An Introduction to
Time-Frequency Analysis
Speaker: Po-Hong Wu
Advisor: Jian-Jung Ding
Digital Image and Signal Processing Lab
GICE, National Taiwan University
NTU, GICE, MD531, DISP Lab
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Outline
•
•
•
•
•
•
•
•
•
•
•
•
Introduction
Short-Time Fourier Transform
Gabor Transform
Wigner Distribution Function
Spectrogram
S Tranform
Cohen’s Class Time-Frequency Distribution
Fractional Fourier Transform
Motion on Time-Frequency Distributions
Hilbert-Huang Transform
Conclusion
Reference
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Introduction
Fourier transform (FT)

X  f    x  t  e  j 2 f t dt
t varies from ∞~∞
Time-Domain  Frequency Domain

f(t)
1
2
0.5
Fourier
transform
1
0
0
-0.5
-1
[A1]
-1
0
5
10
15
20
25
30
-2
-5
0
5
Why do we need time-frequency transform?
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Example: x(t) = cos( t) when t < 10,
x(t) = cos(3 t) when 10  t < 20,
x(t) = cos(2 t) when t  20
-5
-4
-3
-2
-1
0
1
2
3
[B2]
4
5
0
5
10
15
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20
25
30
4
Short Time Fourier Transform

X  t , f    w  t    x   e  j 2 f  d

w(t): mask function
也稱作
windowed Fourier transform
or
time-dependent Fourier transform
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When w(t) is a rectangular function
w(t) = 1 for |t|  B ,
w(t) = 0 , otherwise
X t, f   
tB
t B
x   e  j 2 f  d
STFT
14
12
Hz
10
8
6
4
2
0
[B3]
0
1
2
3
4
5
6
7
8
9
10
11
sec
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Advantage: less computation time
Disadvantage: worse representaion
Application: deal with large data
Ex: real time processing
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Gabor Transform
A specail case of the STFT
2
where w(t )  exp( t )

Gx  t , f    e  ( t ) e  j 2 f  x   d

2
Other definition
1
Gx  t ,   
2



e

( t )2
2
e
t
 j (  )
2
x   d
Gabor
14
12
Hz
10
8
6
4
2
0
[B4]
0
1
2
3
4
5
6
7
8
9
10
11
sec
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Why do we choose the Guassian function?
 Among all functions of w(t), the Gaussian
function has area in time-frequency distribution
is minimal than other STFT.
 Gaussian function is an eigenfunction of Fourier
transform, so the Gabor transform has the same
properties in time domain and in frequency
domain.
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Approximation of the Gabor Transform
Because of e
Gx  t , f   
 a 2
t 1.9143
t 1.9143
Because of e
 0.00001
e
 a2 / 2
1
Gx ,3  t ,   
2
 ( t ) 2  j 2 f 
e
 0.00001
( t )2
t  4.7985 
2
t  4.7985

when |a|>1.9143
e
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x   d
when |a|>4.7985
e
t
 j (  )
2
x   d
10
Generalization of the Gabor Transform

Gx  t , f    e

 ( t ) 2  j 2 f 
e
x   d
For larger σ: higher resolution in the time
domain but lower resolution in the
frequency domain
For smaller σ: higher resolution in the
frequency domain but lower resolution in
the time domain
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Resolution
• Using the generalized Gabor
transform with larger σ
• Using other time unit instead of
second
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Wigner Distribution Function

Wx  t , f    x  t   / 2  x*  t   / 2  e  j 2 f  d

Other definition

Wx (t , f )   X ( f   / 2) X * ( f   / 2)e j 2 d

Wigner distribution
6
frequency (Hz)
4
2
0
-2
-4
-6
[B5]
0
5
10
15
20
25
30
time (sec)
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Signal auto-correlation function
Cx  t ,   x  t   / 2   x  t   / 2 
Spectrum auto-correlation function
S x  , f   X  f   / 2   X   f  / 2 
Ambiguity function (AF)

Ax  ,    x  t   / 2  x*  t   / 2   e  j 2 t  dt

IFTf
IFTf
Wx(t, f )
FTt
FTt
Cx(t,  )
Ax(,  )
FTt
Sx(, f )
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[B6]
IFTf
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Modified Wigner Distribution
Wigner Ville Distribution
For compressing inner interference

