Adaptive Beamforming for Sparse Array Based on Semi-Definite Programming

An adaptive beamforming (ABF) technique for sparse receiving arrays with gain/phase uncertainties is proposed. The basic idea of the proposed method is using the compressed sensing theory to estimate directions and amplitudes of the received signals with sparse array and then obtain the covariance matrix of the signals through the estimated directions and amplitudes. However, on a discrete grid, the accuracy of directions and amplitudes estimation will degrade because of the basis mismatch and the existence of the gain/phase uncertainties. It will influence the performance of the adaptive digital beamforming. In order to eliminate the influence of the gain/phase uncertainties and the basis mismatch, we propose a semi-definite programming-total least squares (SDP-TLS) method in this paper. First, we convert the problem we want to solve into a TLS framework. Then, we develop an alternating descent algorithm to solve this problem. In the algorithm we proposed, the directions and amplitudes are estimated by semi-definite programming. The covariance matrix of the signals, which is used for ABF, is obtained by the estimated directions and amplitudes. Then, the adaptive digital beamforming algorithm is adopted to form a beam with the obtained covariance matrix.