**2. Introduction to sparse arrays**

Sparse arrays have aperture widths equal to regular/filled/full arrays but are sparsely populated. They consist of voids that arise due to missing/inactive sensors. The vacancies are deliberately created and are woven into the design of the sparse array to achieve certain desired characteristics. The span of the array is called its aperture. It is the sum of all the inter-element spacings. A ULA with *N* sensors, each separated by an inter-element spacing of *d* has an aperture distance or aperture length of ð Þ *N* � 1 *d*. The aperture in the units of the inter-element spacing is given by *L* ¼ *N* � 1. Sparse arrays need fewer than *N* sensors (*Ns* < *N*) to offer the same aperture. Failure or absence of either the first sensor or the last sensor in an array of *N* sensors reduces the array aperture by one unit. Failure of both sensors reduces the array aperture by two units. Hence, in the analysis of thinned/sparse arrays or when analyzing arrays with sensor failures, it is generally assumed that the first and the last sensor are always functional, intact and active so that the array aperture is preserved.

Another advantage of sparse arrays is that they are less affected by mutual coupling compared to ULAs. Sparse arrays are generally analyzed in the co-array domain. A difference co-array (DCA) is formed from the physical sparse array by considering all the spatial lags (differences) that can be generated using the available sensors. A missing spatial lag forms a hole. The DCA should be hole-free as the presence of holes introduces ambiguity in the estimation of spatial correlation and hence spatial angles.

There are several types of linear sparse arrays such as minimum hole arrays (MHAs) or Golomb Arrays, minimum redundancy arrays (MRAs), co-prime arrays, nested arrays, super-nested arrays and so on. MRAs and MHAs have been widely used for interferometry in radio astronomy [6, 21, 22, 30, 31]. While MRAs and MHAs existed for more than five decades, other sparse arrays such as co-prime arrays, nested arrays and super-nested arrays have been introduced in the past decade [32–34]. A good review on the properties of these sparse arrays can be found in the initial sections of [27, 35]. The introduction of co-prime array in the past decade can be considered as a watershed moment which has opened doors for modern applications of linear sparse arrays. Following that, nested arrays were introduced. These arrays offer hole-free co-arrays and also have closed-form expressions to determine the sensor locations. Many variants of the co-prime array [36–38] and the nested array [39, 40] have been proposed in the recent past. These arrays either improve the aperture or reduce the number of sensors needed to obtain a given aperture or make the array more immune to mutual coupling or they increase the hole-free region in the DCA.

Following are the desirable characteristics of sparse arrays:


It is desirable that the array has closed-form expressions for sensor positions. Otherwise, there should be a provision to obtain the sensor positions using a lookup table (LUT) or through tabulated entries.
