Transformation between distributions in time–frequency analysis: Difference between revisions
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There are several methods and transforms called "time-frequency distributions" (TFDs), whose interconnections were organized by Leon Cohen.<ref>[[Leon Cohen|L. Cohen]], "Generalized phase-space distribution functions," ''Jour. Math. Phys.'', vol.7, pp. 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. 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. 79–110, 1982.</ref>
The most useful and popular 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), 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".
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''<sub>1</sub>(''t,f'') representing the signal in a joint time–frequency ___domain, another, different, TFD ''ρ''<sub>2</sub>(''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.
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