Convergence analysis of the modified frequency-domain block LMS algorithm with guaranteed optimal steady state performance

Although the bin-normalized frequency domain block LMS (NFBLMS) algorithm has theoretically very fast convergence speed, it suffers from convergence to a biased steady state solution when causality is not met or when the adaptive filter is of deficient length. A modified FBLMS (MFBLMS) algorithm with guaranteed optimal steady state performance has been proposed, but the theoretical analysis on its convergence properties has not been presented. This paper analyzes the convergence behavior of the algorithm by using the theory of asymptotically equivalent matrices. The eigenvalues of the matrix controlling the convergence behavior is proven to have the tendency to be equally distributed, and a theoretical eigenvalue spread is derived based on the first-order autoregressive (AR) signal model, which is significantly lower than that of the time domain LMS (TDLMS) algorithm. Therefore the convergence speed of the MFBLMS is significantly higher than that of the TDLMS algorithm for colored reference signal. Simulations are carried out to validate the convergence behavior predicted from the theoretical analysis.