*3.2.5 Nested array or the two-level nested array*

Nested arrays (NA) can provide hole-free DCAs and were introduced as an alternative to MRAs and as an improvement over co-prime arrays. Nested arrays are better than co-prime arrays as they provide hole-free co-arrays. They too provide CFEs for element positions when the number of sensors is known. Two ULAs are needed to obtain a nested array. The first ULA has *N*<sup>1</sup> sensors with unit spacing and the second ULA has *N*<sup>2</sup> sensors with a spacing of *N*<sup>1</sup> þ 1. The overall nested array is given by the union of these two ULAs [33]. The optimal values of *N*<sup>1</sup> and *N*<sup>2</sup> for a given number of sensors are

$$N\_1 = N\_2 = \frac{N}{2}; N \, even$$

$$N\_1 = \frac{N-1}{2}, N\_2 = \frac{N+1}{2}; N \, odd \, \tag{4}$$

For example, in a 10-element NA, *N*<sup>1</sup> ¼ *N*<sup>2</sup> ¼ 5. The level 1 ULA has sensors at {0, 1, 2, 3, 4} and the level 2 ULA has sensors at {5, 11, 17, 23, 29}. The overall nested array is given by {0, 1, 2, 3, 4, 5, 11, 17, 23, 29}. These arrays are well-known by the name two-level nested arrays (TLNA). Unless otherwise specified, all instances of nested arrays mentioned in this chapter refer to the two-level nested array.
