Gabor analysis over finite Abelian groups

We consider signals and operators in finite dimension which have sparse time-frequency representations. As main result we show that an S-sparse Gabor representation in C n with respect to a random unimodular window can be recovered by Basis Pursuit with high probability provided that S ≤ Cn/ log(n). Our results are applicable to the channel estimation problem in wireless communications and they establish the usefulness of a class of measurement matrices for compressive sensing.

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“…· n d in all of the results within this paper. Details on time-frequency analysis on general finite Abelian groups can be found in [15,21].…”

Section: Theorem 23 Assume X Is An Arbitrary S-sparse Coefficient Vementioning

confidence: 99%

Sparsity in Time-Frequency Representations

Pfander

Rauhut

2009

J Fourier Anal Appl

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“…For related work on locally compact (abelian) groups we refer to the recent papers [2,3,7,8,13,18,29,34] as well as the book [20] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Co-compact Gabor Systems on Locally Compact Abelian Groups

Jakobsen

Lemvig

2015

J Fourier Anal Appl

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In this work we extend classical structure and duality results in Gabor analysis on the euclidean space to the setting of second countable locally compact abelian (LCA) groups. We formulate the concept of rationally oversampling of Gabor systems in an LCA group and prove corresponding characterization results via the Zak transform. From these results we derive non-existence results for critically sampled continuous Gabor frames. We obtain general characterizations in time and in frequency domain of when two Gabor generators yield dual frames. Moreover, we prove the Walnut and Janssen representation of the Gabor frame operator and consider the Wexler-Raz biorthogonality relations for dual generators. Finally, we prove the duality principle for Gabor frames. Unlike most duality results on Gabor systems, we do not rely on the fact that the translation and modulation groups are discrete and co-compact subgroups. Our results only rely on the assumption that either one of the translation and modulation group (in some cases both) are co-compact subgroups of the time and frequency domain. This presentation offers a unified approach to the study of continuous and the discrete Gabor frames.

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“…PINV-matrices) show how to Most of them are downloadable from NuHAG Talk server http://www.univie.ac.at/nuhag-php/program/talks.php obtain implementations in an efficient way^. The algebraic theory is then pursued in [54] in the setting of general finite Abehan groups. Based on this the papers [55,24] provide a refined view on the tools needed to handle the continuous case.…”

Section: Motivation and Introductionmentioning

confidence: 99%

Banach Gelfand Triples for Applications in Physics and Engineering

Feichtinger¹

2009

AIP Conference Proceedings

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The principle of extension is widespread within mathematics. Starting from simple objects one constructs more sophisticated ones, with a kind of natural embedding from the set of old objects to the new, enlarged set. Usually a set of operations on the old set can still be carried out, but maybe also some new ones. Done properly one obtains more completed objects of a similar kind, with additional useful properties. Let us give a simple example: While multiplication and addition can be done exactly and perfectly in the setting of Q, the rational numbers, the field R of real numbers has the advantage of being complete (Cauchy sequences have a limit ...) and hence allowing for numbers Uke : 7r or \/2. Finally the even "more complicated" field C of complex numbers allows to find solutions to equations like z^ = -1. The chain of inclusions of fields, Q C R C C is a good motivating example in the domain of "numbers".The main subject of the present survey-type article is a new theory of Banach Gelfand triples (BGTs), providing a similar setting in the context of (generalized) functions. Test functions are the simple objects, elements of the Hilbert space L^(R'*) are well suited in order to describe concepts of orthogonaUty, and they can be approximated to any given precision (in the 11 • 112-norm) by test functions. Finally one needs an even larger (Banach) space of generalized functions resp. distributions, containing among others pure frequencies and Dirac measures in order to describe various mappings between such Banach Gelfand triples in terms of the most important "elementary building blocks", in a clear analogy to the finite/discrete setting (where Dirac measures correspond to unit vectors).Our concrete Banach Gelfand triple is based on the Segal algebra 5o(R'*), which coincides with the modulation space M^(R'*) = MQ' (R**), and plays a very important and natural role for timefrequency analysis. We will point out that it provides the appropriate setting for a description of many problems in engineering or physics, including the classical Fourier transform or the Kohn-Nirenberg or Weyl calculus for pseudo-differential operators. Particular emphasis will be given to the concept of w*-convergence and w*-continuity of operators which allows to prove conceptual uniqueness results, and to give a correct interpretation to certain formal expressions coming up in various versions of the Dirac formalism.

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Gabor analysis over finite Abelian groups

Cited by 49 publications

References 47 publications

Sparsity in Time-Frequency Representations

Sparsity in Time-Frequency Representations

Co-compact Gabor Systems on Locally Compact Abelian Groups

Banach Gelfand Triples for Applications in Physics and Engineering

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