sep.2008disp lab @md5311 time-frequency analysis 時頻分析 speaker: wen-fu wang 王文阜 ...

Sep.2008 DISP Lab @MD531 1

Time-Frequency Analysis

時頻分析 Speaker: Wen-Fu Wang 王文阜 Advisor: Jian-Jiun Ding 丁建均教授 E-mail: [email protected] Graduate Institute of Communication Engineering National Taiwan University, Taipei, Taiwan, ROC


Outline

Introduction Short Time Fourier Transform Gabor Transform Wigner Distribution Gabor-Wigner Transform Cohen’s Class Time-Frequency

Distribution


Outline

S Transform Hilbert-Huang Transform Applications of Time-Frequency

Analysis Conclusions References


IntroductionFourier Transform (FT)

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

0 5 10 15 20 25 30-1

-0.5

0

0.5

1

-5 0 5-2

-1

0

1

2f(t)

Fouriertransform


Introduction Instantaneous Frequency If

then the instantaneous frequency of f (t) are

If order of >1, then instantaneous frequency varies with time

1

( ) exp( ( ))N

k kk

x t a j t

31 2 '( ) '( )'( ) '( ), , , ,

2 2 2 2Nt tt t

( )k t


Introduction Example :

→ chirp function

Instantaneous frequency =

2cot( )2ej

t

cot( )

2

t


Introduction Time-Frequency analysis Example :

0 5 10 15 20 25 30

-5

-4

-3

-2

-1

0

1

2

3

4

5

t -axis

f -a

xis

-5 0 5-0.5

0

0.5

f -axis

am

plit

ude


Short Time Fourier Transform

The earliest Time-Frequency representation was the short time Fourier transform (STFT)

This scheme divides the temporal signal into a series of small overlapping pieces.

The STFT of a function is defined by1

( , ) ( ) ( )2

jX t x t h t e d



is the window function. The principle of STFT

( )h t

time

s(t)h(t)

FT FT FT

time

frequency

STFT



Example :

The signal of simulation

0 1 2 3 4 5 6 7 8 9 10 11

sec

-1

-0.5

0

0.5

1

Am

plit

ud

e

STFT

0 1 2 3 4 5 6 7 8 9 10 11

sec

0

2

4

6

8

10

12

14

Hz

cos(2 8 ) 1 4

( ) cos(2 4 ) 4 9

cos(2 5 ) 9 12

t t

x t t t

t t



Advantage:

(1) Least computation time for digital implementation compared with other

(2) Its ability to avoid cross-term problem


Gabor Transform

In fact, Gabor transform is a special case of STFT.

When the of STFT, it can be rewritten as

Why does it choose the Gaussian function as a window?

2( ) exp( )h t t

2 2 ( )( ) 2( , ) ( )t

j ftxG t f e e x d


Gabor Transform

The principle of Gabor Transform

time

s(t)h(t)

FT FT FT

time

frequency

STFT


Gabor Transform

Example:

Gabor

0 1 2 3 4 5 6 7 8 9 10 11

sec

0

2

4

6

8

10

12

14

Hz

The signal of simulation

0 1 2 3 4 5 6 7 8 9 10 11

sec

-1

-0.5

0

0.5

1

Am

plit

ud

e

cos(2 8 ) 1 4

( ) cos(2 4 ) 4 9

cos(2 5 ) 9 12

t t

x t t t

t t


Gabor Transform

Advantage:(1) Its ability to avoid cross-term problem(2) The resolution is better than STFT


Wigner Distribution

The Wigner distribution is defined as

In terms of the spectrum, it is

is the Fourier transform of and * means the complex conjugate.

* 2( , ) ( / 2) ( / 2) j fxW t f x t x t e d

* 2( , ) ( / 2) ( / 2) jxW t f X f X f e d

( )X f ( )x t


Wigner Distribution

WD has much better time-frequency resolution

WD is not a linear distribution WD has more computation time If the signal is composed by several

time-frequency components, additional interference will be produced.


Wigner Distribution

Example:

0 5 10 15 20 25 30

-6

-4

-2

0

2

4

6

cos( ) 1 10

( ) cos(3 ) 10 20

cos(2 ) 20 33

t t

x t t t

t t


Wigner Distribution

The inner interference is caused by interference between positive and negative frequency of the signal itself

In order to reduce to inner interference problem, Wigner Ville Distribution can be used


Wigner Distribution

The outer interference is caused by mutual interference of multi-component in signal

In order to reduce to outer interference problem, Modified Wigner Distribution can be used


Wigner Ville Distribution

Due the inner interference is caused by interference between positive and negative frequency of the signal itself

We can use analytic version signal to replace the original signal for filtering out negative frequency


Wigner Ville Distribution

We denote the analytic signal of real valued signal by

is Hilbert transform of We can redefine WD by analytic

signal

( ) ( ) ( )x t x t jx t

* 2( , ) ( / 2) ( / 2) j f

xW t f x t x t e d

( )x t ( )x t

( )x t ( )x t


Modified Wigner Distribution

Due outer interference is caused by mutual interference of multi-component in signal

We can select suitable window function is a way to suppress the outer interference, but retain the sharpness of auto terms.



