Wigner distribution function
The Wigner distribution function (WDF) is used in signal processing as a transform in time-frequency analysis.
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.
for stationary processes, yet it is fully equivalent to the non-stationary autocorrelation function.
Therefore, the Wigner function tells us (roughly) how the spectral density changes in time.
When the input signal is constant, its time-frequency distribution is a horizontal line along the time axis.
For example, if x(t) = 1, then When the input signal is a sinusoidal function, its time-frequency distribution is a horizontal line parallel to the time axis, displaced from it by the sinusoidal signal's frequency.
For example, if then its instantaneous frequency is and its WDF When the input signal is a delta function, since it is only non-zero at t=0 and contains infinite frequency components, its time-frequency distribution should be a vertical line across the origin.
By WDF The Wigner distribution function is best suited for time-frequency analysis when the input signal's phase is 2nd order or lower.
Negative features of the WDF are reflective of the Gabor limit of the classical signal and physically unrelated to any possible underlay of quantum structure.
The following are some examples that exhibit the cross-term feature of the Wigner distribution function.
The Wigner distribution function has several evident properties listed in the following table.
When a signal is not time limited, its Wigner Distribution Function is hard to implement.
