2. Compressive sampling basics

innovative ADCs, specifically when sampling rates reach several gigahertz. This is because employing traditional ADCs would result in very expensive architectures, particularly at high sampling rates; since they require a large die area and consume extremely high power even for

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

To overcome these issues, different methods, such as the time-interleaving structure and the multichannel filter-bank approach, have been proposed in the literature. Unfortunately, these techniques are only suitable for special scenarios or applications because of their high power consumption and their non-ideal characteristics. For further details, the interested reader can refer to [1–4] and the references therein. More recently, a novel sampling strategy based on the emerging technique of compressive sensing (CS) [5–7], is able to reduce the sampling rates considerably. This strategy is also called compressive sampling since it breaks through the

The technique of CS enables the reconstruction of a sparse (compressible) signal vector x ∈R<sup>N</sup>

where Φ ∈R<sup>M</sup>�<sup>N</sup> is the measurement matrix with M < < N. It has been demonstrated that the

During the last decade, CS has been successfully applied in many applications, such as image processing [8], wireless sensor networks and communication networks [9]. Although some recent works have studied the application of CS to radar and localization [10], several important questions have not yet been answered. For example, the application of CS in electronic warfare and passive radar scenarios has not been studied well. Furthermore, the effects of clutter and other structured noises on the performance of CS-based radar systems have not been comprehensively investigated to the knowledge of the authors. The extra computational burden caused by signal reconstruction methods is another practical challenge that should be carefully considered. While all previous works have improved our knowledge, the above shortcomings greatly motivate us to provide a more realistic analysis that addresses these important issues. The main aim of this chapter is to investigate some aspects of CS applied to radar concerning the reduction of ADCs' sampling rates. Also, we present a novel CS-based strategy to decrease the traffic of large-scale localization networks with reduced computational complexity. It is not the aim of this chapter to investigate numerically efficient algorithms but to point out some problems, arising when designing and developing practical CS-based radars

In the literature, the terms "compressive sensing" and "compressive sampling" are used interchangeably. Here, we make the distinction that compressive sensing means a dimension reduction of a data vector using a compressed sensing measurement matrix. Compressive sampling is the reduction of the sampling rate (from the Nyquist rate) in the digitization of an

The organization of this chapter is as follows. In Section 2, we briefly review the basics of CS theory. In Section 3, we first present the radar main concepts. Then, we show how employing

proper selection of the measurement matrix Φ is a key point for the success of CS [6].

y ¼ Φx (1)

resolutions as low as 7-bits [1–4]

limits of the Nyquist sampling theory.

from the measurement vector y∈ RM generated by

and localization systems, as well as possible solutions.

analog signal.

Suppose a signal x ∈RN can be represented as:

$$\mathbf{x} = \boldsymbol{\Psi} \,\mathrm{s},\tag{2}$$

where ψ ¼ ψ1;…; ψ<sup>N</sup> ∈ RN�<sup>N</sup> is a full rank, orthonormal basis matrix, sometimes also known as the basis matrix [5], and the vector <sup>s</sup> <sup>¼</sup> ½ � s1; …; sN <sup>∈</sup>R<sup>N</sup> has <sup>K</sup> non-zero elements. Then the signal x is K-sparse. In fact, the signal x is K-sparse if it is a linear combination of only K basis vectors; that is, only K elements of vector s in (2) are nonzero and the other (N-K) elements are zero. Also, we say that the signal x is compressible if the representation (2) has just a few large coefficients and many small coefficients in s. A sparse signal is compressible as described below. Putting (2) into (1) yields:

$$
\mathbf{y} = \mathbf{O}\boldsymbol{\uppsi}\mathbf{s} = \boldsymbol{\upTheta}\mathbf{s},\tag{3}
$$

where <sup>Θ</sup> <sup>¼</sup> Φψ <sup>∈</sup> RM�<sup>N</sup> is called the sensing matrix. Since <sup>M</sup> <sup>&</sup>lt; N, <sup>y</sup> is a compressed measurement of x. Such compression makes possible the storage and transmission of x at a lower dimension. Hence, CS is an important data compression technique.