The Modified Wigner Distribution is defined as

When the window function , it also become Wigner distribution.

* 2

*

( , ) ( / 2) ( / 2) ( / 2) ( / 2)

( , / 2) ( , / 2)

j fxW t f w w x t x t e d

Y t f Y t f d



Example: WD MWD

0 5 10 15 20 25 30

-6

-4

-2

0

2

4

60 5 10 15 20 25 30

-6

-4

-2

0

2

4

6


Modified Wigner Ville Distribution

Combined form of the WVD and MWD The advantage of WVD is filtering out

inner interference The advantage of MWD is suppressing

the outer interference, but retain the sharpness of auto terms

The MWVD can avoid the inner and outer interference at the same time


Gabor-Wigner Transform

We have compared the properties of the WD and Gabor transform in previous section

The advantage of WD is its high clarity, and the disadvantage of WD is it’s the cross-term problem



In contrast, the advantage of Gabor transform is its ability to avoid cross-term problem, but its clarity is not as good as that of the WDF

The Gabor-Wigner transform (GWT) can achieve the higher clarity and avoiding cross-term problem at the same time.

-10 -5 0 5 10-10

-5

0

5

10

-10 -5 0 5 10-10

-5

0

5

10(c) (d) -10 -5 0 5 10

-10

-5

0

5

10

-10 -5 0 5 10-10

-5

0

5

10(a) (b)



Combined form of the WDF and the Gabor transform

(a)

(b)

(c)

(d)

( , ) ( , ) ( , )f f fC t G t W t 2

( , ) min( ( , ) , ( , ) )f f fC t G t W t

( , ) ( , ) { ( , ) 0.25}f f fC t W t G t

0.7 2.6( , ) ( , ) ( , )f f fC t W t G t


Cohen’s Class Time-Frequency Distribution

The definition of the Cohen class is

is the ambiguity function

How does the Cohen’s class distribution avoid the cross term?

( , ) ( , ) ( , ) exp( 2 ( ))x xC t f A j t f d d

* 2( , ) ( / 2) ( / 2) j txA x t x x e dt


For the ambiguity function:(1) The auto term is always near to the origin (2) The cross-term is always far from the origin

AF WD


t

f

IFTf FTt

IFTf FTt

IFTf FTt



Relationship between Wigner distribution and ambiguity function

( , )xWD t f

( , )xC t

( / 2) ( / 2)x t x t ( , )xA

( , )xS f



How does the Cohen’s class distribution avoid the cross term?

Choi-Williams Distribution

Cone-Shape Distribution

2, exp

2, sin exp 2c



Example:

Ambiguity Function Choi-Williams Distributiontau (sec)

eta

-15 -10 -5 0 5 10 15

-8

-6

-4

-2

0

2

4

6

8

tau(sec)

eta

-15 -10 -5 0 5 10 15

-8

-6

-4

-2

0

2

4

6

8



Example:

Ambiguity Function Wigner Distribution

tau(sec)

eta

-15 -10 -5 0 5 10 15

-5

0

5

t(sec)

freq

uenc

y

-8 -6 -4 -2 0 2 4 6 8

-5

0

5

tau(sec)

eta

-15 -10 -5 0 5 10 15

-5

0

5

t(sec)

freq

uenc

y

-8 -6 -4 -2 0 2 4 6 8

-5

0

5


Cohen’s Class Time-Frequency Distribution Some popular distributions and their kernels

TFD Kernel Formulation

Page

Levin

Kirkwood

Spectrogram

Wigner-Ville 1

Choi-Williams

Cone shape

( , )

/2je 2

'( ') 't j ts t e dt

t

1cos( )

2 *Re ( ) ( ) js t S e

/2je *( ) ( ) js t S e

2

( ) js h t e d *1 1

( ) ( )2 2

js t s t e d

2 2

e

22sin ( ) ec 221

e ( )t



Advantage:The Cohen’s class distribution may avoid the cross term and has higher clarity

Disadvantage:It requires more computation time and lacks of well mathematical properties


S Transform

Unlike STFT, the width of S transform’s window changes with frequency.

Closely related to the wavelet transform

2 22( )

( , ) ( ) exp[ ]22

i ftf f ts f h t e dt


S Transform

Example:

50 100 150 200 250 300

0

50

100

150

cos( ) 1 10

( ) cos(3 ) 10 20

cos(2 ) 20 33

t t

x t t t

t t


Hilbert-Huang Transform

Traditional data analysis methods are all based on linear and stationary assumptions

The HHT consists of two parts: empirical mode decomposition (EMD) and Hilbert spectral analysis (HSA).


