Xampling at the Rate of Innovation

TL;DR

This paper introduces a general framework for signal recovery from samples taken at their rate of innovation, encompassing nonlinear and complex signal classes, and proves convergence of the proposed least-squares method under stability conditions.

Contribution

It extends sampling theory to broader nonlinear signal classes without restrictive assumptions, providing a stable recovery method with convergence guarantees.

Findings

Proves convergence of the least-squares approach under stability.

Demonstrates effectiveness in recovering pulse streams.

Shows improved noise robustness over existing algorithms.

Abstract

We address the problem of recovering signals from samples taken at their rate of innovation. Our only assumption is that the sampling system is such that the parameters defining the signal can be stably determined from the samples, a condition that lies at the heart of every sampling theorem. Consequently, our analysis subsumes previously studied nonlinear acquisition devices and nonlinear signal classes. In particular, we do not restrict attention to memoryless nonlinear distortions or to union-of-subspace models. This allows treatment of various finite-rate-of-innovation (FRI) signals that were not previously studied, including, for example, continuous phase modulation transmissions. Our strategy relies on minimizing the error between the measured samples and those corresponding to our signal estimate. This least-squares (LS) objective is generally non-convex and might possess many local minima. Nevertheless, we prove that under the stability hypothesis, any optimization method designed to trap a stationary point of the LS criterion necessarily converges to the true solution. We demonstrate our approach in the context of recovering pulse streams in settings that were not previously treated. Furthermore, in situations for which other algorithms are applicable, we show that our method is often preferable in terms of noise robustness.