Sub-Nyquist Sampling of Short Pulses

TL;DR

This paper introduces a stable, low-rate, multichannel sampling method for superpositions of unknown pulses, leveraging Gabor frames and compressed sensing, even in multiband scenarios, filling a key gap in signal processing.

Contribution

It presents a novel sub-Nyquist sampling scheme for unknown pulse superpositions using Gabor frames and compressed sensing, applicable to multiband signals without prior pulse location knowledge.

Findings

Effective reconstruction of unknown pulse superpositions at sub-Nyquist rates.

Robustness of the sampling scheme in noisy environments.

Theoretical error bounds for signal reconstruction.

Abstract

We develop sub-Nyquist sampling systems for analog signals comprised of several, possibly overlapping, finite duration pulses with unknown shapes and time positions. Efficient sampling schemes when either the pulse shape or the locations of the pulses are known have been previously developed. To the best of our knowledge, stable and low-rate sampling strategies for continuous signals that are superpositions of unknown pulses without knowledge of the pulse locations have not been derived. The goal in this paper is to fill this gap. We propose a multichannel scheme based on Gabor frames that exploits the sparsity of signals in time and enables sampling multipulse signals at sub-Nyquist rates. Moreover, if the signal is additionally essentially multiband, then the sampling scheme can be adapted to lower the sampling rate without knowing the band locations. We show that, with proper preprocessing, the necessary Gabor coefficients, can be recovered from the samples using standard methods of compressed sensing. In addition, we provide error estimates on the reconstruction and analyze the proposed architecture in the presence of noise.