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

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