Performance Analysis for Uniform Linear Arrays Exploiting Two Coprime Frequencies

Abstract

Sparse arrays can achieve a higher number of degrees of freedom (DOFs) compared with uniform linear array (ULA) counterparts. To further reduce the number of physical sensors while keeping a high number of DOFs, a direction of arrival (DOA) estimation algorithm by exploiting coprime frequencies base on a sparse ULA is recently proposed. However, the performance of such approach is not properly analyzed. In this letter, we analyze the Cramer–Rao bound (CRB) as the lower bound of the DOA estimation performance. The difference between the results presented in this letter and the recent CRB results on sparse arrays lies primarily in the additional phases occurred when utilizing different frequencies. It is shown in this letter that the phases affect the covariance matrix of the received data vector and, as a result, change the number of resolvable sources and alter the achieved CRB. We first demonstrate the effect of the additional phases with an example of two closely spaced sources, and the CRB for a sparse ULA exploiting two coprime frequencies is then derived. Numerical simulations are provided to validate the analyses.