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In the field of time–frequency analysis, several signal formulations are used to represent the signal in a joint time–frequency domain.〔L. Cohen, "Time–Frequency Analysis," ''Prentice-Hall'', New York, 1995. ISBN 978-0135945322〕 There are several methods and transforms called "time-frequency distributions" (TFDs), whose interconnections were organized by Leon Cohen.〔L. Cohen, "Generalized phase-space distribution functions," ''Jour. Math. Phys.'', vol.7, pp. 781–786, 1966.〕 〔L. Cohen, "Quantization Problem and Variational Principle in the Phase Space Formulation of Quantum Mechanics," ''Jour. Math. Phys.'', vol.7, pp. 1863–1866, 1976.〕〔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. 79–110, 1982.〕〔E. Sejdić, I. Djurović, J. Jiang, “Time-frequency feature representation using energy concentration: An overview of recent advances,” ''Digital Signal Processing'', vol. 19, no. 1, pp. 153-183, January 2009.〕 The most useful and popular methods form a class referred to as "quadratic" or bilinear time–frequency distributions. A core member of this class is the Wigner–Ville distribution (WVD), as all other TFDs can be written as a smoothed or 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".〔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.〕 The scope of this article is to illustrate 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 this article is "signal processing". Noting that a signal can recovered from a particular distribution under certain conditions, given a certain TFD ''ρ''1(''t,f'') representing the signal in a joint time–frequency domain, another, different, TFD ''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 in Cohen's book. ==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–frequency representation, such as Wigner distribution function (WDF) and other bilinear time–frequency distributions, can be expressed as : (1) where 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). The kernel of the Wigner distribution function (WDF) is one. However, no particular significance should be attached to that, since it is possible 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Transformation between distributions in time–frequency analysis」の詳細全文を読む スポンサード リンク
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