A Toeplitz Covariance Matrix Reconstruction Approach for Direction-of-Arrival Estimation

Abstract

It is known that there exist two kinds of methods for direction-of-arrival (DOA) estimation in the literature: the subspace-based method and the sparsity-based method. However, pervious works reveal that the former method cannot address the case in which the number of signals is larger than that of sensors, whereas the latter one always suffers from the influence of basis mismatch. In this paper, to overcome these two shortcomings, we propose a new method called covariance matrix reconstruction approach (CMRA) for both uniform linear array and sparse linear array. In particular, by exploiting the Toeplitz structure of the covariance matrix of the array output, we formulate a low-rank matrix reconstruction (LRMR) problem for covariance matrix recovery. The nonconvex LRMR problem is then relaxed by replacing the rank norm with the nuclear norm and solved using the optimization toolbox. Next, we retrieve the DOAs from the recovered covariance matrix by using the subspace-based methods and obtain an estimated number of signals as a byproduct. We also provide two algorithm implementations for the LRMR problem based on duality and alternating direction method of multipliers, respectively. It is shown that CMRA can be regarded as an atomic norm minimization model or a gridless version of the sparsity-based methods and can recover more signals than sensors with a well-designed array. Numerical experiments are provided to validate the effectiveness of the proposed method, in comparison with some of the existing methods.