Sampling and super resolution of sparse signals beyond the Fourier domain
File(s)Sparse Sampling_FinalSubmission.pdf (3.82 MB)
Accepted version
Author(s)
Bhandari, A
Eldar, YC
Type
Journal Article
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.
Date Issued
2019-03-15
Date Acceptance
2018-11-28
ISSN
1053-587X
Publisher
Institute of Electrical and Electronics Engineers
Start Page
1508
End Page
1521
Journal / Book Title
IEEE Transactions on Signal Processing
Volume
67
Issue
6
Copyright Statement
© 2018 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.
Identifier
http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000457991600002&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=1ba7043ffcc86c417c072aa74d649202
Subjects
Science & Technology
Technology
Engineering, Electrical & Electronic
Engineering
Finite rate of innovation
fractional Fourier domain
sampling
sparsity
special affine Fourier transform
super resolution
TIME-DELAY ESTIMATION
FRACTIONAL FOURIER
FINITE-RATE
PHASE RETRIEVAL
TRANSFORM
SUPERRESOLUTION
RECONSTRUCTION
CHIRP
DECONVOLUTION
CONVOLUTION
MD Multidisciplinary
Networking & Telecommunications
Publication Status
Published
Date Publish Online
2018-12-27