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26994 Closed-form expressions for time-frequency operations involving Hermite functions
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Korevaar, C.W. and Oude Alink, M.S. and de Boer, P.T. and Kokkeler, A.B.J. and Smit, G.J.M. (2016) Closed-form expressions for time-frequency operations involving Hermite functions. IEEE transactions on signal processing, 64 (6). 1383 -1390. ISSN 1053-587X *** ISI Impact 2,624 ***

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Official URL: http://dx.doi.org/10.1109/TSP.2015.2488580

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Abstract

The product, convolution, correlation, Wigner distribution function (WDF) and ambiguity function (AF) of two Hermite functions of arbitrary order n and m are derived and expressed as a bounded, weighted sum of n+m Hermite functions. It was already known that these mathematical operations performed on Gaussians (Hermite functions of the zeroth-order) lead to a result which can be expressed as a Gaussian function again. We generalize this reciprocity to Hermite functions of arbitrary order. The product, convolution, correlation, WDF, and AF operations performed on two Hermite functions of arbitrary order lead to remarkably similar closed-form expressions, where the difference between the operations is primarily determined by distinct phase changes of the weights of the Hermite functions in the result. The closed-form expressions are generalized to the class of square-integrable functions. A key insight from the closed-form expressions is applied to the design of orthogonal, time-frequency localized communication signals which are characterized by an AF with rotational symmetry. In addition to this application, the theoretical expressions may prove useful for signal analysis in fields ranging from communications, radar and image processing to quantum mechanics.

Item Type:Article
Research Group:EWI-CAES: Computer Architecture for Embedded Systems, EWI-DACS: Design and Analysis of Communication Systems
Research Program:CTIT-General
Research Project:TSP: Tracking, tracing, Sensoring Platform
Uncontrolled Keywords:Hermite functions, Wigner distribution function, Ambiguity function, Closed-form solutions, Correlation functions, Signal analysis, Signal detection, Closed-form solutions, Convolution, Correlation, Eigenvalues and eigenfunctions, Fourier transforms, Polynomials, Time-frequency analysis
ID Code:26994
Status:Published
Deposited On:03 May 2016
Refereed:Yes
International:Yes
ISI Impact Factor:2,624
More Information:statisticsmetis

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