Exponential asymptotic stability of time-varying inverse prediction error filters

It is a classical result of linear prediction theory that as long as the minimum prediction error variance is nonzero, the transfer function of the optimum linear prediction error filter for a stationary process is minimum phase, and therefore, its inverse is exponentially stable. Here, extensions of this result to the case of nonstationary processes are investigated. In that context, the filter becomes time-varying, and the concept of “transfer function” ceases to make sense. Nevertheless, we prove that under mild condition on the input process, the inverse system remains exponentially stable. We also consider filters obtained in a deterministic framework and show that if the time-varying coefficients of the predictor are computed by means of the recursive weighted least squares algorithm, then its inverse remains exponentially stable under a similar set of conditions