4. Application of CS to localization

#### 4.1. Introduction

Determining the location of unknown sources, often called localization, is an important problem in many different applications, sometimes as the first step toward solving more complicated tasks [26]. As stated in [27], a common approach to localize unknown sources in wireless systems is to collect the ranging information from radio signals traveling between the unknown source and a number of reference nodes. Usually, ranging information is measured based on one or more physical parameters of the radio signal, such as the time of arrival (TOA), time-difference-of-arrival (TDOA), received signal strength (RSS), and angle of arrival (AOA). Among these techniques, TOA-based range estimates are inherently more accurate than others, while the RSS-based is low-cost and easily implementable.

Localization of unknown sources is not possible unless the required number of reference nodes exist in the neighborhood of the unknown source. For example, in the 2-D case, at least three reference nodes are required for localization. Therefore, usually a network of reference nodes is utilized for the positioning of unknown sources in the area of interest. In such an architecture, all reference points send their ranging measurements to a special reference point, called Fusion Centre, which performs the localization.

In most practical cases, due to limitations such as low data-rate links between reference points and high network traffic, it is not possible to send all measurements to the Fusion Centre. This significantly affects the localization accuracy. To address these issues, some recent works have applied CS to localization; for example, see [28], and the references therein. Applying CS is a promising approach to handle the aforementioned problems since it significantly reduces the amount of data generated by reference nodes. However, the extra computational burden caused by CS reconstruction methods becomes prohibitive as the number of sensors (network size) and/or sampling rate (bandwidth) increases. Hence, the challenge is to find a lowcomplexity CS based framework for practical localization networks.

In the following, we focus on the TDOA-based localization scenario and develop a novel CSbased localization framework that estimates the TDOAs directly from CS measurements without reconstructing the full-scale signals. It is worth mentioning that the developed method solves the computational cost issue since it eliminates the reconstruction step. Although this approach is specially designed for TDOA-based localization, it can be developed for other applications.

#### 4.2. TDOA-based localization

Sparse Bayesian Learning (SBL), which was first proposed by Tipping [22], is one of the most important families of Bayesian algorithms. In the last decade, this method was the focus of numerous studies and it was greatly extended by many other researchers; for more details please see [23] and the references therein. Having noticed the benefits of SBL, [14] has recently applied SBL to EW receiver design. Also, [20] applied SBL to the scenario where the observed data is represented as the superposition of signal plus noise plus interference. These works can be considered as a good start for the most general scenario where data measured by a radar are

Applying CS to radar and other systems, on one hand, reduces the volume of generated data and Nyquist sampling rate, but on the other hand, results in additional computational cost [24]. In some practical scenarios, the extra computational burden caused by CS reconstruction methods appears as a new design challenge. For example, when the number of sensors (network size) increases, this extra computational cost becomes prohibitive. Hence, finding low-complexity CS recovery methods has become one of the most popular topics in CS theory. More recently, [25] introduced a general framework, called compressive signal processing, in which signal processing problems are solved directly in the compressive measurement domain. This methodology is in sharp contrast to the standard CS problem where full signals are first recovered from compressed measurements and then signal processing approaches are performed on the reconstructed signals. Applying such an interesting strategy enables us to take advantage of CS benefits without any extra cost. In the next Section, we will provide a similar compressive signal processing-based foundation for the localization of unknown

Determining the location of unknown sources, often called localization, is an important problem in many different applications, sometimes as the first step toward solving more complicated tasks [26]. As stated in [27], a common approach to localize unknown sources in wireless systems is to collect the ranging information from radio signals traveling between the unknown source and a number of reference nodes. Usually, ranging information is measured based on one or more physical parameters of the radio signal, such as the time of arrival (TOA), time-difference-of-arrival (TDOA), received signal strength (RSS), and angle of arrival (AOA). Among these techniques, TOA-based range estimates are inherently more accurate

Localization of unknown sources is not possible unless the required number of reference nodes exist in the neighborhood of the unknown source. For example, in the 2-D case, at least three reference nodes are required for localization. Therefore, usually a network of reference nodes is

than others, while the RSS-based is low-cost and easily implementable.

contaminated by clutter and perfect knowledge of the basis matrix is not available.

