Minimum rate sampling of signals with arbitrary frequency support

We examine the problem of reconstructing a signal from periodic non-uniform samples, i.e. a uniform train from which samples are deleted in some periodic fashion. We develop the necessary and sufficient conditions for reconstruction, both for one and multiple dimensions. We prove that one-dimensional multiband signals which have arbitrary frequency support can be sampled without loss arbitrarily close to the theoretically minimum rate. An important advantage of our approach is the existence of an efficient design procedure for the reconstruction system. We show that the algorithm of projection on convex sets can be used to design the reconstruction filters efficiently. Once the filters are designed, the reconstruction algorithm is non-iterative. We give illustrative examples.