Tracking capability of the least mean square algorithm: Application to an asynchronous echo canceller

The tracking capability of a time-varying system by an adaptive filter through the least mean square (LMS) algorithm is studied in the particular case of a white, zero-mean reference input. This case is suitable for echo cancellation achieving full-duplex data transmission. A difference is made according to the statistical characteristics of the echo channel variations to be tracked. The usual but unrealistic assumption concerning the independence between the input sequence and the adaptive filter deviation from optimality is not used. Results concerning the residual mean square error are recalled for the case of random, white, and zero-mean channel variations, whereas new computations are proposed for the case of deterministic and bounded time variations. Then an optimum step size is derived. Moreover, in the case of "slow variations," these latter results are compared to those obtained with the independence assumption; we find that they agree perfectly. This theoretical study is applied to a particular case of timing jitter occurring in an asynchronous echo canceller, which makes the echo channel time varying.