*3.2.6 Super-nested arrays*

Though nested arrays are better than co-prime arrays in terms of the ability to provide hole-free co-arrays, they are severely affected by mutual coupling. This is due to the dense ULA portion at the beginning (level 1). Super-nested arrays were introduced to overcome this drawback of nested arrays [34]. In super-nested arrays, the level 1 elements of the NA are re-arranged (interleaved) to different positions within the span of the array so that the number of sensors with unit spacing gets reduced, thereby making the array less susceptible to mutual coupling. A 10-element super-nested array has sensors at {0, 2, 4, 7, 9, 11, 17, 23, 28, 29}. It can be observed that the level 1 elements of the NA are interleaved to different positions. Super-nested arrays too provide the same aperture as nested arrays for a given number of sensors and have CFEs for element positions. The formulation of supernested arrays is slightly complicated and is, therefore, not explained here.

As a continuation, augmented nested arrays (ANAs) [49] were formulated. In ANAs, the level 1 dense sub-array of NA is split into several parts and is re-arranged to the left and right of the level 2 sparse array. The design of ANAs is elegant as they provide larger apertures, higher DOFs, and are less susceptible to mutual coupling than nested and super-nested arrays.

### *3.2.7 Yang's improved nested array*

An improved nested array (INA) that provides larger aperture than the nested array for the same number of sensors has been proposed [50]. The improved nested array has a hole-free co-array. However, like the original nested array, the Yang's nested array is also vulnerable to the effects of mutual coupling as it too has a dense ULA portion at the beginning. This array has a total of *N* ¼ *N*<sup>1</sup> þ *N*<sup>2</sup> þ 1 sensors of which *N*<sup>1</sup> sensors at level 1 with unit spacing, *N*<sup>2</sup> sensors at level 2 with an interelement spacing of *N*<sup>1</sup> þ 2 and a separate sensor at ð*N*1*N*<sup>2</sup> þ 2*N*<sup>2</sup> þ *N*<sup>1</sup> � 1).

The values of *N*<sup>1</sup> and *N*<sup>2</sup> are given by

$$N\_1 = \frac{N}{2} - \mathbf{1}, N\_2 = \frac{N}{2}; N \text{ even}$$

$$N\_1 = \frac{N-1}{2} - \mathbf{1}, N\_2 = \frac{N+1}{2}; N \text{ odd} \tag{5}$$

It follows that a 10-element Yang's nested array has *N*<sup>1</sup> ¼ 4 and *N*<sup>2</sup> ¼ 5. The separate sensor at the end is located at the position 33. The overall 10-element Yang's improved nested array is given by {0, 1, 2, 3, 4, 10, 16, 22, 28, 33}. It can be observed that the first few sensors are adjacent to each other and have unit spacing.

An extended nested array was also proposed in 2016 [51]. However, it does not offer apertures as large as the Yang's INA described here.

### *3.2.8 Huang's nested array*

A nested array configuration that provides larger aperture than the two-level nested array has been recently proposed [52]. This array provides larger aperture than MRAs (obviously than nested, super-nested, improved nested, and co-prime arrays) for a given number of sensors. However, Huang's nested array suffers from holes in the co-array. The construction is similar to that of the Yang's nested array in that there is a level 1 ULA, level 2 ULA with increased spacing and a separate sensor at the end. However, the number of sensors at each level and the element spacing in level 2 determine the sensor locations and the overall behavior of the array.

#### *3.2.9 Triply primed array*

The triply primed array (TPA) is the union of three ULAs with different interelement spacings. Three mutually prime numbers *N*1, *N*<sup>2</sup> and *N*<sup>3</sup> must be selected [53]. The total sensors in the TPA are *N*<sup>1</sup> þ *N*<sup>2</sup> þ *N*<sup>3</sup> � 2. The first ULA has *N*<sup>1</sup> elements separated by an inter-element spacing of *N*2*N*3. The second ULA has *N*<sup>2</sup> elements separated by an inter-element spacing of *N*1*N*3. The third ULA has *N*<sup>3</sup> elements separated by an inter-element spacing of *N*1*N*2. For example, *N*<sup>1</sup> ¼ 3, *N*<sup>2</sup> ¼ 4, *N*<sup>3</sup> ¼ 5 represents a 10-element TPA. The problem with TPAs is that their DCA has a smaller number of continuous lags. Therefore, fourth order statistics are made use of (i.e., the DCA of the DCA is used for DOA estimation).

**Table 1** lists out the optimum sensor positions for different 10-element linear sparse arrays. The optimum MHA configuration for 10 sensors has been obtained through table look-up [46]. The sensor positions are shown in **Figure 2**. The continuous part of the DCAs of these sparse arrays are shown in **Table 2**.
