Wigner distribution function - AbsoluteAstronomy.com

The Wigner distribution function (WDF) was first proposed to account for quantum corrections to classical statistical mechanics in 1932 by Eugene Wigner, cf. Wigner quasi-probability distribution

Wigner quasi-probability distribution

The Wigner quasi-probability distribution is a quasi-probability distribution. It was introduced by Eugene Wigner in 1932 to study quantum corrections to classical statistical mechanics...

.

Given the shared algebraic structure between position-momentum and time-frequency pairs, it may also usefully serve as a transform in time-frequency analysis

Time-frequency analysis

In signal processing, time–frequency analysis comprises those techniques that study a signal in both the time and frequency domains simultaneously, using various time–frequency representations...

. Compared to a short-time Fourier transform

Short-time Fourier transform

The short-time Fourier transform , or alternatively short-term Fourier transform, is a Fourier-related transform used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time....

, such as the Gabor transform

Gabor transform

The Gabor transform, named after Dennis Gabor, is a special case of the short-time Fourier transform. It is used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time...

, the Wigner distribution function can furnish higher clarity in some cases.

Mathematical definition

There are several different definitions for the Wigner distribution function. The definition given here is specific to time-frequency analysis. The Wigner distribution function is

where

is the imaginary unit

Imaginary unit

In mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...

. The WDF is essentially the Fourier transform of the input signal’s autocorrelation

Autocorrelation

Autocorrelation is the cross-correlation of a signal with itself. Informally, it is the similarity between observations as a function of the time separation between them...

function — the Fourier spectrum of the product between the signal and its delayed, time reversed copy, as a function of the delay.

Time frequency analysis example

Here are some examples to illustrate 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 on to the frequency axis. For example, if x(t) = 1, then

Sinusoidal input signal

When the input signal is a sinusoidal function, its time-frequency distribution is a horizontal line parallel to the frequency axis, at its sinusoidal frequency. For example, if , then

Chirp input signal

When the input signal is a 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 by 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.

Performance of Wigner distribution function

Here are some examples to show performance features of the Wigner distribution function preferable to the Gabor transform.

rectangular function

Cross term property

The Wigner distribution function is not a linear transform. A cross term ("time beats") 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

Wigner quasi-probability distribution

The Wigner quasi-probability distribution is a quasi-probability distribution. It was introduced by Eugene Wigner in 1932 to study quantum corrections to classical statistical mechanics...

, this term has important and useful physics consequences. The short-time Fourier transform does not have this feature. The following are some examples that show the cross term feature of the Wigner distribution function.

In order to reduce the cross term problem, many other transforms have been proposed, including the modified Wigner distribution function, the Gabor–Wigner transform, and Cohen’s class distribution.

Properties of the Wigner distribution function

The Wigner distribution function has several evident properties listed in the following table.

	Remarks
1	Projection property
2	Energy property
3	Recovery property
4	Mean condition frequency and mean condition time
5	Moment properties
6	Real properties
7	Region properties
8	Multiplication theory
9	Convolution theory
10	Correlation theory
11	Time-shifting property
12	Modulation property