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Due to the good filter performance for the non-Gaussian noise, the adaptive filters with error nonlinearities have received increasing attention recently. From the viewpoint of the weighted function, in this paper, the existing least mean square (LMS)-based adaptive algorithms with error nonlinearities are divided into three types, i.e., V-shaped, A-shaped, and M-shaped algorithms. Then, to obtain the merits of the V-shaped and A-shaped algorithms, a new family of robust M-shaped error weighted LMS algorithms is proposed. Their steady-state mean square deviation (MSD) analyses are made, which reveal the learning abilities of error nonlinearities: 1) for the V-shaped algorithm, it can achieve smaller steady state MSD for sub-Gaussian noise than that for super-Gaussian noise; 2) the A-shaped algorithm can be used more effectively for super-Gaussian noise than that for sub-Gaussian noise; and 3) the M-shaped algorithm combines the characteristics of the V-shaped and A-shaped algorithms. Furthermore, based on the proposed robust M-shaped function, a proportionate normalized robust M-shaped algorithm is presented for echo cancellation application. Finally, Monte Carlo simulations are conducted to verify the theoretical results and to demonstrate the efficiency of the proposed algorithms in different environments.

A Family of Robust M-Shaped Error Weighted Least Mean Square Algorithms: Performance Analysis and Echo Cancellation Application