in particular for engineers, academics and

I would like to take this opportunity to thank all the authors, who made valuable contributions with the chapters in this book. Without their hard work, it would be impossible to publish this book. I would also like to thank Ms Jana Sertic for her

**Professor Wuqiang Yang**

United Kingdom

Professor of Electronic Instrumentation,

University of Manchester, Manchester,

School of Electrical and Electronic Engineering,

postgraduate students studying instrumentation and measurement.

## Preface

It is my great pleasure to be an editor for InTech Open Access publisher, one of the most successful Open Access book publishers in the fields of science, technology and medicine.

Sensor arrays are used to overcome the limits of simple and/or individual conventional sensors. Obviously, it is more complicated to deal with some issues related to sensor arrays, e.g. signal processing, than the conventional sensors. Some of the issues are addressed in this book, with emphasis on signal processing, calibration and some advanced applications, e.g. how to place sensors as an array for accurate measurement, how to calibrate a sensor array by experiment, how to use a sensor array to track non-stationary targets efficiently and effectively, how to use an ultrasonic sensor array for shape recognition and position measurement, how to use sensor arrays to detect chemical agents, and applications of gas sensor arrays, including e-nose. This book should be useful for those who would like to learn about recent developments in sensor arrays, in particular for engineers, academics and postgraduate students studying instrumentation and measurement.

I would like to take this opportunity to thank all the authors, who made valuable contributions with the chapters in this book. Without their hard work, it would be impossible to publish this book. I would also like to thank Ms Jana Sertic for her efficient management.

> **Professor Wuqiang Yang**  Professor of Electronic Instrumentation, School of Electrical and Electronic Engineering, University of Manchester, Manchester, United Kingdom

**1** 

*China* 

Junli Liang and Ding Liu

*Xi'an University of Technology, Xi'an,* 

**Passive Source Localization of Sensor Arrays**

Passive source localization is a key issue in sensor array signal processing such as sonar, radar, wireless communication, microphone array speech processing, seismology, electronic surveillance and medical imaging, and thus receives significant attention. Although a variety of advanced algorithms, for example MUltiple SIgnal Classification (MUSIC), Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT), and Propagator Method (PM), have been developed, there are still some problems: (i)For twodimensional (2D) directions-of-arrival (DOA) estimation, the failure in pairing causes severe performance degradation; (ii) In some practical applications, the signals received by a sensor array may come from multiple near-field sources or multiple far-field sources or their mixture. Due to different signal models for near-field and far-field sources, the existing algorithms cannot deal with them simultaneously well; and (iii) For joint azimuth and elevation direction finding, the existing estimators often encounter an estimation failure problem especially when elevation angles are between 70 and 90 degrees. In this chapter,

several high-resolution methods are presented to overcome these difficulties.

method can be used to form the rank-reduction propagator method. electric angle

In Section 1.3, a common signal model for "any-field" sources (i.e., near-field sources or farfield sources or their mixture) is given and a two-stage MUSIC algorithm is developed to localize "any-field" sources. In the first stage, one special cumulant matrix is derived and the related virtual "steering vector" is the function of the common electric angle in both

In Section 1.2, a novel 2D DOA estimation algorithm without match procedure in the Lshaped array geometry is proposed. It is well known that two matched electric angles (functions of elevation and azimuth angles) must be obtained before elevation and azimuth angles are estimated. However, the failure in pairing would cause severe performance degradation. By introducing a novel electric angle, the L-shaped array configuration without any rotational invariance property between two orthogonal uniform linear sub-arrays evolves into some particular rotational invariance geometry. Thus, the steering vector is separated into two parts. One can be estimated by the rank-reduction ESPRIT algorithm and the other is obtained from the eigenvalue decomposition of one particular matrix. Finally, the elevation and azimuth angles can be easily obtained from the recovered steering vector to avoid pairing. Although it is developed for the L-shaped array configuration, the proposed algorithm can be easily extended to other array geometries such as two parallel linear sub-arrays, the rectangular array, and the symmetric circular array. In addition, the

**1. Introduction** 

## **Passive Source Localization of Sensor Arrays**

### Junli Liang and Ding Liu

*Xi'an University of Technology, Xi'an, China* 

#### **1. Introduction**

Passive source localization is a key issue in sensor array signal processing such as sonar, radar, wireless communication, microphone array speech processing, seismology, electronic surveillance and medical imaging, and thus receives significant attention. Although a variety of advanced algorithms, for example MUltiple SIgnal Classification (MUSIC), Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT), and Propagator Method (PM), have been developed, there are still some problems: (i)For twodimensional (2D) directions-of-arrival (DOA) estimation, the failure in pairing causes severe performance degradation; (ii) In some practical applications, the signals received by a sensor array may come from multiple near-field sources or multiple far-field sources or their mixture. Due to different signal models for near-field and far-field sources, the existing algorithms cannot deal with them simultaneously well; and (iii) For joint azimuth and elevation direction finding, the existing estimators often encounter an estimation failure problem especially when elevation angles are between 70 and 90 degrees. In this chapter, several high-resolution methods are presented to overcome these difficulties.

In Section 1.2, a novel 2D DOA estimation algorithm without match procedure in the Lshaped array geometry is proposed. It is well known that two matched electric angles (functions of elevation and azimuth angles) must be obtained before elevation and azimuth angles are estimated. However, the failure in pairing would cause severe performance degradation. By introducing a novel electric angle, the L-shaped array configuration without any rotational invariance property between two orthogonal uniform linear sub-arrays evolves into some particular rotational invariance geometry. Thus, the steering vector is separated into two parts. One can be estimated by the rank-reduction ESPRIT algorithm and the other is obtained from the eigenvalue decomposition of one particular matrix. Finally, the elevation and azimuth angles can be easily obtained from the recovered steering vector to avoid pairing. Although it is developed for the L-shaped array configuration, the proposed algorithm can be easily extended to other array geometries such as two parallel linear sub-arrays, the rectangular array, and the symmetric circular array. In addition, the method can be used to form the rank-reduction propagator method. electric angle

In Section 1.3, a common signal model for "any-field" sources (i.e., near-field sources or farfield sources or their mixture) is given and a two-stage MUSIC algorithm is developed to localize "any-field" sources. In the first stage, one special cumulant matrix is derived and the related virtual "steering vector" is the function of the common electric angle in both

Passive Source Localization of Sensor Arrays 3

**Aa a a** = [ ( , ) (,) ( , )

*l L* **<sup>s</sup>** *k s k sk s k* <sup>=</sup> , ( 1) ( 1) ( , ) 1 *l l l l ll*

 *l ll* = −2 sin cos / π

> *l l* = −2 cos / π

,0 1,0 1,0 0,0 0, 0, 1 0,1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) *<sup>T</sup>* **n** *k n kn k n kn k n k n k n k* = *M M*− − *M M* .

*z*

• •

• •

β*l*

The auto-correlation matrix of **r**( ) *<sup>k</sup>* can be expressed as [ ( ) ( )] *<sup>H</sup>* **R rr** <sup>=</sup> *Ek k* , 2

where **V** is the diagonal matrix with the eigen-values arranged as

( 1) ( 1) ( 1) ( , ) 1 *l l l l l l l ll*

 θ

γ

*l l e e e e e e e ee*

<sup>1</sup> *LL M* 1 21 *v vv v* ≥ ≥ <sup>&</sup>gt; + + ≥ ≥ , the diagonal matrix (2 1 ) (2 1 ) *<sup>M</sup> LML*

 γ

1 21 1 21 1 21 [ , , ]diag[ , , ][ , , ] *HH H <sup>H</sup>* **R UVU U V U U V U u u** == + = *ss s nn n <sup>M</sup>*+ ++ *v v <sup>M</sup>* **u u** *<sup>M</sup>* , (2)

<sup>−</sup> = , and thus the steering vector be written in another form as:

 γ

<sup>−</sup> − − <sup>=</sup> ×× × **<sup>a</sup>** (3)

 γ

*<sup>T</sup> jM j M j jM jM j M j M j j*

2 1 *<sup>T</sup> j jM j M <sup>j</sup> e ee e*

<sup>+</sup> , where [ ( ) ( )] *<sup>H</sup>* **R ss** *<sup>s</sup>* = *Ek k* , and its eigen-value decomposition (EVD)

α*l*

o

( ) 0,0,*d*

 α*d*

 γφ

γ

 αλ*d*

 γ φ

> γ

λ

*<sup>T</sup> jM j M j jM j M j*

*y*

*<sup>n</sup> <sup>R</sup>* <sup>+</sup> − × <sup>+</sup> <sup>−</sup> **<sup>V</sup>** <sup>∈</sup> is composed of

 θγ

> θ

= ,

**a aa**

γ

 γ

> γ φ

*<sup>n</sup> C* <sup>+</sup> <sup>×</sup> <sup>+</sup> <sup>−</sup> **U** ∈ consists of the eigenvectors related

 θ

can be separated into two parts, i.e. (,) 1 2 ( )( ) *j j e e*

 γγ<sup>−</sup> <sup>=</sup> **<sup>a</sup>** and (2 1) ( 1) *M M* <sup>+</sup> <sup>×</sup> <sup>+</sup>

 φ  φφ

*l l ee e ee e*

− − <sup>=</sup> **<sup>a</sup>** 

( ) *<sup>l</sup> <sup>s</sup> <sup>k</sup>*

 γ

 *ll LL* ]

β

γ 1 1 φ

γ φ

φ

γ

•

( ) ( 1) ,0,0 *M d* −

*x*

Fig. 1. L-shaped sensor array configuration

φ γ

γ

γ φ

2 1

*<sup>H</sup>* = + **AR A I** *s nM* σ

Let's define ( ) *l ll j j e e* θ

> γ φ

Furthermore, **a**(,)


yields

( ) *Md*,0,0

eigen-values 1 2 21 , ,, *vv v LL M* ++ + ; (2 1) (2 1 ) *<sup>M</sup> M L*

to 1 2 21 , ,, *vv v LL M* ++ + , spanning the noise subspace of **R** .

 γ

where ( 1) *<sup>M</sup>* <sup>+</sup> -dimensional vector ( ) ( 1)

•

•

( ) *d*,0,0

( ) 0,0,( 1) *M* − *d*

( ) 0,0,*Md*

[ ] <sup>1</sup> ( ) ( ), , ( ), , ( ) *<sup>T</sup>*

near-field and far-field signal models so that DOA of near-field or far-field can be obtained from this electric angle using the conventional high-resolution MUSIC algorithm. In the second stage, another particular cumulant matrix is derived, in which the virtual "steering matrix" has full column rank no matter whether the received signals are multiple near-field sources or multiple far-field ones or their mixture. More importantly, the virtual "steering vector" can be separated into two parts, in which the first one is the function of the common electric angle in both signal models, whereas the second part is the function of the electric angle that exists only in the near-field signal model. Furthermore, by substituting the common electric angle, which is estimated in the first stage into one special Hermitian matrix formed from another MUSIC spectral function, the range of near-field sources can be obtained from the eigenvector of the Hermitian matrix. Although it is developed for azimuth angle (and range) estimation only, it can be developed further for the joint azimuth and elevation angles (as well as range) estimation.

In Section 1.4, a novel high-accuracy estimator for elevation angle is developed to avoid the estimation failure problem encountered in the conventional elevation estimators. Firstly, three cumulant matrices are constructed using fourth-order cumulants of some properly chosen array outputs of a specially designed volume array to increase the array aperture. Secondly, a parallel factor (PARAFAC) model of cumulant matrices in the cumulant domain is formed to avoid pairing parameters. Finally, a flexible and high-resolution elevation angle estimator is derived from multiple electric angles, which are solved from the above steps.

#### **2. 2D DOA estimation without match procedure**

Estimation of 2-D DOA is a key issue in sensor array signal processing such as radar, sonar, radio astronomy, and mobile communication systems [1-4]. Similar to other array geometries such as the parallel uniform linear array, the rectangular array and the circular array, there is an un-avoidable parameter association problem in the L-shaped array configuration because the failure in pairing would cause severe performance degradation. This section will give a novel 2-D DOA estimation algorithm, which does not require match procedure.

