Sampling and Super-resolution of Sparse Signals Beyond the Fourier Domain

TL;DR

This paper extends super-resolution techniques to the Special Affine Fourier Transform domain, providing new sampling theorems for sparse signals in various transform domains beyond Fourier, with applications in optics and signal processing.

Contribution

It introduces a unifying framework for sparse signal recovery in the SAFT domain, generalizing Fourier, Fresnel, and Fractional Fourier domain results, including new sampling theorems and convolution properties.

Findings

Established sampling theorems for sparse signals in the SAFT domain.

Developed the SAFT series and short time SAFT for signal analysis.

Demonstrated the convolution-multiplication property in the SAFT domain.

Abstract

Recovering a sparse signal from its low-pass projections in the Fourier domain is a problem of broad interest in science and engineering and is commonly referred to as super-resolution. In many cases, however, Fourier domain may not be the natural choice. For example, in holography, low-pass projections of sparse signals are obtained in the Fresnel domain. Similarly, time-varying system identification relies on low-pass projections on the space of linear frequency modulated signals. In this paper, we study the recovery of sparse signals from low-pass projections in the Special Affine Fourier Transform domain (SAFT). The SAFT parametrically generalizes a number of well known unitary transformations that are used in signal processing and optics. In analogy to the Shannon's sampling framework, we specify sampling theorems for recovery of sparse signals considering three specific cases: (1) sampling with arbitrary, bandlimited kernels, (2) sampling with smooth, time-limited kernels and, (3) recovery from Gabor transform measurements linked with the SAFT domain. Our work offers a unifying perspective on the sparse sampling problem which is compatible with the Fourier, Fresnel and Fractional Fourier domain based results. In deriving our results, we introduce the SAFT series (analogous to the Fourier series) and the short time SAFT, and study convolution theorems that establish a convolution--multiplication property in the SAFT domain.