Time Frequency Analysis Tutorial
Cohen’s Class Distribution
Professor: Jian-Jiun Ding
R00942039 胥吉友
Outline
Abstract
Chapter 1
Introduction-----------------------------------------------------3
Chapter 2
Wigner Distribution Function------------------------------4
Chapter 3
Cohen’s Class Distribution---------------------------------10
Chapter 4
Several Types of Cohen’s Class Distribution----------15
Chapter 5
Conclusion------------------------------------------------------18
References
Abstract
In the field of time-frequency analysis, the Cohen’s Class Distribution is a
useful method to obtain the time-frequency representation. In this tutorial, we
first introduce why these methods of time-frequency analysis be invented. In the
chapter 2, we introduce the prototype of Cohen’s Class Distribution: the Wigner
Distribution. In the chapter 3, we introduce the mathematical definition of the
Cohen’s Class Distribution and two important part of the Cohen’s Class
Distribution: the Ambiguity Function and the Filter-like Function Kernel. Then
we introduce several types of Cohen’s Class Distribution in chapter 4. In the end,
we analyze the advantage and disadvantage of Cohen’s Class Distribution.
Chapter 1
Introduction
Although the standard Fourier analysis plays an important role in signal
processing, because that it allows the decomposition of a signal into individual
frequency components and establishes the relative intensity of each component.
However, it cannot tell us when those frequency components occurred. So the
concept of time-frequency analysis is proposed. By the development of these
years, nowadays, we have many tools to do the time-frequency analysis, like the
STFT, the Gabor transform, the Wigner distribution……etc., and our main topic,
the Cohen’s Class Distribution.
Chapter 2
Wigner Distribution Function
The Wigner distribution function (WDF) was first proposed in physics to
account for quantum corrections to classical statistical mechanics in 1932 by
Eugene Wigner. Because of the shared algebraic structure between
position-momentum and time-frequency pairs, it is also useful in time-frequency
signal processing.
2.1 Definition of Wigner Distribution Function
The definition of the Wigner Distribution Function is as follows:

Wx  t , f    x  t   / 2  x*  t   / 2  e j 2 f  d

It is the Fourier transform of the input signal’s autocorrelation function.
2.2 Time-Frequency Analysis Example
There are some examples in the below to illustrate how the WDF is used in
time-frequency analysis, and we can see why the WDF has higher clarity.
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

Wx  t , f    e j 2 f  d    f 

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 x(t) = ei2πht, then




Wx  t , f    e j 2 ht  /2e j 2 ht  /2e j 2 f  d   e j 2  f h d    f  h 
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

Wx  t , f    e j 2 k  t  /2 e  j 2 k t  /2 e  j 2 f  d   e j 4 kt e  j 2 f  d
2
2


 e
 j 2  f  2 kt 


d    f  2kt 
2
1 d  2 kt 
We can check that the instantaneous frequency of input is
 2kt
2
dt
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, then

Wx  t , f      t   / 2    t   / 2  e  j 2 f  d


 4    2t      2t    e  j 2 f  d

 4  4t  e j 4 tf    t  e j 4 tf    t 
From the computation above, we can see that 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, it can exactly generate the time-frequency
distribution of the input signal.
2.3 Cross-Term Problem
Since the Wigner Distribution Function is not a linear transform, the
Cross-Term will occur when there is more than one component in the input
signal. We have a demonstration that the cross-term exists as follows:
If x  t    g t    s t 

Wx  t , f    x  t   / 2  x*  t   / 2  e  j 2 f  d


   g  t   / 2    s  t   / 2    * g *  t   / 2    * s*  t   / 2   e  j 2 f  d



2
2
    g  t   / 2  g *  t   / 2   e  j 2 f  d     s  t   / 2  s*  t   / 2   e  j 2 f  d


 
 

   * g  t   / 2  s*  t   / 2    *  g *  t   / 2  s  t   / 2   e  j 2 f  d

  Wg  t , f    Ws  t , f 
2
2

   * g  t   / 2  s*  t   / 2    *  g *  t   / 2  s  t   / 2   e  j 2 f  d

we can see that the cross-term is



 * g  t   / 2 s*  t   / 2    * g *  t   / 2  s t   / 2  e j 2 f  d
Examples of the WDF
2

exp  jt /10  j 3t  , if  9  t  1
s t   

0, otherwise
2
r  t   exp  jt 2 / 2  j 6t  exp   t  4  /10


f t   s t   r t 
4
2
0
-2
-4
-10
-5
0
5
10
5
10
WDF of s(t)
4
2
0
-2
-4
-10
-5
0
WDF of r(t)
4
2
0
-2
-4
-10
-5
0
5
10
WDF of f (t) = s(t) + r(t)
Horizontal axis: t-axis, Vertical axis: f -axis
2.4 Properties of the Wigner Distribution Function
The Wigner distribution function has several evident properties listed in the
following.
Projection property

x  t    Wx  t , f  df
2
X  f  
2



W
x ,t f
dt
Energy property

 