#### **2.1 Description of the proposed algorithm**

Let's consider an L-shaped sensor array with 2 1 *M* + omni-directional sensors, as shown in Fig. 1. The element placed at the origin is set for the referencing point. The array in the *x z* − plane consists of two uniform linear sub-arrays with element spacing *d* , each being composed of *M* elements. Assume that *L* far-field, no-coherent, narrowband sources impinging on this antenna array. Let α*l* and β*<sup>l</sup>* be the elevation and azimuth angles of the *l* -th source, and thus the wave vector *<sup>l</sup>* containing DOA information can be defined as *l ll ll l* = [sin cos ,sin sin ,cos α β α β α ] , *l L* = 1, , . After being sampled, the signals received by the sensor array can be expressed as

$$\mathbf{r}(k) = \mathbf{A}\mathbf{s}(k) + \mathbf{n}(k) \; , \; k = 0, \dots, K - 1 \; , \tag{1}$$

where ,0 1,0 1,0 0,0 0, 0, 1 0,1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) *<sup>T</sup> k r kr k r kr k r kr k r k* <sup>=</sup> *M M*− − *M M* **<sup>r</sup>** 

near-field and far-field signal models so that DOA of near-field or far-field can be obtained from this electric angle using the conventional high-resolution MUSIC algorithm. In the second stage, another particular cumulant matrix is derived, in which the virtual "steering matrix" has full column rank no matter whether the received signals are multiple near-field sources or multiple far-field ones or their mixture. More importantly, the virtual "steering vector" can be separated into two parts, in which the first one is the function of the common electric angle in both signal models, whereas the second part is the function of the electric angle that exists only in the near-field signal model. Furthermore, by substituting the common electric angle, which is estimated in the first stage into one special Hermitian matrix formed from another MUSIC spectral function, the range of near-field sources can be obtained from the eigenvector of the Hermitian matrix. Although it is developed for azimuth angle (and range) estimation only, it can be developed further for the joint azimuth

In Section 1.4, a novel high-accuracy estimator for elevation angle is developed to avoid the estimation failure problem encountered in the conventional elevation estimators. Firstly, three cumulant matrices are constructed using fourth-order cumulants of some properly chosen array outputs of a specially designed volume array to increase the array aperture. Secondly, a parallel factor (PARAFAC) model of cumulant matrices in the cumulant domain is formed to avoid pairing parameters. Finally, a flexible and high-resolution elevation angle estimator is derived from multiple electric angles, which are solved from the above steps.

Estimation of 2-D DOA is a key issue in sensor array signal processing such as radar, sonar, radio astronomy, and mobile communication systems [1-4]. Similar to other array geometries such as the parallel uniform linear array, the rectangular array and the circular array, there is an un-avoidable parameter association problem in the L-shaped array configuration because the failure in pairing would cause severe performance degradation. This section will give a novel 2-D DOA estimation algorithm, which does not require match

Let's consider an L-shaped sensor array with 2 1 *M* + omni-directional sensors, as shown in Fig. 1. The element placed at the origin is set for the referencing point. The array in the *x z* − plane consists of two uniform linear sub-arrays with element spacing *d* , each being composed of *M* elements. Assume that *L* far-field, no-coherent, narrowband sources

*l* -th source, and thus the wave vector *<sup>l</sup>* containing DOA information can be defined as

where ,0 1,0 1,0 0,0 0, 0, 1 0,1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) *<sup>T</sup> k r kr k r kr k r kr k r k* <sup>=</sup> *M M*− − *M M* **<sup>r</sup>** 

*<sup>l</sup>* be the elevation and azimuth angles of the

] , *l L* = 1, , . After being sampled, the signals received

**r As n** () () () *k kk* = + , 0, , 1 *k K* = − , (1)

α*l* and β

and elevation angles (as well as range) estimation.

**2. 2D DOA estimation without match procedure** 

**2.1 Description of the proposed algorithm** 

impinging on this antenna array. Let

α

by the sensor array can be expressed as

β

α

*l ll ll l* = [sin cos ,sin sin ,cos

β

α

procedure.

$$\mathbf{A} = \left[\mathbf{a}(\boldsymbol{\gamma}\_{1}, \boldsymbol{\phi}\_{1}) \cdots \mathbf{a}(\boldsymbol{\gamma}\_{l}, \boldsymbol{\phi}\_{l}) \cdots \mathbf{a}(\boldsymbol{\gamma}\_{1}, \boldsymbol{\phi}\_{l})\right]$$

$$\mathbf{s}(k) = \left[s\_{1}(k), \cdots, s\_{l}(k), \cdots, s\_{L}(k)\right]^{T}, \ \mathbf{a}(\boldsymbol{\gamma}\_{l}, \boldsymbol{\phi}\_{l}) = \left[e^{jM\boldsymbol{\gamma}\_{l}} \ e^{j(\boldsymbol{\gamma}\_{l} - 1)\boldsymbol{\gamma}\_{l}} \ \cdots e^{j\boldsymbol{\gamma}\_{l}} \ \mathbf{1} \ \ e^{j(\boldsymbol{\Lambda}\_{l} - 1)\boldsymbol{\phi}\_{l}} \ \cdots e^{j\boldsymbol{\phi}\_{l}}\right]^{T}$$

$$\boldsymbol{\gamma}\_{l} = -2\pi d \sin \alpha\_{l} \cos \beta\_{l} / \ \lambda$$

$$\boldsymbol{\phi}\_{l} = -2\pi d \cos \alpha\_{l} / \ \lambda$$

$$\mathbf{n}(k) = \left[n\_{M,0}(k) \ n\_{M-1,0}(k) \ \cdots \ n\_{1,0}(k) \ n\_{0,0}(k) \ n\_{0,M}(k) \ n\_{0,M-1}(k) \ \cdots \ n\_{0,1}(k)\right]^{T}.$$

$$\sum\_{(0,0,Md)} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \oint\_{1}^{\boldsymbol{\theta}} s\_{i}(k)$$

