**4.1 Sparse arrays with special properties**

Many modifications to coprime arrays have been proposed in the Literature. For example, the thinned coprime array, the coprime array with compressed interelement spacing (CACIS), coprime array with displaced subarrays (CADiS), coprime array with reduced sensors (CARS) etc. [37, 38, 62]. Most recent is the generalized thinned coprime array (GTCA) [63], of which, the above ones can be considered as special cases. A multi-level prime array which extends the concept of coprime arrays has also been proposed [36]. The three-level prime array (3LPA) is a special case of the multi-level prime array and should not be confused with the triply primed array described in Section 3.2.9.

Recently, a unified array geometry in the form of a generalized nested subarray (GNSA) was proposed [55]. The geometry has a nested structure with two prototype arrays A and B. Specifically, if the prototype arrays A and B are MRAs, then the geometry leads to the nested MRA (NMRA) thereby providing the largest sparse array and associated hole-free coarray. On the other hand, when A and B are ULAs, the design provides a sparse array with the least aperture and DOFs. It can be said that the GNSA has revolutionized the development of sparse arrays in recent times. The prototype arrays could be nested arrays or super-nested arrays, or for that matter, any sparse array that has a hole-free coarray [55].

### *4.1.1 Sparse arrays for active sensing*

Active sensing applications need hole-free sum co-arrays. Symmetric sparse arrays are also useful for certain applications. The concatenated nested array (CNA) is one such array which is obtained by appending the level 1 elements of the nested
