7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By...
-
date post
20-Dec-2015 -
Category
Documents
-
view
221 -
download
2
Transcript of 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By...
![Page 1: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/1.jpg)
7th IEEE Technical Exchange Meeting 2000
Hybrid Wavelet-SVD based Filtering of Noise in Harmonics
By
Prof. Maamar Bettayeb Prof. Maamar Bettayeb andand
Syed Faisal Ali ShahSyed Faisal Ali Shah
King Fahd University of Petroleum & Minerals
Electrical Engineering Department
![Page 2: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/2.jpg)
2
Overview
MotivationProblem FormulationNoise Filtering MethodsSVD(Singular Value Decomposition) based Noise
FilteringWavelet DenoisingHybrid Wavelet-SVDSimulation ResultsConclusion
![Page 3: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/3.jpg)
3
Motivation ...
Quality of Power
Sources of Harmonics
Harmonics deteriorate Quality of Power
Harmonics Filtering
Noise Filtering
...
![Page 4: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/4.jpg)
4
Noise Filtering: Problem Formulation
A signal with harmonics embedded in additive noise
The problem is to recover noise free harmonic signal X from the observation Z.
N
nnon kknA
kkXkZ
1
)()sin(
)()()(
![Page 5: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/5.jpg)
5
Methods of Noise Filtering
Conventional Filters LS RLS LAV etc...
Classical Methods
Modern Methods
Singular Value Singular Value Decomposition Decomposition (SVD)(SVD)
WaveletsWavelets
![Page 6: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/6.jpg)
6
Singular Value Decomposition(SVD)
The SVD of an m x n matrix A of rank r is defined as
A=UVT
where U=[u1 ... um], V=[v1 ... vn] and
=diag [1 ... r ]
Number of singular values determine the
rank of the matrix.
![Page 7: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/7.jpg)
7
SVD based Noise Filtering
Singular Values are robust. Little perturbation with noise. Larger Singular Values (SV) corresponds to
the Signal.Smaller SV corresponds to noise.Truncate small SV to get Noise Filtered
Data.
![Page 8: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/8.jpg)
8
SVD based Noise Filtering Algorithm
OBTAINSAMPLES OFNOISY DATA
CONSTRUCTHANKEL MATRIX
APPLYSINGULAR
VALUEDECOMPOSITON
ESTABLISHREDUCED RANK
MATRIX
RESTORE HANKELMATRIX
TO OBTAIN NOISEFILTERED DATA
![Page 9: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/9.jpg)
9
Hankel Matrix Structure
The Data Matrix Z in Hankel Structure:
)1()()1(
)()2()1()1()1()0(
TzNzNz
MzzzMzzz
Z
where N+M=T+1, NMThe reduced rank matrix can be constructed
by taking a definite number of Singular Values.
![Page 10: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/10.jpg)
10
Establishment of Reduced Rank Matrix
In case of Harmonics each frequency Component (sinusoid) corresponds to 2 singular values.
Thus for a signal having r frequency components, the reduced rank matrix (noise filtered) is
Zr=U2r2rV2rT=
r
i
Tiii vu
2
1
![Page 11: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/11.jpg)
11
Reconstruction of Noise Filtered Data
The reduced rank matrix Zr is not Hankel anymore.
We can restore the Hankel Structure by averaging the antidiagonal elements.
)1(ˆ)(ˆ)1(ˆ
)(ˆ)2(ˆ)1(ˆ)1(ˆ)1(ˆ)0(ˆ
ˆ
TzNzNz
MzzzMzzz
Z
![Page 12: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/12.jpg)
12
Wavelet Denoising
Besides other applications of Wavelets, they are widely used in Denoising.
Donoho proposed the formal interpretation of Denoising in 1995.
Denoising StepsApply Wavelet DecompositionThreshold the Wavelet CoefficientsUse Wavelet reconstruction to obtain the estimate of
the signal.
![Page 13: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/13.jpg)
13
0 200 400 600 800 1000 1200-10
-8
-6
-4
-2
0
2
4
6
8
10
Wavelet Denoising In Action
![Page 14: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/14.jpg)
14
0 200 400 600 800 1000 1200-10
0
10
App.
4
0 200 400 600 800 1000 1200-5
0
5
Det. 4
0 200 400 600 800 1000 1200-5
0
5
Det. 3
0 200 400 600 800 1000 1200-5
0
5
Det. 2
0 200 400 600 800 1000 1200-5
0
5
Det. 1
Approximation and Details
Before Denoising
0 200 400 600 800 1000 1200-10
0
10
App.
4
0 200 400 600 800 1000 1200-5
0
5
Det. 4
0 200 400 600 800 1000 1200-2
0
2
Det. 3
0 200 400 600 800 1000 1200-2
0
2
Det. 2
0 200 400 600 800 1000 1200-0.05
0
0.05
Det. 1
After Denoising
![Page 15: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/15.jpg)
15
0 200 400 600 800 1000 1200-8
-6
-4
-2
0
2
4
6
8
Wavelet Denoising In Action (contd.)
0 200 400 600 800 1000 1200-10
-8
-6
-4
-2
0
2
4
6
8
10
Before Denoising After Denoising
![Page 16: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/16.jpg)
16
n
kjjk nnzd )()( ,
Wavelet Denoising Steps
Wavelet Decomposition
Coefficient Thresholding
)|)(|sgn( jkjknewjk ddd
Reconstruction (Inverse Wavelet
Transform)
j kkj
newjk
k
nd
nkcnZ
)(
)()()(ˆ
,
![Page 17: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/17.jpg)
17
Hybrid Wavelet-SVD based Denoising
Hybrid Techniques
SVD-Wavelet Wavelet-SVD
Improved results are obtained at Low SNR’s.
