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

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In [[time-frequency analysis]], there should be a procedure to transform one distribution into another. It has been shown that a signal can recovered from a particular distribution if the kernel is not zero in a finite region. Given a distribution for which the signal can be recovered, the recovered signal can be taken to calculate any other distribution, so in these cases a relationship to expected to exist between them.

==Introduction==

In the field of [[time-frequency signal processing]] (including [[time-frequency analysis]]), the goal is to define signal formulations that are used for representing the signal in a joint time-frequency ___domain (see also [[time-frequency representations]]<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)<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 used methods form a class referred to as [[quadratic time-frequency distributions]] or [[bilinear time-frequency distributions]]. 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 version of the WVD. Another popular member of this class is the spectrogram (square magnitude of the short-time Fourier transform) which has the advantage of being positive and therefore easy to interpret (but has other disadvantages). The theory and methodology for defining a TFD that verifies certain desirable properties is given in <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 0080443354.</ref>. This article outlines some elements of the procedure to transform one distribution into another, using an approach and terminology that is borrowed from Quantum Mechanics although the subject matter of the article is "signal processing". Noting that a signal can recovered from a particular distribution under certain conditions, given a certain TFD ρ1(t,f) representing the signal in a joint time-frequency ___domain, another different TFD ρ2(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 is given from a signal processing perspective in <ref>B. Boashash, editor, “Time-Frequency Signal Analysis and Processing – A Comprehensive Reference”, Elsevier Science, Oxford, 2003; ISBN 0080443354</ref>.

==General class==

OnlyIf bilinearwe use the variable ω=2πf, then, borrowing the notations used in the field of quantum mechanics, we can show that time-frequency representation, such as [[Wigner distribution function]] (WDF) and other [[Cohen'squadratic class distribution functionTFDs]], can be expressed as

: <math>C(t,\omega) = \dfrac{1}{4\pi^2}\iiint s^*(u-\dfrac{1}{2}\tau)s(u+\dfrac{1}{2}\tau)\phi(\theta,\tau)e^{-j\theta t-j\tau\omega+j\theta u}\, du\,d\tau\,d\theta</math> (1)

where <math>\phi(\theta,\tau)</math> is a two dimensional function called the kernel, which determines the distribution and its properties (for a signal processing terminology and treatment of this question, the reader is referred to the references already cited in the introduction).

For the kernel of the [[Wigner distribution function]] (WDF) is one. However, it is no particular significance should be attached to that since it is to write the general form so that the kernel of any distribution is one, in which case the kernel of the [[Wigner distribution function]] (WDF) would be something else.

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==References==

{{Reflist}}

[1] L. Cohen, "TIME-FREQUENCY ANALYSIS," ''Prentice-Hall'', New York, 1995.