WLB-CANUN: Widely Linear Beamforming in Coprime Array With Non-Uniform Noise

The performance of widely linear beamforming (WLB) is superior to adaptive beamforming, but it is limited by the uniform linear array geometry and non-uniform noise. In this paper, to overcome these limitations together, we propose a framework for widely linear beamforming in coprime array with non-uniform noise (WLB-CANUN). We subtract the non-uniform noise component from the coprime array sample covariance matrix, and vectorize the resulted matrix to create the difference co-array (DCA). Since the DCA is not uniform, we interpolate it and recover its signal by formulating the atomic norm minimization problem with the Toeplitz and orthogonal subspace constraints.The pseudo sample covariance matrix of coprime array does not contain the non-uniform noise component, which can be directly vectorized to create the sum co-array (SCA). Due to the non-uniformity of SCA, we interpolate it and recover its signal by formulating another atomic norm minimization problem with the Hankel and orthogonal subspace constraints. The directions of non-circular signals can be estimated by the traditional subspace method, which are utilized to estimate their non-circular coefficients. A least square optimization problem using the sample and pseudo sample covariance matrices of coprime array is formulated and solved to estimate the powers of non-circular signals. The interference-plus-noise covariance matrix (INCM), pseudo INCM and augmented INCM of coprime array are reconstructed, so that the ultimate augmented weight vector can be calculated. Simulation results indicate that the proposed WLB-CANUN method overcomes the limitations of WLB in coprime array with non-uniform noise, and enhances the performance compared to the existing WLB methods.