3. Application of CS to radar

In this section, we seek to provide a more realistic analysis of the application of CS in practical radar systems. We follow an alternative approach instead of that used in previous published works in the context of CS radar, which is a generalization of a canonical CS formulation. Although our approach is designed for the radar scenario, it is also capable of accommodating other practical scenarios in which the basis matrix is (partially) unknown and/or the observed data is contaminated by structured noises (interference).

#### 3.1. Introduction

Radar (radio detection and ranging) systems are present in many different civilian, military, and biomedical applications [13]. Radar systems are used to detect and determine the range, angle, and velocity of different objects, such as aircraft, missiles, ships, tanks, helicopters, and ground stations. Air traffic control, mapping of ground contours, detecting weather formations, and automotive traffic enforcement are some civilian applications of Radar systems.

Traditional radar systems consist of a transmitter, a transmitting antenna, a receiving antenna, and a receiver (powerful processor). The transmitter sends probing pulses of electromagnetic waves toward the areas of interest. The properties of the transmitted waves change when they are reflected by the potential targets. This enables the radar to locate the unknown targets (threats). This kind of detection is usually called active detection. Passive radars, which are essentially receive-only radars, do not transmit any probing signal. Instead, passive radars perform detection and estimation from signals that come from sources such as radar, radio and television (TV) stations [14].

A single-input single-output (SISO) radar consists of a single transmitter and a single receiver. A few decades ago, multiple-input multiple-output (MIMO) radar systems have been proposed as an extension of SISO radar systems. MIMO radar systems employ multiple elements on the transmitting and receiving sides, while SISO radars employ one element on each side. It is demonstrated that employing multiple transmit and receive elements significantly improves the performance. Since an SISO radar can be considered as a special case of MIMO radars, most recent works have focused on the MIMO scenario.

Broadly speaking, there are two main groups of MIMO radars: Co-located MIMO radars and distributed MIMO radars [15]. In the co-located MIMO radar all the antennas are closely spaced, while in the distributed MIMO radar, the antennas are widely separated. To be more precise, a distributed MIMO radar views the potential target from different angles. Hence, if the received signal from any specific path is weak, it can be compensated by signals received from other paths. Although all transmit-receive antenna pairs in a co-located MIMO radar see the potential target from the same view, transmitters use different probing waveforms. In summary, a distributed MIMO radar exploits the spatial diversity, while a co-located MIMO radar exploits the waveform diversity.

#### 3.2. CS for radar

<sup>x</sup> <sup>¼</sup> ½ � <sup>x</sup>ð Þ<sup>1</sup> ;…; x 1040 ð Þ <sup>T</sup> (7)

<sup>y</sup> <sup>¼</sup> <sup>y</sup>ð Þ<sup>1</sup> ;…; y 80 ð Þ � �<sup>T</sup> (8)

(9)

0 ⋮ A<sup>20</sup>

A<sup>1</sup> 0 … 0

0 A<sup>2</sup> … ⋮ ⋮⋱ 0 0 …

where Ai <sup>∈</sup> <sup>R</sup><sup>4</sup>�52contains elements of �1. Hence, the sampling rate is reduced by a factor of 13,

In this section, we seek to provide a more realistic analysis of the application of CS in practical radar systems. We follow an alternative approach instead of that used in previous published works in the context of CS radar, which is a generalization of a canonical CS formulation. Although our approach is designed for the radar scenario, it is also capable of accommodating other practical scenarios in which the basis matrix is (partially) unknown and/or the observed

Radar (radio detection and ranging) systems are present in many different civilian, military, and biomedical applications [13]. Radar systems are used to detect and determine the range, angle, and velocity of different objects, such as aircraft, missiles, ships, tanks, helicopters, and ground stations. Air traffic control, mapping of ground contours, detecting weather formations, and automotive traffic enforcement are some civilian applications of Radar systems.

Traditional radar systems consist of a transmitter, a transmitting antenna, a receiving antenna, and a receiver (powerful processor). The transmitter sends probing pulses of electromagnetic waves toward the areas of interest. The properties of the transmitted waves change when they are reflected by the potential targets. This enables the radar to locate the unknown targets (threats). This kind of detection is usually called active detection. Passive radars, which are essentially receive-only radars, do not transmit any probing signal. Instead, passive radars perform detection and estimation from signals that come from sources such as radar, radio and

A single-input single-output (SISO) radar consists of a single transmitter and a single receiver. A few decades ago, multiple-input multiple-output (MIMO) radar systems have been proposed as an extension of SISO radar systems. MIMO radar systems employ multiple elements on the transmitting and receiving sides, while SISO radars employ one element on each side. It

Φ ¼

as is the number of samples, i.e., <sup>N</sup>

3. Application of CS to radar

3.1. Introduction

television (TV) stations [14].

data is contaminated by structured noises (interference).

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

<sup>M</sup> ¼ 13:

Usually, radar detection and classification tasks require the transmission of wide-bandwidth probing signals during short observation times. Employing wideband probing pulses necessitates using fast ADCs with high sampling rates, which in turn leads to the generation of a huge amount of data. In most cases, data processing becomes one of the most important design issues. Recently, the emerging technique of compressive sampling has been proposed to alleviate the identified practical problems [16]. As mentioned previously, CS exploits the sparsity (compressibility) of received signals in different spaces to reduce the sampling rate as well as the volume of generated data and hence, is a promising technique for sophisticated radar systems.

The idea of using the CS technique in the context of radar systems was initially proposed by Herman and Strohmer in [17]. They showed that since the number of targets is typically much smaller than the number of range-Doppler cells, the prerequisite on the signal sparsity is often met in most radar scenarios, and hence CS can be efficiently used in radar systems. It is worth mentioning that [17] just focused on the simple SISO radar scenario. Then, Chen and Vaidyanathan in [18] extended the work in [17] to the MIMO radar case. During the last decade, different aspects of employing CS in both SISO and MIMO radars have been investigated; please see [15, 19] and the references therein.

Although CS has been applied in radar problems, it has not been comprehensively studied with respect to clutter and other structured noises. To be more precise, all related works modeled the observed signal by radar as a signal contaminated by additive noise. However, it is more realistic to add another term into the model to account for the clutter. Also, the proposed methods in the literature are suitable only for the case where radar transmitting waveforms are completely known, and hence, are not applicable to some important practical cases, such as electronic surveillance and threat recognition cases. To the best of our knowledge, none of the published works, on application of CS to radar, studied the general case where the signal is contaminated by clutter and the basis matrix is (partially) unknown.

The above shortcomings motivate us to provide a more realistic model that addresses these important issues. To this end, in the next subsection, we first present a high-level block diagram of CS-based radar architecture to show how CS can be employed to reduce the sampling rate of traditional radar systems. Then, in the next subsections, we study the case where perfect knowledge of the transmitted waveforms is not available, and when the received signals are contaminated by clutter and other structured noises. Finally, we introduce our solution for such a complex problem.

#### 3.3. Sub-Nyquist radar system

We now explain how CS is able to relax the required Nyquist-sampling rate of traditional radar systems. Let x(t) represent the signal received by a radar system with bandwidth B. As depicted in Figure 2, x(t) is first multiplied by the sensing matrix Φð Þt in the time domain at Nyquist frequency f S. The modulated signal is then integrated for a duration of <sup>N</sup> f S :The output

is sampled at sub-Nyquist rate <sup>f</sup> <sup>S</sup> <sup>N</sup>. One can easily show that

$$y\_i = \int\_t^{t + \frac{N}{k\_\xi}} \Phi\_i(t) \mathbf{x}(t) dt, \quad i = 1, \ldots, M\_\prime \tag{10}$$

<sup>Φ</sup>ið Þj <sup>t</sup> <sup>t</sup><sup>¼</sup> <sup>j</sup> f S <sup>t</sup>¼j�<sup>1</sup> f S

> y1 y2 ⋮ yM

, x ¼ ½ � x1; …; xN

assumption is not valid in some important radar applications.

yi <sup>¼</sup> <sup>X</sup> N

j¼1

Φ1,<sup>1</sup> Φ1, <sup>2</sup> … Φ1,N

our purposes, it is more convenient to write (3) in matrix form as:or, equivalently, as (excluding

<sup>T</sup>, and <sup>Φ</sup> <sup>¼</sup>

The architecture presented in Figure 2 can be adapted to any application. The main idea is that the input signal is first compressed in the analog domain and then, traditional ADCs are used

Remark: Exploiting the sparsity of the received signal x in different spaces, we have: x ¼ ψs, where ψ represents the basis matrix in which signal x is sparse. Taking into account the noise

The standard equation (15) is the starting point of existing literature in the context of CS-based radars. To be more precise, exploiting the sparsity of radar signals in various spaces, radar problems are first transformed into the CS context (radar problems are reformulated as a recovery of sparse vectors). Then the CS problems are solved by conventional sparse recovery methods. All conventional recovery methods assume that both sensing ð Þ Φ and basis ð Þ ψ matrices are known and available. As we will discuss in Subsection 3.4, this fundamental

Φ2,<sup>1</sup> Φ2, <sup>2</sup> … ⋮ ⋮ ⋱ ΦM,<sup>1</sup> ΦM,<sup>2</sup> …

Then, yi in (10) can be represented as

where xj <sup>¼</sup> <sup>Ð</sup><sup>t</sup><sup>þ</sup> <sup>j</sup>

where y ¼ y1;…; yM

for sampling.

� �<sup>T</sup>

reduces the sampling rate to Mf <sup>S</sup>

(denoted by e), y can be expressed as:

noise):

f S <sup>t</sup>þð Þ <sup>j</sup>�<sup>1</sup> f S

¼ Φi,j ∈f g þ1; �1 , i ¼ 1, …, M, j ¼ 1, …, N: (11)

Applications of Compressive Sampling Technique to Radar and Localization

Φ2,N ⋮ ΦM,N

Φið Þt x tð Þdt and xj are simply the Nyquist samples of the input signal x(t). For

<sup>N</sup> , which is much lower than the Nyquist rate f <sup>S</sup>.

Φi,jxj, i ¼ 1, …, M, (12)

x1 x2 ⋮ xN

y ¼ Φx, (14)

Φ1,<sup>1</sup> Φ1,<sup>2</sup> … Φ1,N

Φ2,N

. Thus, CS

⋮

ΦM,N

Φ2,<sup>1</sup> Φ2,<sup>2</sup> …

⋮ ⋮ ⋱

ΦM,<sup>1</sup> ΦM, <sup>2</sup> …

y ¼ Φψs þ e: (15)

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

(13)

117

where N is the number of integration samples per compression block. The selection of the sensing matrix that multiplies the input vector is a key point for the success of every CS approach. It has been shown that in most applications, the use of random sensing matrices provides a good performance. Let us assume that

Figure 2. Block diagram of compressive sampling architecture for radar systems.

Applications of Compressive Sampling Technique to Radar and Localization http://dx.doi.org/10.5772/intechopen.75072 117

$$\left. \Phi\_i(t) \right|\_{t=\frac{j-1}{f\_S}}^{t=\frac{j}{f\_S}} = \Phi\_{i,j} \in \{+1, -1\}, \quad i = 1, \ldots, M, \ j = 1, \ldots, N. \tag{11}$$

Then, yi in (10) can be represented as

The above shortcomings motivate us to provide a more realistic model that addresses these important issues. To this end, in the next subsection, we first present a high-level block diagram of CS-based radar architecture to show how CS can be employed to reduce the sampling rate of traditional radar systems. Then, in the next subsections, we study the case where perfect knowledge of the transmitted waveforms is not available, and when the received signals are contaminated by clutter and other structured noises. Finally, we introduce our

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

We now explain how CS is able to relax the required Nyquist-sampling rate of traditional radar systems. Let x(t) represent the signal received by a radar system with bandwidth B. As depicted in Figure 2, x(t) is first multiplied by the sensing matrix Φð Þt in the time domain at

<sup>N</sup>. One can easily show that

where N is the number of integration samples per compression block. The selection of the sensing matrix that multiplies the input vector is a key point for the success of every CS approach. It has been shown that in most applications, the use of random sensing matrices

f S

Φið Þt xð Þt dt, i ¼ 1, …, M, (10)

:The output

Nyquist frequency f S. The modulated signal is then integrated for a duration of <sup>N</sup>

yi ¼

Figure 2. Block diagram of compressive sampling architecture for radar systems.

provides a good performance. Let us assume that

ð<sup>t</sup>þ<sup>N</sup> f S t

solution for such a complex problem.

3.3. Sub-Nyquist radar system

is sampled at sub-Nyquist rate <sup>f</sup> <sup>S</sup>

$$y\_i = \sum\_{j=1}^{N} \Phi\_{i,j} \mathbf{x}\_{j\prime} \quad \mathbf{i} = \mathbf{1}, \ldots, M,\tag{12}$$

$$
\begin{bmatrix} y\_1 \\ y\_2 \\ \vdots \\ y\_M \end{bmatrix} = \begin{bmatrix} \Phi\_{1,1} & \Phi\_{1,2} & \dots & \Phi\_{1,N} \\ \Phi\_{2,1} & \Phi\_{2,2} & \dots & \Phi\_{2,N} \\ \vdots & \vdots & \ddots & \vdots \\ \Phi\_{M,1} & \Phi\_{M,2} & \dots & \Phi\_{M,N} \end{bmatrix} \begin{bmatrix} x\_1 \\ x\_2 \\ \vdots \\ x\_N \end{bmatrix} \tag{13}
$$

where xj <sup>¼</sup> <sup>Ð</sup><sup>t</sup><sup>þ</sup> <sup>j</sup> f S <sup>t</sup>þð Þ <sup>j</sup>�<sup>1</sup> f S Φið Þt x tð Þdt and xj are simply the Nyquist samples of the input signal x(t). For our purposes, it is more convenient to write (3) in matrix form as:or, equivalently, as (excluding noise):

$$y = \Phi \mathbf{x},\tag{14}$$

$$\begin{aligned} \text{where } \mathbf{y} = \begin{bmatrix} y\_1, \dots, y\_M \end{bmatrix}^T \text{ : } \mathbf{x} = \begin{bmatrix} \mathbf{x}\_1, \dots, \mathbf{x}\_N \end{bmatrix}^T \text{ and } \mathbf{O} = \begin{bmatrix} \Phi\_{1,1} & \Phi\_{1,2} & \dots & \Phi\_{1,N} \\\\ \Phi\_{2,1} & \Phi\_{2,2} & \dots & \Phi\_{2,N} \\\\ \vdots & \vdots & \ddots & \vdots \\\\ \Phi\_{M,1} & \Phi\_{M,2} & \dots & \Phi\_{M,N} \end{bmatrix} . \text{ Thus, } \mathbf{C} \mathbf{S}^{-1} \text{ is the } \mathbf{O} \text{-th row of } \mathbf{O}. \end{aligned}$$

reduces the sampling rate to Mf <sup>S</sup> <sup>N</sup> , which is much lower than the Nyquist rate f <sup>S</sup>.

The architecture presented in Figure 2 can be adapted to any application. The main idea is that the input signal is first compressed in the analog domain and then, traditional ADCs are used for sampling.

Remark: Exploiting the sparsity of the received signal x in different spaces, we have: x ¼ ψs, where ψ represents the basis matrix in which signal x is sparse. Taking into account the noise (denoted by e), y can be expressed as:

$$
\Delta y = \Phi \psi s + e.\tag{15}
$$

The standard equation (15) is the starting point of existing literature in the context of CS-based radars. To be more precise, exploiting the sparsity of radar signals in various spaces, radar problems are first transformed into the CS context (radar problems are reformulated as a recovery of sparse vectors). Then the CS problems are solved by conventional sparse recovery methods. All conventional recovery methods assume that both sensing ð Þ Φ and basis ð Þ ψ matrices are known and available. As we will discuss in Subsection 3.4, this fundamental assumption is not valid in some important radar applications.

#### 3.4. Blind compressive sampling

In some radar scenarios, knowledge of the basis matrix ð Þ ψ is available. For example, in MIMO radar scenario, the transmitted waveforms are known a priori. Therefore, a receiver can use this knowledge to construct the basis matrix locally. However, in some practical cases like passive radars, the transmitters are not part of the radar system, and hence, perfect knowledge of the transmitted waveforms is not available. Therefore, a receiver is only able to reconstruct from a noisy version of the basis matrix ð Þ ψ : As discussed in the related literature, erroneous basis matrices can lead to a significant performance loss (huge decrease in probability of detection), which is not acceptable in practice.

unwanted objects. Therefore, it is realistic to introduce a third term in the CS model (15) to account for clutter and other interfering signals. In the presence of clutter, the signal measured

where c represents the clutter. As stated in [20], merging the clutter ð Þc and the additive (unstructured) noise ð Þe components allows us to address this kind of problems. However, it is important to highlight that merging the noise ð Þe and the clutter components ð Þc or ignoring the weaker component (usually additive noise) leads to poor performance in most practical cluttered environments. Therefore, it is important to derive a novel framework for recovering the sparse signals from corrupted measurements by noise and clutter. The developed method should be able to obtain knowledge on the structure of clutter. This knowledge can be used to

However, adding clutter as in (16) makes the sparse recovery problem much more challenging since, as stated in [21], the clutter covariance matrix is usually unknown in radar applications and has to be estimated from the observed data. This means that in addition to the sparse vector s, the noise and the clutter covariance are other unknowns that should be estimated

To be more precise, the ultimate goal is to determine the non-zero elements of vector s from far fewer measured samples y generated by (16), while the covariance matrix of the clutter c and the variance of the noise e are both unknown. Similar to the Blind CS problem, reconstruction

As we discussed in previous subsections, reconstruction of sparse radar signals under the scenario where the received vector is contaminated by clutter and/or perfect knowledge of the basis (dictionary) matrix is not available, is very complicated. In the canonical CS formulation, i.e., y ¼ Φψsþe, only two unknowns exist: sparse vector s and e and its variance. However, in both Blind CS and cluttered environment cases, in addition to the sparse vector s and variance of the noise, the basis matrix (covariance matrix of the clutter) is another unknown that should be estimated from the measurement vector y:Unfortunately, none of the proposed approaches

Applying a probabilistic approach seems to be the best method to efficiently handle these complex problems. As mentioned in [20], most probabilistic approaches for sparse signal recovery are Bayesian. It is well known that the Bayesian methods regularize the underdetermined problem by employing priors on the regression coefficients. In fact, since Bayesian methods estimate the posterior distribution of the unknown coefficients instead of their point estimates, they gain more information than other methods. Also, it is worth mentioning that the performance of Bayesian approaches is better than that of other techniques, especially when a probabilistic model is a reasonable representation of the physical process that gener-

discriminate the intended signal from the contaminating sources.

of sparse radar signals in the presence of clutter is a complicated problem.

for conventional CS formulation is applicable to these complex scenarios.

y ¼ Φψs þ e þ c, (16)

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by the radar receiver can be written as

from measurements y.

3.6. Proposed solution

ates the observations [20].

Also, in some other applications such as electronic surveillance and threat recognition, electronic warfare (EW) [14] receivers are preferred to both passive and active radars. Particularly, EW receivers need neither probing pulses nor illuminator signals, since they listen to the electromagnetic radiations of the potential threats instead of weak reflected radar (illuminator) signals. More specifically, EW receivers detect potential targets through sensing the electromagnetic spectrum, and extracting (and/or analyzing) the characteristics of their transmissions. Therefore, their detection range is much higher than that of radars, while they remain electronically silent and undetectable. However, applying standard CS techniques in the context of EW receivers is not possible. This is because a priori knowledge about the transmitted signal by unknown source is not available at all, which means a receiver is not able to build the basis matrix ð Þ ψ .

The above-mentioned scenarios motivated us to study the CS problem for the case where perfect knowledge of the basis matrix ð Þ ψ is not available (ψ is completely unknown or noisy). This interesting scenario can be referred to as the blind CS problem [11]. In summary, the blind CS problem aims to recover the sparse vector s from measurements y obtained from y ¼ Φψsþe, while the basis matrix ψ is unknown. This is in sharp contrast to the existing CS literature which is based on a perfect knowledge of the basis matrix. In general, solving such a problem is very hard since we have three unknowns: sparse vector s, basis matrix ψ, and noise vector e. An efficient CS-based method which iteratively solves this problem has been proposed in [14].

#### 3.5. CS-based radars in cluttered environments

Clutter, a term used for unwanted echoes, can cause serious performance issues with radar systems. Although CS has been applied to different radar problems, it has not yet been studied with respect to clutter and other structured noises. It is not possible to neglect the effects of clutter since clutter is produced from nearly all surfaces when illuminated by a radar, such as ground, sea, rain, animals/insects, chaff and atmospheric turbulences. In order to build reliable practical CS-based radars, it is necessary to investigate the harmful effects of clutter and other environmental factors on the CS performance. Up to now, all existing works in the field of CS radar limited their studies to the ideal clutter-free scenario.

The definition of clutter depends on the mission and function of the intended radar system. Usually, in the context of radar, clutter is modeled as the superposition of echoes from all unwanted objects. Therefore, it is realistic to introduce a third term in the CS model (15) to account for clutter and other interfering signals. In the presence of clutter, the signal measured by the radar receiver can be written as

$$y = \Phi \psi \mathbf{s} + \mathbf{c} + \mathbf{c},\tag{16}$$

where c represents the clutter. As stated in [20], merging the clutter ð Þc and the additive (unstructured) noise ð Þe components allows us to address this kind of problems. However, it is important to highlight that merging the noise ð Þe and the clutter components ð Þc or ignoring the weaker component (usually additive noise) leads to poor performance in most practical cluttered environments. Therefore, it is important to derive a novel framework for recovering the sparse signals from corrupted measurements by noise and clutter. The developed method should be able to obtain knowledge on the structure of clutter. This knowledge can be used to discriminate the intended signal from the contaminating sources.

However, adding clutter as in (16) makes the sparse recovery problem much more challenging since, as stated in [21], the clutter covariance matrix is usually unknown in radar applications and has to be estimated from the observed data. This means that in addition to the sparse vector s, the noise and the clutter covariance are other unknowns that should be estimated from measurements y.

To be more precise, the ultimate goal is to determine the non-zero elements of vector s from far fewer measured samples y generated by (16), while the covariance matrix of the clutter c and the variance of the noise e are both unknown. Similar to the Blind CS problem, reconstruction of sparse radar signals in the presence of clutter is a complicated problem.

## 3.6. Proposed solution

3.4. Blind compressive sampling

basis matrix ð Þ ψ .

posed in [14].

detection), which is not acceptable in practice.

3.5. CS-based radars in cluttered environments

radar limited their studies to the ideal clutter-free scenario.

In some radar scenarios, knowledge of the basis matrix ð Þ ψ is available. For example, in MIMO radar scenario, the transmitted waveforms are known a priori. Therefore, a receiver can use this knowledge to construct the basis matrix locally. However, in some practical cases like passive radars, the transmitters are not part of the radar system, and hence, perfect knowledge of the transmitted waveforms is not available. Therefore, a receiver is only able to reconstruct from a noisy version of the basis matrix ð Þ ψ : As discussed in the related literature, erroneous basis matrices can lead to a significant performance loss (huge decrease in probability of

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

Also, in some other applications such as electronic surveillance and threat recognition, electronic warfare (EW) [14] receivers are preferred to both passive and active radars. Particularly, EW receivers need neither probing pulses nor illuminator signals, since they listen to the electromagnetic radiations of the potential threats instead of weak reflected radar (illuminator) signals. More specifically, EW receivers detect potential targets through sensing the electromagnetic spectrum, and extracting (and/or analyzing) the characteristics of their transmissions. Therefore, their detection range is much higher than that of radars, while they remain electronically silent and undetectable. However, applying standard CS techniques in the context of EW receivers is not possible. This is because a priori knowledge about the transmitted signal by unknown source is not available at all, which means a receiver is not able to build the

The above-mentioned scenarios motivated us to study the CS problem for the case where perfect knowledge of the basis matrix ð Þ ψ is not available (ψ is completely unknown or noisy). This interesting scenario can be referred to as the blind CS problem [11]. In summary, the blind CS problem aims to recover the sparse vector s from measurements y obtained from y ¼ Φψsþe, while the basis matrix ψ is unknown. This is in sharp contrast to the existing CS literature which is based on a perfect knowledge of the basis matrix. In general, solving such a problem is very hard since we have three unknowns: sparse vector s, basis matrix ψ, and noise vector e. An efficient CS-based method which iteratively solves this problem has been pro-

Clutter, a term used for unwanted echoes, can cause serious performance issues with radar systems. Although CS has been applied to different radar problems, it has not yet been studied with respect to clutter and other structured noises. It is not possible to neglect the effects of clutter since clutter is produced from nearly all surfaces when illuminated by a radar, such as ground, sea, rain, animals/insects, chaff and atmospheric turbulences. In order to build reliable practical CS-based radars, it is necessary to investigate the harmful effects of clutter and other environmental factors on the CS performance. Up to now, all existing works in the field of CS

The definition of clutter depends on the mission and function of the intended radar system. Usually, in the context of radar, clutter is modeled as the superposition of echoes from all As we discussed in previous subsections, reconstruction of sparse radar signals under the scenario where the received vector is contaminated by clutter and/or perfect knowledge of the basis (dictionary) matrix is not available, is very complicated. In the canonical CS formulation, i.e., y ¼ Φψsþe, only two unknowns exist: sparse vector s and e and its variance. However, in both Blind CS and cluttered environment cases, in addition to the sparse vector s and variance of the noise, the basis matrix (covariance matrix of the clutter) is another unknown that should be estimated from the measurement vector y:Unfortunately, none of the proposed approaches for conventional CS formulation is applicable to these complex scenarios.

Applying a probabilistic approach seems to be the best method to efficiently handle these complex problems. As mentioned in [20], most probabilistic approaches for sparse signal recovery are Bayesian. It is well known that the Bayesian methods regularize the underdetermined problem by employing priors on the regression coefficients. In fact, since Bayesian methods estimate the posterior distribution of the unknown coefficients instead of their point estimates, they gain more information than other methods. Also, it is worth mentioning that the performance of Bayesian approaches is better than that of other techniques, especially when a probabilistic model is a reasonable representation of the physical process that generates the observations [20].

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

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

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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 low-

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

• Step 1: All reference points (also known as observing receivers) send their collected

• Step 2: Fusion Centre estimates TDOAs between different reference points involved in the

• Step 3: Fusion Centre solves the equations that relate the unknown source position to the

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

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

complexity CS based framework for practical localization networks.

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

Centre, which performs the localization.

applications.

4.2. TDOA-based localization

estimated TDOAs.

accuracy substantially.

samples to the Fusion Centre.

positioning of unknown sources.

#### 3.7. Complexity analysis

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 sources.
