Proportionate affine projection tanh algorithm and its step-size optimization

The problem of sparse adaptive system identification such as acoustic echo cancellation (AEC) needs robust adaptive filtering algorithms in the situation where the system is often corrupted by impulsive noise. To solve this problem this work proposes a robust proportionate constrained optimization method based on the proportionate matrix weighted norm, and a proportionate affine projection tanh algorithm (PAPTA) is derived. To keep both fast convergence speed and low misalignment, a variable step -size is derived for the proposed algorithm. In addition, by expanding the tanh function and considering the dependence between the adaptive filter weight vector and the noise vectors, the weighted energy conservation method is utilized to analyze the steady-state performance and stability of the proposed algorithm. Finally, through simulations, the superior performance of the proposed algorithm is demonstrated in terms of convergence speed and robustness in sparse systems, and the accuracy of the theoretical steady-state analysis is verified.