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The fractional fourier transform and the wigner distribution

Published online by Cambridge University Press:  17 February 2009

David Mustard
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Abstract

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The Wigner distribution and many other members of the Cohen class of generalized phase-space distributions of a signal all share certain translation properties and the property that their two marginal distributions of energy density along the time and along the frequency axes equal the signal power and the spectral energy density. A natural generalization of this last property is shown to be a certain relationship through the Radon transform between the distribution and the signal's fractional Fourier transform. It is shown that the Wigner distribution is now distinguished by being the only member of the Cohen class that has this generalized property as well as a generalized translation property. The inversion theorem for the Wigner distribution is then extended to yield the fractional Fourier transforms.

Type
Research Article
Information
The ANZIAM Journal , Volume 38 , Issue 2 , October 1996 , pp. 209 - 219
Copyright
Copyright © Australian Mathematical Society 1996

References

[1]Bargmann, V., “On a Hilbert space of analytic functions and an associated integral transform. Part I.”, Comm. Pure Appl. Math. 14 (1961) 187–214.CrossRefGoogle Scholar
[2]Bouachache, B., Escudié, B., Flandrin, P. and Gréa, J., “Sur une condition nécessaire et suffisante de positivité de la représentation conjointe en temps et fréquence des signaux d'énergie finie”, C. R. Acad. Sc. Paris serie A 288 (1979) 307–309.Google Scholar
[3]Claasen, T. A. C. M. and Mecklenbräuker, W. F. G., “The Wigner distribution – a tool for time-frequency signal analysis III”, Philips J. Res. 35 (1980) 372–389.Google Scholar
[4]Cohen, L., “Generalized phase-space distribution functions”, J. of Math. Phys. 7 (1966) 781–786.CrossRefGoogle Scholar
[5]Condon, E. U., “Immersion of the Fourier transform in a continuous group of functional transformations”, Proc. Nat. Acad. Sci. USA 23 (1937) 158–164.CrossRefGoogle Scholar
[6]Deans, S. R., The Radon Transform and Some of its Applications (Wiley, New York, 1983).Google Scholar
[7]Dym, H. and McKean, H. P., Fourier Series and Integrals (Academic Press, New York, 1972).Google Scholar
[8]Erdélyi, A, Magnus, W., Oberhettinger, F. and Tricomi, F. G., Higher Transcendental Functions, Vol. II (McGraw Hill, New York, 1953).Google Scholar
[9]Flandrin, P. and Escudié, B., “Time and frequency representation of finite-energy signals: a physical property as a result of an Hilbertian condition”, Signal Processing 2 (1980) 93–100.CrossRefGoogle Scholar
[10]Fock, V., “Verallgemeinerung und Lösung der Diracschen statistischen Gleichung”, Z. Physik 49 (1928) 339–357.CrossRefGoogle Scholar
[11]Folland, G. B., Harmonic analysis in phase space, Annals of Mathematics Studies (Princeton Univ. Press, Princeton, 1989).CrossRefGoogle Scholar
[12]Helgason, S., The Radon transform, Progress in Mathematics Vol. 5 (Birkhaüser, Basel, 1980).CrossRefGoogle Scholar
[13]Miller, W. Jr, Lie Theory and Special Functions (Academic Press, New York, 1968).Google Scholar
[14]Mustard, D., “The fractional Fourier transform and the Wigner distribution”, Applied Mathematics Preprint AM89/6 School of Mathematics, UNSW (1989).Google Scholar
[15]Mustard, D., “Uncertainty principles invariant under the fractional Fourier transform”, J. Austral. Math. Soc. Ser. B 33 (1991) 180–191.CrossRefGoogle Scholar
[16]Mustard, D., “Lie group imbeddings of the Fourier transform”, Applied Mathematics Preprint AM87/13 School of Mathematics, UNSW (1987).Google Scholar
[17]Mustard, D., “Lie group imbeddings of the Fourier transform and a new family of uncertainty principles”, in Proceedings of the Centre for Mathematical Analysis, Australian National University, Volume 16, Miniconferences on Harmonic Analysis and Operator Algebras (Canberra: 5–8 August and 2–3 December 1987) (1987) 211–222.Google Scholar
[18]Ozaktas, H. M., Barshan, B., Mendlovic, D. and Onural, L., “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms”, J. Opt. Soc. Am. A 11 (1994) 547–559.CrossRefGoogle Scholar
[19]Nievergelt, Y., SI AM Review 28 (1986) 79–84.CrossRefGoogle Scholar
[20]Papoulis, A., Signal Analysis (McGraw-Hill, New York, 1977).Google Scholar
[21]Proc. IEEE Special Issue on Computerized Tomography (03 1983).Google Scholar
[22]Taylor, M., “Noncommutative microlocal analysis, Pt. I”, Memoirs of the AMS 52 (1984), No. 313.CrossRefGoogle Scholar
[23]Vilenkin, N. Ja., Special Functions and Theory of Group Representations (Izd. Nauka., Moscow, 1965 (in Russian)) Special Functions and the Theory of Group Representations (AMS Transl. Vol.22, Providence, R.I., 1968 (English Transl.)).Google Scholar
[24]Wigner, E., “On the quantum correction for thermodynamic equilibrium”, Phys. Rev. 40 (1932) 749–759.CrossRefGoogle Scholar