Investigations on n-tuple coprime arrays

Abstract

Coprime arrays so far combine two sparsely-spaced subarrays, undersampled by factors of M and N, in order to achieve MN degrees of freedom. To ensure that the grating lobes of each subarray can be largely eliminated, M and N must be coprime. In number theory, sets of pairwise coprime numbers can exceed just two numbers. The current work extends the theory to include coprime linear arrays with an arbitrary number, n, of subarrays. A triple coprime array comprised of n = 3 equal-aperture subarrays with M, N, and O elements, undersampled by factors of NO, MO, and MN, respectively, may use just M + N+O-1 shifts to observe MNO directions. The design frequency of such an array not only exceeds the Nyquist spatial sampling limit, but is also greater than that of a standard (double) coprime array with equivalent aperture and number of elements. A triple coprime array is constructed with subarrays of M = 3, N = 4, and O = 5 and measured in a simulated free field condition. Experimental validation of the array confirms that the triple coprime array can also observe lower frequencies up to the design frequency. This paper discusses advantages and practical significance of n-tuple coprime microphone arrays over conventional double coprime linear arrays.