FrFT and Time-Frequency Distribution 分數傅立葉轉換與時頻分析 Guo-Cyuan Guo 郭國銓 指導教授 :Jian Jiun Ding 丁建均 Institute of Communications Engineering National

FrFT and Time-FrequencyDistribution

分數傅立葉轉換與時頻分析

Guo-Cyuan Guo 郭國銓指導教授 :Jian Jiun Ding丁建均

Institute of Communications EngineeringNational Taiwan University

Feb., 2008

DISP LAB 2

Outline

Introduction FrFT & LCT Time-Frequency Distribution Applications DFrFT Acoustics & Music Signals Conclusions and Future works Reference

DISP LAB 3

Outline

Introduction FrFT & LCT Time-Frequency Distribution Applications DFrFT Acoustics & Music Signals Conclusions and Future works Reference

DISP LAB 4

Introduction

Fourier Transform(18-th century):

Fractional Fourier Transform (FrFT): 1980 Victor Namias (Quantum mechanics) 1994 Almeida (Signal Processing) Ozaktas (Optics)

LCT 1970 matrix optics— Fresnel transform Mathematics

1( ) ( )

2j tf t F e d

1( ) ( )

2j tF f t e dt

DISP LAB 5

Introduction

FT FrFT LCT

DISP LAB 6

Outline

Introduction FrFT & LCT Time-Frequency Distribution Applications DFrFT Acoustics & Music Signals Conclusions and Future works Reference

DISP LAB 7

Fractional Fourier Transform

-5 -4 -3 -2 -1 0 1 2 3 4 5-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

w

F(w

)

0

-5 -4 -3 -2 -1 0 1 2 3 4 5

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

w

F(w

)

/ 2

FT

0.1 FT

?

DISP LAB 8

FrFT & Linear Canonical Transform Definition:

2 2

cot cot csc2 21 cot

( )2

u tj j jutje e e f t dt

( )F u f u

f u

if N , N is integer

if 2N , N is integer

if (2N+1) , N is integer

2 2

2

2

2

, , ,

2

, b 0( )

, b=0

j d au j u t j tb b b

a b c d jcd u

j f t dtF u

d f d u

e e e

e

DISP LAB 9

FrFT (cont’)

-5 -4 -3 -2 -1 0 1 2 3 4 5-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

w

F(w

)

-5 -4 -3 -2 -1 0 1 2 3 4 5-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

w

F(w

)-5 -4 -3 -2 -1 0 1 2 3 4 5

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

w

F(w

)

0 0.01 0.2

-5 -4 -3 -2 -1 0 1 2 3 4 5

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

w

F(w

)

/ 2

-5 -4 -3 -2 -1 0 1 2 3 4 5

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

w

F(w

)

-5 -4 -3 -2 -1 0 1 2 3 4 5

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

wF

(w)

/ 4 3 / 4

DISP LAB 10

Outline

Introduction FrFT & LCT Time-Frequency Distribution Applications DFrFT Acoustics & Music Signals Conclusions and Future works Reference

DISP LAB 11

Time-Frequency Distribution Short Time Fourier Transform(STFT)

Gabor transform

Wigner Distribution(WD)

2( )

2( , )t

jG t e e x d

*1, / 2 / 2

2j

gW t g t g t e d

( , ) jG t w t e x d

DISP LAB 12

T-F Distribution(cont’) Input: cos( ) , 0 10

( ) cos(3 ) ,10 20

cos(2 ) , 20 30

t t

f t t t

t t

time (sec)

freq

uenc

y

0 5 10 15 20 25 30-20

-15

-10

-5

0

5

10

15

20

time (sec)

freq

uenc

y

0 5 10 15 20 25 30-10

-8

-6

-4

-2

0

2

4

6

8

10

Gabor WDF

DISP LAB 13

T-F Distribution(cont’)

Gabor WDF

time (sec)

freq

uenc

y

WDF

-10 -8 -6 -4 -2 0 2 4 6 8 10-10

-8

-6

-4

-2

0

2

4

6

8

10

time (sec)

freq

uenc

y

Gabor1.5 WDF0.8

-10 -8 -6 -4 -2 0 2 4 6 8 10-10

-8

-6

-4

-2

0

2

4

6

8

10

time (sec)

freq

uenc

y

Gabor

-10 -8 -6 -4 -2 0 2 4 6 8 10-10

-8

-6

-4

-2

0

2

4

6

8

10

Gabor-Wigner

DISP LAB 14

Outline

Introduction FrFT & LCT Time-Frequency Distribution Applications DFrFT Acoustics & Music Signals Conclusions and Future works Reference

