CHAOTIC BEHAVIOR OF ADAPTIVE RECURSIVE FILTERS - APPLICATION TO TELEPHONE SIGNAL COMPRESSION

Adaptive filtering gives simple iterative methods for extracting a useful signal by linearly filtering an observation correlated with this useful signal. When the observation contains past values of the filter output - as is the case for a recursive structure - the adaptive filter is a nonlinear operator which can generate complex behaviors even though adaptation is a standard derivative of the gradient algorithm. Such behaviors are observed in a particular case: the ARMA adaptive prediction with a sinusoidal input. A slow adaptation speed involves a quasi-periodic behavior due to the locally unstable character of the predictor. This phenomenon called self-stabilization persists as the adaptation speed increases. Its study is more complicated: the behavior becomes chaotic. As an illustration, the digitization of telephone signals through ARMA adaptive prediction shows the necessity of controlling the qualitative behavior of the adaptive filtering solutions.