1. Introduction

The well-known Nyquist sampling theorem, which has served as a starting point for development of traditional analog-to-digital converters (ADCs), states that the sampling rate needs to be at least twice as high as the bandwidth of the input signal to achieve aliasing-free sampling. Moving to higher communication throughputs and carrier frequencies motivates the search of

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 resolutions as low as 7-bits [1–4]

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 limits of the Nyquist sampling theory.

The technique of CS enables the reconstruction of a sparse (compressible) signal vector x ∈R<sup>N</sup> from the measurement vector y∈ RM generated by

$$\mathbf{y} = \boldsymbol{\Phi} \mathbf{x} \tag{1}$$

CS can reduce the sampling rate substantially. Next, we introduce two main scenarios where the standard CS radar formulation is not applicable. Finally, we propose an efficient technique that can be used to address the shortcomings of existing methods. In Section 4, we first briefly review the main concepts of localization. Then, we explain how the reconstruction step of the CS technique significantly increases the complexity of localization algorithms. Finally, we introduce a novel method which eliminates the complex recovery step of CS-based localizers

∈ 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

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

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

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

There are mathematical conditions on Θ, and the numbers of measurements M, needed to

x ¼ ψ s, (2)

Applications of Compressive Sampling Technique to Radar and Localization

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

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y ¼ Φψs ¼ Θs, (3)

and reduces the data traffic between the sensors and the fusion center.

dimension. Hence, CS is an important data compression technique.

ensure recovery of s. They are given in the next three subsections.

2. Compressive sampling basics

where ψ ¼ ψ1;…; ψ<sup>N</sup>

recover x via (2).

positions 3 and 5.

Suppose a signal x ∈RN can be represented as:

described below. Putting (2) into (1) yields:

where Φ ∈R<sup>M</sup>�<sup>N</sup> is the measurement matrix with M < < N. It has been demonstrated that the proper selection of the measurement matrix Φ is a key point for the success of CS [6].

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 and localization systems, as well as possible solutions.

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 analog signal.

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 CS can reduce the sampling rate substantially. Next, we introduce two main scenarios where the standard CS radar formulation is not applicable. Finally, we propose an efficient technique that can be used to address the shortcomings of existing methods. In Section 4, we first briefly review the main concepts of localization. Then, we explain how the reconstruction step of the CS technique significantly increases the complexity of localization algorithms. Finally, we introduce a novel method which eliminates the complex recovery step of CS-based localizers and reduces the data traffic between the sensors and the fusion center.