Fig. 1. L-shaped sensor array configuration

The auto-correlation matrix of **r**( ) *<sup>k</sup>* can be expressed as [ ( ) ( )] *<sup>H</sup>* **R rr** <sup>=</sup> *Ek k* , 2 2 1 *<sup>H</sup>* = + **AR A I** *s nM* σ <sup>+</sup> , where [ ( ) ( )] *<sup>H</sup>* **R ss** *<sup>s</sup>* = *Ek k* , and its eigen-value decomposition (EVD) yields

$$\mathbf{R} = \mathbf{U}\mathbf{V}\mathbf{U}^{H} = \mathbf{U}\_{s}\mathbf{V}\_{s}\mathbf{U}\_{s}^{H} + \mathbf{U}\_{n}\mathbf{V}\_{n}\mathbf{U}\_{n}^{H} = [\mathbf{u}\_{1\prime}\cdots, \mathbf{u}\_{2M+1}]\text{diag}[v\_{1\prime}, \cdots, v\_{2M+1}][\mathbf{u}\_{1\prime}\cdots, \mathbf{u}\_{2M+1}]^{H},\tag{2}$$

where **V** is the diagonal matrix with the eigen-values arranged as <sup>1</sup> *LL M* 1 21 *v vv v* ≥ ≥ <sup>&</sup>gt; + + ≥ ≥ , the diagonal matrix (2 1 ) (2 1 ) *<sup>M</sup> LML <sup>n</sup> <sup>R</sup>* <sup>+</sup> − × <sup>+</sup> <sup>−</sup> **<sup>V</sup>** <sup>∈</sup> is composed of eigen-values 1 2 21 , ,, *vv v LL M* ++ + ; (2 1) (2 1 ) *<sup>M</sup> M L <sup>n</sup> C* <sup>+</sup> <sup>×</sup> <sup>+</sup> <sup>−</sup> **U** ∈ consists of the eigenvectors related to 1 2 21 , ,, *vv v LL M* ++ + , spanning the noise subspace of **R** .

Let's define ( ) *l ll j j e e* θ φ γ<sup>−</sup> = , and thus the steering vector be written in another form as:

$$\mathbf{a}(\boldsymbol{\gamma}\_{l},\boldsymbol{\phi}\_{l}) = \begin{bmatrix} e^{jM\boldsymbol{\gamma}\_{l}} \ e^{j(M-1)\boldsymbol{\gamma}\_{l}} & \cdots \ e^{j\boldsymbol{\gamma}\_{l}} & \mathbf{1} \ e^{jM\boldsymbol{\theta}\_{l}} \times e^{j(M-1)\boldsymbol{\theta}\_{l}} \ \times e^{j(M-1)\boldsymbol{\gamma}\_{l}} & \cdots \ e^{j\boldsymbol{\theta}\_{l}} \times e^{j\boldsymbol{\gamma}\_{l}} \end{bmatrix}^{T} \tag{3}$$

Furthermore, **a**(,) γ φ can be separated into two parts, i.e. (,) 1 2 ( )( ) *j j e e* θ γ **a aa** γ φ = , where ( 1) *<sup>M</sup>* <sup>+</sup> -dimensional vector ( ) ( 1) 2 1 *<sup>T</sup> j jM j M <sup>j</sup> e ee e* γ γ γγ <sup>−</sup> <sup>=</sup> **<sup>a</sup>** and (2 1) ( 1) *M M* <sup>+</sup> <sup>×</sup> <sup>+</sup> - dimensional matrix

Passive Source Localization of Sensor Arrays 5

To verify the effectiveness of the proposed algorithm, let's consider an L-shaped array with

π

error (RMSE) is used as the performance measure. All results provided are based on 500

<sup>500</sup> , <sup>1</sup> RMSE(the th signal) ˆ*il l <sup>i</sup> <sup>l</sup>*

*i l* (in unit of degree) stands for the estimation of the *l* -th elevation

th trial. For comparison, the propagator method [6, 9] and the ESPRIT method [8, 11] with

In the first experiment, the effect of signal-to-noise (SNR) on the performance of the proposed algorithm is investigated. The number of snapshots is set to 400 and the SNR varies from 0 to 30 dB. The averaged performances (RMSE of elevation and azimuth angle estimations versus SNR for two sources) over 500 Monte Carlo runs are shown in Figs. 2 and 3. As expected, when the SNR increases, the RMSE of the estimated parameters decrease. In addition, it is observed that the proposed algorithm improves the performance slightly compared to the conventional ESPRIT algorithm, which must have a precise association

In the second experiment, the influence of snapshot number on the performance of the proposed algorithm is explored. The same parameters as that of the second experiment are used, except that the SNR is fixed at 10 dB and the number of snapshots varies from 200 to 2000. The averaged performances (RMSE of elevation and azimuth angle estimations versus snapshot number for two sources) over 500 Monte Carlo runs are shown in Figs. 4 and 5. From these figures, it can be seen that RMSE of the elevation and azimuth estimations decrease as snapshot number increases. In addition, the proposed algorithm has higher

), respectively with DOAs of o o

 and *<sup>j</sup>*0.25 *<sup>k</sup> e* π

= = , impinge on this array. The root mean square

( ) <sup>500</sup> <sup>2</sup> <sup>1</sup>

α α

<sup>=</sup> <sup>=</sup> <sup>−</sup> (8)

λ

α

. Two

*<sup>l</sup>* in the *i* -

13 elements as shown in Fig.1. These sensor locations are in unit of / 2 *d* =

2 2 ( 40 , 55 )

independent runs. The RMSE for DOA estimation is defined as

 β

**2.2 Simulation results** 

1 1 ( 60 , 35 )

in which , ˆα

procedure.

 β

α

uncorrelated equivalent-power sources ( *<sup>j</sup>*0.2 *<sup>k</sup> e*

α

= = and o o

correct pairing are simultaneously executed.

estimation accuracy than the ESPRIT method.

