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Wigner Distribution

The Wigner distribution of a square-integrable complex function f is the time-frequency function

W_f(x,xi)=int_(-infty)^inftyf(x+t/2)f(x-t/2)^_e^(-2piixit)dt.
(1)

Its marginal integrals, interpreted weakly when necessary, recover the position and frequency energy densities,

int_(-infty)^inftyW_f(x,xi)dxi=|f(x)|^2
(2)
int_(-infty)^inftyW_f(x,xi)dx=|f^^(xi)|^2,
(3)

where f^^ is the Fourier transform with kernel e^(-2piixix). If ||f||_2=1, both marginals are probability density functions. Under this normalization, the real-valued Wigner distribution is a quasiprobability distribution, since it need not be nonnegative and therefore need not be a joint probability density function.

See also

Fourier Transform, Fractional Fourier Transform, Probability Density Function

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References

Cohen, L. "Generalized Phase-Space Distribution Functions." J. Math. Phys. 7, 781-786, 1966. https://doi.org/10.1063/1.1931206.Wigner, E. "On the Quantum Correction for Thermodynamic Equilibrium." Phys. Rev. 40, 749-759, 1932. https://doi.org/10.1103/PhysRev.40.749.

Cite this as:

Weisstein, Eric W. "Wigner Distribution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/WignerDistribution.html

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