Gabor Analysis and Algorithms [electronic resource] : Theory and Applications /
In his paper Theory of Communication [Gab46], D. Gabor proposed the use of a family of functions obtained from one Gaussian by time-and frequency shifts. Each of these is well concentrated in time and frequency; together they are meant to constitute a complete collection of building blocks into which more complicated time-depending functions can be decomposed. The application to communication proposed by Gabor was to send the coeffi cients of the decomposition into this family of a signal, rather than the signal itself. This remained a proposal-as far as I know there were no seri ous attempts to implement it for communication purposes in practice, and in fact, at the critical time-frequency density proposed originally, there is a mathematical obstruction; as was understood later, the family of shifted and modulated Gaussians spans the space of square integrable functions [BBGK71, Per71] (it even has one function to spare [BGZ75] . . . ) but it does not constitute what we now call a frame, leading to numerical insta bilities. The Balian-Low theorem (about which the reader can find more in some of the contributions in this book) and its extensions showed that a similar mishap occurs if the Gaussian is replaced by any other function that is "reasonably" smooth and localized. One is thus led naturally to considering a higher time-frequency density.
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Boston, MA : Birkhäuser Boston : Imprint: Birkhäuser,
1998
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Subjects: | Mathematics., Functional analysis., Applied mathematics., Engineering mathematics., Applications of Mathematics., Signal, Image and Speech Processing., Appl.Mathematics/Computational Methods of Engineering., Functional Analysis., |
Online Access: | http://dx.doi.org/10.1007/978-1-4612-2016-9 |
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Mathematics. Functional analysis. Applied mathematics. Engineering mathematics. Mathematics. Applications of Mathematics. Signal, Image and Speech Processing. Appl.Mathematics/Computational Methods of Engineering. Functional Analysis. Mathematics. Functional analysis. Applied mathematics. Engineering mathematics. Mathematics. Applications of Mathematics. Signal, Image and Speech Processing. Appl.Mathematics/Computational Methods of Engineering. Functional Analysis. |
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Mathematics. Functional analysis. Applied mathematics. Engineering mathematics. Mathematics. Applications of Mathematics. Signal, Image and Speech Processing. Appl.Mathematics/Computational Methods of Engineering. Functional Analysis. Mathematics. Functional analysis. Applied mathematics. Engineering mathematics. Mathematics. Applications of Mathematics. Signal, Image and Speech Processing. Appl.Mathematics/Computational Methods of Engineering. Functional Analysis. Feichtinger, Hans G. editor. Strohmer, Thomas. editor. SpringerLink (Online service) Gabor Analysis and Algorithms [electronic resource] : Theory and Applications / |
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In his paper Theory of Communication [Gab46], D. Gabor proposed the use of a family of functions obtained from one Gaussian by time-and frequency shifts. Each of these is well concentrated in time and frequency; together they are meant to constitute a complete collection of building blocks into which more complicated time-depending functions can be decomposed. The application to communication proposed by Gabor was to send the coeffi cients of the decomposition into this family of a signal, rather than the signal itself. This remained a proposal-as far as I know there were no seri ous attempts to implement it for communication purposes in practice, and in fact, at the critical time-frequency density proposed originally, there is a mathematical obstruction; as was understood later, the family of shifted and modulated Gaussians spans the space of square integrable functions [BBGK71, Per71] (it even has one function to spare [BGZ75] . . . ) but it does not constitute what we now call a frame, leading to numerical insta bilities. The Balian-Low theorem (about which the reader can find more in some of the contributions in this book) and its extensions showed that a similar mishap occurs if the Gaussian is replaced by any other function that is "reasonably" smooth and localized. One is thus led naturally to considering a higher time-frequency density. |
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Mathematics. Functional analysis. Applied mathematics. Engineering mathematics. Mathematics. Applications of Mathematics. Signal, Image and Speech Processing. Appl.Mathematics/Computational Methods of Engineering. Functional Analysis. |
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Feichtinger, Hans G. editor. Strohmer, Thomas. editor. SpringerLink (Online service) |
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Feichtinger, Hans G. editor. Strohmer, Thomas. editor. SpringerLink (Online service) |
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Feichtinger, Hans G. editor. |
