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Transformation between distributions in time–frequency analysis
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<p>In the field of <a href="/facts/Time%E2%80%93frequency_analysis/oeysrIKN">time–frequency analysis</a>, several signal formulations are used to represent the signal in a joint time–frequency domain. </p><p>There are several methods and transforms called "time-frequency distributions" (TFDs), whose interconnections were organized by Leon Cohen. The most useful and popular methods form a class referred to as "quadratic" or <a href="/facts/Bilinear_time%E2%80%93frequency_distribution/zS9J0gHy">bilinear time–frequency distributions</a>. A core member of this class is the <a href="/facts/Wigner%E2%80%93Ville_distribution/5tyVfdFF">Wigner–Ville distribution</a> (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 <a href="/facts/Spectrogram/Jt8zHI0H">spectrogram</a> which is the square of the magnitude of the <a href="/facts/Short-time_Fourier_transform/XHo1t2m6">short-time Fourier transform</a> (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". </p><p>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 <a href="/facts/Phase_space_formulation/ogvtdMNs">phase space formulation</a> of <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>, even though the subject matter of this article is "signal processing". Noting that a signal can be recovered from a particular distribution under certain conditions, given a certain TFD <i>ρ</i>1(<i>t</i>,<i>f</i>) representing the signal in a joint time–frequency domain, another, different, TFD <i>ρ</i>2(<i>t</i>,<i>f</i>) 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. </p>