A Unified Approach to the Statistical Convergence Analysis of Frequency-Domain Adaptive Filters

The frequency-domain adaptive filter (FDAF) algorithms are used in many applications due to their computational efficiency and good convergence performance. Many efforts have been made to analyze the convergence behavior of FDAF in the past. However, the previous analyses are based on coarse approximations of overlap-save procedures or small step-size assumptions and hence came to inaccurate predictions of the transient and steady-state performance. Moreover, the rigorous step-size bound in the mean-square sense has not been provided so far. To address these problems, we carry out an extensive analysis of the convergence behaviors for a family of FDAFs based on the overlap-save structure. Using a unified update equation of four FDAFs, the state recursions of the mean weight-error vector and the weight-error covariance matrix are worked out rigorously in the frequency domain, which are then used to investigate the mean-square deviation (MSD) and mean-square error (MSE) during the transient phase. In addition, we obtain the analytical results on the steady-state MSD and MSE, and the bound on the step size for both the mean and mean-square stabilities. Specifically, the analysis presented here does not restrict the regression data to being Gaussian or white. Computer simulations in a system identification scenario confirmed that the proposed theoretical results are much more accurate than the previous approaches.