  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.
