**1. Introduction**

Array processing research has flourished and raked-in much attention in the past five to six decades. It has been an evergreen topic that has fancied many researchers due to the sheer variety of its applications. Array Processing is a scientific field of study which involves the processing of information-bearing signals received by an array of sensors operating in an environment of interest [1], for example, on the ground, above ground or under water. An array has two or more sensors which are arranged in a specific geometrical layout. An array has better directional properties than an individual sensor.

Sensor arrays find application in diverse fields such as radar (radio detection and ranging), space exploration, sonar (sound navigation and ranging), seismology, chemical sensing, medical imaging, wireless communications, navigation, source localization etc. Depending on the application, the sensors could be antennas, microphones, hydrophones, geophones, ultrasonic probes etc. For example, hydrophone arrays are used in sonar; acoustic arrays are used for audio source localization; piezoelectric sensors are used in medical ultrasound; geophone arrays are used in seismology etc. More specifically, antenna arrays are used for electromagnetic applications such as radar, radio astronomy, remote sensing, wireless communications, positioning and navigation [2]. Exhaustive reading for the topic of array processing can be found in [3–5]. A mention of the multidisciplinary nature of this topic is given in the introductory chapter of [4] and final chapters of [3, 5]. A thorough coverage of phased array theory and array processing applications in the modern era is provided in [6]. Arrays can have different geometries – such as linear, planar, circular, hexagonal, spherical etc. An array spanning across more dimensions can extract more details about the scene of interest.

In many of the above applications, it is of primary interest to sense the directions from which the source signals impinge the array. The signal of interest may be an electromagnetic wave, a sound wave, an underwater acoustic wave, a gas diffused into the surroundings, the location of a tumor etc. The use of multiple sensors bestows the array with a sense of direction. Individual sensors cannot sense direction.

A uniform linear array (array in which the antenna elements are arranged along a straight line) with an inter-element spacing of *d* is given in **Figure 1**.

Direction of Arrival (DOA) estimation involves determining the directions/ angles at which electromagnetic sources are located with reference to the receiver array or the directions from which electromagnetic signals strike the array [7]. DOA estimation methods have been classified into three broad approaches, namely, classical methods, subspace methods and maximum-likelihood (ML) methods [8]. A new class of DOA estimation algorithms based on compressed sensing (CS) and sparse recovery have received much attention in the recent past [9–12]. Dealing with coherent arrivals is a main problem in DOA estimation of practical sources. Two signals are said to be coherent if one is a scaled and shifted version of the other. Multipath interference and intentional jamming are the main causes for sources being coherent to each other [13]. When coherent sources arrive at the array, the array covariance matrix becomes non-diagonal, singular and rank-deficient. That is, its rank would be less than the number of incoming signals [14]. Hence, when subspace methods are used for DOA estimation, an additional step of spatial smoothing would be needed to restore the rank of the covariance matrix. Also, a major drawback of subspace methods is that they need prior information about the number of source angles to be detected, which is often impossible in practical

**Figure 1.** *A uniform linear array (ULA) with* N *sensors.*
