an introduction to time-frequency analysis advisor : jian-jiun ding, ph. d. presenter : ke-jie liao...

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

Signal

Mask

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 t

H 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 fvx 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 y

G t f G t f G t f

0

00

2( ) ( , ) ( , ) j f

xt

tx tG t tf G t f e

2 00

( )( , ) ( , )j tf x

x t eG 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

0

21

012

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

0

21

012

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 1

2 5( )1( 2 6) 1 9

2

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 af

a aa

t tx W af

a 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

( ) ( )t

jay t e x t

( , ) ( , )

( , ) ( , )

y x

y x

aS t f S t f t

W t f W t taf

( , ) ( , )

( , ) ( , )

y x

y x

S t f S t f f

W t f W t f

a

fa

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 t

W 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 v

G u v G u v u v

Counterclockwise rotation matrix

NewOld'

'

cos sin

sin cos

u u

vv

'

'

cos sin

sin cos

u u

v 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

ff

( , , , )

( , , , )

( , ) ( , )

( , ) ( , )

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 u

c a vv

Old New

Inverse exist since ad-bc=1

a b

c 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

f

1

2

1t0t

Nois

e

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 WDF

Complexity ㊣㊣㊣ ㊣㊣ ㊣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.