Transformation between distributions in time–frequency analysis: Difference between revisions

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{{Expert-subject|Mathematics|date=January 2010}}

In the field of [[time–frequency analysis]], the goal is to defineseveral signal formulations that are used forto representingrepresent the signal in a joint time–frequency ___domain.<ref> L. Cohen, "Time–Frequency Analaysis," ''Prentice-Hall'', New York, 1995.</ref>(seeSee also [[time–frequency representation]]s<ref>B. Boashash, “Time-Frequency Concepts”, Chapter 1, pp. 3–28, in B. Boashash, ed,, Time-Frequency Signal Analysis & Processing: A Comprehensive Reference, Elsevier Science, Oxford, 2003; ISBN 008044335.</ref>).

There are several methods and transforms called "time-frequency distributions" (TFDs), whose interconnections were organized by L. Cohen.<ref>L. Cohen, "Generalized phase-space distribution functions," ''Jour. Math. Phys.'', vol.7, pp.&nbsp;781–786, 1966.</ref>
<ref>L. Cohen, "Quantization Problem and Variational Principle in the Phase Space Formulation of Quantum Mechanics," ''Jour. Math. Phys.'', vol.7, pp.&nbsp;1863–1866, 1976.</ref><ref>A. J. E. M. Janssen, "On the locus and spread of pseudo-density functions in the time frequency plane," ''Philips Journal of Research'', vol. 37, pp.&nbsp;79–110, 1982.</ref><ref>B. Boashash, “Heuristic Formulation of Time-Frequency Distributions”, Chapter 2, pp. 29–58, in B. Boashash, editor, Time-Frequency Signal Analysis and Processing: A Comprehensive Reference, Elsevier Science, Oxford, 2003; ISBN 008044335.</ref>

The most useful and usedpopular methods form a class referred to as "quadratic" or [[bilinear time–frequency distribution]]s. A core member of this class is the [[Wigner–Ville distribution]] (WVD),<ref>B. Boashash, "Note on the Use of the Wigner Distribution for Time Frequency Signal Analysis", IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. 36, No. 9, pp. 1518–1521, Sept. 1988</ref> as all other TFDs can be written as a smoothed versionor convolved versions of the WVD. Another popular member of this class is the [[spectrogram]] which is the square of the magnitude of the [[short-time Fourier transform]] (STFT). The spectrogram has the advantage of being positive and is easy to interpret, but also has disadvantages, like being irreversible, which means that once the spectrogram of a signal is computed, the original signal can't be extracted from the spectrogram. The theory and methodology for defining a TFD that verifies certain desirable properties is given in the "Theory of Quadratic TFDs" <ref>B. Boashash, “Theory of Quadratic TFDs”, Chapter 3, pp. 59–82, in B. Boashash, editor, Time-Frequency Signal Analysis & Processing: A Comprehensive Reference, Elsevier, Oxford, 2003; ISBN 0-08-044335-4.</ref>.

The scope of this article is to outlineillustrate some elements of the procedure to transform one distribution into another. The method used to transform a distribution is borrowed from the [[phase space formulation]] of [[quantum mechanics]], even though the subject matter of thethis article is "signal processing". Noting that a signal can recovered from a particular distribution under certain conditions, given a certain TFD ρ1''ρ''₁(''t,f'') representing the signal in a joint time–frequency ___domain, another, different, TFD ρ2''ρ''₂(''t,f'') of the same signal can be obtained to calculate any other distribution, by simple smoothing or filtering; some of these relationships are shown below. A full treatment of the question can be given from ain signalCohen's processing perspective.<ref>B. Boashash, editor, “Time-Frequency Signal Analysis and Processing – A Comprehensive Reference”, Elsevier Science, Oxford, 2003; ISBN 0-08-044335-4</ref>book.

==General class==

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==References==

{{Reflist}}

[1] L. Cohen, "Time–Frequency Analaysis," ''Prentice-Hall'', New York, 1995.

[2] L. Cohen, "Generalized phase-space distribution functions," ''Jour. Math. Phys.'', vol.7, pp.&nbsp;781–786, 1966.

[3] L. Cohen, "Quantization Problem and Variational Principle in the Phase Space Formulation of Quantum Mechanics," ''Jour. Math. Phys.'', vol.7, pp.&nbsp;1863–1866, 1976.

[4] A. J. E. M. Janssen, "On the locus and spread of pseudo-density functions in the time frequency plane," ''Philips Journal of Research'', vol. 37, pp.&nbsp;79–110, 1982.

[[Category:Time–frequency analysis]]