Slow asymptotic convergence of LMS acoustic echo cancelers

In most acoustic echo canceler (AEC) applications, an adaptive finite impulse response (FIR) filter is employed with coefficients that are computed using the LMS algorithm. The paper establishes a theoretical basis for the slow asymptotic convergence that is often noted in practice for such applications. The analytical approach expresses the mean-square error trajectory in terms of eigenmodes and then applies the asymptotic theory of Toeplitz matrices to obtain a solution that is based on a general characterization of the actual room impulse response. The method leads to good approximations even for a moderate number of taps (N>16) and applies to both full-band and subband designs. Explicit mathematical expressions of the mean-square error convergence are derived for bandlimited white noise, a first-order Markov process, and, more generally, pth-order rational spectra and a direct power-law model, which relates to lowpass FIR filters. These expressions show that the asymptotic convergence is generally slow, being at best of order 1/t for bandlimited white noise. It is argued that input filter design cannot do much to improve slow convergence. However, the theory suggests postfiltering as a remedy that would be useful for the full-band LMS AEC and may also be applicable to subband designs