**Table 1.**

*Sparse array configurations for 10 physical sensors.*

#### **Figure 2.**

*Sensor positions for various sparse arrays (arrays are considered in the same order as listed in Table 1 from top to bottom).*


#### **Table 2.**

*Continuous portion of the difference co-array of the sparse arrays listed in Table 1.*

The proliferation of linear sparse arrays in the past decade has led to the development of coarray-based DOA estimation methods. Coarray methods are based on the concept of difference co-array (DCA) and are well-suited for angle estimation in sparse arrays. As the physical array has missing sensors due to the sparseness, the array correlation matrix does not represent a Toeplitz structure and is not suitable for estimation of spatial correlation. Therefore, the analysis is shifted to the co-array domain. Due to the continuity of the DCA, the co-array correlation matrix represents a complete Hermitian Toeplitz structure and can be used to estimate spatial angles. Co-array MUSIC algorithm is widely used for DOA estimation in sparse arrays [32, 33]. More recently, other algorithms such as the co-array root-MUSIC [54, 55] and the co-array ESPRIT [56] have been introduced. The Khatri Rao (KR-MUSIC) algorithm which is applicable only to quasi-stationary sources (i.e., the sources which can be assumed to be stationary for short time durations) was introduced prior to the co-array MUSIC [57]. Recently, many algorithms based on compressed sensing have been introduced for DOA estimation in sparse arrays [58, 59]. In summary, DOA estimation algorithms that operate (i) when the number of sources is unknown, (ii) in the presence of coherent arrivals, (iii) under unknown mutual coupling, (iv) under low signal-to-noise ratio (low SNR) conditions, (v) under low snapshot conditions, (vi) in the presence of non-uniform or random noise, and (vii) in a short computational time; are largely sought-after for practical applications [60, 61].
