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

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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.<ref>L. Cohen, "Time–Frequency Analysis," ''Prentice-Hall'', New York, 1995. {{ISBN |978-0135945322}}</ref>

There are several methods and transforms called "time-frequency distributions" (TFDs), whose interconnections were organized by Leon Cohen.<ref>L. Cohen, "Generalized phase-space distribution functions," ''J. Math. Phys.'', '''7''' (1966) pp.&nbsp;781–786, [http://dx.doi.org/10.1063/1.1931206 doi:10.1063/1.1931206] </ref>

<ref>L. Cohen, "Quantization Problem and Variational Principle in the Phase Space Formulation of Quantum Mechanics," ''J. Math. Phys.'', '''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>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.</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".<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 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.