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Gabor Analysis and Algorithms [electronic resource] : Theory and Applications / |
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Gabor Analysis and Algorithms [electronic resource] : Theory and Applications / |
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Gabor Analysis and Algorithms [electronic resource] : Theory and Applications / |
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Gabor Analysis and Algorithms [electronic resource] : Theory and Applications / |
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Gabor Analysis and Algorithms [electronic resource] : Theory and Applications / |
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gabor analysis and algorithms [electronic resource] : theory and applications / |
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Boston, MA : Birkhäuser Boston : Imprint: Birkhäuser, |
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1998 |
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http://dx.doi.org/10.1007/978-1-4612-2016-9 |
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KOHA-OAI-TEST:2248522018-07-31T00:04:35ZGabor Analysis and Algorithms [electronic resource] : Theory and Applications / Feichtinger, Hans G. editor. Strohmer, Thomas. editor. SpringerLink (Online service) textBoston, MA : Birkhäuser Boston : Imprint: Birkhäuser,1998.engIn his paper Theory of Communication [Gab46], D. Gabor proposed the use of a family of functions obtained from one Gaussian by time-and frequency shifts. Each of these is well concentrated in time and frequency; together they are meant to constitute a complete collection of building blocks into which more complicated time-depending functions can be decomposed. The application to communication proposed by Gabor was to send the coeffi cients of the decomposition into this family of a signal, rather than the signal itself. This remained a proposal-as far as I know there were no seri ous attempts to implement it for communication purposes in practice, and in fact, at the critical time-frequency density proposed originally, there is a mathematical obstruction; as was understood later, the family of shifted and modulated Gaussians spans the space of square integrable functions [BBGK71, Per71] (it even has one function to spare [BGZ75] . . . ) but it does not constitute what we now call a frame, leading to numerical insta bilities. The Balian-Low theorem (about which the reader can find more in some of the contributions in this book) and its extensions showed that a similar mishap occurs if the Gaussian is replaced by any other function that is "reasonably" smooth and localized. One is thus led naturally to considering a higher time-frequency density.1 The duality condition for Weyl-Heisenberg frames -- 2 Gabor systems and the Balian-Low Theorem -- 3 A Banach space of test functions for Gabor analysis -- 4 Pseudodifferential operators, Gabor frames, and local trigonometric bases -- 5 Perturbation of frames and applications to Gabor frames -- 6 Aspects of Gabor analysis on locally compact abelian groups -- 7 Quantization of TF lattice-invariant operators on elementary LCA groups -- 8 Numerical algorithms for discrete Gabor expansions -- 9 Oversampled modulated filter banks -- 10 Adaptation of Weyl-Heisenberg frames to underspread environments -- 11 Gabor representation and signal detection -- 12 Multi-window Gabor schemes in signal and image representations -- 13 Gabor kernels for affine-invariant object recognition -- 14 Gabor’s signal expansion in optics.In his paper Theory of Communication [Gab46], D. Gabor proposed the use of a family of functions obtained from one Gaussian by time-and frequency shifts. Each of these is well concentrated in time and frequency; together they are meant to constitute a complete collection of building blocks into which more complicated time-depending functions can be decomposed. The application to communication proposed by Gabor was to send the coeffi cients of the decomposition into this family of a signal, rather than the signal itself. This remained a proposal-as far as I know there were no seri ous attempts to implement it for communication purposes in practice, and in fact, at the critical time-frequency density proposed originally, there is a mathematical obstruction; as was understood later, the family of shifted and modulated Gaussians spans the space of square integrable functions [BBGK71, Per71] (it even has one function to spare [BGZ75] . . . ) but it does not constitute what we now call a frame, leading to numerical insta bilities. The Balian-Low theorem (about which the reader can find more in some of the contributions in this book) and its extensions showed that a similar mishap occurs if the Gaussian is replaced by any other function that is "reasonably" smooth and localized. One is thus led naturally to considering a higher time-frequency density.Mathematics.Functional analysis.Applied mathematics.Engineering mathematics.Mathematics.Applications of Mathematics.Signal, Image and Speech Processing.Appl.Mathematics/Computational Methods of Engineering.Functional Analysis.Springer eBookshttp://dx.doi.org/10.1007/978-1-4612-2016-9URN:ISBN:9781461220169 |