Hilbert-Huang Transform---EMD

Intrinsic Mode Functions (IMF)


Hilbert-Huang Transform--EMD

The Sifting Process

10 20 30 40 50 60 70 80 90 100 110 120

-2

-1

0

1

2

IMF 1; iteration 0

10 20 30 40 50 60 70 80 90 100 110 120

-2

-1

0

1

2

IMF 1; iteration 0



10 20 30 40 50 60 70 80 90 100 110 120

-2

-1

0

1

2

IMF 1; iteration 0

10 20 30 40 50 60 70 80 90 100 110 120

-2

-1

0

1

2

IMF 1; iteration 0



10 20 30 40 50 60 70 80 90 100 110 120

-2

-1

0

1

2

IMF 1; iteration 0

10 20 30 40 50 60 70 80 90 100 110 120

-1.5

-1

-0.5

0

0.5

1

1.5

residue


Hilbert-Huang Transform--HSA

We have obtained the intrinsic mode function components by EMD process method

Then we will do the Hilbert transform to each IMF component.


Applications of Time-Frequency Analysis

(1) Finding the Instantaneous Frequency

(2) Sampling Theory (3) Modulation and Multiplexing (4) Filter Design (5) Signal Representation (6) Random Process Analysis


Applications of Time-Frequency Analysis

(7) Acoustics (8) Data Compression (9) Spread Spectrum Analysis (10) Radar Signal Analysis (11) Biomedical Engineering (12) Economic Data Analysis


Conclusions

Advantage Disadvantage

STFT

and

Gabor transform

1. Low computation

2. The range of the integration is

limited

3. No cross term

4. Linear operation

1. Complex value

2. Low resolution

Wigner

distribution

function

1. Real

2. High resolution

3. If the time/frequency limited,

time/frequency of the WDF is

limited with the same range

1. High computation

2. Cross term

3. Non-linear operation

Cohen’s

class

distribution

1. Avoid the cross term

2. Higher clarity

1. High computation

2. Lack of well

mathematical

properties

Gabor-Wigner

distribution

function

1. Combine the advantage of the

WDF and the Gabor transform

2. Higher clarity

3. No cross-term

1. High computation


Conclusions Compare with Fourier, wavelet and HHT analyses

Fourier Wavelet Hilbert

Basis A priori A priori Adaptive

Frequency Convolution: global Convolution: regional Differentiation: local

Presentation Energy-frequency Energy-time-frequency Energy-time-frequency

Non-linear No No Yes

Non-stationary No Yes Yes

Uncertainty Yes Yes No

Harmonics Yes Yes No

Feature extraction No Discrete: no

Continuous: yes

Yes

Theoretical base Theory complete Theory complete Empirical


Conclusions

Advantage: The instantaneous frequency can be

observed Disadvantage:

Higher complexity for computation Which method is better?


References

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

N. E. Huang, S. Shen, ＂Hilbert-Huang Transform and its Applications＂ , World scientific, Singapore, 2005.

http://tec.earth.sinica.edu.tw/tec/images/Huang_etal98.pdf





http://books.google.com/books?hl=zh-TW&lr=&id=VhItWRmtoEUC&oi=fnd&pg=PR5&dq=%22Hilbert-Huang+Transform+and+its+Applications%22&ots=-BEZ5ng-kr&sig=N49h5TcNAwDktoi0YWv65jQJ5H8#PPA205,M1


References S. C. Pei and J. J. Ding, “

Relations between the fractional operations and the Wigner distribution, ambiguity function,” IEEE Trans. Signal Processing, vol. 49, no. 8, pp. 1638-1655, Aug. 2001.

L. Cohen, ＂Time-Frequency distributions-A review” Proc. IEEE, Vol. 77, No. 7, pp. 941-981, July 1989.

C. H. Page, “Instantaneous Power Spectra,” National Bureau of Standards, Washington, D. C., 1951

http://ieeexplore.ieee.org/iel5/78/4305424/04305455.pdf?tp=&arnumber=4305455&isnumber=4305424

http://ieeexplore.ieee.org/iel5/78/4305424/04305455.pdf?tp=&arnumber=4305455&isnumber=4305424

http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=30749

http://scitation.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=JAPIAU000023000001000103000001&idtype=cvips&prog=normal


References 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, Oct. 2007

R. G. Stockwell, L. Mansinha, and R. P. Lowe, “Localization of the complex spectrum: the S transform,” IEEE Trans. Signal Processing, vol. 44, no. 4, pp. 998–1001, Apr. 1996.

sep.2008disp lab @md5311 time-frequency analysis 時頻分析 speaker: wen-fu wang 王文阜 ...

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