120 Advanced Electronic Circuits - Principles, Architectures and Applications on Emerging Technologies

3.7. Complexity analysis

sources.

4.1. Introduction

4. Application of CS to localization

Typically, TDOA-based localization consists of three main steps:


Therefore, estimating TDOAs is an essential first step for localization of unknown sources [29], and affects the accuracy of the positioning. Most practical localizers obtain TDOA estimates by cross-correlating the received signals from different reference points. This method is usually called generalized cross-correlation (GCC). However, large distances between reference points and Fusion Centre impose limitations on the data rate between the nodes [28]. Therefore, it is not possible to estimate TDOAs of all sensor pair combinations. This limitation reduces the accuracy substantially.

Some recent works employ CS to overcome the identified issues. The block diagram of this strategy is shown in Figure 3. In particular, reference points first apply the technique of CS on their observed samples and then transmit CS-based version of their collected samples to the Fusion Centre. Fusion Centre then applies one of the CS recovery methods on the received CS measurements to reconstruct the Nyquist samples from CS measurements sent by the

Figure 3. Block diagram of CS-based TDOA localizer.

reference points. Next, it estimates TDOAs by cross-correlating all reconstructed signal pairs. Although applying such a strategy significantly reduces the traffic of localization networks (the amount of data that need to be sent from the reference receivers to Fusion Centre), the reconstruction step introduces a prohibitive extra computational burden on the Fusion Centre. Also, it should be highlighted that all the existing recovery algorithms are subject to errors, especially in the presence of noise, interference, and clutter, which affect the accuracy of positioning. The above shortcomings motivate us to provide a new framework that addresses these important issues.

#### 4.3. TDOA estimation without reconstruction

Consider the estimation of the TDOA between two N � 1 vectors

$$\mathbf{x}\_1 = [\mathbf{x}(\mathbf{0}), \dots, \mathbf{x}(\mathbf{N} - \mathbf{1})]^T \tag{17}$$

(20), via CS reconstruction techniques. But the computations could be time-consuming and the

A novel way to obtain CS measurements that preserve the time-shift relationship uses a Φ that

<sup>Φ</sup> <sup>¼</sup> <sup>½</sup> <sup>I</sup> <sup>j</sup> I Ij j <sup>I</sup> �∈R<sup>M</sup>�<sup>M</sup> (21)

Applications of Compressive Sampling Technique to Radar and Localization

http://dx.doi.org/10.5772/intechopen.75072

123

y1ð Þ¼ 1 xð Þþ 1 x 5ð Þþ xð Þþ 9 xð Þ 13 (22)

y1ð Þ¼ 0 xð Þþ 0 x 4ð Þþ xð Þþ 8 xð Þ 12

y1ð Þ¼ 2 xð Þþ 2 x 6ð Þþ xð Þþ 10 xð Þ 14

y1ð Þ¼ 3 xð Þþ 3 x 7ð Þþ xð Þþ 11 xð Þ 15

y2ð Þ¼ 0 x Dð Þþ x Dð Þþ þ 4 x Dð Þþ þ 8 x Dð Þ þ 12

y2ð Þ¼ 1 x Dð Þþ þ 1 x Dð Þþ þ 5 x Dð Þþ þ 9 x Dð Þ þ 13

y2ð Þ¼ 2 x Dð Þþ þ 2 x Dð Þþ þ 6 x Dð Þþ þ 10 x Dð Þ þ 14

Suppose D = 2, then, the circular cross correlation of y<sup>1</sup> and y<sup>2</sup> will peak at a shift of 2. In this example, Φ has compressed the measurements from N = 16 to M = 4, and TDOA estimation is

