The Wigner distribution function 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. 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 uncertainty in quantum wave theory. WDF spectrograms are visually distinctly different than FFT spectrograms. WDF spectrograms are too slow for streaming audio compared to FFT ones: they take about 50 times longer to compute. WDF is a better choice than FFT when studying audio in one detail, where the highest quality TF graph is required, e.g. for a neural network; WDF is computationally too expensive for streaming audio, e.g. speech recognition. To generate a sample-accurate WDF spectrogram in real time would require about 16 cores of a modern desktop PC.
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, its non-stationary autocorrelation function is given by where denotes the average over all possible realizations of the process and is the mean, which may or may not be a function of time. The Wigner function is then given by first expressing the autocorrelation function in terms of the average time and time lag, and then Fourier transforming the lag. So for a single time series, the Wigner function is simply given by The motivation for the Wigner function is that it reduces to the spectral density function at all times for stationary processes, yet it is fully equivalent to the non-stationary autocorrelation function. Therefore, the Wigner function tells us how the spectral density changes in time.
Time-frequency analysis example
Here are some examples illustrating how the WDF is used in time-frequency analysis.
Constant input signal
When the input signal is constant, its time-frequency distribution is a horizontal line along the time axis. For example, if x = 1, then
Sinusoidal input signal
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
Chirp input signal
When the input signal is a linear chirp function, the instantaneous frequency is a linear function. This means that the time frequency distribution should be a straight line. For example, if then its instantaneous frequency is and its WDF
Delta input signal
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. This means that the time frequency distribution of the delta function should also be a delta function. By WDF The Wigner distribution function is best suited for time-frequency analysis when the input signal's phase is 2nd order or lower. For those signals, WDF can exactly generate the time frequency distribution of the input signal.
The Wigner distribution function is not a linear transform. A cross term occurs when there is more than one component in the input signal, analogous in time to frequency beats. In the ancestral physics Wigner quasi-probability distribution, this term has important and useful physics consequences, required for faithful expectation values. By contrast, the short-time Fourier transform does not have this feature. 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. In order to reduce the cross-term difficulty, several approaches have been proposed in the literature, some of them leading to new transforms as the modified Wigner distribution function, the Gabor–Wigner transform, the Choi-Williams distribution function and Cohen's class distribution.
Properties of the Wigner distribution function
The Wigner distribution function has several evident properties listed in the following table. ; Projection property ; Energy property ; Recovery property ; Mean condition frequency and mean condition time ; Moment properties ; Real properties ; Region properties ; Multiplication theorem ; Convolution theorem ; Correlation theorem ; Time-shifting covariance ; Modulation covariance ; Scale covariance