Fig. 2. RMSE of elevation angle estimations versus SNR

$$\mathbf{a}\_1(e^{j\theta}) = \begin{bmatrix} \mathbf{I}\_{(M+1)\times(M+1)} \\\\ \text{diag}\left\{ e^{jM\theta}, \dots, e^{j\theta} \right\} \ \mathbf{0}\_{M\times 1} \end{bmatrix} \tag{4}$$

Based on the theory that the noise subspace **U***n* is orthogonal to the range space of **A** , (,) *<sup>H</sup> n ll* **Ua 0** γ φ = , *l L* = 1, , , the electric angle pair {,} *l l* γ φ , *l L* = 1, , can be found from the *L* deepest minima of the following MUSIC spectral function:

$$\begin{aligned} f\_1(\boldsymbol{\chi}, \boldsymbol{\phi}) &= \mathbf{a}^H(\boldsymbol{\gamma}\_l, \boldsymbol{\phi}\_l) \mathbf{U}\_n \mathbf{U}\_n^H \mathbf{a}(\boldsymbol{\gamma}\_l, \boldsymbol{\phi}\_l) = \mathbf{a}\_2^H \left( e^{j\boldsymbol{\gamma}\_l} \right) \mathbf{a}\_1^H \left( e^{j\boldsymbol{\theta}\_l} \right) \mathbf{U}\_n \mathbf{U}\_n^H \mathbf{a}\_1 \left( e^{j\boldsymbol{\theta}\_l} \right) \mathbf{a}\_2 \left( e^{j\boldsymbol{\gamma}\_l} \right) \\ &= \mathbf{a}\_2^H \left( e^{j\boldsymbol{\gamma}\_l} \right) \mathbf{C} \left( e^{j\boldsymbol{\theta}\_l} \right) \mathbf{a}\_2 \left( e^{j\boldsymbol{\gamma}\_l} \right) = 0 \end{aligned} \tag{5}$$

where

$$\mathbf{C}(e^{j\theta}) = \mathbf{a}\_1^H \left( e^{j\theta} \right) \mathbf{U}\_n \mathbf{U}\_n^H \mathbf{a}\_1 \left( e^{j\theta} \right) \tag{6}$$

is an ( 1) ( 1) *M M* + × + -dimensional Hermitian matrix.

Note that 1 ( )*<sup>l</sup> <sup>j</sup> <sup>e</sup>* θ **a 0** <sup>≠</sup> and 2 2 () () () 0 *ll l <sup>H</sup> jj j ee e* γθ γ **a Ca** = , 1,2, , *l L* = . From Eq. (5)-(6), it can be seen that if and only if *<sup>l</sup> j j e e* θ θ = , the matrix ( ) *<sup>j</sup> e* θ **C** drops rank, or equivalently, when the polynomial of *<sup>j</sup> x e* θ = , *fx x* <sup>2</sup> () () = = det 0 {**C** } . Obviously, *x*ˆ , lying inside the unit circle and being closest to the unit circle, is actually the signal root.

Eq. (5) implies that by substituting the estimated ˆ *l j e* θ into ( ) *<sup>j</sup> e* θ **C** in Eq. (5), ˆ *l* γ can be found from the minima of the following function:

$$\hat{\mathcal{Y}}\_l = \min\_{\mathcal{Y}} \quad \mathbf{a}\_2^H \left( \boldsymbol{\varepsilon}^{j\boldsymbol{\gamma}} \right) \mathbf{C}(\boldsymbol{\varepsilon}^{j\hat{\boldsymbol{\theta}}}) \mathbf{a}\_2 \left( \boldsymbol{\varepsilon}^{j\boldsymbol{\gamma}} \right) \tag{7}$$

the minima of which indicates estimation.

When 2 *ppqq* φ −≠− γ φγ π + *h* **,** *h* ∈ −{ 1,0,1} **,** *p*, {1, , } *q L* ∈ **, i.e.,** θ θ *p q* ≠ **,** Eq. (7) implies that ( ) ˆ <sup>2</sup> *<sup>p</sup> <sup>j</sup> e* γ **<sup>a</sup>** is just the unique eigenvector corresponding to the smallest eigen-value of ˆ ( ) *<sup>p</sup> <sup>j</sup> e* θ **C** . However, when 2 *ppqq* φγ φγ − = − + *h*π , ( ) ˆ <sup>2</sup> *<sup>p</sup> <sup>j</sup> e* γ **a** is no longer the unique eigenvector corresponding to the smallest eigen-value of ˆ ( ) *<sup>p</sup> <sup>j</sup> e* θ **C** . The eigen-value decomposition (EVD) of ˆ ( ) *<sup>p</sup> <sup>j</sup> e* θ **<sup>C</sup>** yields ˆ <sup>1</sup> 1 1 1 11 1 ( ) [ , , ]diag[ , , ][ , , ] *<sup>p</sup> <sup>j</sup> e M MM v v* θ <sup>−</sup> **C uu** = + ++ **u u** , where the eigen-values are arranged as 1 2 *M M* <sup>1</sup> *vv v v* ≥ ≥ > = <sup>+</sup> . It is obvious that under the case 2 *ppqq* φ − γ φγ π <sup>=</sup> <sup>−</sup> <sup>+</sup> *<sup>h</sup>* ( ) ˆ <sup>2</sup> *<sup>p</sup> <sup>j</sup> e* γ **<sup>a</sup>** and ( ) ˆ <sup>2</sup> *<sup>q</sup> <sup>j</sup> e* γ **a** are the linear combinations of two eigenvectors {**u u** *M M* , <sup>+</sup>1} **,** which are orthogonal to **U uu u** s 12 1 <sup>=</sup> { , ,, *<sup>M</sup>*<sup>−</sup> } . Obviously, both ˆ *p j e* γ and ˆ *q j e* γ are the roots of 3 2 s2 ( ) () () <sup>0</sup> *H H <sup>s</sup> fx x x* = = **a UU a** .