This chapter has introduced the concept and advantages of compressive sampling (CS). Applying CS to radar signals with a high bandwidth can significantly reduce the sampling rate and the power required by the Analog-to-Digital converter. It has also been shown that even when the transmitted signal and the basis matrix ψ are unknown, such as in passive radars and Electronic Warfare receivers, or when clutter with an unknown covariance matrix is present, CS can be used with Sparse Bayesian Learning to reduce the sampling rate and the amount of data generated. Finally, another application of CS to localization of unknown sources has been described. When estimating the TDOA between signals, using CS can reduce the data traffic between the reference points and Fusion Centre. Furthermore, a method to estimate the TDOA without reconstructing the signals at Fusion Centre has been developed to avoid the huge computational cost and possible errors that other CS-based techniques using

obtained directly from y<sup>1</sup> and y2, without the need for reconstruction.

y2ð Þ¼ 3 x Dð Þþ þ 3 x Dð Þþ þ 7 x Dð Þþ þ 11 x Dð Þ þ 15 (23)

reconstruction will have errors if noise is present.

and for y<sup>2</sup>

5. Conclusion

reconstruction require.

sums the elements of a vector. As an example, let M = 4, N = 16, and

where <sup>I</sup><sup>∈</sup> RM�<sup>M</sup> is an identity matrix. Then, the elements of <sup>y</sup>1are

and

$$\mathbf{x}\_2 = \left[ \mathbf{x}(D), \dots, \mathbf{x}(D + \mathbf{N} - 1) \right]^T \tag{18}$$

where D, an integer, is the TDOA between x<sup>1</sup> and x2:

The circular cross-correlation of x<sup>1</sup> and x<sup>2</sup> will yield a peak at a correlation shift of D samples. Now with CS, the measurements become

$$y\_1 = \Phi \mathbf{x}\_1 \tag{19}$$

and

$$y\_2 = \Phi \mathbf{x}\_2.\tag{20}$$

where <sup>y</sup><sup>1</sup> <sup>∈</sup>RM, <sup>y</sup><sup>2</sup> <sup>∈</sup>RM, <sup>Φ</sup> <sup>∈</sup> RM�<sup>N</sup>, and <sup>M</sup> <sup>&</sup>lt; <sup>N</sup>. In general, the transformations <sup>Φ</sup>x<sup>1</sup> and <sup>Φ</sup>x<sup>2</sup> will break up the time-shift relationship between x<sup>1</sup> and x2. A cross correlation of y<sup>1</sup> and y<sup>2</sup> will not give a peak at a shift of D. A common solution is to reconstruct x<sup>1</sup> and x<sup>2</sup> from (19) and (20), via CS reconstruction techniques. But the computations could be time-consuming and the reconstruction will have errors if noise is present.

A novel way to obtain CS measurements that preserve the time-shift relationship uses a Φ that sums the elements of a vector. As an example, let M = 4, N = 16, and

$$\mathbf{OP} = \left[\mathbf{I} \mid \mathbf{I} \mid \mathbf{I} \mid \mathbf{I}\right] \in \mathbb{R}^{M \times M} \tag{21}$$

where <sup>I</sup><sup>∈</sup> RM�<sup>M</sup> is an identity matrix. Then, the elements of <sup>y</sup>1are

$$y\_1(0) = \mathbf{x}(0) + \mathbf{x}(4) + \mathbf{x}(8) + \mathbf{x}(12)$$

$$y\_1(1) = \mathbf{x}(1) + \mathbf{x}(5) + \mathbf{x}(9) + \mathbf{x}(13)\tag{22}$$

$$y\_1(2) = \mathbf{x}(2) + \mathbf{x}(6) + \mathbf{x}(10) + \mathbf{x}(14)$$

$$y\_1(3) = \mathbf{x}(3) + \mathbf{x}(7) + \mathbf{x}(11) + \mathbf{x}(15)$$

and for y<sup>2</sup>

reference points. Next, it estimates TDOAs by cross-correlating all reconstructed signal pairs. Although applying such a strategy significantly reduces the traffic of localization networks (the amount of data that need to be sent from the reference receivers to Fusion Centre), the reconstruction step introduces a prohibitive extra computational burden on the Fusion Centre. Also, it should be highlighted that all the existing recovery algorithms are subject to errors, especially in the presence of noise, interference, and clutter, which affect the accuracy of positioning. The above shortcomings motivate us to provide a new framework that addresses