DISP LAB 15

Filter Design

time (sec)

wOriginal Signal

-10 -5 0 5 10-10

-5

0

5

10

time (sec)

w

-10 -5 0 5 10-10

-5

0

5

10

time (sec)

w

Original Signal plus noise

-10 -5 0 5 10-10

-5

0

5

10

-10 -8 -6 -4 -2 0 2 4 6 8 100

0.5

1Normal FT, NMSE =7.8923%

-10 -8 -6 -4 -2 0 2 4 6 8 100

0.5

1Normal FT, NMSE =1.073%

DISP LAB 16

Filter Design(cont’)

u

v

u

v

u

v

u

v

u

v

u

v

2 0

0 0.5

a b

c d

1 0

1 1

a b

c d

1 1

0 1

a b

c d

0 1

1 0

a b

c d

cos( / 4) sin( / 4)

sin( / 4) cos( / 4)

a b

c d

DISP LAB 17

Fourier Optics

output planeinput plane

1 0z 2z f 3 2z f

output planeinput plane

DISP LAB 18

Fourier Optics(cont’) Through free space:

output planeinput plane

z

DISP LAB 19

Fourier Optics(cont’) Through thin lens

output plane

input plane

DISP LAB 20

Fourier Optics(cont’) Through the gradient-index medium (GRIN)

d

DISP LAB 21

Fourier Optics(cont’)

output planeinput plane

1 0z 2z f 3 2z f

DISP LAB 22

Outline

Introduction FrFT & LCT Time-Frequency Distribution Applications DFrFT Acoustics & Music Signals Conclusions and Future works Reference

DISP LAB 23

DFrFT Definition1:

Definition2:

2 22 2 cot

2cotsin2

1 cot

2

jp tu tu

jj M p qq

u tp M

jF q e e e f p

0 1 2 30 1 2 3

tF a t F a t F a t F a t F 4

/2

1

1

4j t i k

ik

a t e

DISP LAB 24

DFrFT

2 / 2 /

1

0

2

0

ˆ ˆ

ˆ ˆ for N is odd

ˆ ˆ ˆ ˆ for N is even

T

Njk T

k kk

Njk T jN T

k k N Nk

F UD U

e u u

e u u e u u

0

2 /

2

1

0

0

j

j

j N

j N

e

e

D

e

e

Definition3:

DISP LAB 25

DFrFT

-30 -20 -10 0 10 20 30-1

-0.5

0

0.5

1

1.5

2

2.5

n

F(n

)

-30 -20 -10 0 10 20 30-1

-0.5

0

0.5

1

1.5

2

2.5

n

F(n

)

0 0.05

-30 -20 -10 0 10 20 30-1

-0.5

0

0.5

1

1.5

2

2.5

n

F(n

)

0.2

-30 -20 -10 0 10 20 30-1

-0.5

0

0.5

1

1.5

2

2.5

n

F(n

)

-30 -20 -10 0 10 20 30-1

-0.5

0

0.5

1

1.5

2

2.5

n

F(n

)

-30 -20 -10 0 10 20 30-1

-0.5

0

0.5

1

1.5

2

2.5

n

F(n

)

/ 4 / 2 3 / 4

DISP LAB 26

Outline

Introduction FrFT & LCT Time-Frequency Distribution Applications DFrFT Acoustics & Music Signals Conclusions and Future works Reference

DISP LAB 27

Pronounce

Pulmonary alveolus Resonant cavity voice

Random sequence generator

voiced

Periodic pulse train generator

unvoiced

x[n]Vocal Tract Model

G

DISP LAB 28

Hearing

Frequency

……

Weighting

Bark Scale

DISP LAB 29

Masking Effect

Sound Pressure Level

Frequency

Masking signal

Masked signals

Unmasked signal

Hearing threshold

Masking threshold

DISP LAB 30

MFCCSpeech signalx(n)

