FrFT and Time-Frequency Distribution 分數傅立葉轉換與時頻分析 Guo-Cyuan Guo 郭國銓...
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Transcript of FrFT and Time-Frequency Distribution 分數傅立葉轉換與時頻分析 Guo-Cyuan Guo 郭國銓...
![Page 1: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/1.jpg)
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
![Page 2: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/2.jpg)
DISP LAB 2
Outline
Introduction FrFT & LCT Time-Frequency Distribution Applications DFrFT Acoustics & Music Signals Conclusions and Future works Reference
![Page 3: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/3.jpg)
DISP LAB 3
Outline
Introduction FrFT & LCT Time-Frequency Distribution Applications DFrFT Acoustics & Music Signals Conclusions and Future works Reference
![Page 4: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/4.jpg)
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
![Page 5: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/5.jpg)
DISP LAB 5
Introduction
FT FrFT LCT
![Page 6: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/6.jpg)
DISP LAB 6
Outline
Introduction FrFT & LCT Time-Frequency Distribution Applications DFrFT Acoustics & Music Signals Conclusions and Future works Reference
![Page 7: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/7.jpg)
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
?
![Page 8: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/8.jpg)
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
![Page 9: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/9.jpg)
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
![Page 10: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/10.jpg)
DISP LAB 10
Outline
Introduction FrFT & LCT Time-Frequency Distribution Applications DFrFT Acoustics & Music Signals Conclusions and Future works Reference
![Page 11: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/11.jpg)
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
![Page 12: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/12.jpg)
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
![Page 13: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/13.jpg)
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
![Page 14: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/14.jpg)
DISP LAB 14
Outline
Introduction FrFT & LCT Time-Frequency Distribution Applications DFrFT Acoustics & Music Signals Conclusions and Future works Reference
![Page 15: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/15.jpg)
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%
![Page 16: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/16.jpg)
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
![Page 17: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/17.jpg)
DISP LAB 17
Fourier Optics
output planeinput plane
1 0z 2z f 3 2z f
output planeinput plane
![Page 18: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/18.jpg)
DISP LAB 18
Fourier Optics(cont’) Through free space:
output planeinput plane
z
![Page 19: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/19.jpg)
DISP LAB 19
Fourier Optics(cont’) Through thin lens
output plane
input plane
![Page 20: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/20.jpg)
DISP LAB 20
Fourier Optics(cont’) Through the gradient-index medium (GRIN)
d
![Page 21: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/21.jpg)
DISP LAB 21
Fourier Optics(cont’)
output planeinput plane
1 0z 2z f 3 2z f
![Page 22: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/22.jpg)
DISP LAB 22
Outline
Introduction FrFT & LCT Time-Frequency Distribution Applications DFrFT Acoustics & Music Signals Conclusions and Future works Reference
![Page 23: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/23.jpg)
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
![Page 24: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/24.jpg)
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:
![Page 25: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/25.jpg)
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
![Page 26: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/26.jpg)
DISP LAB 26
Outline
Introduction FrFT & LCT Time-Frequency Distribution Applications DFrFT Acoustics & Music Signals Conclusions and Future works Reference
![Page 27: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/27.jpg)
DISP LAB 27
Pronounce
Pulmonary alveolus Resonant cavity voice
Random sequence generator
voiced
Periodic pulse train generator
unvoiced
x[n]Vocal Tract Model
G
![Page 28: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/28.jpg)
DISP LAB 28
Hearing
Frequency
……
Weighting
Bark Scale
![Page 29: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/29.jpg)
DISP LAB 29
Masking Effect
Sound Pressure Level
Frequency
Masking signal
Masked signals
Unmasked signal
Hearing threshold
Masking threshold
![Page 30: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/30.jpg)
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
![Page 31: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/31.jpg)
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
![Page 32: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/32.jpg)
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
![Page 33: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/33.jpg)
DISP LAB 33
Problems
The computation problem Real time Resolution Harmonics
![Page 34: 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](https://reader036.fdocument.pub/reader036/viewer/2022062422/56649f1b5503460f94c316c3/html5/thumbnails/34.jpg)
DISP LAB 34
Acoustics Signals
ㄞㄟㄠㄡ
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0 1 2 3 4 5 6 7 8 9-0.2
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DISP LAB 35
Problems
Computation Resolution Frame decision Correlation
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DISP LAB 36
Outline
Introduction FrFT & LCT Time-Frequency Distribution Applications DFrFT Acoustics & Music Signals Conclusions and Future works Reference
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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
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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.