Wigner distribution function

Summary

The 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. Its 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. It can take negative values, so it is not a true probability density. Eugene Wigner introduced it in 1932 in quantum mechanics, as a quasi-probability distribution on phase space. It is also a basic tool of time–frequency analysis.