MCMC methods for restoration of quantised time series

In digital systems, the amplitude of a time series is quantised with finite resolution. This is a nonlinear process which introduces distortion. We develop a Bayesian, model-based approach to reducing the quantisation distortion when moving a time series, such as an audio signal, to a higher resolution medium. The signal is modelled as a discrete-time, continuous-valued autoregressive (AR) process of unknown order. The model parameters and reconstructed signal are estimated using Markov chain Monte Carlo (MCMC) techniques. This requires samples to be drawn from a truncated multivariate Gaussian distribution, for which a MetropolisHastings approach is developed. 1. QUANTISATION DISTORTION For digital processing, transmission, or storage, a signal is represented as discrete in both time (due to sampling) and value (due to quantisation). The quantisation process introduces an error component, often referred to as ‘quantisation noise’. Since this quantisation error is signal-dependent, it is perhaps better described as quantisation distortion. The distortion consists of a series of added harmonics, together with aliased images of them. For complicated signals, this can be quite innocuous, behaving very much like white noise. When the signal is simple, however, the structure becomes clear. Furthermore, the relative levels of the added harmonics (and their inharmonic aliases) vary significantly with small changes in the level of the input signal. This behaviour can be disturbing in music signals, especially with those instruments, such as pianos, whose note waveforms become more sinusoidal as they decay. This effect produces what is known as ‘granulation noise’ [1]. Quantisation occurs both during analogue-to-digital conversion and during any subsequent manipulation of the digital signal that increases the word length, such as multiplication. This work is supported by the Engineering & Physical Sciences Research Council, U.K. e AR k,a x y