General Derivation of Frequency-Domain Adaptive Filtering

Adaptive filters [60] play an important role in echo cancellation because we need to identify and track unknown and time-varying channels [24J . There are roughly two classes of adaptive algorithms. One class includes filters that are updated in the time domain, sample-by-sample in general, like the classical least mean square (LMS) [134] and recursive least-squares (RLS) [4], [66] algorithms. The other class contains filters that are updated in the frequency domain, block-by-block in general, using the fast Fourier transform (FFT) as an intermediary step. As a result of this block processing, the arithmetic complexity of the algorithms in the latter category is significantly reduced compared to time-domain adaptive algorithms. Use of the FFT is appropriate to the Toeplitz structure, which results from the time-shift properties of the filter input signal. Consequently, deriving a frequency-domain (FD) adaptive algorithm is just a matter of rewriting the time-domain error criterion in a way that Toeplitz and circulant matrices are explicitly shown.