Wigner distribution function

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 ] , its non-stationary autocorrelation function is given by

C x ( t 1 , t 2 ) = ⟨ ( x [ t 1 ] − μ [ t 1 ] ) ( x [ t 2 ] − μ [ t 2 ] ) ∗ ⟩ ,

where ⟨ ⋯ ⟩ denotes the average over all possible realizations of the process and μ ( t ) is the mean, which may or may not be a function of time. The Wigner function 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 and time lag τ = 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 .

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 .

The motivation for the Wigner function is that it reduces to the spectral density function at all times t 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.

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(t) = 1, then

W x ( t , f ) = ∫ − ∞ ∞ e − i 2 π τ f d τ = δ ( f ) .

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 x(t) = e ^i2πkt, then

W x ( t , f ) = ∫ − ∞ ∞ e i 2 π k ( t + τ 2 ) e − i 2 π k ( t − τ 2 ) e − i 2 π τ f d τ = ∫ − ∞ ∞ e − i 2 π τ ( f − k ) d τ = δ ( f − k ) .

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

x ( t ) = e i 2 π k t 2 ,

then its instantaneous frequency is

1 2 π d ( 2 π k t 2 ) d t = 2 k t   ,

and its WDF

W x ( t , f ) = ∫ − ∞ ∞ e i 2 π k ( t + τ 2 ) 2 e − i 2 π k ( t − τ 2 ) 2 e − i 2 π τ f d τ = ∫ − ∞ ∞ e i 4 π k t τ e − i 2 π τ f d τ = ∫ − ∞ ∞ e − i 2 π τ ( f − 2 k t ) d τ = δ ( f − 2 k t )   .

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

W x ( t , f ) = ∫ − ∞ ∞ δ ( t + τ 2 ) δ ( t − τ 2 ) e − i 2 π τ f d τ = 4 ∫ − ∞ ∞ δ ( 2 t + τ ) δ ( 2 t − τ ) e − i 2 π τ f d τ = 4 δ ( 4 t ) e i 4 π t f = δ ( t ) e i 4 π t f = δ ( t ) .

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.

Boxcar function

x ( t ) = { 1 | t | < 1 / 2 0 otherwise ,

the rectangular function   ⇒

W x ( t , f ) = { 1 π f sin ⁡ ( 2 π f { 1 − 2 | t | } ) | t | < 1 / 2 0 otherwise

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, 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. The following are some examples that exhibit the cross-term feature of the Wigner distribution function.

x ( t ) = { cos ⁡ ( 2 π t ) t ≤ − 2 cos ⁡ ( 4 π t ) − 2 < t ≤ 2 cos ⁡ ( 3 π t ) t > 2

x ( t ) = e i t 3

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.

Projection property

| x ( t ) | 2 = ∫ − ∞ ∞ W x ( t , f ) d f | X ( f ) | 2 = ∫ − ∞ ∞ W x ( t , f ) d t

Energy property

∫ − ∞ ∞ ∫ − ∞ ∞ W x ( t , f ) d f d t = ∫ − ∞ ∞ | x ( t ) | 2 d t = ∫ − ∞ ∞ | X ( f ) | 2 d f

Recovery property

∫ − ∞ ∞ W x ( t 2 , f ) e i 2 π f t d f = x ( t ) x ∗ ( 0 ) ∫ − ∞ ∞ W x ( t , f 2 ) e i 2 π f t d t = X ( f ) X ∗ ( 0 )

Mean condition frequency and mean condition time

X ( f ) = | X ( f ) | e i 2 π ψ ( f ) , x ( t ) = | x ( t ) | e i 2 π ϕ ( t ) , if  ϕ ′ ( t ) = | x ( t ) | − 2 ∫ − ∞ ∞ f W x ( t , f ) d f  and  − ψ ′ ( f ) = | X ( f ) | − 2 ∫ − ∞ ∞ t W x ( t , f ) d t

Moment properties

∫ − ∞ ∞ ∫ − ∞ ∞ t n W x ( t , f ) d t d f = ∫ − ∞ ∞ t n | x ( t ) | 2 d t ∫ − ∞ ∞ ∫ − ∞ ∞ f n W x ( t , f ) d t d f = ∫ − ∞ ∞ f n | X ( f ) | 2 d f

Real properties

W x ∗ ( t , f ) = W x ( t , f )

Region properties

If  x ( t ) = 0  for  t > t 0  then  W x ( t , f ) = 0  for  t > t 0 If  x ( t ) = 0  for  t < t 0  then  W x ( t , f ) = 0  for  t < t 0

Multiplication theorem

If  y ( t ) = x ( t ) h ( t ) then  W y ( t , f ) = ∫ − ∞ ∞ W x ( t , ρ ) W h ( t , f − ρ ) d ρ

Convolution theorem

If  y ( t ) = ∫ − ∞ ∞ x ( t − τ ) h ( τ ) d τ then  W y ( t , f ) = ∫ − ∞ ∞ W x ( ρ , f ) W h ( t − ρ , f ) d ρ

Correlation theorem

If  y ( t ) = ∫ − ∞ ∞ x ( t + τ ) h ∗ ( τ ) d τ  then  W y ( t , ω ) = ∫ − ∞ ∞ W x ( ρ , ω ) W h ( − t + ρ , ω ) d ρ

Time-shifting covariance

If  y ( t ) = x ( t − t 0 ) then  W y ( t , f ) = W x ( t − t 0 , f )

Modulation covariance

If  y ( t ) = e i 2 π f 0 t x ( t ) then  W y ( t , f ) = W x ( t , f − f 0 )

Scale covariance

If  y ( t ) = a x ( a t )  for some  a > 0  then  then  W y ( t , f ) = W x ( a t , f a )

Windowed Wigner Distribution Function

When a signal is not time limited, its Wigner Distribution Function is hard to implement. Thus we add a new function(mask) to its integration part, so that we only have to implement part of the original function instead of integrate all the way from negative infinity to positive infinity. Original function: Function with mask: is real and time-limited

Implementation

According to definition:

And then we compare the difference between two conditions.

3 Conditions

Then we consider the condition with mask function:

We can see that have value only between –B to B, thus conducting with can remove cross term of the function. But if x(t) is not a Delta function nor a narrow frequency function, instead, it is a function with wide frequency or ripple. The edge of the signal may still exist between –B and B, which still cause the cross term problem. for example: