Sparse Fractal Array Design with Increased Degrees of Freedom

Abstract

Sparse arrays are of great interest since they can identify O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) uncorrelated sources with N physical sensors. This stems from their large difference coarray, defined as the differences between sensor locations. In a recent study, desired array properties such as closed-form expression for sensor locations, symmetry and large hole-free difference coarray were considered and it was shown that most existing sparse arrays do not exhibit these characteristics simultaneously. Standard Cantor arrays were shown to satisfy all the criteria above, however, their difference coarrays are of size O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">log2 3</sup> ) which is smaller than that obtained with minimum redundancy arrays and nested arrays. In this paper, we introduce a fractal array design where a generator array is extended in a simple recursive fashion. In contrast to previous work, the generator is assumed to be a sparse array with a hole-free difference coarray. We study the resulting arrays and prove they inherit their properties from the generator. Thus, this approach can be used to extend any known sparse configuration to an arbitrarily large array. A small-scale array, which meets all design criteria, can be created and then expanded to generate a symmetric fractal array with a difference coarray of size O(N2), unlike Cantor arrays.