Signal Recovery from Time and Frequency Samples

TL;DR

This paper investigates a two-sided sampling approach for signal recovery, utilizing samples from both a signal and its Fourier transform, which enhances reconstruction quality under bandwidth and sensing constraints.

Contribution

It extends classical one-sided sampling methods by formulating a two-sided recovery framework in finite and infinite-dimensional spaces, demonstrating improved conditioning and reconstruction.

Findings

Two-sided sampling often results in better conditioned systems.

Supplementing time samples with frequency information improves reconstruction.

Numerical experiments confirm advantages over traditional methods.

Abstract

We analyze signal recovery when samples are taken concomitantly from a signal and its Fourier transform. This two-sided sampling framework extends classical one-sided reconstruction and is particularly useful when measurements in either domain alone are insufficient because of sensing, storage, or bandwidth constraints. We formulate the resulting recovery problem in finite-dimensional spaces and reproducing kernel Hilbert spaces, and illustrate the infinite-dimensional setting in a Fourier-symmetric Sobolev space. Numerical experiments with sinc- and Hermite-based schemes indicate that, under a fixed sampling budget, two-sided sampling often yields better conditioned systems than one-sided approaches. A simplified spectrum-monitoring example further demonstrates improved reconstruction when limited time samples are supplemented with frequency-domain information.