an introduction to time-frequency analysis
DESCRIPTION
An Introduction to Time-Frequency Analysis. Advisor : Jian-Jiun Ding, Ph. D. Presenter : Ke-Jie Liao Taiwan ROC Taipei National Taiwan University GICE DISP Lab MD531. Outline. Introduction STFT Rectangular STFT Gabor Transform Wigner Distribution Function - PowerPoint PPT PresentationTRANSCRIPT
An Introduction to Time-Frequency Analysis
Advisor : Jian-Jiun Ding, Ph. D.Presenter : Ke-Jie Liao
Taiwan ROCTaipei
National Taiwan UniversityGICE
DISP LabMD531
2
Outline Introduction STFT
Rectangular STFT Gabor Transform
Wigner Distribution Function Motions on the Time-Frequency Distribution
FRFT LCT
Applications on Time-Frequency AnalysisSignal Decomposition and Filter Design Sampling Theory Modulation and Multiplexing
3
Introduction
Frequency? Another way to consider things.
Frequency related applications FDM Sampling Filter design , etc ….
4
Introduction
Conventional Fourier transform 1-D Totally losing time information Suitable for analyzing stationary signal ,i.e. frequency does not vary with time.
[1]
5
Introduction
Time-frequency analysis Mostly originated form FT Implemented using FFT
[1]
6
Short Time Fourier Transform
Modification of Fourier Transform Sliding window, mask function, weighting function
Mathematical expression
Reversing Shifting FT
2( ) ( )( , ) ( ) { ( )}j fX t f x e d F xw t w t
w(t)
7
Short Time Fourier Transform
Requirements of the mask function w(t) is an even function. i.e. w(t)=w(-t). max(w(t))=w(0),w(t1) w(t2) if |t1|<|t2|. when |t| is large.
An example of window functions( ) 0w t
t
Window width K
8
Short Time Fourier Transform
Requirements of the mask function w(t) is an even function. i.e. w(t)=w(-t). max(w(t))=w(0),w(t1) w(t2) if |t1|<|t2|. when |t| is large.
An illustration of evenness of mask functions( ) 0w t
SignalMask
t0
9
Short Time Fourier Transform
Effect of window width K Controlling the time resolution and freq. resolution.
Small K Better time resolution, but worse in freq. resolution
Large K Better freq. resolution, but worse in time resolution
10
Short Time Fourier Transform
The time-freq. area of STFTs are fixed
K decreases
tt
f fMore details in time More details in freq.
11
Rectangular STFT
Rectangle as the mask function Uniform weighting
Definition Forward
Inverse
where
2( , ) ( )t
jB
B
f
t
X t f x e d
21( ) ( , ) j ftx t X t f e df
1t B t t B
2B
1
12
Rectangular STFT
Examples of Rectangular STFTs
cos(4 ) ,0 10
( ) cos(2 ) ,10 20
cos( ) , 20 30
t t
x t t t
t t
2 ,0 10
( ) 1 ,10 20
0.5 ,20 30i
t
f t t
t
B=0.25 B=0.5
t
f
t
f
10 20 30010 20 300
221100
13
Rectangular STFT
Examples of Rectangular STFTs
cos(4 ) ,0 10
( ) cos(2 ) ,10 20
cos( ) , 20 30
t t
x t t t
t t
2 ,0 10
( ) 1 ,10 20
0.5 ,20 30i
t
f t t
t
B=1 B=3
t
f
t
f
0 10 20 30 0 10 20 30
0122
10
14
Rectangular STFT
Properties of rec-STFTs Linearity
Shifting
Modulation
( ) ( ) ( )( , ) ( , ) ( , )
h t x t y tH t f X t f Y t f
00
20
2( ) ( , )t B
j f j f
t B
x e d X t f e
020
2[ ( ) ] ( , )t B
j f
t
j f
B
e fx e d X t f
15
Rectangular STFT
Properties of rec-STFTs Integration
Power integration
Energy sum
2 ( ),( , )
0 ,j fv x v v B t v B
X t f e dfotherwise
* *( , ) ( )( , ) ( )t B
t B
X t f df x dY t f y
* *(( , ) () (), )X t f df dt B xf y dY t
16
Gabor Transform
Gaussian as the mask function
Mathematical expression
Since where
GT’s time-freq area is the minimal against other STFTs!
2
( ) tew t
2 221.9143
( ) ( )
1.9
2
143
( , ) ( ) ( )t
j ft j f
t
txG t f e x ded e xe
2
0.00001ae | | 1.9143a
17
Gabor Transform
Compared with rec-STFTs Window differences Resolution – The GT has better clarity Complexity
Discontinuity Weighting differences
18
Gabor Transform
Compared with rec-STFTs Resolution – GT has better clarity
Example of 2
( ) ty t e
The rec-STFT The GT
Better resolution!
t
f
t
f
00
00
19
Gabor Transform
Compared with the rec-STFTs Window differences Resolution – GT has better clarity
Example of 2
( ) ty t e
The rec-STFT The GT
GT’s area is minimal!
