Cascaded all-pass sections for LMS adaptive filtering

The behaviour of the LMS adaptive algorithm is analyzed for a class of adaptive filters that is based on a cascade of identical N-th order all-pass sections. The well-known tapped-delay-line is a special case of this class. We look at the rate of convergence and the steady-state weight fluctuations. It is shown that in the steady state the weight-error correlation matrix satisfies a Lyapounov equation for sufficiently small values of the step-size. Sometimes a priori knowledge of the unknown reference system is available that can be used to select the N parameters of the all-pass section. In these cases the LMS adaptive filter based on a cascade of identical all-pass sections can outperform the LMS adaptive tapped-delay-line.