Estimating Signals with Finite Rate of Innovation from Noisy Samples: A Stochastic Algorithm

TL;DR

This paper introduces a stochastic Gibbs sampling algorithm to accurately reconstruct signals with finite rate of innovation from noisy samples, including those acquired with Gaussian kernels that do not satisfy traditional conditions.

Contribution

The paper presents a novel MCMC-based method for reconstructing finite rate of innovation signals from noisy samples with Gaussian kernels, extending beyond previous kernel restrictions.

Findings

Algorithm achieves high accuracy in noisy conditions

Demonstrates robustness across various sampling kernels

Outperforms existing methods in simulations

Abstract

As an example of the recently-introduced concept of rate of innovation, signals that are linear combinations of a finite number of Diracs per unit time can be acquired by linear filtering followed by uniform sampling. However, in reality, samples are rarely noiseless. In this paper, we introduce a novel stochastic algorithm to reconstruct a signal with finite rate of innovation from its noisy samples. Even though variants of this problem has been approached previously, satisfactory solutions are only available for certain classes of sampling kernels, for example kernels which satisfy the Strang-Fix condition. In this paper, we consider the infinite-support Gaussian kernel, which does not satisfy the Strang-Fix condition. Other classes of kernels can be employed. Our algorithm is based on Gibbs sampling, a Markov chain Monte Carlo (MCMC) method. Extensive numerical simulations demonstrate the accuracy and robustness of our algorithm.