Performance Analysis of Frequency-Domain Adaptive Filter for Cyclostationary Inputs

The Frequency-Domain Adaptive Filter （FDAF） is widely used in audio signal processing owing to its high computational efficiency and fast convergence speed. Existing performance analyses of FDAF are generally based on the assumption that the input signal is a wide-sense stationary random process. This paper examines the mean and mean-square convergence properties of both unconstrained and constrained FDAF algorithms for cyclostationary input within the framework of system identification. The cyclostationary input signal is modeled as a random process with periodically time-varying power， without restricting the input distribution to be Gaussian or white. First， by transforming the filter coefficients from the frequency domain back to the time domain， time-domain equivalent of the FDAF algorithm is presented. Based on the evolution of the weight-error vector， recursive formulas for the transient mean-square deviation （MSD） and excess mean-square error （EMSE） are derived. Second， using the theory of first-order periodic matrix difference equations， the convergence conditions of the algorithm are established， and closed-form expressions for the steady-state MSD and EMSE are further obtained. Theoretical analysis demonstrates that the steady-state MSD and EMSE of the FDAF algorithm exhibit periodic fluctuation， whose period is determined by the filter length and period of the power variation of the input signal. Finally， the derived theoretical model is rigorously validated through a sequence of numerical simulations and comparisons with measurement data.