122 Advanced Electronic Circuits - Principles, Architectures and Applications on Emerging Technologies

The circular cross-correlation of x<sup>1</sup> and x<sup>2</sup> will yield a peak at a correlation shift of D samples.

where <sup>y</sup><sup>1</sup> <sup>∈</sup>RM, <sup>y</sup><sup>2</sup> <sup>∈</sup>RM, <sup>Φ</sup> <sup>∈</sup> RM�<sup>N</sup>, and <sup>M</sup> <sup>&</sup>lt; <sup>N</sup>. In general, the transformations <sup>Φ</sup>x<sup>1</sup> and <sup>Φ</sup>x<sup>2</sup> will break up the time-shift relationship between x<sup>1</sup> and x2. A cross correlation of y<sup>1</sup> and y<sup>2</sup> will not give a peak at a shift of D. A common solution is to reconstruct x<sup>1</sup> and x<sup>2</sup> from (19) and

<sup>x</sup><sup>1</sup> <sup>¼</sup> ½ � <sup>x</sup>ð Þ<sup>0</sup> ;…; x Nð Þ � <sup>1</sup> <sup>T</sup> (17)

<sup>x</sup><sup>2</sup> <sup>¼</sup> ½ � x Dð Þ;…; x Dð Þ <sup>þ</sup> <sup>N</sup> � <sup>1</sup> <sup>T</sup> (18)

y<sup>1</sup> ¼ Φx<sup>1</sup> (19)

y<sup>2</sup> ¼ Φx2, (20)

these important issues.

and

and

4.3. TDOA estimation without reconstruction

Figure 3. Block diagram of CS-based TDOA localizer.

where D, an integer, is the TDOA between x<sup>1</sup> and x2:

Now with CS, the measurements become

Consider the estimation of the TDOA between two N � 1 vectors

$$y\_2(0) = \mathbf{x}(D) + \mathbf{x}(D+4) + \mathbf{x}(D+8) + \mathbf{x}(D+12)$$

$$y\_2(1) = \mathbf{x}(D+1) + \mathbf{x}(D+5) + \mathbf{x}(D+9) + \mathbf{x}(D+13)$$

$$y\_2(2) = \mathbf{x}(D+2) + \mathbf{x}(D+6) + \mathbf{x}(D+10) + \mathbf{x}(D+14)$$

$$y\_2(3) = \mathbf{x}(D+3) + \mathbf{x}(D+7) + \mathbf{x}(D+11) + \mathbf{x}(D+15) \tag{23}$$

Suppose D = 2, then, the circular cross correlation of y<sup>1</sup> and y<sup>2</sup> will peak at a shift of 2. In this example, Φ has compressed the measurements from N = 16 to M = 4, and TDOA estimation is obtained directly from y<sup>1</sup> and y2, without the need for reconstruction.

#### 5. Conclusion

This chapter has introduced the concept and advantages of compressive sampling (CS). Applying CS to radar signals with a high bandwidth can significantly reduce the sampling rate and the power required by the Analog-to-Digital converter. It has also been shown that even when the transmitted signal and the basis matrix ψ are unknown, such as in passive radars and Electronic Warfare receivers, or when clutter with an unknown covariance matrix is present, CS can be used with Sparse Bayesian Learning to reduce the sampling rate and the amount of data generated. Finally, another application of CS to localization of unknown sources has been described. When estimating the TDOA between signals, using CS can reduce the data traffic between the reference points and Fusion Centre. Furthermore, a method to estimate the TDOA without reconstructing the signals at Fusion Centre has been developed to avoid the huge computational cost and possible errors that other CS-based techniques using reconstruction require.