 
Wx  t , f  dtdf  


x  t  dt  
2


X  f  df
2
Recovery property



Wx  t / 2, f  e j 2 ft df  x  t  x*  0



Wx  t , f / 2  e j 2 ft dt  X  f  X *  0 
Mean condition frequency and mean condition time
x t   x t  e
Xf  Xf e
j 2  t 
If    t   x  t 
2



j 2  f 
fWx  t , f  df and    f   X  f 
2



tWx  t , f  dt
Moment properties
 




 
 
 

t nWx  t , f  dtdf   t n x  t  dt
2

f nWx  t , f  dtdf  


f n X  f  df
2
Real properties
Wx*  t , f   Wx  t , f 
Region properties
If x  t   0 for t  t0 then Wx  t , f   0 for t  t0
If x  t   0 for t  t0 then Wx  t , f   0 for t  t0
Multiplication theory

If y  t   x  t  h  t  then Wy  t , f    Wx  t ,  Wh  t , f    d 

Convolution theory




If y  t    x  t    h   d then Wy  t , f    Wx   , f Wh  t   , f  d 
Correlation theory




If y  t    x  t    h*   d then Wy  t , f    Wx   , f Wh  t   , f  d 
Time-shifting property
If y  t   x  t  t0  then Wy  t , f   Wx  t  t0 , f 
Modulation property
If y  t   e j 2 f0t x  t  then Wy  t , f   Wx  t , f  f 0 
Chapter 3
Cohen’s Class Distribution
The Cohen's Class Distribution function was first proposed in 1966 in the
context of quantum mechanics by L. Cohen. This distribution function is
mathematically similar to a generalized time–frequency representation which
utilizes bilinear transformations.
3.1 Definition of Cohen’s Class Distribution
The definition of the class of Cohen time–frequency distributions is as follows:
Cx  t , f   



 
Ax  ,    ,  exp  j 2 (t   f )  d d
where Ax   is the ambiguity function
which will be further discussed later,

Ax  ,    x  t   / 2  x*  t   / 2   e  j 2 t  dt

and    is the kernel function which is usually a low-pass function and is
used to mask out the interference.
3.2
Ambiguity Function
Consider the well-known power spectral density Px  f  and the signal
auto-correlation function Rx   in the case of a stationary process. The
relationship between these functions is as follows:

Px  f    Rx   e j 2 f  d


Rx     x  t   / 2  x*  t   / 2  dt

For a non-stationary signal x  t  , these relations can be generalized using a
time-dependent power spectral density or equivalently the famous Wigner
distribution function of x  t  as follows:

Wx  t , f    Rx  t ,  e j 2 f  d


Rx  t ,    x  t   / 2 x*  t   / 2 d

If the Fourier transform of the auto-correlation function is taken with respect to t
instead of τ, we get the ambiguity function as follows:

Ax     x  t   / 2 x*  t   / 2  e j 2 t dt

The relationship between the Wigner distribution function, the auto-correlation
function and the ambiguity function can then be illustrated by the following
figure.
For the signal with only 1 term
If x  t   exp   t  t0   j 2 f0t 


2

2
2
Ax     exp    t   / 2  t0   j 2 f 0  t   / 2   exp    t   / 2  t0   j 2 f 0  t   / 2   e  j 2 t dt








2
  exp   2  t  t0    2 / 2  j 2 f 0  e  j 2 t dt



  exp    2t 2   2 / 2   j 2 f 0  e  j 2 t0 e  j 2 t dt



   2  2  
1
exp   

  exp  j 2  f 0  t0  
2
  2 2  
We can see the distribution of WDF and AF as follows.
For the signal with 2 terms
2
2
x  t   exp  1  t  t1   j 2 f1t   exp   2  t  t2   j 2 f 2t 




2
If x1  t   exp  1  t  t1   j 2 f1t 


2
x2  t   exp   2  t  t2   j 2 f 2t 


Ax    Ax1    Ax 2    Ax1x 2    Ax 2 x1  
Ax1   
    2  2 
1
exp    1 
  exp  j 2  f1  t1  
21
21  
  2
Ax 2   
    2  2 
exp    2 
  exp  j 2  f 2  t2  
2 2
2
2


