Transcript Learning Wavelet Transform by MATLAB Toolbox

Learning Wavelet Transform
by MATLAB Toolbox
Professor : R.J. Chang
Student : Tsung-Lin Wu
Date :2012/12/14
Outline
1. Wavelet evolution
2. Wavelet transform toolbox in MATLAB
3. Wavelet function in MATLAB
4. Wavelet packet transform toolbox in MATLAB
5. Wavelet packet function in MATLAB
6. Definition of SNR
1. Wavelet evolution
• Fourier transform:
F ( ) 

i t

f (t ) e

dt
• Time-frequency tile for Fourier transform:
f
t
• Poor time-localization
1. Wavelet evolution(count.)
• Short-time Fourier transform:
S ( ,  ) 



f (t ) w (t   ) e
 i t
dt
• Time-frequency tile for STFT:
f
f
f
t
t
• w(t-τ) is the window function.
t
1. Wavelet evolution(count.)
• Wavelet transform:
s( t ) 
C
j,k
 j,k ( t ),
k  Z j Z
 a ,b ( t ) 
1
a
(
tb
)
a, b  Z
a
• Time-frequency tile for wavelet transform:
Translations and Scaling of a Wavelet
f
t
1. Wavelet evolution(count.)
• Compare these three methods:
Fourier transform
STFT
Wavelet
Resolution in time domain
and frequency domain
No resolution in
time domain
Low resolution in
time domain.
High resolution in
time domain.
Can analyze the nonstationary signal?
No
Yes
Good
2. Wavelet transform toolbox in MATLAB
• Decomposition and reconstruction :
2
h[n ]
~
h [n]
2
x[n ]
~y [ n ]
H
y H [ n ]
y H [n]
g[n]
~y [ n ]
L
y L [ n ]
y L [n]
2
2
~
x [n]
g~ [ n ]
• This structure contains for J = 3 the terminal nodes of the following tree.
Input signal
Level 1
Level 2
Level 3
2. Wavelet transform toolbox in MATLAB(count.)
• Step 1:
Type command “wavemenu” at
command window and hit the
Wavelet 1-D button.
• Step 2:
Load signal from “*.mat files”
or “workspace”.
• Step 3:
Select the mother wavelet and
levels, for example: db7, level 5.
• Step 4:
Push “Analyze” button.
2. Wavelet transform toolbox in MATLAB(count.)
• The signals include approximations(a) and Details(d):
s＝a5+d5+d4+d3+d2+d1
d1
d2
d3
d4
a5 d5
Wavelet transform toolbox in MATLAB(count.)
• Step 6:
Push “Statistics” button to show the statistics of signal of each levels.
Wavelet transform toolbox in MATLAB(count.)
• Step 7:
Push “De-noise” button to remove the detail parts.
Wavelet transform toolbox in MATLAB(count.)
3. Wavelet function in MATLAB
• 1. Single-level discrete 1-D wavelet transform:
[cA,cD] = dwt(X,'wname')
Example:
load noissin
s = noissin(1:1000);
[ca1,cd1] = dwt(s,'haar');
plot(s)
subplot(121),plot(ca1)
subplot(122),plot(cd1)
1.5
1
0.5
0
-0.5
-1
-1.5
0
100
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1000
s
2.5
0.8
2
0.6
1.5
0.4
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0.5
0
0
-0.2
-0.5
-0.4
-1
-0.6
-1.5
-2
0
50
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300
ca1
350
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-0.8
500 0
50
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250
cd1
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500
3. Wavelet function in MATLAB(count.)
• 2. Single-level inverse discrete 1-D wavelet transform:
X = idwt(cA,cD,'wname')
Example:
load noissin
s = noissin(1:1000);
[ca1,cd1] = dwt(s,'haar');
X = idwt(ca1,cd1,'haar');
plot(X)
2.5
2
1.5
1
0.5
0
-0.5
1.5
1.5
-1
-1.5
1
-2
1
0
50
100
150
200
250
300
350
400
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500
ca1
+
0.5
0.8
0.5
0
0
0.6
0.4
-0.5
-0.5
0.2
0
-1
-1
-0.2
-0.4
-1.5
-1.5
-0.6
0
100
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500
s
600
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1000
-0.8
0
50
100
150
200
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cd1
350
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0
100
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400
500
X
600
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800
900
1000
3. Wavelet function in MATLAB(count.)
• 3. Multilevel 1-D wavelet decomposition:
[C,L] = wavedec(X,N,'wname')
Example:
load sumsin
s = sumsin;
[c,l] = wavedec(s,3, 'db1');
3. Wavelet function in MATLAB(count.)
• 4. Reconstruct single branch from 1-D wavelet coefficients:
X = wrcoef('type',C,L,'wname',N)
Example:
load sumsin
s = sumsin;
[c,l] = wavedec(s,5, 'sym4');
a5 = wrcoef('a',c,l, 'sym4',5);
4. Wavelet packet transform toolbox in MATLAB
• Step 1:
Type command “wavemenu” at
command window and hit the
Wavelet Packet 1-D button.
• Step 2:
Load signal from “*.mat files”
or “workspace”.
• Step 3:
Select the mother wavelet and
levels, for example: db7, level 5,
shannon.
• Step 4:
Push “Analyze” button.
4. Wavelet packet transform toolbox in MATLAB
5. Wavelet packet function in MATLAB
1. Single-level discrete 1-D wavelet packet transform:
[T,D] = wpdec(X,N,'wname',E)
Example:
load noissin
s = noissin(1:1000);
wpt = wpdec(s,3,'haar','shannon');
plot(wpt)
5. Wavelet packet function in MATLAB
2. Single-level denoise discrete 1-D wavelet packet transform:
[XD,TREED,PERF0,PERFL2]=wpdencmp(X,SORH,N,'wname',CRIT,PAR,K
EEPAPP)
Example:
load noissin
s = noissin(1:1000);
n = length(s);
thr = sqrt(2*log(n*log(n)/log(2)));
xwpd = wpdencmp(s,'s',3,'haar','sure',thr,1);
plot(t,xx,'r',t,xwpd)
30
20
10
0
-10
-20
-30
-40
0
0.1
0.2
0.3
0.4
0.5
0.6
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0.9
1
6. Definition of SNR
•
第一種方法為時域計量，為信噪比最基本的定義，信號的平均功率除以雜訊
平均功率，以下稱為Time Domain 1 。
1
SN R1  10 log 1 0
•
N 
N
0
N
0
2
y S (t )
2
y N (t )
第二種方法也是為時域計量，為 Mitaim與Kosko所使用之定義，以下稱為
Time Domain 2。
SN R 2  10 log 10
•
1
N 
2 Y [k0 ]