High freq. due to discon.
t
f
t0
0
0
0
20
Gabor Transform
Properties of the GT Linearity
Shifting
Modulation
( ) ( ) ( )( , ) ( , ) ( , )z x y
z x yG t f G t f G t f
0
00
2( ) ( , ) ( , ) j f
xt
tx tG t tf G t f e
2 0 0( )
( , ) ( , )j tf xx t e
G f G t ft f
Same as the rec-STFT!
21
Gabor Transform
Properties of the GT Integration
Power integration Energy sum Power decayed
2 22 ( 1)( , ) ( )kj tf tkxG t f e df e x kt
K=1-> recover original signal
22
Gabor Transform
Gaussian function centered at origin
Generalization of the GT Definition 2 2
1.91
( ) 2 ( ) 2
43
1.9143
( , ) ( ) ( )
t
t j f t j fx
t
G t f e e x d e e x d
0
1 2
( ) tw t e 2
( ) tw t e
23
Gabor Transform
plays the same role as K,B.(window width) increases -> window width decreases decreases -> window width increases
Examples : Synthesized cosine wave
1 0.1
t
f
t
f
0 10 20 30 0 10 20 30
0210
12
24
Gabor Transform
plays the same role as K,B.(window width) increases -> window width decreases decreases -> window width increases
Examples : Synthesized cosine wave
1.5 5
t
f
t
f
0 10 20 30 0 10 20 30
0210
12
25
Wigner Distribution Function
Definition
Auto correlated -> FT Good mathematical properties
Autocorrelation Higher clarity than GTs But also introduce cross term problem!
* *2( ) ( ) ( ) ( )2
( , ) {2
}2 2
j fx x t x tW t f e d F x t x t
26
Wigner Distribution Function
Cross term problem WDFs are not linear operations. ( ) ( ) ( )h t g t s t
2 2( , ) | | ( , ) | | ( , )h g sW t f W t f W t f
* 2* * *( ) ( ) ( )[ ( )2
]2 2 2
j fg t s t g t s e dt
Cross term!
( ) ( )g t s t
* * * *( ) ( )g t s t
n(n-1) cross term!!
27
Wigner Distribution Function
An example of cross term problem 2
2
( 4 )10
( 6 )
9 1( )1 9
tj t
j t t
e tx te t
1 1( 4) 9 12 5( )1 ( 2 6) 1 92
i
t tf t
t t
Without cross term With cross term
t
ff
t
0 0
00
28
Wigner Distribution Function
Compared with the GT Higher clarity Higher complexity
An example WDF GT
4 4
( ) cos(4 )2
j t j te ex t t
t
f
t
f
00
0 0
29
Wigner Distribution Function
But clarity is not always better than GTDue to cross term problem Functions with phase degree higher than 2
WDF GT
Indistinguishable!!
3( ) exp( ( 5) 6 )x t j t j t
[1]t
f
t
f
0
0
0
0
30
Wigner Distribution Function
Properties of WDFs Shifting
Modulation
Energy property
00( ) ( , ) ( , )xx t tW t f W t t f
2 0e (0
)( , ) ( , )j f t x
x tW t f W t f f
2 2( , ) | ( ) | | ( ) |xW t f dtdf x t dt X f df
31
Wigner Distribution Function
Properties of WDFs Recovery property is real Energy property Region property Multiplication Convolution Correlation Moment Mean condition frequency and mean condition time
( , )xW t f
32
Motions on the Time-Frequency Distribution
Operations on the time-frequency domain Horizontal Shifting (Shifting on along the time axis)
Vertical Shifting (Shifting on along the freq. axis)
0,0 0
0 0
2( ) ( , )
( ) ( , )
jSTFT GTx
WDFx
ftx t t S t t f
x t t W t t f
e
t
f
0
0
2 ,0
20
( ) ( , )
( ) ( , )
j f t STFT GTx
j f t WDFx
e x t S t f f
e x t W t f f
t
f
33
Motions on the Time-Frequency Distribution
Operations on the time-frequency domain Dilation
Case 1 : a>1
Case 2 : a<1
,1 ( ) ( , )| |1 ( ) ( , )| |
STFT GTx
WDFx
t tx S afa aat tx W afa aa
34
Motions on the Time-Frequency Distribution
Operations on the time-frequency domain Shearing - Moving the side of signal on one
direction Case 1 :
Case 2 :
2
( ) ( )j aty t e x t
2
( ) ( )tjay t e x t
( , ) ( , )( , ) ( , )
y x
y x
aS t f S t f tW t f W t taf
( , ) ( , )( , ) ( , )
y x
y x
S t f S t f fW t f W t f
afa
t
f
t
f
Moving this sidea>0
a>0
Moving this side
35
Motions on the Time-Frequency Distribution
Rotations on the time-frequency domain Clockwise 90 degrees – Using FTs
t
f( ) { ( )}X f FT x t
2
| ( , ) | | ( , ) |
( , ) ( , )( , ) ( , )
X
j ftX
x
x
X x
S t f S f t
G t f G f tW t f W f t
e
Clockwise rotation 90
36
Motions on the Time-Frequency Distribution
Rotations on the time-frequency domain Generalized rotation with any angles – Using WDFs
or GTs via the FRFT Definition of the FRFT
Additive property
2 2cot 2 csc cot( ) [ ( )] 1 cot ( )j u j ut j tFX u O x t j e e e x t du
{ [ ( )]} [ ( )]F F FO O x t O x t
37
Motions on the Time-Frequency Distribution
Rotations on the time-frequency domain [Theorem]
The fractional Fourier transform (FRFT) with angle is equivalent to the clockwise rotation operation with angle for the WDF or GT.
