Paul T. Troughton and Simon J. Godsill. MCMC methods for restoration of nonlinearly distorted autoregressive signals. Signal Processing, 81(1), pp. 83-97, 2001.

We approach the problem of restoring distorted autoregressive (AR) signals by using a cascade model, in which the observed signal is modelled as the output of a nonlinear AR (NAR) process excited by the linear AR signal we are attempting to recover. The Volterra expansion of the NAR model has a very large number of possible terms even when truncated at fairly small maximum polynomial degrees and lags. We address the problem of subset selection and uncertainty in the nonlinear stage and model order uncertainty in the linear stage through a hierarchical Bayesian approach. A Markov chain Monte Carlo (MCMC) approach is used for implementation, with reversible-jump moves for the linear model order and a rapidly mixing Gibbs sampler for subset selection in the nonlinear model stage, exploiting the partially analytic properties of the model. We demonstrate the method using both synthetic AR data and an audio extract, and extend the approach to process a long distorted audio time series, for which the source model cannot be considered to be time-invariant.