Robust Widely Linear Affine Projection M-Estimate Adaptive Algorithm: Performance Analysis and Application

The widely-linear affine projection algorithm (WL-APA) based on the orthogonal affine projection principle and the WL model utilizes the past multiple input signal vectors for updating the tap weights of the adaptive filter and achieves fast convergence with colored noncircular input signals while maintaining a certain steady-state estimation accuracy. However, the WL-APA suffers from significant performance degradation under impulsive interference. In this article, we propose a WL affine projection M-estimate (WL-APM) algorithm to enhance the robustness of the WL-APA against impulsive noise, which is obtained by solving a robust constrained minimization problem based on the complex-valued modified Huber (MH) function using the Lagrange multiplier method and Wirtinger Calculus (or called CR calculus). To achieve a reduction in the computational complexity of the WL-APM algorithm to facilitate its hardware implementation, a low-complexity variant based on fast recursive filtering and complex-valued leading dichotomous coordinate descent (DCD) iteration, namely DCD-WL-APM, is proposed. Then, the convergence behavior of the WL-APM algorithm is characterized by mean and mean-square stability, transient and steady-state mean-square deviation (MSD), and the stability step size conditions and MSD expressions are derived. The advantages of the proposed DCD-WL-APM algorithm in terms of computational complexity are also demonstrated in the form of picture and table. Finally, the accuracy of the theoretical analysis results and the superiority of our proposed WL-APM algorithm and its low-complexity version are verified by extensive simulations, including system identification and stereo acoustic echo cancellation (SAEC).