L 1
k 0
k 0  f 0 L Ts
2
2
Y [k ]  2 Y [k0 ]
2
L 1
Y [k ] 

i
yt e
2 k
t
L
t0
第三種方法為頻域計量，其定義為信號頻譜在信號頻率下的平均功率除以雜
訊頻譜在信號頻率下的平均功率，以下稱為Narrow Band 1 。
  
lim
SN R  10 log 1 0
  0

S S ( ) d 
  
S N ( )
6. Definition of SNR
• 第四種方法也是為頻域計量，其定義為信號和雜訊頻譜在信號頻率下
的平均功率除以雜訊頻譜在信號頻率下的平均功率，以下稱為Narrow
Band 2。
  
lim
S N R  1 0 lo g 1 0
  0

S ( ) d 
  
S N ( )
• 第五種方法也是為頻域計量，以下稱為Wide Band，其定義為總信號
頻譜的平均功率除以總雜訊頻譜的平均功率，以下稱為Wide Band 。
k   


SN R w  10 log 10
k 1

lim
  0
S S ( ) d 
k   

S
0
N
( ) d 
6. Definition of SNR
以此範例來觀察上述信噪比定義的不同，模擬方法為使用Monte Carlo
法。
35
Time Domain 1
Time Domain 2
Wide Band
Narrow Band 1
Narrow Band 2
30
25
資料規格表
0.2
弦波頻率(f)
0.02Hz
噪音強度(2D) 0.05, 0.05~0.8
5Hz
SNR (dB)
15
振幅(A)
取樣頻率(fs)
20
10
5
0
-5
-10
-15
-20
0.05
0.1
0.15
0.2
0.25
0.3
Noise Intensity
0.35
Time Domain 1，Time Domain 2和Wide Band 會重合在一起。
Narrow Band 1和Narrow Band 2會重合在一起。
0.4
0.45
0.5
Thank you for your attention.