From the estimates { }ˆ ˆ , *l l j j e e* γ θ , the elevation and azimuth angle estimates can be given as ( ) ( ) ( ) ˆ ˆ ˆ arccos / 2 *l l j j <sup>l</sup> ee d* θ γ α = −∠ × λ π and ( ) ( ) ( ) ˆ ˆ arccos / 2 sin ˆ *<sup>l</sup> <sup>j</sup> l l e d* γ β = − ∠ λπ α , respectively. Since ( ) ˆ <sup>2</sup> *<sup>l</sup> <sup>j</sup> <sup>e</sup>* γ **<sup>a</sup>** is related to ˆ ( )*<sup>l</sup> <sup>j</sup> <sup>e</sup>* θ **<sup>C</sup>** (i.e., corresponding to ( ) ˆ <sup>1</sup> *<sup>l</sup> <sup>j</sup> <sup>e</sup>* θ **a** ), the proposed algorithm can avoid pairing parameters. In addition, it avoids the spectral search because both ˆ *l j e* γ and ˆ *l j e* θare estimated by solving polynomial roots.

#### **2.2 Simulation results**

4 Sensor Array

( ) diag , , *M M <sup>j</sup>*

1 2 1 12

*ll nn ll n n*

**a UU a a a UU a a**

( ) 1 1 () () *jj j H H n n ee e*

> γ

> > θ

*l j e* θ

() () ˆ

θ

> ( ) *<sup>p</sup> <sup>j</sup> e* θ

<sup>−</sup> **C uu** = + ++ **u u** , where the

, the elevation and azimuth angle estimates can be given as

<sup>1</sup> *<sup>l</sup> <sup>j</sup> <sup>e</sup>* θ

 and ( ) ( ) ( ) ˆ ˆ arccos / 2 sin ˆ *<sup>l</sup> <sup>j</sup> l l e d* γ

= − ∠

**<sup>C</sup>** (i.e., corresponding to ( ) ˆ

can avoid pairing parameters. In addition, it avoids the spectral search because both ˆ

1 1 1 11 1 ( ) [ , , ]diag[ , , ][ , , ] *<sup>p</sup> <sup>j</sup> e M MM v v*

2 2 ˆ min ( )*<sup>l</sup> <sup>H</sup> jj j <sup>l</sup> ee e* γ

+ *h* **,** *h* ∈ −{ 1,0,1} **,** *p*, {1, , } *q L* ∈ **, i.e.,** 

**<sup>a</sup>** is just the unique eigenvector corresponding to the smallest eigen-value of ˆ

**<sup>C</sup>** yields ˆ <sup>1</sup>

eigen-values are arranged as 1 2 *M M* <sup>1</sup> *vv v v* ≥ ≥ > = <sup>+</sup> . It is obvious that under the case

= , *fx x* <sup>2</sup> () () = = det 0 {**C** } . Obviously, *x*ˆ , lying inside the unit circle and

*f e e ee*

0

γθ

γ

π , ( ) ˆ <sup>2</sup> *<sup>p</sup> <sup>j</sup> e* γ

<sup>2</sup> *<sup>q</sup> <sup>j</sup> e* γ

{**u u** *M M* , <sup>+</sup>1} **,** which are orthogonal to **U uu u** s 12 1 <sup>=</sup> { , ,, *<sup>M</sup>*<sup>−</sup> } . Obviously, both ˆ

β

θθ

= , the matrix ( ) *<sup>j</sup> e*

 γ φ

**I <sup>a</sup> <sup>0</sup>**

1

= , *l L* = 1, , , the electric angle pair {,} *l l*

= =

 γ

**a 0** <sup>≠</sup> and 2 2 () () () 0 *ll l <sup>H</sup> jj j ee e*

γ

θ

deepest minima of the following MUSIC spectral function:

( )( ) ( )

*ee e*

= =

*ll l*

is an ( 1) ( 1) *M M* + × + -dimensional Hermitian matrix.

θ

being closest to the unit circle, is actually the signal root. Eq. (5) implies that by substituting the estimated ˆ

(,) ( , ) ( , )

2 2

γθ

γφ

θ

be seen that if and only if *<sup>l</sup> j j e e*

θ

from the minima of the following function:

the minima of which indicates estimation.

However, when 2 *ppqq* φγ

> π<sup>=</sup> <sup>−</sup> <sup>+</sup> *<sup>h</sup>* ( ) ˆ

are the roots of 3 2 s2 ( ) () () <sup>0</sup> *H H*

γ θ

> λ π

( ) *<sup>p</sup> <sup>j</sup> e* θ

From the estimates { }ˆ ˆ , *l l j j e e*

( ) ( ) ( ) ˆ ˆ ˆ arccos / 2 *l l j j <sup>l</sup> ee d* θ γ

**<sup>a</sup>** is related to ˆ ( )*<sup>l</sup> <sup>j</sup> <sup>e</sup>*

 π

 φγ− = − + *h*

corresponding to the smallest eigen-value of ˆ

<sup>2</sup> *<sup>p</sup> <sup>j</sup> e* γ

θ

*<sup>s</sup> fx x x* = = **a UU a** .

θ

are estimated by solving polynomial roots.

**<sup>a</sup>** and ( ) ˆ

*H jj j*

 γ φ

(,) *<sup>H</sup> n ll* **Ua 0** γ φ

where

Note that 1 ( )*<sup>l</sup> <sup>j</sup> <sup>e</sup>*

polynomial of *<sup>j</sup> x e*

When 2 *ppqq*

 −≠− γ φγ

φ

(EVD) of ˆ

2 *ppqq*

= −∠ ×

Since ( ) ˆ <sup>2</sup> *<sup>l</sup> <sup>j</sup> <sup>e</sup>* γ

( ) ˆ <sup>2</sup> *<sup>p</sup> <sup>j</sup> e* γ

φ − γ φγ

α

and ˆ *l j e* θ *e*

θ

{ } ( 1) ( 1)

> γ φ

*H H HH H j j jj*

**a Ca** (5)

γ

*e e*

θ

Based on the theory that the noise subspace **U***n* is orthogonal to the range space of **A** ,

*jM j*

+ × +

 <sup>=</sup>

 θ

1

( ) ( ) ( )( )

*l l l l*

, *l L* = 1, , can be found from the *L*

 θ

**C** drops rank, or equivalently, when the

**C** in Eq. (5), ˆ

*l* γ

*p q* ≠ **,** Eq. (7) implies that

*p j e* γ and ˆ *q j e* γ

, respectively.

