Paul T. Troughton. Simulation Methods for Linear and Nonlinear Time Series Models with Application to Distorted Audio Signals. Ph.D. thesis, University of Cambridge, 1999.

This dissertation is concerned with the development of Markov chain Monte Carlo (MCMC) methods for the Bayesian restoration of degraded audio signals. First, the Bayesian approach to time series modelling is reviewed, then established MCMC methods are introduced.

The first problem to be addressed is that of model order uncertainty. A reversible-jump sampler is proposed which can move between models of different order. It is shown that faster convergence can be achieved by exploiting the analytic structure of the time series model.

This approach to model order uncertainty is applied to the problem of noise reduction using the simulation smoother. The effects of incorrect autoregressive (AR) model orders are demonstrated, and a mixed model order MCMC noise reduction scheme is developed.

Nonlinear time series models are surveyed, and the advantages of linear-in-the-parameters models explained. A nonlinear AR (NAR) model, based on the Volterra polynomial expansion, is described, in which the model selection problem becomes one of subset selection. Subset selection methods are reviewed, including Bayesian MCMC methods. A new MCMC approach is formulated, using latent indicator variables in a Gibbs sampler. It is shown that using analytic results to create a multi-move sampler leads to better performance.

The effects, and some sources, of distortion in audio recordings are described. The few previous attempts to remove these types of distortion are reviewed. A general method is proposed, based on a cascade model in which the signal is modelled as an AR process, and the nonlinear channel as an NAR process. The model structure, order and parameters are jointly estimated in a MCMC scheme. The method is extended to process long sequences, in which the audio signal cannot be modelled as stationary, by estimating the nonlinear model structure and parameters jointly across all the blocks.

The quantisation distortion present in limited word length digital audio is examined. A model-based framework is proposed for restoring such quantised signals. In order to implement this, methods are investigated for drawing samples from truncated multivariate Gaussian distributions. The restoration is improved by the use of sinusoidal modelling with AR residuals.

ZIPPED POSTSCRIPT file Download full version (802K Zipped postscript)