A wave theory of long adaptive filters

Long LMS filters of the tapped-delay line type are in widespread use, particularly in acoustic applications. For the limiting case of an infinite line length it is shown that the behavior of such filters is governed by remarkably simple laws. This is true for the steady state, where for small stepsizes the weight-error correlations become independent of the input signal, but also for the transient behavior, where the spatial Fourier transform of the weight-error distribution decays exponentially. Moreover, a necessary and (probably) sufficient stability bound for the stepsize is derived. The "wave theory" developed for the infinite line length also predicts the behavior of rather short filters with sufficient accuracy, particularly for a moderately colored input signal. No independence assumption is required and no assumption concerning the spectral distribution of the additive noise. Under steady-state conditions, the weight-error correlation between two line taps is solely determined by the noise autocorrelation, with the time delay replaced by the tap distance.