Wx (t , f )   x(t   / 2) x (t   / 2)e j 2 f d
*

Analytic signal
x(t )  x(t )  jx(t )
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 Pseudo Wigner Distribution
For surpressing outer interference

Wx (t , f )   w( / 2) w( / 2)x(t   / 2) x* (t   / 2)e  j 2 f d


  Y (t , f   / 2)Y * (t , f   / 2) d
where


Y (t , f )   w( ) x(t   )e j 2 f  d

Pseudo L-Wigner distribution
6
4
frequency (Hz)
2
0
-2
[B7]
-4
-6
0
5
10
15
20
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DISP Lab
time (s)
25
30
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Gabor-Wigner Distribution
C f (t , w)  p(G f (t , w),W f (t , w)),
(a) 10
(b) 10
5
5
0
0
-5
-5

-10
-10
-5
0
5
10
-10
-10
p( x, y )  xy
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-5
0
5
[B8]
10
p ( x, y )  min( x , y )
2
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Spectrogram

SP(t , f ) 


x( )h(t   )e
 j 2 f 
d
2
Another form
SP(t , f )  



 
Wh (  t , f  )Wx ( , )d d
Spectrogram
14
12
Hz
10
8
6
4
2
0
0
1
2
3
4
5
6
7
sec
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9
10
11
[B9]
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S-Transform
Original S-Transform
 f 2 (  t )2 i 2 ft
ST ( , f )   x(t )
exp[
]e
dt

2
2
f

 f 2 (  t )2
exp[
]
Where w(t)=
2
2
f
S Transform
4
3
2
frequency
cos( t ) when 1  t  10

x(t )  cos(3 t ) when 10  t  20
cos(2 t) when 20  t  30

1
0
-1
-2
-3
[B10]
-4
0
5
10
15
20
25
30
time (sec)
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Generalized S-Transform

GS ( , f , p)   x(t )w(  t , f , p)ei 2 t dt

Another definition

GS ( , f , p)   X (  f )W ( , f , p)ei 2 d

Ristriction



w(  t , f , p)d  1
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Novel S-Transform with the Special
Varying Window
wS (  t , f ) 
F ( f )(  t ) 2
exp(
)
2
2
F( f )
Restriction



ws (  t , f )d  1
2
F
(
f
)

1/

When
When F ( f ) 
, it becomes the Gabor transform.
f 2 , it becomes the original S-trnasform.
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Cohen’s Class Time-Frequency Distribution
 
Cx (t , f )    Ax (, )(, )exp( j 2 (t  f ))d d
Ambiguity function

Ax ( , )   x(t   / 2) x* ( x   / 2)e j 2 t dt

Cx (t , )
FTt
IFTf
FTt
IFTf
WDx (t , f )
Ax ( , )
x(t   / 2) x (t   / 2)
[B11]
FTt
S x ( , f )
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IFTf
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For the ambiguity function
The auto terms are always near to
the origin.
The cross terms are always from the
origin.
[B12]
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Kernel function  ( , )
• Choi-Williams Distribution
   exp    
2
 ,
Choi-Williams distribution
10
8
6
4
eta
2
0
-2
-4
[B13]
-6
-8
-10
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-15
-10
-5
0
tau (sec)
5
10
15
24
• Cone-Shape Distribution
2 2
(, )  sin c( )e
 >0
Cone Shape distribution
10
8
6
4
eta
2
0
-2
-4
-6
-8
-10
-15
-10
-5
0
5
10
15
tau (sec)
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Fractional Fourier Transform
FT  x  t   X  f 
FT FT  x  t   x  t 
FT  FT  FT  x  t    X   f   IFT  f  t 
FT  FT  FT FT  x  t    x  t 
How to rotate the time-frequency distribution by
the angle other than /2, , and 3/2?
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• Zero rotation:
R I
0
• Consistency with Fourier transform:
 /2
R = FT
• Additivity of rotation:
R  R  R  
• rotation:
2
R I
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X   u   1  j cot 
2
1

