Mean-square error and stability analysis of a subband structure for the rapid identification of sparse impulse responses

In a context of supervised adaptive filtering, the sparsity of the impulse response to be identified can be employed to accelerate the convergence rate of the algorithm. This idea was first explored by the so-called proportionate NLMS (PNLMS) algorithm, where the adaptation step-sizes are made larger for the coefficients with larger magnitudes. Whereas fast initial adaptation convergence rate is obtained with the PNLMS algorithm for white-noise input, slow convergence is observed for colored input signals. The combination of the PNLMS approach and a subband structure results in an algorithm with better convergence rate for sparse systems and colored input signals. In this paper, the steady-state mean-square error (MSE) and the maximum value of the step-size β that allows convergence of the subband PNLMS-type algorithm are analyzed. Theoretical results are confirmed by simulations.