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# Poisson's Summation Formula in Radar Imaging

Fischer, Jens and Rommel, Tobias and Stens, Rudolf (2021) Poisson's Summation Formula in Radar Imaging. In: Proceedings of the European Conference on Synthetic Aperture Radar, EUSAR, pp. 424-429. VDE Verlag. European Conference on Synthetic Aperture Radar (EUSAR), 2021-03-29 - 2021-04-01, Leipzig (virtually), Germany. ISSN 2197-4403.

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## Abstract

Poisson’s Summation Formula (PSF) has numerous applications, not only in radar imaging. It is the bridge between continuous (infinite integrals) and discrete settings (finite sums) and, hence, links the Fourier transform, Fourier series, the Discrete-Time Fourier Transform (DTFT) and the Discrete Fourier Transform (DFT) to one another. Most importantly, however, is the fact that Poisson’s Summation Formula is the actual core statement hidden in any sampling theorem. In this paper, we show that two operations, discretization and periodization, are needed to come from infinite integrals to finite sums. These operations are moreover involved in any imaging process, in particular, in radar imaging. We use two operations, comb and rep, to better understand Poisson’s Summation Formula, both introduced by Woodward who is also known for introducing the sinc function and Woodward’s ambiguity function. With these operations applied to functions it can be shown that "sampling a function with 1/T" means to "periodize its Fourier transform with T" and, vice versa, "periodizing a function with T" means to "sample its Fourier transform with 1/T". This simple statement turns out to be both, the sampling theorem and Poisson’s summation formula. However, there are functions which cannot be sampled and functions which cannot be periodized. Constant functions or constantly growing functions, for example, can be sampled but cannot be periodized. Vice versa, a Dirac delta can be periodized, it yields a Dirac comb, but cannot be sampled. So, another outcome in this paper is an easy-to-apply rule for distinguishing those functions which can be sampled or periodized or sampled and periodized and those which cannot.

Item URL in elib:https://elib.dlr.de/133956/
Document Type:Conference or Workshop Item (Speech, Poster)
Title:Poisson's Summation Formula in Radar Imaging
Authors:
AuthorsInstitution or Email of AuthorsAuthor's ORCID iDORCID Put Code
Fischer, JensUNSPECIFIEDhttps://orcid.org/0000-0002-8987-0859UNSPECIFIED
Rommel, TobiasUNSPECIFIEDhttps://orcid.org/0000-0003-1864-7585UNSPECIFIED
Stens, RudolfUNSPECIFIEDUNSPECIFIEDUNSPECIFIED
Date:April 2021
Journal or Publication Title:Proceedings of the European Conference on Synthetic Aperture Radar, EUSAR
Refereed publication:Yes
Open Access:Yes
Gold Open Access:No
In SCOPUS:Yes
In ISI Web of Science:No
Page Range:pp. 424-429
Publisher:VDE Verlag
ISSN:2197-4403
Status:Published
Event Title:European Conference on Synthetic Aperture Radar (EUSAR)
Event Location:Leipzig (virtually), Germany
Event Type:international Conference
Event Start Date:29 March 2021
Event End Date:1 April 2021
Organizer:VDE
HGF - Research field:Aeronautics, Space and Transport
HGF - Program:Space
HGF - Program Themes:Earth Observation
DLR - Research area:Raumfahrt
DLR - Program:R EO - Earth Observation
DLR - Research theme (Project):R - Aircraft SAR
Location: Oberpfaffenhofen
Institutes and Institutions:Microwaves and Radar Institute > SAR Technology
Deposited By: Fischer, Jens
Deposited On:03 Feb 2020 07:41