*l j e* γ

can be found

( ) *<sup>p</sup> <sup>j</sup> e* θ**C** .

γ

(4)

×

*M*

θ

 θ**C a UU a** = (6)

**a Ca** = , 1,2, , *l L* = . From Eq. (5)-(6), it can

 into ( ) *<sup>j</sup> e* θ

γ

= **a C a** (7)

θ θ

**a** are the linear combinations of two eigenvectors

λπ

**a** is no longer the unique eigenvector

**C** . The eigen-value decomposition

 α

**a** ), the proposed algorithm

To verify the effectiveness of the proposed algorithm, let's consider an L-shaped array with 13 elements as shown in Fig.1. These sensor locations are in unit of / 2 *d* = λ . Two uncorrelated equivalent-power sources ( *<sup>j</sup>*0.2 *<sup>k</sup> e* π and *<sup>j</sup>*0.25 *<sup>k</sup> e* π ), respectively with DOAs of o o 1 1 ( 60 , 35 ) α β = = and o o 2 2 ( 40 , 55 ) α β = = , impinge on this array. The root mean square error (RMSE) is used as the performance measure. All results provided are based on 500 independent runs. The RMSE for DOA estimation is defined as

$$\text{RMSE(the } l \text{th signal)} = \sqrt{\frac{1}{500} \sum\_{i=1}^{500} \left( \dot{\alpha}\_{i,l} - \alpha\_l \right)^2} \tag{8}$$

in which , ˆα*i l* (in unit of degree) stands for the estimation of the *l* -th elevation α*<sup>l</sup>* in the *i* th trial. For comparison, the propagator method [6, 9] and the ESPRIT method [8, 11] with correct pairing are simultaneously executed.

In the first experiment, the effect of signal-to-noise (SNR) on the performance of the proposed algorithm is investigated. The number of snapshots is set to 400 and the SNR varies from 0 to 30 dB. The averaged performances (RMSE of elevation and azimuth angle estimations versus SNR for two sources) over 500 Monte Carlo runs are shown in Figs. 2 and 3. As expected, when the SNR increases, the RMSE of the estimated parameters decrease. In addition, it is observed that the proposed algorithm improves the performance slightly compared to the conventional ESPRIT algorithm, which must have a precise association procedure.

In the second experiment, the influence of snapshot number on the performance of the proposed algorithm is explored. The same parameters as that of the second experiment are used, except that the SNR is fixed at 10 dB and the number of snapshots varies from 200 to 2000. The averaged performances (RMSE of elevation and azimuth angle estimations versus snapshot number for two sources) over 500 Monte Carlo runs are shown in Figs. 4 and 5. From these figures, it can be seen that RMSE of the elevation and azimuth estimations decrease as snapshot number increases. In addition, the proposed algorithm has higher estimation accuracy than the ESPRIT method.

Fig. 2. RMSE of elevation angle estimations versus SNR

Passive Source Localization of Sensor Arrays 7

In some practical applications, the signals received by an array are often the mixture of nearfield and far-field sources, such as speaker localization using microphone arrays and guidance (homing) systems [12-19]. For example, in the application of speaker localization using microphone arrays, each speaker may be in the near-field or far-field of the array [16]. In this case, either existing near-field source localization methods or far-field source those may fail in localizing mixed near-field and far-field sources. This section will give a new passive source localization algorithm, which can localize near-field sources or far-field

Consider that *L* (near-field1 or far-field) narrowband, independent radiating sources, impinge on the uniform linear array (ULA) with 2 1 *N* + elements as shown in Fig.6. Let the 0 th sensor be the phase reference point. After sampled with a proper rate that satisfies the

• • • • • • • • 

signal wavelength and array dimension, respectively (see [4] for details). Whereas far-field means the

<sup>−</sup> *<sup>N</sup>* <sup>−</sup> <sup>2</sup> <sup>−</sup><sup>1</sup> <sup>0</sup> 1 2 *i <sup>N</sup>*

θ*l*

= + , − ≤≤ *NiN* , *k K* <sup>=</sup> 0, , 1 <sup>−</sup> , (9)

 *l*

•

<sup>2</sup> [ ,2 ] *D* π λ λ

, where

λ

and *D* are

*lr*


Nyquist rate, the signal received by the *i* th sensor can be expressed as [5-11]

<sup>2</sup> *<sup>l</sup> <sup>j</sup> <sup>e</sup>* γ

**a** ), the proposed algorithm can avoid pairing parameters; and (ii)

**<sup>a</sup>** is related to ˆ ( )*<sup>l</sup> <sup>j</sup> <sup>e</sup>*

*l j e* γ and ˆ *l j e* θare

θ**C** (i.e.

From the above experiments, it can be seen that (i) since ( ) ˆ

the proposed algorithm avoids the spectral search due to that both ˆ

**3. Passive localization of mixed near-field and far-field sources** 

corresponding to ( ) ˆ

sources or their mixture.

<sup>1</sup> *<sup>l</sup> <sup>j</sup> <sup>e</sup>* θ

**3.1 Description of the proposed algorithm** 

Fig. 6. Uniform linear array configuration

λ.

radiating zone beyond <sup>1</sup> <sup>2</sup> [0, 2 ] *D*

1

=

*L*

() () () *il*

*x k s ke n k* τ

*j il i l*

1 Note that *Fresnel* zone (i.e. near-field) lies in the radiating zone 1 1 <sup>2</sup>

estimated by solving polynomial roots.

Fig. 3. RMSE of azimuth angle estimations versus SNR

Fig. 4. RMSE of elevation angle estimations versus snapshot number

Fig. 5. RMSE of azimuth angle estimations versus snapshot number

Fig. 3. RMSE of azimuth angle estimations versus SNR

Fig. 4. RMSE of elevation angle estimations versus snapshot number

Fig. 5. RMSE of azimuth angle estimations versus snapshot number

From the above experiments, it can be seen that (i) since ( ) ˆ <sup>2</sup> *<sup>l</sup> <sup>j</sup> <sup>e</sup>* γ **<sup>a</sup>** is related to ˆ ( )*<sup>l</sup> <sup>j</sup> <sup>e</sup>* θ **C** (i.e. corresponding to ( ) ˆ <sup>1</sup> *<sup>l</sup> <sup>j</sup> <sup>e</sup>* θ **a** ), the proposed algorithm can avoid pairing parameters; and (ii) the proposed algorithm avoids the spectral search due to that both ˆ *l j e* γ and ˆ *l j e* θ are estimated by solving polynomial roots.

