The Generalized Multidelay Adaptive Filter: Structure and Convergence Analysis

recognized as an attractive alternative to time-domain algorithms when dealing with systems with large impulse response and/or correlated input. Recently, new frequency-domain LMS adaptive schemes have been proposed. These algorithms essentially retain the attractive features of frequency-domain implementations, while requiring a processing delay considerably smaller than the length of the impulse response. Frequency-domain adaptive filters have long been The first purpose of this contribution is to show that these algorithms can be seen as particular implementations of a more general scheme, the generalized multidelay filter (GMDF). Within this general class of algorithms, we focus on implementations based on the weighted overlap and add reconstruction algorithms; these variants, overlooked in previous contributions, provide an independent control of the overall processing delay and of the rate of update of the filter coefficients, allowing a trade-off between the computational complexity and the rate of convergence. The second purpose of this work is to present a comprehensive analysis of the performance of this new scheme and to provide insight into the influence of impulse response segmentation on the behavior of the adaptive algorithm. Exact analytical expressions for the steady-state mean-square error are first derived. Necessary and sufficient conditions for the convergence of the algorithm to the optimal solution within finite variance are then obtained, and are translated into bounds for the stepsize parameter. Simulations are presented to support our analysis and to demonstrate the practical usefulness of the GMDF algorithm in applications where large impulse response has to be processed.