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==Introduction==
In the field of [[time-frequencytime–frequency analysis]], the goal is to define signal formulations that are used for representing the signal in a joint time-frequencytime–frequency ___domain (see also [[time-frequencytime–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).<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" 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 version 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 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 0080443354.</ref>.
The scope of this article is to outline some elements of the procedure to transform one distribution into another. The method used to transform a distribution is borrowed from [[quantum mechanics]], even though 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-frequencytime–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 can be given from a signal processing perspective.<ref>B. Boashash, editor, “Time-Frequency Signal Analysis and Processing – A Comprehensive Reference”, Elsevier Science, Oxford, 2003; ISBN 0080443354</ref>.
==General class==
If we use the variable ''ω''=2''πf'', then, borrowing the notations used in the field of quantum mechanics, we can show that time-frequencytime–frequency representation, such as [[Wigner distribution function]] (WDF) and other [[bilinear time-frequencytime–frequency distribution]]s, 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)
{{Reflist}}
[1] L. Cohen, "TIME-FREQUENCYTime–Frequency ANALYSISAnalaysis," ''Prentice-Hall'', New York, 1995.
[2] L. Cohen, "Generalized phase-space distribution functions," ''Jour. Math. Phys.'', vol.7, pp. 781–786, 1966.
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