The Wigner distribution function (WDF) is used in signal processing as a transform in time-frequency analysis. The WDF was first proposed in physics to account for quantum corrections to classical statistical mechanics in 1932 by Eugene Wigner, and it is of importance in quantum mechanics in phase space (see, by way of comparison: Wigner quasi-probability distribution, also called the Wigner function or the Wigner–Ville distribution). Given the shared algebraic structure between position-momentum and time-frequency conjugate pairs, it also usefully serves in signal processing, as a transform in time-frequency analysis, the subject of this article. Compared to a short-time Fourier transform, such as the Gabor transform, the Wigner distribution function provides the highest possible temporal vs frequency resolution which is mathematically possible within the limitations of the uncertainty principle. The downside is the introduction of large cross terms between every pair of signal components and between positive and negative frequencies, which makes the original formulation of the function a poor fit for most analysis applications. Subsequent modifications have been proposed which preserve the sharpness of the Wigner distribution function but largely suppress cross terms.
Mathematical definition There are several different definitions for the Wigner distribution function. The definition given here is specific to time-frequency analysis. Given the time series x [ t ] {\displaystyle x[t]} , its non-stationary auto-covariance function is given by
C x ( t 1 , t 2 ) = ⟨ ( x [ t 1 ] − μ [ t 1 ] ) ( x [ t 2 ] − μ [ t 2 ] ) ∗ ⟩ , {\displaystyle C_{x}(t_{1},t_{2})=\left\langle \left(x[t_{1}]-\mu [t_{1}]\right)\left(x[t_{2}]-\mu [t_{2}]\right)^{*}\right\rangle ,}
where ⟨ ⋯ ⟩ {\displaystyle \langle \cdots \rangle } denotes the average over all possible realizations of the process and μ ( t ) {\displaystyle \mu (t)} is the mean, which may or may not be a function of time. The Wigner function W x ( t , f ) {\displaystyle W_{x}(t,f)} is then given by first expressing the autocorrelation function in terms of the average time t = ( t 1 + t 2 ) / 2 {\displaystyle t=(t_{1}+t_{2})/2} and time lag τ = t 1 − t 2 {\displaystyle \tau =t_{1}-t_{2}} , and then Fourier transforming the lag.
W x ( t , f ) = ∫ − ∞ ∞ C x ( t + τ 2 , t − τ 2 ) e − 2 π i τ f d τ . {\displaystyle W_{x}(t,f)=\int _{-\infty }^{\infty }C_{x}\left(t+{\frac {\tau }{2}},t-{\frac {\tau }{2}}\right)\,e^{-2\pi i\tau f}\,d\tau .}
So for a single (mean-zero) time series, the Wigner function is simply given by
W x ( t , f ) = ∫ − ∞ ∞ x ( t + τ 2 ) x ∗ ( t − τ 2 ) e − 2 π i τ f d τ . {\displaystyle W_{x}(t,f)=\int _{-\infty }^{\infty }x\left(t+{\frac {\tau }{2}}\right)\,x^{*}\left(t-{\frac {\tau }{2}}\right)\,e^{-2\pi i\tau f}\,d\tau .}
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