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  "body": "\n## Summary\n\nThe Wigner distribution of a function $f$ on $\\R$ is a real-valued function on the time–frequency plane built from $f$ and its complex conjugate.\nIts integral over frequency gives $\\abs{f}^2$, and its integral over time gives $\\abs{\\hat f}^2$, so it behaves like a joint density of time and frequency.\nIt can take negative values, so it is not a true probability density.\nEugene Wigner introduced it in 1932 in quantum mechanics, as a quasi-probability distribution on phase space.\nIt is also a basic tool of [[time–frequency analysis|GQFPVXEWK1]].\n\n## Explanations\n\n### L1 human\n\nOne would like a picture that tells how much of a signal sits at each time and each frequency.\nThe Wigner distribution compares the signal at times a little before and a little after each moment, and takes a Fourier transform of that comparison.\nSumming it over all frequencies gives back the energy at each time, and summing over all times gives the energy at each frequency.\nFor a Gaussian pulse it is a Gaussian bump in the plane.\nFor a sum of two pulses it shows an extra wavy pattern between them, which can be negative.\n",
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