#### **3. Passive localization of mixed near-field and far-field sources**

In some practical applications, the signals received by an array are often the mixture of nearfield and far-field sources, such as speaker localization using microphone arrays and guidance (homing) systems [12-19]. For example, in the application of speaker localization using microphone arrays, each speaker may be in the near-field or far-field of the array [16]. In this case, either existing near-field source localization methods or far-field source those may fail in localizing mixed near-field and far-field sources. This section will give a new passive source localization algorithm, which can localize near-field sources or far-field sources or their mixture.

#### **3.1 Description of the proposed algorithm**

Consider that *L* (near-field1 or far-field) narrowband, independent radiating sources, impinge on the uniform linear array (ULA) with 2 1 *N* + elements as shown in Fig.6. Let the 0 th sensor be the phase reference point. After sampled with a proper rate that satisfies the Nyquist rate, the signal received by the *i* th sensor can be expressed as [5-11]

$$\mathbf{x}\_{i}(k) = \sum\_{l=1}^{L} s\_{l}(k)e^{j\tau\_{il}} + n\_{i}(k) \quad , \text{ } -\mathbf{N} \le i \le \mathbf{N} \text{ } \; , \; k = \mathbf{0} \text{ } \cdots \text{ } \text{} \text{ } \tag{9}$$

Fig. 6. Uniform linear array configuration

<sup>1</sup> Note that *Fresnel* zone (i.e. near-field) lies in the radiating zone 1 1 <sup>2</sup> <sup>2</sup> [ ,2 ] *D* π λ λ , where λ and *D* are signal wavelength and array dimension, respectively (see [4] for details). Whereas far-field means the radiating zone beyond <sup>1</sup> <sup>2</sup> [0, 2 ] *D* λ.

Passive Source Localization of Sensor Arrays 9

To construct a matrix with full rank for arbitrary-field sources , let *n m* = − and *q* = 0 . Thus,

*x kx kxkxk c e e m p N N* γ

Let *mmN* =++ 1 and *p pN* =+ + 1 , and thus *mp N* , [1,2 1] ∈ + . Based on the idea from (11)- (12), a special (2 1) (2 1) *N N* + × + -dimensional cumulant matrix **C** can be defined, the

*ce e mp N*

Note that the (2 1) (2 1) *N N* + × + matrix **C** can be represented in a compact matrix form

− −− **b** = -

<sup>4</sup> [ 1 21 1 21 1 21 , , dia ] g( , , )[ , , ] *H H <sup>H</sup>* **C BC A W** = == *<sup>s</sup>* **Zww** *N NN* + ++

where is the diagonal matrix with the singular values arranged as

**<sup>B</sup>** , consists of the left singular vectors 1 2 , ,, **ww w** *<sup>L</sup>* . Similarly, (2 1) (2 1 ) *N NL*

( ) ( , ) /2 *<sup>l</sup>*

21 12 ˆ min ( , ) ( , ) min ( ) ( ) ( ) ( ), 1, , ˆ ˆ ˆˆ *H H HH H <sup>l</sup> l nn l l nn l l L*

== = **a ZZ a a a ZZ a a** 

 φ

N 1

**0**

×

 φ

> γ

 γ

<sup>=</sup> is the eigen-value matrix # **W***s s* 1 2 **W** , i.e.

 γ

which is orthogonal to **A** , consists of the right singular vectors 1 2 21 , ,, *LL N* ++ + **zz z** .

γ= ∠ **-**

 into **a**(,) γ φ

{ }

*j jN e e*


*jN j*

 

diag , , ,1

γ

*e e*

 

<sup>=</sup>

diag , ,

γ

<sup>4</sup> diag 41 4 4 [ ,, ,, ] *<sup>s</sup> ,s ,sl ,sL* **C** = *c cc* , virtual "steering matrix" 1 [ ( ), , ( ), , ( )] *l L* **Bb b b** =

*<sup>l</sup> ee e* γγ

*<sup>H</sup>* **C BC A** = *<sup>s</sup>* , where the superscript *H* stands for the Hermitian transpose,

γφ

<sup>=</sup> <sup>∈</sup> <sup>+</sup>

( ) <sup>2</sup> \* \* \* 2( ) 4, 1 cum{ ( ), ( ), ( ), ( )} , , [ , ] *l ll L*

*l*

( ) <sup>2</sup>

2( 1) ( 1) ( 1)

−− −− + − −

*j m N jp N jp N*

virtual "steering vector" 2 (2 2) <sup>2</sup> ( ) [ , , 1, , ] *ll l jN j N j N <sup>T</sup>*

Based on the first 2*N* lines **W***s*1 and last 2*N* lines **W***s*<sup>2</sup> , *<sup>l</sup>*

estimated from the eigen-values of the following matrix [3] :

 γγ

*l* γ

where { }

<sup>1</sup> anti

 γφ

where 1 <sup>222</sup> diag[ , , , , ] *l L j jj eee* γ

> γφ

( )

**a**

γ

**T** .

By substituting the estimate ˆ

φ

**-**

γ

The singular value decomposition (SVD) of **C** yields

 σ<sup>1</sup> ≥ ≥ *LL N* <sup>&</sup>gt; + + 1 21 ≥ ≥ . Let (2 1) *N L*

γ

( , )=cum{ ( ), ( ), ( ), ( )}

*mp x k x k x k x k*

*mN mN pN*

−− − + + − −

\* \* 1 1 10

=

*m mpq sl*

−

( ,) *m p* th element of which can be given by

4, 1

*sl*

*L*

**C**

*l*

as 4

σ

 σσ