This means that instead of measuring the N-point signal x directly, the CS framework acquires the information from far fewer measurements (M ≪ N) than traditional methods. Notice that since Θ has far fewer rows than columns, (3) is non-invertible and underdetermined, rendering the CS problem ill-posed [11]. The main question therefore is: "how to recover x from y?"

In general, (3) has no unique solution since it has more unknowns (N) than equations (M). However, since x is K-sparse, we know that N-K unknown elements in s are zero. It is then possible to recover s from (3) using a technique known as l<sup>1</sup> minimization [6]. From s we recover x via (2).

The signal x is thus compressible and recoverable. It is well known that images and speech signals are compressible signals. Another example is a vector x whose elements are sums of samples of two sinusoids. Its ψ in (2) consists of columns of sinusoidal samples of frequencies w1, …, wN: If the sinusoids have frequencies w<sup>3</sup> and w5, then s has only non-zero elements at positions 3 and 5.

There are mathematical conditions on Θ, and the numbers of measurements M, needed to ensure recovery of s. They are given in the next three subsections.

#### 2.1. Number of measurements

Choosing the number of measurements M is a trade-off: While a small M is desirable for high compression, it must be sufficiently large to enable reconstruction. Generally M should be in the order of log <sup>2</sup> N <sup>K</sup> . A rule of thumb is M ffi 4K ½ � 6 provided that Θ satisfies both the conditions of Incoherence (Subsection 2.2) and Restricted isometry property (Subsection 2.3).

#### 2.2. Incoherence

From (3), it is seen that the elements of y are a linear combination (l.c.) of the elements of s, via the matrix Φψ. If Φ is highly correlated to ψ, the probability of having independent l.c. (or measurements) of s decreases.

To see this, suppose Φψ has a column, say the i-th column that contains all zeros. Then y is missing a measurement of the i-th element in s, and if this element is non-zero, we cannot recover s from y. This will happen if a row of Φ is orthogonal to a column of ψ, i.e., there is a strong correlation between Φ and ψ. In CS theory, there is a theorem that relates the required number of measurements M, for perfect reconstruction, to the coherency (a numerical number) of Φ and ψ. The higher the coherency, the higher the required M is.

#### 2.3. Restricted isometry property (RIP)

For Θ ¼ Φψ, the RIP requires that for perfect reconstruction, Θ must satisfy the inequality

$$\alpha \|\mathbf{s}\| \le \|\boldsymbol{\Theta}\mathbf{s}\| \le \boldsymbol{\beta} \|\mathbf{s}\|,\tag{4}$$

is a non-sparse vector; in this case, the recovered vector will consist of the K most significant

Checking if a chosen Θ satisfies the RIP criteria is computationally expensive and cannot be realized in practice. It has been shown in [6] that with high probability, random Gaussian,

Finally, in practice, noise is commonly present in the measurement. Therefore, (3) becomes

where e ∈R<sup>M</sup> represents an unknown noise vector. Recovering s from y will yield additional errors due to e. A bound on this error, as a function of the power of e, can be obtained [7].

In some applications, for example in an electronic warfare (EW) receiver, a high sampling rate is required because the receiver must scan for signals with high bandwidths. High rate ADCs have low accuracy and consume high power. In addition, the high number of samples can fill up the available memory quickly. As we will show, compressive sampling can essentially reduce the sampling rate and number of samples via a system known as the random-

A simplified block diagram of the sampling scheme given in [12] is shown in Figure 1. An

Here, x(t) represents the input signal with bandwidth B Hz. Also, note that the Nyquist

<sup>4</sup>�<sup>52</sup> Hz: So, the actual sampling rate is <sup>2</sup><sup>B</sup>

<sup>2</sup><sup>B</sup> seconds of the product of x(t) and PRBS, then resets and repeats. We have:

y ¼ Φψs þ e ¼ Θs þ e, (6)

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<sup>2</sup><sup>B</sup> second. Since the duration of 52 bits

<sup>52</sup> Hz and each integrator

Bernoulli, and partial Fourier matrices do satisfy the RIP condition.

elements.

is <sup>52</sup>

sums <sup>52</sup>

2.4. Sampling rate reduction by CS

modulation pre-integrator (RMPI) [12].