2
f(t): rectangle
0
-1
-5
2
0
5
 j 2 csc  u t
2

1
1
0
0
-1
-5
2


e j cot u  e
2
0
5
1
1
1
0
0
0
-1
-5
-1
-5
-1
-5
x  t  dt

-1
-5
2

e
j cot  t 2
0
5

F(w): sinc function
0
5
0
5
0
5
[A3]
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Application
Decomposition in the time-frequency distribution
1.5
1
2
x(t) = signal + noise
Fourier transform of x(t)
1
0.5
0
0
-0.5
-10
2
1
(non-separable)
-5
0
5
10
-1
-10
0
5
10
5
10
1.5
fractional Fourier
transform
of x(t)
1

0
(separable)
-5
recovered signal
0.5
0
-1
-10
-5
0
5
10
-0.5
-10
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-5
0
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f-axis
Signa
l
noise
Signal
noise
FRFT
FRFT
t-axis
noise
Signa
l
cutoff line
cutoff line
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Modulation and Multiplexing
5
5
unfilled
T-F slot
0
-5
-20
0
-5
0
20
-20
(c) WDF of G(u)
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0
20
(d) GWT of G(u)
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•
Time domain
•
Modulation
•
•
•
Frequency domain fractional domain
Shifting
Shifting
Modulation
Differentiation
 −j2f
Modulation + Shifting
Modulation + Shifting
 j2f
Differentiation and  j2f
Differentiation
Differentiation and  −j2f
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Motion on Time-Frequency Distributions
Horizontal Shifting
x (t  t0 )  S x (t  t0 , f ) e
 j 2 f t0
 Wx ( t  t 0 , f )
,STFT, Gabor
,Wigner
Vertical Shifting
e
j 2 f 0 t
x ( t )  S x ( t , f  f0 ) ,STFT,Gabor
 Wx ( t , f  f0 ) ,Wigner
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Dilation
1
t
t
x ( )  S x ( , af ) ,STFT,Gabor
|a|
a
a
 Wx (
t
, af ) ,WDF
a
Shearing
2
x (t )  e
j at
y (t )
S x (t , f )  S y (t , f  at ) ,STFT,Gabor
Wx (t , f )  W y (t , f  at ) ,WDF
2
j t
x (t )  e a  y (t )
S x (t , f )  S y (t  af , f ) ,STFT,Gabor
Wx (t , f )  W y (t  af , f ) ,WDF
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Rotation
If F{x(t)}=X(f), then F{X(t)}=x(-f).
We can derive:
| S X (t , f ) || S x ( f , t ) |
G X (t , f )  Gx ( f , t )e
W X (t , f )  Wx ( f , t )
,STFT
 j 2 ft
,Gabor
,WDF
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Hilbert-Huang Transform
 Introduction
Most of distribution are designed for
stationary and linear signals, but, In the real
world, most of signals are non-stationary and
non-linear.
HHT consists two parts:
 empirical mode decomposition (EMD)
 Hilbert spectral analysis (HSA)
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Empirical decomposition function
Any complicated data can be decomposed into a
finite and small number of intrinsic mode
functions (IMF) by sifting processing.
 Intrinsic mode function
(1)In the whole data set, the number of
extrema and the number of zero-crossing
must either equal or differ at most by one.
(2)At any point, the mean value of the
envelope defined by the local maxima and
the envelope defined by the local minima
is zero.
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 Sifting Process
(1) First, find all the local maxima extrema of x(t).
IMF 1; iteration 0
2
1
0
-1
-2
10
20
30
40
50
60
70
80
90
100
110
120
(2) Interpolate (cubic spline fitting) between all the
maxima extrema ending up with some upper
envelope emax (t ) .
IMF 1; iteration 0
2
1
0
-1
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-2
10
20
30
40
50
60
70
38
80
90
100
110
120
(3) Find all the local minima extrema.
IMF 1; iteration 0
2
1
0
-1
-2
10
20
30
40
50
60
70
80
90
100
110
120
(4) Interpolate (cubic spline fitting) between all the
minima extrema ending up with some lower
envelope emin (t ) .
IMF 1; iteration 0
2
1
0
-1
-2
10
20
30
40
50
60
70
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80
90
100
110
120
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(5) Compute the mean envelope between upper
envelope and lower envelope. m(t )  emin (t )  emax (t )
2
IMF 1; iteration 0
2
1
0
-1
-2
10
20
30
40
50
60
70
80
90
100
110
120
(6) Compute the residue h(t )  x(t )  m(t )
residue
1.5
1
0.5
0
-0.5
-1
-1.5
10
20
30
40
50
60
70
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90
100
110
120
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(7) Repeat the above procedure (step (1) ~ step (6))
on the residue until the residue is a monotonic
function or constant.
The original signal equals the sum of the various
IMFs plus the residual trend.
n
x(t )   ck (t )  rn (t )
k 1
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EX:
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
1
2
3
4
5
6
4
x 10
IMF1
0.1
0.05
0
-0.05
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.5
0.6
0.7
IMF2
0.2
0.1
0
-0.1
-0.2
0
0.1
0.2
0.3
0.4
Time
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IMF3
0.4
0.2
0
-0.2
-0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.5
0.6
0.7
0.5
0.6
0.7
0.5
0.6
0.7
IMF4
0.4
0.2
0
-0.2
-0.4
0
0.1
0.2
0.3
0.4
Time
IMF5
0.1
0.05
0
-0.05
-0.1
0
0.1
0.2
0.3
0.4
IMF6
0.04
0.02
0
-0.02
-0.04
0
0.1
0.2
0.3
0.4
Time
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IMF7
0.01
0.005
0
-0.005
-0.01
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.5
0.6
0.7
0.5
0.6
0.7
0.5
0.6
0.7
IMF8
0.01
0.005
0
-0.005
-0.01
0
0.1
0.2
0.3
-3
10
0.4
Time
IMF9
x 10
5
0
-5
0
0.1
0.2
0.3
-3
10
0.4
IMF10
x 10
5
0
-5
0
0.1
0.2
0.3
0.4
Time
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-3
4
IMF11
x 10
2
0
-2
-4
0
0.1
0.2
0.3
-3
0
0.4
0.5
0.6
0.7
0.5
0.6
0.7
IMF12
x 10
-0.5
-1
-1.5
0
0.1
0.2
0.3
0.4
Time
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Hilbert Spectral Anaysis
1 1
  x( ) 
y (t )  H {x(t )}  x(t )   PV  
d 

