Sampling based on timing: Time encoding machines on shift-invariant subspaces

Sampling information using timing is an approach that has received renewed attention in sampling theory. The question is how to map amplitude information into the timing domain. One such encoder, called time encoding machine, was introduced by Lazar and Tóth (2004 [23]) for the special case of band-limited functions. In this paper, we extend their result to a general framework including shift-invariant subspaces. We prove that time encoding machines may be considered as non-uniform sampling devices, where time locations are unknown a priori. Using this fact, we show that perfect representation and reconstruction of a signal with a time encoding machine is possible whenever this device satisfies some density property. We prove that this method is robust under timing quantization, and therefore can lead to the design of simple and energy efficient sampling devices.