2


1
     t 2    t   j 2   f   2  
1
d
d
d 
d
  exp  j 2  f   t  f d t  
Ax1x 2   
exp    



 
2 
2
 


 
t   t1  t2  / 2, f    f1  f 2  / 2,    1   2  / 2
td  t1  t2 , f d  f1  f 2 ,  d  1   2
Ax 2 x1    A* x1x 2  
We can see the distribution of WDF and AF as follows.
3.3 Filter-like Function Kernel
From the previous section, we know in the ambiguity function, the
distribution of the auto-term is always near to the origin, and the cross-term is
always far from the origin.
With this property, the cross-term can be filtered out effortlessly if a proper
low-pass kernel function is applied in η,τ domain. It means
    1 for small  , 
    0 for large  , 
The following is an example that demonstrates how the cross-term is filtered out.
By Choosing (, ) different types of function, we can get several types of
Cohen’s Class Distribution, we will introduce this part in the next chapter.
3.4 Implementation for the Cohen’s Class Distribution
Since the complexity of Cohen’s Class Distribution is high, we have some
methods to simplify the following expression.
Cx  t , f   



 


Ax      exp  j 2  t   f   d d
  x  u   / 2  x  u   / 2     exp   j 2 u  j 2 t   f   dud d


*
  
Simplify Method 1: Not all value of Ax   should be computed
If    = 0 for || > B or || > C,
Cx  t , f   
  x u   / 2 x u  / 2    exp   j 2 u  j 2 t  f  dudd
C
B

*
C  B 
Simplify Method 2: The parameter  is unrelated to input and output
Cx  t , f   
C
 x u   / 2 x u   / 2   

*
 C 

C

B
B
   exp  j 2  t  u   d  exp   j 2 f  dud

 x  u   / 2 x  u   / 2    , t  u  exp   j 2 f  dud
*
 C 
where   , t       exp  j 2t  d , since   ,t  is unrelated to the
B
B
input, we can evaluate it beforehand, the original three integral will be simplify to
two integral.
Chapter 4
Several Types of Cohen’s Class
Distribution
4.1 Wigner Distribution Function
With the kernel function     1 , it means an all-pass filter; we will get
the Wigner Distribution Function which we have introduced in chapter 2.

Wx  t , f    x  t   / 2  x*  t   / 2  e j 2 f  d

4.2 Choi–Williams distribution function
Choi–Williams distribution function was first proposed by Hyung-Ill Choi
and William J. Williams in 1989. This distribution function adopts exponential
kernel to suppress the cross-term. However, the kernel gain does not decrease
along the η,τ axes in the ambiguity domain. Consequently, the kernel function of
Choi–Williams distribution function can only filter out the cross-terms result
from the components differ in both time and frequency center.
The kernel of Choi–Williams distribution is defined as follows:
2
    exp    


where α is an adjustable parameter. The figure of mask function likes follows
4.3 Cone-shape distribution function
Cone-shape distribution function was first proposed by Yunxin Zhao, Les E.
Atlas, and Robert J. Marks in 1990. The reason why this distribution is so named
is because its kernel function in t,τ domain looks like two cones. The advantage of
this special kernel function is that it can completely remove the cross-term
between two components that have same center frequency, but on the other
hand, the cross-term results from components with the same time center cannot
be removed by the cone-shape kernel.
The kernel of cone-shape distribution function is defined as follows:
   
sin  

exp  2 2 
where α is an adjustable parameter. The figure of mask function likes follows
4.4 Other Types of Cohen’s Class Distribution
There are some other types of Cohen’s Class Distribution, we listed the
kernel function they used in the below.
Page distribution function
    exp  j 

Levin (Margenau-Hill) distribution function
    cos  
Kirkwood distribution function
    exp  j 
Born-Jordan distribution function
    sin c  
Chapter 5
Conclusion
Compare to the Wigner Distribution Function, the Cohen’s class distribution
may avoid the cross term. Compare to the STFT or Gabor transform, it may has
higher clarity. However, it requires more computation time and lacks of well
mathematical properties. Moreover, there is a tradeoff between the quality of the
auto term and the ability of removing the cross terms.
Reference
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the Department of Electrical Engineering, National Taiwan University (NTU),
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[Ref] S. Qian and D. Chen, Joint Time-Frequency Analysis: Methods and
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[Ref] L. Cohen, Time-Frequency Analysis, Prentice-Hall, New York, 1995.
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[Ref] Y. Zhao, L. E. Atlas, and R. J. Marks, “The use of cone-shape kernels for
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[Ref] B. Boashash, "Note on the Use of the Wigner Distribution for Time
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