Robust adaptive filter with lncosh cost

In this paper, a least lncosh (Llncosh) algorithm is derived by utilizing the lncosh cost function. The lncosh cost is characterized by the natural logarithm of hyperbolic cosine function, which behaves like a hybrid of the mean square error (MSE) and mean absolute error (MAE) criteria depending on adjusting a positive parameter λ. Hence, the Llncosh algorithm performs like the least mean square (LMS) algorithm for small errors and behaves as the sign-error LMS (SLMS) algorithm for large errors. It provides comparable performance to the LMS algorithm in Gaussian noise. When compared with several existing robust approaches, the superior steady-state performance and stronger robustness can be attained in impulsive noise. The mean behavior, mean-square behavior and steady-state performance analyses of the proposed algorithm are also provided. In addition, aiming to acquire a compromise between fast initial convergence rate and satisfactory steady-state performance, we introduce a variable-λ Llncosh (VLlncosh) scheme. Lastly, in order to resist the sparsity of the acoustic echo path, an improved proportionate least lncosh (PLlncosh) algorithm is presented. The good performance against impulsive noise and theoretical results of the proposed algorithm are validated by simulations.