MCMC methods for restoration of nonlinearly distorted autoregressive signals

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 process (NAR) 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 orders and lags. We address the problem of subset selection and uncertainty in the nonlinear stage and model length uncertainty in the linear stage through a hierarchical Bayesian approach, using reversible jump Markov chain Monte Carlo (MCMC) and Gibbs sampling. We demonstrate the method using synthetic AR data, and extend the approach to process a long distorted audio time series, for which the source model cannot be considered to be stationary.