Improving convergence of the PNLMS algorithm for sparse impulse response identification

A proportionate normalized least mean square (PNLMS) algorithm has been proposed for sparse impulse response identification. It provides fast initial convergence, but it begins to slow down dramatically after the initial period. We analyze the coefficient adaptation process of the steepest descent algorithm and derive how to calculate the optimal proportionate step size in order to achieve the fastest convergence. The results bring forward a novel view of the concept of proportion. We propose a μ-law PNLMS (MPNLMS) algorithm using an approximation of the optimal proportionate step size. Line segment approximation and partial update techniques are discussed to bring down the computational complexity.