Sampling and Reconstruction of Sparse Signals in Shift-Invariant Spaces: Generalized Shannon's Theorem Meets Compressive Sensing
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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 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 compressive sensing (CS) framework for recovery of the SI samples from a reduced set of measurements. The samples are subsequently filtered by a discrete-time correction filter in order to reconstruct expansion coefficients of an observed signal. Furthermore, we offer a generalization of the proposed framework to other sampling kernels that lie in arbitrary SI spaces. The generalized method embeds the correction filter in a CS optimization problem which directly reconstructs expansion coefficients of the signal. Both approaches recast an inherently infinite-dimensional inverse problem as a finite-dimensional CS problem in an exact way. Finally, we conduct numerical experiments on signals in 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 signal, obtained from a reduced set of measurements. Such continuous signal representations are particularly suitable for signal processing without converting them into samples.
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