Pre-emphasis

Window

DFT

Mel filter bank

DCT

Energy

Derivatives

MFCC

2 2

,

,

,

t t

t t t

t t

y j e

y y j e

y j e

2log

DISP LAB 31

Music Sim.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

sec

-100 0 100 200 300 400 500 600 700 800 900 1000-60

-40

-20

0

20

40

60

frequency

dB

Fs=8000 window size=200ms

time (sec)

freq

uenc

y 10

0Hz

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

9

10

Fs=8000 window size=200ms

time (sec)

freq

uenc

y 10

0Hz

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

9

10

DISP LAB 32

Music Sim.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

sec

-100 0 100 200 300 400 500 600 700 800 900 1000-40

-20

0

20

40

60

80

frequency (Hz)

dB

Fs=44100 window size=200ms

time (sec)

freq

uenc

y 10

0Hz

0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

1

2

3

4

5

6

7

8

9

10

Fs=44100 window size=200ms

time (sec)

freq

uenc

y 10

0Hz

0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

1

2

3

4

5

6

7

8

9

10

DISP LAB 33

Problems

The computation problem Real time Resolution Harmonics

DISP LAB 34

Acoustics Signals

ㄞㄟㄠㄡ

Fs=10000 window size=100ms

time (sec)

norm

aliz

ed fr

eque

ncy

0 1 2 3 4 5 6 7 8

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 1 2 3 4 5 6 7 8 9-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

sec

DISP LAB 35

Problems

Computation Resolution Frame decision Correlation

DISP LAB 36

Outline

Introduction FrFT & LCT Time-Frequency Distribution Applications DFrFT Acoustics & Music Signals Conclusions and Future works Reference

DISP LAB 37

Conclusions and Future works FrFT & LCT &DFrFT Time-Frequency Distribution Applications Acoustics & Music Signals Fractional Fourier Series Discrete Time Fourier Transform Time-Frequency Resolution and Computation Music Autoscore

DISP LAB 38

Reference [1] H.M. Ozaktas, Z. Zalevsky and M. A. Kutay, The fractional Fourier transform with Applications

in Optics and Signal Processing, John Wiley & Sons, 2001. [2] J. J. Ding, Research of Fractional Fourier Transform and Linear Canonical Transform, Ph.D.

thesis, National Taiwan University, Taipei, Taiwan, R.O.C, 2001. [3] S. Qian and D. Chen, Joint Time-Frequency Analysis: Methods and Applications, Prentice Hall,

N.J., 1996. [4] R. L. Allen and D. W. Mills, Signal Analysis: Time, Frequency, Scale, and Structure, Wiley-

Interscience, NJ, 2004. [5] S. C. Pei and J. J. Ding, “Relations between Gabor Transforms and Fractional Fourier Tran

sforms and Their Applications for Signal Processing,” Revised Version: T-SP-04763- 2006.R1.

[6] X. G. Xia, “On Bandlimited Signals with Fractional Fourier Transform,” IEEE Signal Processing Letters, Vol. 3, No. 3, March 1996.

[7] P. Andres, W. D. Furlan and G. Saavedra, “Variable Fractional Fourier Processor: A Simple Implementation,” J. Opt. Soc. Am. A, Vol. 14, p.853-858, No. 4 , April 1997.

[8] H. M. Ozaktas and D. Mendlovic, “Fractional Fourier Optics,” J. Opt. Soc. Am. A, Vol. 12, p.743-751, No. 4, April 1995.

[9] D. Mendlovic, R. G. Dorsch, A. W. Lohmann, Z. Zalevsky, and C. Ferreira, “Optical Illustration of a Varied Fractional Fourier Transform Order and the Radon-Wigner Display,” Appl.

Opt. Vol. 35, No. 20, 10, p.3925-3929, July 1996. [10] L. R. Rabiner and R. W. Schafer, Digital Processing of Speech Signals, Pren-tice-Hall, 1978. [11] 王小川 , 語音訊號處理 , 全華科技圖書股份有限公司 , Taipei, 2004. [12] A. Klapuri , “Signal Processing Methods for the Automatic Transcription of Mu-sic,” Ph. D

thesis, Tampere University of Technology, Tampere, March 2004.