t 
 t 

z (t )  x(t )  iy(t )  a(t )ei (t )
y (t )
 (t )  arg( z (t ))  tan (
)
x(t )
1
f (t ) 
1
1 d (t )
 (t ) 
2
2 dt
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Advantage
Disadvantage
STFT
1.
Low computation
1.
Complex value
and
2.
The range of the integration is limited
2.
Low resolution
Gabor transform
3.
No cross term
4.
Linear operation
Real
1.
High computation
2.
High resolution
2.
Cross term
3.
If the time/frequency limited, time/frequency of the 3.
Wigner
function
distribution 1.
Non-linear operation
WDF is limited with the same range
Cohen’s
1.
Avoid the cross term
1.
High computation
class
2.
Higher clarity
2.
Lack of well mathematical properties
distribution
Gabor-Wigner
distribution 1.
function
Combine the advantage of the WDF and the Gabor 1.
High computation
transform
2.
Higher clarity
3.
No cross-term
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Conclusion
 We introduce many distributions here and put most attention
on computation time and representations. We can find that
the representation with higher clarity cost more computation
time for all methods.
Resolution
Computation time
 The Hilbert-Huang transform is the most power method to
deal with non-linear and non-stationary signals but lacks of
physical background.
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Reference
[1][A]J. J. Ding, “Time-Frequency Analysis and Wavelet Transform,” National
Taiwan University, 2009. [Online].Available:
http://djj.ee.ntu.edu.tw/TFW.htm.
[2][B]W. F. Wang, “Time-Frequency Analyses and Their Fast Implementation
Algorithm,” Master Thesis, National Taiwan University, June, 2009.
[3]Luis B. Almeida, Member, IEEE, “The Fractional Fourier Transform and
Time-Frequency Representations,” IEEE Transaction On Signal Processing,
vol. 42, no. 11, November 1994.
[4]M. R. Spiegel, Mathematical Handbook of Formulas and Tables, McGrawHill, 1990.
[5]N. E. Huang, Z. Shen and S. R. Long, et al., “The empirical mode
decomposition and the Hilbert spectrum for nonlinear and non-stationary
time Series Analysis＂, Proc. Royal Society, vol. 454, pp.903-995, London,
1998.
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