An Introduction to Time-Frequency Analysis

1

An Introduction to Time-Frequency Analysis

Speaker: Po-Hong WuAdvisor: Jian-Jung Ding

Digital Image and Signal Processing LabGICE, National Taiwan University

NTU, GICE, MD531, DISP Lab

NTU, GICE, MD531, DISP Lab 2

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


Introduction

Fourier transform (FT) t varies from ∞~∞Time-Domain Frequency Domain

[A1]

Why do we need time-frequency transform?

2j f tX f x t e dt

0 5 10 15 20 25 30-1

-0.5

0

0.5

1

-5 0 5-2

-1

0

1

2f(t)

Fouriertransform


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

[B2]

0 5 10 15 20 25 30

-5

-4

-3

-2

-1

0

1

2

3

4

5


Short Time Fourier Transform

w(t): mask function

也稱作 windowed Fourier transform

or

time-dependent Fourier transform

2, j fX t f w t x e d


When w(t) is a rectangular functionw(t) = 1 for |t| B , w(t) = 0 , otherwise

[B3]

2,t B j f

t BX t f x e d

STFT

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

sec

0

2

4

6

8

10

12

14

Hz


Advantage: less computation timeDisadvantage: worse representaionApplication: deal with large data Ex: real time processing


Gabor TransformA specail case of the STFTwhere

Other definition

[B4]

2( ) 2, t j fxG t f e e x d

2( ) exp( )w t t

Gabor

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

sec

0

2

4

6

8

10

12

14

Hz

2( )

( )2 21

,2

t tj

xG t e e x d


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.


Approximation of the Gabor Transform

Because of when |a|>1.9143

Because of when |a|>4.7985

21.9143 ( ) 2

1.9143,

t t j fx t

G t f e e x d

2

0.00001ae

2( )

( )4.79852 2

,3 4.7985

1,

2

t tjt

x tG t e e x d

2 / 2 0.00001ae


Generalization of the Gabor Transform

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

2( ) 2, t j fxG t f e e x d


Resolution• Using the generalized Gabor

transform with larger σ • Using other time unit instead of

second


Wigner Distribution Function

Other definition

[B5]

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

time (sec)

frequency

(H

z)

Wigner distribution

0 5 10 15 20 25 30-6

-4

-2

0

2

4

6

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


Signal auto-correlation function

Spectrum auto-correlation function

Ambiguity function (AF)

[B6]

, / 2 / 2xC t x t x t

, / 2 / 2xS f X f X f

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

Ax(, )

IFTf FTt

IFTf FTt

Sx(, f )FTt IFTf

Cx(t, )

Wx(t, f )


Modified Wigner DistributionWigner Ville DistributionFor compressing inner interference

Analytic signal ( ) ( ) ( )x t x t jx t

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

xW t f x t x t e d


Pseudo Wigner DistributionFor surpressing outer interference

where

[B7]

* 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

2( , ) ( ) ( ) j fY t f w x t e d

Pseudo L-Wigner distribution

time (s)

freq

uenc

y (H

z)

0 5 10 15 20 25 30-6

-4

-2

0

2

4

6


Gabor-Wigner Distribution

[B8]

( , ) ( ( , ), ( , )),f f fC t w p G t w W t w

-10 -5 0 5 10-10

-5

0

5

10

-10 -5 0 5 10-10

-5

0

5

10(a) (b)

( , )p x y xy 2( , ) min( , )p x y x y


Spectrogram

Another form

[B9]

22( , ) ( ) ( ) j fSP t f x h t e d

( , ) ( , ) ( , )h xSP t f W t f W d d

Spectrogram

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

sec

0

2

4

6

8

10

12

14

Hz


S-TransformOriginal S-Transform

Where w(t)=

[B10]

2 22( )

( , ) ( ) exp[ ]22

i ftf f tST f x t e dt

2 2( )exp[ ]

22

f f t

cos( ) 1 10

( ) cos(3 ) 10 20

cos(2 t) 20 30

t when t

x t t when t

when t

time (sec)

freq

uenc

y

S Transform

0 5 10 15 20 25 30-4

-3

-2

-1

0

1

2

3

4


Generalized S-Transform

Another definition

Ristriction

2( , , ) ( ) ( , , ) i tGS f p x t w t f p e dt

2( , , ) ( ) ( , , ) iGS f p X f W f p e d

( , , ) 1w t f p d


Novel S-Transform with the Special Varying Window

Restriction

When , it becomes the Gabor transform.When , it becomes the original S-trnasform.

2( ) ( )( )( , ) exp( )

22S

F f F f tw t f

( , ) 1sw t f d

2( ) 1/F f

2( )F f f


Cohen’s Class Time-Frequency Distribution

Ambiguity function

[B11]

( , ) ( , ) ( , ) exp( 2 ( ))C t f A j t f d dx x

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

( , )xWD t f

( , )xC t

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

IFTf FTt

IFTf FTt

IFTf FTt

( , )xS f


For the ambiguity functionThe auto terms are always near to

the origin.The cross terms are always from the

origin.

