Monte Carlo Deconvolution of Digital Signals Guided by the Inverse Filter

Abstract

Digital deconvolution concerns the restoration of an underlying discrete signal from a blurred noisy observation sequence. The problem can be formulated in a Bayesian framework. As is usual in the Bayesian context, the computation of relevant posterior quantities is the major challenge. Previous work made substantial progress toward making this computation feasible, most notably through the Gibbs sampling approach of Chen and Li and the sequential importance sampling approach of Liu and Chen. Yet, there is room for improvement, because both global Monte Carlo strategies can be very slow to reach the target distribution. We propose two new sampling strategies for efficient restoration of digital signals. The key idea is to use inverse filtering to transform the posterior integration into a problem that is substantially more local than the direct formulation, leading to stable Monte Carlo procedures that are very fast to converge. In the final section, we consider extensions to the case of blind deconvolution, where the filter coefficients are unknown and must be estimated as the signal is being restored.