Stochastic analysis of frequency-domain adaptive filters

This study investigates the convergence behaviors of a family of frequency-domain adaptive filters (FDAFs) under both exact- and under-modeling situations. The stochastic analysis is conducted by transforming the frequency-domain equations into their time-domain counterparts. We discuss the transient and steady-state convergence behaviors of four FDAF versions, i.e., the constrained FDAFs with and without step-normalization, the unconstrained FDAFs with and without step-normalization, and we also present the upper bounds of step size for mean stability and mean-square stability. Starting from the expression for the steady-state mean weight vector, this study investigates whether the FDAFs can converge to unknown system impulse responses and optimum Wiener solutions. Moreover, we provide the closed-form minimum mean-square error (MMSE) that each FDAF can achieve. The difference between the current work and our previous one is threefold. First, the presented time-domain analysis is much easier to handle and has a more explicit physical meaning than that in the frequency domain. Second, we here consider an arbitrary overlap factor between consecutive blocks, while our previous analysis only focuses on 50% overlap. Third, the presented MMSE expressions and excess mean-square error (EMSE) approximations have not been given before. Simulations reveal high consistency between the experimental and theoretical results.