Frequency-domain adaptation of causal digital filters

The adaptation of causal FIR digital filters in the discrete frequency domain is considered, and it is shown how the bin-normalized form of the LMS algorithm can converge to a biased solution for problems such as linear prediction. A discrete frequency-domain version of Newton's algorithm is derived, and it is demonstrated how this can converge to the optimal causal solution, even for linear prediction problems. The algorithm employs a spectral factorization of the estimated power spectral density of the reference signal, the entirely noncausal part of which is used before the causality constraint in the adaptation algorithm, and the entirely causal part is applied after the causality constraint. The spectral factors can be calculated online from a recursive estimate of the power spectral density without too great a loss of convergence speed. The extension of the algorithm to the adaptation of feedforward controllers is also described, in which case, the spectral factors of the reference signals filtered by the plant response are required, and these are shown to be equal to the spectral factors of the reference signal multiplied by the minimum phase part or the plant frequency response.