**1.1 Motivation for sparse arrays**

Though there are many definitions of sparse arrays in the array processing literature, the focus of this book chapter would be only on grid-based sparse arrays.

In grid-based sparse arrays, the sensors are assumed to be located on the grid points defined by an integer multiple of the basic inter-element spacing. That is, the sensors are assumed to lie at 0, *d*, 2*d*, and so on. In general, the inter-element spacing would be half wavelength. The element positions are normalized to the half wavelength. Accordingly, a sparse array with sensor positions {0, 2, 4, 5} means that the array has an overall aperture of five units and its sensors are located at the grid points 0, 2*d*, 4*d* and 5*d*. There are no sensors at grid points *d*, 3*d*. As the array is sparsely populated with sensors compared to a regular/full array, it is called as a sparse array. Sparse arrays need fewer sensors or active elements than ULAs to realize a given aperture length. Hence, they provide huge savings in the system costs associated thereof (e.g., feed, power consumption, and radio frequency chains). They are a type of green technology in array processing.

Different types of linear sparse arrays have been studied in the past. Minimum Redundancy Arrays (MRAs) have numerous useful properties and had been primarily studied in the past in relation to radio astronomy [21, 22]. MRAs have also been applied in digital communications [23–25]. Numerous properties and modern applications of MRAs can be found in [26–28].

The use of sparse or non-uniform arrays in close future communication systems such as fifth generation (5G) telecom and the Internet of Things (IoT) is appealing as sparse arrays need fewer active elements than ULAs and operate with little or no degradation in performance [29].

#### **1.2 Contribution of this chapter**

The area of sparse arrays is continuously evolving and there are at least 50 types of linear sparse arrays in the current literature. A comprehensive understanding of various sparse array configurations is essential in the current scenario as more and more sparse arrays are being rapidly introduced into the literature. To the best of our knowledge, a comprehensive review on the properties of 1D sparse arrays has not been taken up in the past (barring the works by Liu and Vaidyanathan, Cohen and Eldar). This is a sincere attempt to bring a few well-known sparse arrays under one roof so that their characteristics could be compared.

The rest of the chapter is organized as follows. Section 2 gives an introduction to linear sparse arrays. Section 3 explains some terminology related to sparse arrays and discusses the characteristics of a few well-known sparse arrays. Section 4 gives an overview of sparse arrays with special properties and also discusses the effect of sensor failures on array performance. Section 5 provides a few future directions and Section 6 concludes the chapter.