<sup>2</sup>B, an ADC samples at <sup>2</sup><sup>B</sup>

Figure 1. Block diagram of RMPI with four channels.

RMPI with four output channels generates a CS vector y∈R<sup>80</sup>:

sampling rate is 2B Hz, and one-bit duration is equal to <sup>1</sup>

with 0 < α < 1 and 1 < β < 2:

Notice that the isometry is the length of a vector. The inequality (4) limits the amount by which Euclidean distance k k Θs can differ from k ks : The lower bound in (4) ensures a perfect recovery. Suppose k ks 6¼0 but k k Θs ¼ 0. This violates the lower bound of (4). Indeed, if y ¼ Φψs ¼ 0, we cannot recover s from y. Note that k k Θs ¼ 0 when k ks 6¼0 implies that s is in the null space of Θ, meaning that at least two columns of Θ are linearly dependent. Further to this point, consider two K-sparse vector s and s<sup>∗</sup>. The maximum sparsity of s-s<sup>∗</sup>is 2 K. To be able to distinguish Θs from Θs<sup>∗</sup>, we must have

$$
\Theta(\mathbf{s} - \mathbf{s}^\*) \neq 0. \tag{5}
$$

i.e., any 2 K columns of Θ must be linearly independent. This ensures that Θ satisfy the lower bound of (4).

The upper bound is necessary to keep the lower bound meaningful. Otherwise, Θ can be arbitrary scaled to satisfy the lower bound.

Note that incoherency and RIP both give conditions for perfect reconstruction. But the incoherency condition is valid only for a K-sparse vector. In contrast, RIP applies even when s is a non-sparse vector; in this case, the recovered vector will consist of the K most significant elements.

Checking if a chosen Θ satisfies the RIP criteria is computationally expensive and cannot be realized in practice. It has been shown in [6] that with high probability, random Gaussian, Bernoulli, and partial Fourier matrices do satisfy the RIP condition.

Finally, in practice, noise is commonly present in the measurement. Therefore, (3) becomes

$$
\mathbf{y} = \mathbf{O}\boldsymbol{\uptheta}\mathbf{s} + \mathbf{e} = \boldsymbol{\upTheta}\mathbf{s} + \mathbf{e},\tag{6}
$$

where e ∈R<sup>M</sup> represents an unknown noise vector. Recovering s from y will yield additional errors due to e. A bound on this error, as a function of the power of e, can be obtained [7].

#### 2.4. Sampling rate reduction by CS

2.1. Number of measurements

measurements) of s decreases.

2.3. Restricted isometry property (RIP)

distinguish Θs from Θs<sup>∗</sup>, we must have

arbitrary scaled to satisfy the lower bound.

bound of (4).

with 0 < α < 1 and 1 < β < 2:

N

the order of log <sup>2</sup>

2.2. Incoherence

Choosing the number of measurements M is a trade-off: While a small M is desirable for high compression, it must be sufficiently large to enable reconstruction. Generally M should be in

From (3), it is seen that the elements of y are a linear combination (l.c.) of the elements of s, via the matrix Φψ. If Φ is highly correlated to ψ, the probability of having independent l.c. (or

To see this, suppose Φψ has a column, say the i-th column that contains all zeros. Then y is missing a measurement of the i-th element in s, and if this element is non-zero, we cannot recover s from y. This will happen if a row of Φ is orthogonal to a column of ψ, i.e., there is a strong correlation between Φ and ψ. In CS theory, there is a theorem that relates the required number of measurements M, for perfect reconstruction, to the coherency (a numerical number)

