*Sparse Linear Antenna Arrays: A Review DOI: http://dx.doi.org/10.5772/intechopen.99444*

array just after the level 2 elements such that the overall array is symmetric [40]. However, this array heavily suffers from the effects of mutual coupling owing to the closely spaced elements at both the ends. The Interleaved Wichmann Array (IWA) was proposed to overcome the mutual coupling problem of the CNA by rearranging the sensors such that the number of sensor pairs with unit spacing is reduced [39]. On similar lines, a nested structure using two CNAs, namely, the Kløve array has been introduced with hole-free sum and difference coarray and is suitable for active as well as passive sensing [64]. More recently, low redundancy arrays with nested arrays and Kløve-Mossige as basis were proposed with hole-free sum coarray [65]. The sum coarray is defined as

$$\mathbb{P} = \{\mathbf{z}\_i + \mathbf{z}\_j | i, j = \mathbf{1}, \mathbf{2}, \dots, N\_s\}, \tag{7}$$

where all the cross summations between sensor positions are considered.

## *4.1.2 Sparse arrays for DOA estimation of non-circular sources*

While the difference coarray approach is well suited for the DOA estimation of circular sources, many non-circular source signals exist in practice. For example, binary phase shift keying (BPSK), minimum shift keying (MSK), unbalanced quadrature phase shift keying (UQPSK) etc. Non-circular sources have non-zero pseudo covariances (non-zero ellipse covariance matrix) which can be used to enhance the aperture of the virtual array to further exploit the received information for parameter estimation [66, 67]. To fully leverage the special properties of non-circular sources, the DOA estimation is performed using the sum difference co-array (SDCA) which is defined as

$$
\mathbb{R} = \mathbb{H} \cup \mathbb{P} \cup \mathbb{Q} \tag{8}
$$

where is the difference co-array, is the sum co-array and is the mirrored sum co-array and equal to - . In short, for a sparse array with sensors at ¼ *z*1, *z*2, … , *zNs* f g, the locations of the virtual sensors in the SDCA are given by

$$\mathbb{R} = \left\{ \mathbf{z}\_i - \mathbf{z}\_j, \mathbf{z}\_i + \mathbf{z}\_j, -\mathbf{z}\_i - \mathbf{z}\_j | \mathbf{i}, j = \mathbf{1}, \mathbf{2}, \cdots, N\_t \right\} \tag{9}$$

The use of difference co-array along with the sum co-array increases the virtual array span and leads to increased DOFs than possible by using the DCA alone. In SDCA-based designs, the vectorized conjugate augmented MUSIC (VCAM) algorithm is generally used for DOA estimation [68].

One of the prominent designs for sparse arrays that are suitable for non-circular sources is the nested array with displaced subarray (NADiS) as it provides CFEs for element positions, virtual apertures, and DOFs. The NADiS array has a large central continuous portion in the SDCA thereby providing large uniform DOFs. However, the SDCA is not completely hole-free. As an improvement, sparse array for noncircular sources (SANC) array was proposed with a hole-free SDCA [69]. A drawback of the SANC array is that it has no CFE for element positions. Therefore, the sensor positions in SANC have to be determined through exhaustive searching. Though an improved nested array with SDCA (INAwSDCA) [68] was proposed recently, there is no comparison with the NADiS and SANC arrays. An ultimate design of the nested array for non-circular signals is the translated nested array [67] which has CFEs for sensor positions and provides larger apertures than the NADis and the SANC. In addition, the SDCA of translated nested array is hole-free.

### *4.1.3 Array motion*

In recent years, moving array platforms or array motion are being exploited to obtain higher DOFs in sparse arrays. The synthetic coprime array [70], dilated nested array [71] and the multi-level dilated nested array [72] are few examples of sparse array designs that leverage platform motion to fill the holes in the DCAs. This topic is pretty new and is widely being explored.

## **4.2 Sensor failures in antenna arrays**