( , ) ( cos sin , sin cos )
( , ) ( cos sin , sin cos )X
X
x
x
W u v W u v u vG u v G u v u v
Counterclockwise rotation matrix
NewOld'
'
cos sinsin cos
u uvv
'
'
cos sinsin cos
u uv v
New Old
Clockwise rotation matrix
38
Motions on the Time-Frequency Distribution
Rotations on the time-frequency domain [Theorem]
The fractional Fourier transform (FRFT) with angle is equivalent to the clockwise rotation operation with angle for the WDF or GT.
Examples (Via GTs)
-5 0 5-5
0
5
-5 0 5-5
0
5
-5 0 5-5
0
5
(a) G (t, ) f (b) G (t, ) F (c) G (t, )
F [1]
39
Motions on the Time-Frequency Distribution
Rotations on the time-frequency domain [Theorem]
The fractional Fourier transform (FRFT) with angle is equivalent to the clockwise rotation operation with angle for the WDF or GT.
Examples (Via GTs)
-5 0 5-5
0
5
-5 0 5-5
0
5
-5 0 5-5
0
5
(d) G (t, ) F (e) G (t, ) F (f) G (t, ) F [1]
40
Motions on the Time-Frequency Distribution
Twisting operations on the time-frequency domain LCT s
t t
f f
( , , , )
( , , , )
( , ) ( , )
( , ) ( , )
X xa b c d
X xa b c d
W u v W du bv cu av
W au bv cu dv W u v
The area is unchanged
'
'
u d b uc a vv
Old New
Inverse exist since ad-bc=1a bc d
LCT
41
Applications on Time-Frequency Analysis
Signal Decomposition and Filter Design A signal has several components - > separable in
time ->
separable in freq. ->
separable in time-freq.
t f t
f
Vertical cut off line on the t-f domain
Horizontal cut off line on the t-f domain
42
Applications on Time-Frequency Analysis
Signal Decomposition and Filter Design An example
t
f1
2
1t0t
Noise
Signals
Rotation -> filtering in the FRFT domain
43
Applications on Time-Frequency Analysis
Signal Decomposition and Filter Design An example The area in the t-f domain isn’t finite!
[1]
44
Applications on Time-Frequency Analysis
Signal Decomposition and Filter Design An example
[1]
2 2 20.23 0.3 8.5 0.46 9.6( ) 0.5 0.5 0.5j t j t j t j t j tn t e e e
45
Applications on Time-Frequency Analysis
Signal Decomposition and Filter Design An example
[1]
2 2 20.23 0.3 8.5 0.46 9.6( ) 0.5 0.5 0.5j t j t j t j t j tn t e e e
46
Applications on Time-Frequency Analysis
Signal Decomposition and Filter Design An example
[1]
The area in the t-f domain isn’t finite!
47
Applications on Time-Frequency Analysis
Sampling Theory Nyquist theorem : , B Adaptive sampling
2sf B
[1]
48
Conclusions and Future work
Comparison among STFT,GT,WDF
Time-frequency analysis apply to image processing?
rec-STFT GT WDFComplexity ㊣㊣㊣ ㊣㊣ ㊣Clarity ㊣ ㊣㊣ ㊣㊣㊣
勝 !勝 !
49
References
[1] Jian-Jiun Ding, Time frequency analysis and wavelet transform class note, the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2007.
[2] S. C. Pei and J. J. Ding, ”Relations between Gabor transforms and fractional Fourier transforms and their applications for signal processing”, IEEE Trans. Signal Processing, vol.55,no. 10,pp.4839-4850.
[3] S. Qian and D. Chen, Joint Time-Frequency Analysis: Methods and Applications, Prentice-Hall, 1996.
[4] D. Gabor, ”Theory of communication”, J. Inst. Elec. Eng., vol. 93, pp.429-457, Nov. 1946.
[5] L. B. Almeida, ”The fractional Fourier transform and time-frequency representations, ”IEEE Trans. Signal Processing, vol. 42,no. 11, pp. 3084-3091, Nov. 1994.
[6] K. B. Wolf, “Integral Transforms in Science and Engineering,” Ch. 9: Canonical transforms, New York, Plenum Press, 1979.
50
References
[7] X. G. Xia, “On bandlimited signals with fractional Fourier transform,” IEEE Signal Processing Letters, vol. 3, no. 3, pp. 72-74, March 1996.
[8] L. Cohen, Time-Frequency Analysis, Prentice-Hall, New York, 1995.
[9] T. A. C. M. Classen and W. F. G. Mecklenbrauker, “The Wigner distributiona tool for time-frequency signal analysis; Part I,” Philips J. Res., vol. 35, pp. 217-250, 1980.