For Θ ¼ Φψ, the RIP requires that for perfect reconstruction, Θ must satisfy the inequality

Notice that the isometry is the length of a vector. The inequality (4) limits the amount by which Euclidean distance k k Θs can differ from k ks : The lower bound in (4) ensures a perfect recovery. Suppose k ks 6¼0 but k k Θs ¼ 0. This violates the lower bound of (4). Indeed, if y ¼ Φψs ¼ 0, we cannot recover s from y. Note that k k Θs ¼ 0 when k ks 6¼0 implies that s is in the null space of Θ, meaning that at least two columns of Θ are linearly dependent. Further to this point, consider two K-sparse vector s and s<sup>∗</sup>. The maximum sparsity of s-s<sup>∗</sup>is 2 K. To be able to

i.e., any 2 K columns of Θ must be linearly independent. This ensures that Θ satisfy the lower

The upper bound is necessary to keep the lower bound meaningful. Otherwise, Θ can be

Note that incoherency and RIP both give conditions for perfect reconstruction. But the incoherency condition is valid only for a K-sparse vector. In contrast, RIP applies even when s

αk ks ≤ k k Θs ≤ βk ks , (4)

<sup>Θ</sup> <sup>s</sup> � <sup>s</sup><sup>∗</sup> ð Þ 6¼ <sup>0</sup>: (5)

of Φ and ψ. The higher the coherency, the higher the required M is.

tions of Incoherence (Subsection 2.2) and Restricted isometry property (Subsection 2.3).

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

<sup>K</sup> . A rule of thumb is M ffi 4K ½ � 6 provided that Θ satisfies both the condi-

In some applications, for example in an electronic warfare (EW) receiver, a high sampling rate is required because the receiver must scan for signals with high bandwidths. High rate ADCs have low accuracy and consume high power. In addition, the high number of samples can fill up the available memory quickly. As we will show, compressive sampling can essentially reduce the sampling rate and number of samples via a system known as the randommodulation pre-integrator (RMPI) [12].

A simplified block diagram of the sampling scheme given in [12] is shown in Figure 1. An RMPI with four output channels generates a CS vector y∈R<sup>80</sup>:

Here, x(t) represents the input signal with bandwidth B Hz. Also, note that the Nyquist sampling rate is 2B Hz, and one-bit duration is equal to <sup>1</sup> <sup>2</sup><sup>B</sup> second. Since the duration of 52 bits is <sup>52</sup> <sup>2</sup>B, an ADC samples at <sup>2</sup><sup>B</sup> <sup>4</sup>�<sup>52</sup> Hz: So, the actual sampling rate is <sup>2</sup><sup>B</sup> <sup>52</sup> Hz and each integrator sums <sup>52</sup> <sup>2</sup><sup>B</sup> seconds of the product of x(t) and PRBS, then resets and repeats. We have:

Figure 1. Block diagram of RMPI with four channels.

$$\mathbf{x} = \begin{bmatrix} \mathbf{x}(1), \dots, \mathbf{x}(1040) \end{bmatrix}^T \tag{7}$$

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,

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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

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

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 investi-

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.

most recent works have focused on the MIMO scenario.

radar exploits the waveform diversity.

gated; please see [15, 19] and the references therein.

3.2. CS for radar

systems.

$$\mathbf{y} = \begin{bmatrix} y(1), \dots, \mathbf{y}(80) \end{bmatrix}^T \tag{8}$$

$$\mathbf{OP} = \begin{bmatrix} \mathbf{A}\_1 \ 0 \ \dots \ & \mathbf{0} \\ 0 \ \mathbf{A}\_2 \ \dots & \mathbf{0} \\ \vdots & \vdots & \ddots & \vdots \\ 0 \ 0 \ \dots & \mathbf{A}\_{20} \end{bmatrix} \tag{9}$$

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, as is the number of samples, i.e., <sup>N</sup> <sup>M</sup> ¼ 13:
