Convergence analysis of a deficient-length LMS filter and optimal-length sequence to model exponential decay impulse response

This letter analyzes the mean-square convergence of a deficient-length least mean-square adaptive filter whose length is less than that of the unknown system and proves that filter length and the envelope of impulse response are the important factors in convergence rate control. For the impulse response with an exponential decay envelope, which covers a large set of physical systems, e.g., acoustic echo path, an optimal filter length sequence is figured out to achieve the fastest convergence. The simulations of an exact exponential decay envelope and of a real-life echo path in a car environment are performed via computer, and the results validate our analysis well.