Steady-state analysis of biased filtered-x algorithms for adaptive room equalization

This paper provides an analysis of the steady-state behavior of two biased adaptive algorithms recently introduced for listening room compensation, the biased filtered-x normalized least mean squares (Fx-BNLMS) and the biased filtered-x improved proportionate NLMS (Fx-BIPNLMS). We give theoretical results that show that the biased algorithms can outperform the unbiased ones in terms of the mean square error, especially in low signal-to-noise ratio (SNR) scenarios. Moreover, for impulse responses exhibiting high sparse-ness, the improved proportionate algorithms achieve faster convergence than the standard NLMS. Thereby, the advantages of the Fx-BIPNLMS algorithm are justified theoretically in terms of the excess mean square error. Simulation results show that there is a relatively good match between theory and practice, especially for low μ values.