[B12]

24

Kernel function• Choi-Williams Distribution

[B13]

( , )

2, exp

tau (sec)

eta

Choi-Williams distribution

-15 -10 -5 0 5 10 15-10

-8

-6

-4

-2

0

2

4

6

8

10



• Cone-Shape Distribution22( , ) sin ( )e >0c

tau (sec)

eta

Cone Shape distribution

-15 -10 -5 0 5 10 15-10

-8

-6

-4

-2

0

2

4

6

8

10

26

Fractional Fourier Transform

How to rotate the time-frequency distribution by the angle other than /2, , and 3/2?

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



• Zero rotation: • Consistency with Fourier transform: = FT• Additivity of rotation:

• rotation:

0R I

/2R

R R R

2R I


[A3]

-5 0 5-1

0

1

2

-5 0 5-1

0

1

2

-5 0 5-1

0

1

2

-5 0 5-1

0

1

2

-5 0 5-1

0

1

2

-5 0 5-1

0

1

2

f(t): rectangle

F(w): sinc function

22 2 csc cotcot1 cot

tj u j tj uX u j x t dte e e


ApplicationDecomposition in the time-frequency distribution

-10 -5 0 5 10-0.5

0

0.5

1

1.5

-10 -5 0 5 10-1

0

1

2

-10 -5 0 5 10-1

0

1

2

-10 -5 0 5 10-0.5

0

0.5

1

1.5

Fourier transform of x(t) x(t) = signal + noise

fractional Fourier transform of x(t)

(separable)

(non-separable)

recovered signal


f-axis

Signal noise

t-axis

FRFT FRFT

noise Signal

cutoff line

Signal

cutoff line

noise


Modulation and Multiplexing

-20 0 20

-5

0

5

-20 0 20

-5

0

5

(c) WDF of G(u) (d) GWT of G(u)

unfilledT-F slot


• Time domain Frequency domain fractional domain

• Modulation Shifting Modulation + Shifting

• Shifting Modulation Modulation + Shifting

• Differentiation j2f Differentiation and j2f

• −j2f Differentiation Differentiation and −j2f


Motion on Time-Frequency Distributions

Horizontal Shifting

Vertical Shifting

2 0( ) ( , ) ,STFT, Gabor0 0

( , ) ,Wigner0

j f tx t t S t t f ex

W t t fx

2 0 ( ) ( , ) ,STFT,Gabor0

( , ) ,Wigner0

j f te x t S t f fx

W t f fx


Dilation

Shearing

1( ) ( , ) ,STFT,Gabor

| |

( , ) ,WDF

t tx S afx

a a a

tW afx

a

2( ) ( )

( , ) ( , ) ,STFT,Gabor

( , ) ( , ) ,WDF

j atx t e y t

S t f S t f atx y

W t f W t f atx y

2

( ) ( )

( , ) ( , ) ,STFT,Gabor

( , ) ( , ) ,WDF

tjax t e y t

S t f S t af fx y

W t f W t af fx y


RotationIf F{x(t)}=X(f), then F{X(t)}=x(-f).We can derive:

| ( , ) | | ( , ) | ,STFT

2( , ) ( , ) ,Gabor

( , ) ( , ) ,WDF

S t f S f txXj ftG t f G f t exX

W t f W f txX


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)


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.


Sifting Process(1) First, find all the local maxima extrema of x(t).

(2) Interpolate (cubic spline fitting) between all the maxima extrema ending up with some upper envelope .

max ( )e t

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


(3) Find all the local minima extrema.

(4) Interpolate (cubic spline fitting) between all the minima extrema ending up with some lower envelope . min ( )e t

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


(5) Compute the mean envelope between upper envelope and lower envelope.

(6) Compute the residue ( ) ( ) ( )h t x t m t

min max( ) ( )( )

2

e t e tm t

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


(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.

1

( ) ( ) ( )n

k nk

x t c t r t


EX:

0 1 2 3 4 5 6

x 104

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-0.1

-0.05

0

0.05

0.1IMF1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-0.2

-0.1

0

0.1

0.2

Time

IMF2


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-0.4

-0.2

0

0.2

0.4IMF3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-0.4

-0.2

0

0.2

0.4

Time

IMF4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-0.1

-0.05

0

0.05

0.1IMF5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-0.04

-0.02

0

0.02

0.04

Time

IMF6


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-0.01

-0.005

0

0.005

0.01IMF7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-0.01

-0.005

0

0.005

0.01

Time

IMF8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-5

0

5

10x 10

-3 IMF9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-5

0

5

10x 10

-3

Time

IMF10


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-4

-2

0

2

4x 10

-3 IMF11

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-1.5

-1

-0.5

0x 10

-3

Time

IMF12


Hilbert Spectral Anaysis1 1 ( )

( ) { ( )} ( )x

y t H x t x t PV dt t

( )( ) ( ) ( ) ( ) i tz t x t iy t a t e

1 ( )( ) arg( ( )) tan ( )

( )

y tt z t

x t

1 1 ( )( ) ( )

2 2

d tf t t

dt


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


ConclusionWe 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.


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, McGraw-Hill, 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.

http://djj.ee.ntu.edu.tw/TFW.htm

Documents

An Introduction to Time-Frequency Analysis