Sampling and Reconstruction of Sparse Signals in Shift-Invariant Spaces: Generalized Shannon's Theorem Meets Compressive Sensing

TL;DR

This paper develops a unified framework combining shift-invariant space sampling with compressive sensing to efficiently reconstruct sparse signals, generalizing Shannon's theorem for continuous signals.

Contribution

It introduces a novel approach that integrates CS with SI spaces, extending the classical Shannon sampling theorem to sparse, continuous-domain signals with generalized kernels.

Findings

Effective reconstruction of sparse signals in SI spaces using reduced measurements

Generalized framework applies to various compactly supported kernels

Numerical validation on polynomial B-spline signals demonstrates practical viability

Abstract

This paper introduces a novel framework and corresponding methods for sampling and reconstruction of sparse signals in shift-invariant (SI) spaces. We reinterpret the random demodulator, a system that acquires sparse bandlimited signals, as a system for the acquisition of linear combinations of the samples in the SI setting with the box function as the sampling kernel. The sparsity assumption is exploited by the compressive sensing (CS) paradigm for a recovery of the SI samples from a reduced set of measurements. The SI samples are subsequently filtered by a discrete-time correction filter to reconstruct expansion coefficients of the observed signal. Furthermore, we offer a generalization of the proposed framework to other compactly supported sampling kernels that span a wider class of SI spaces. The generalized method embeds the correction filter in the CS optimization problem which directly reconstructs expansion coefficients of the signal. Both approaches recast an inherently continuous-domain inverse problem in a set of finite-dimensional CS problems in an exact way. Finally, we conduct numerical experiments on signals in polynomial B-spline spaces whose expansion coefficients are assumed to be sparse in a certain transform domain. The coefficients can be regarded as parametric models of an underlying continuous-time signal, obtained from a reduced set of measurements. Such continuous signal representations are particularly suitable for signal processing without converting them into samples.