WignerVilledistribution

The Wigner–Ville distribution is a time–frequency analysis tool used primarily in signal processing to represent the spectral content of a signal as it evolves over time. It is defined for a continuous‑time signal \(x(t)\) by the double integral

\[

W_x(t,f)=\int_{-\infty}^{\infty}x\!\left(t+\frac{\tau}{2}\right)\!x^*\!\left(t-\frac{\tau}{2}\right) e^{-j2\pi f\tau}\,d\tau,

\]

where \(x^*(t)\) denotes the complex conjugate. The result is a function of time \(t\) and frequency \(f\) that combines both time and frequency information in a single representation.

Unlike the short‑time Fourier transform, the Wigner–Ville distribution is quadratic in the signal, which gives it the desirable property of high resolution in both time and frequency. However, this quadratic nature also introduces interference terms (sometimes called cross‑terms) when the signal contains multiple components, which can obscure the true energy distribution. Various smoothing or modified versions, such as the Choi–Williams or pseudo Wigner–Ville distributions, have been developed to mitigate these cross‑term effects.

The distribution preserves the total signal energy in the sense that the integral of the Wigner–Ville distribution over all frequencies yields the instantaneous energy \( |x(t)|^2 \), and the integral over all times yields the total energy of the signal. It is often used in applications ranging from radar signal analysis and seismology to biomedical signal processing and quantum mechanics, where time‑dependent spectral characteristics are of interest.