Performance of multiple LMS adaptive filters in tandem

The tandem of adaptive filters can occur in practice such as in echo cancellation application for voice communications. This paper analyzes the performance of a number of adaptive filters in tandem. The adaptation algorithm is assumed to be least mean square (LMS). The analysis includes learning trajectory, steady-state excess error due to noise, tracking lag bias, and tracking lag variance. Recursive formulae for their computation are derived. The analysis is exact under Gaussian input and independency assumption. It does not restrict the step size of the filters in tandem to be identical. The validity of the theoretical development is corroborated by simulations. The results indicate that in the special case of equal and small step size, both the steady-state excess error due to noise and the tracking lag variance increase approximately linearly with the number of filters in tandem, whereas the tracking lag bias decreases approximately exponentially with the number of filters in tandem. Consequently, the tandem of adaptive filters can improve the tracking capability of an adaptive system in the situation where the step size is small or the dynamics of an unknown system to be modeled is high.