Widely linear complex-value improved-proportionate affine projection algorithm based on M-estimation for underwater acoustic channel estimation

To address the performance degradation of conventional real-valued adaptive filtering algorithms for channel estimation under noncircular signal conditions in underwater acoustic (UWA) channels, a widely linear complex-value improved-proportionate affine projection algorithm is proposed. First, the second-order statistical information of the noncircular signals is fully exploited by jointly modeling the signal and its conjugate. Then, a proportionate factor is introduced to adaptively adjust the step size for updating the filter coefficients, enabling rapid tracking of UWA channels while reducing steady-state estimation error. Furthermore, to enhance the robustness of the proposed algorithm in complex noise environments, a complex-value modified Huber function is employed to correct the a posteriori estimation error vector, and a robust constrained minimum perturbation problem is formulated. In addition, the steady-state mean and mean-square convergence properties of the proposed algorithm are theoretically analyzed, and the sufficient step-size condition for convergence as well as the expression for steady-state mean square deviation are derived. Simulation results demonstrate that, under noncircular signal and complex noise conditions, the proposed algorithm achieves faster convergence, lower steady-state estimation error, and improved robustness compared with existing methods.