Closed-form expression for resolution limit of direction-of-arrival estimation in co-prime array

Abstract

Utilizing co-prime linear arrays (CLA) in place of uniform linear arrays can greatly improve the direction-of-arrival (DOA) resolution with the same number of elements. However, the explicit DOA resolution limit (DRL) of the CLA is unavailable and the resolution gain has not been investigated sufficiently. In this work, Shannon’s information theory is utilized to establish the closed-form expression of the DRL of the CLA. For complex Gaussian sources, we derive the scattering information of duo-source whose amplitudes are the same. A critical state is picked in which the scattering information’s quadrature part equals 1 bit, and the DOA separation is determined to be the DRL. The explicit DRL is then obtained by a Taylor expansion, which is algorithm-independent and can be applied to all signal-to-noise ratios. The expression illustrates that the DRL is approximately inversely proportional to the direction cosine, the root-mean-square aperture width, and the square root of the signal-to-noise ratio. In addition, the quantitative relationship between the number of elements, sparse array aperture, and the optimal resolution limits of three kinds of common-used sparse arrays is obtained, which is of practical significance to the sparse array design.