Time delay estimation: Compressed sensing over an infinite union of subspaces

Sampling theorems for signals that lie in a union of subspaces have been receiving growing interest. A recent model that describes analog signals over a union is that of a union of shift-invariant (SI) subspaces. Until now, sampling and recovery algorithms have been developed only for a finite union of SI subspaces. Here we extend this paradigm to a special case of an infinite union, in which the SI subspaces are generated by pulses with unknown delays, taken from a continuous interval. We develop a unified approach to time delay recovery of the pulses, from low rate samples of the signal taken at the lowest possible rate. In particular, we derive sufficient conditions on the pulses and the sampling filters in order to ensure perfect recovery of the signal. We then show that by properly manipulating the low-rate samples, the time delays can be recovered using the well-known ESPRIT algorithm.