DataWavelet
DenoisingSVD Filtered
Data
![Page 18: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/18.jpg)
![Page 19: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/19.jpg)
19
Performance Comparison
Different filtering techniques are compared on the basis of
Relative Mean Square ErrorRelative Mean Square Error
N
ii
N
iii
x
xxRMSE
1
2
1
2~
![Page 20: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/20.jpg)
20
Simulation -- Test Signal
Standard Test Signal
It is a distorted voltage
signal in a 3- full
wave six pulse bridge
rectifier.
T ra n sferIm p ed a n ce
L o a dB u s
L o a dB u s
L o a d
S ix P u lseR ectifier
G en era to r
![Page 21: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/21.jpg)
21
Simulation -- Test Signal Contents
Harmonic Order
Amplitude Phase
Fund. (60Hz.) 0.95 -2.02
5th (300Hz.) 0.09 82.1
7th (420Hz.) 0.043 7.9
11th (660Hz.) 0.03 -147.1
13th (780Hz.) 0.033 162.6
![Page 22: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/22.jpg)
22
Simulation -- Issues
Two cases of harmonic filtering are considered; Filtering of Noise (keeping all Harmonics)
• First 10 singular values are kept
• Very low Threshold (0.3 - 0.008)
Filtering of Noise and higher order Harmonics• First 2 singular values are kept
• High Threshold (4-5)
![Page 23: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/23.jpg)
RMSE vs Denoising Threshold
0 1 2 3 4 5 60
0.1
0.2
0.3
0.4
0.5
0.6
0.7Relative Mean Sqaure Error vs Threshold(SNR=0dB)
Denoising Threshold
Rel
ativ
e M
ean
Squ
are
Err
or
WL
WL+SVD
SVD
SVD + WL
![Page 24: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/24.jpg)
24
Simulation -- Details
Noise has Gaussian distribution.Results are generated for three different
Noise Levels corresponding to 20dB, 10dB and 0dB SNR.
The original signal is decomposed to 4 levels by using ‘dB8’ wavelet.
![Page 25: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/25.jpg)
25
Results---Tabular Form
Filtering of Noise only (Low Threshold)Filtering of Noise only (Low Threshold)
SNR SVD WL WL-SVD
0dB 10.10% 49.85% 6.63%
10dB 1.08% 7.38% 0.93%
20dB 0.048% 0.89% 0.05%
RM
SE
![Page 26: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/26.jpg)
26
Results---Tabular Form
Filtering of Noise and Higher HarmonicsFiltering of Noise and Higher Harmonics(High Threshold)(High Threshold)
SNR SVD WL WL-SVD
0dB 0.93% 5.42% 0.93%
10dB 0.084% 0.81% 0.084%
20dB 0.0097% 0.32% 0.0098%
RM
SE
![Page 27: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/27.jpg)
27
Original and Noisy Signal(10dB)
0 10 20 30 40 50 60 70-1.5
-1
-0.5
0
0.5
1
1.5
Noisy Signal, SNR= 10dB
Time Index
0 10 20 30 40 50 60 70-1.5
-1
-0.5
0
0.5
1
1.5
Original Signal
Time Index
Am
plitu
de
in p
u
![Page 28: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/28.jpg)
Original Signal and Filtered Signal (10dB)
0 10 20 30 40 50 60 70-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1Filtering by SVD only
Time Index
Original SignalFiltered Signal
Filtering of Noise and Higher Harmonics--Filtering by SVD
![Page 29: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/29.jpg)
Original Signal and Filtered Signal (0dB)
Filtering of Noise and Higher Harmonics--Filtering by SVD
0 10 20 30 40 50 60 70-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1Filtering by SVD only
Time Index
Filtered SignalOriginal Signal
![Page 30: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/30.jpg)
Original Signal and Filtered Signal (0dB)
Filtering of Noise only --Filtering by SVD
0 10 20 30 40 50 60 70-1.5
-1
-0.5
0
0.5
1
1.5Filtering by SVD only
Time Index
Filtered SignalOriginal Signal
![Page 31: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/31.jpg)
Original Signal and Filtered Signal (0dB)
Filtering of Noise only --Wavelet Denoising
0 10 20 30 40 50 60 70-1.5
-1
-0.5
0
0.5
1
1.5Wavelet Denoising
Time Index
Filtered SignalOriginal Signal
![Page 32: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/32.jpg)
Original Signal and Filtered Signal (0dB)
Filtering of Noise only --Wavelet-SVD Denoising
0 10 20 30 40 50 60 70-1.5
-1
-0.5
0
0.5
1
1.5Wavelet Denoising then SVD
Time Index
Filtered SignalOriginal Signal
![Page 33: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/33.jpg)
Original Signal and Filtered Signal (10dB)
Filtering of Noise only --Filtering by SVD
0 10 20 30 40 50 60 70-1.5
-1
-0.5
0
0.5
1
1.5Filtering by SVD only
Time Index
Filtered SignalOriginal Signal
![Page 34: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/34.jpg)
Original Signal and Filtered Signal (10dB)
Filtering of Noise only --Wavelet Denoising
0 10 20 30 40 50 60 70-1.5
-1
-0.5
0
0.5
1
1.5Wavelet Denoising
Time Index
Filtered SignalOriginal Signal
![Page 35: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/35.jpg)
Original Signal and Filtered Signal (10dB)
Filtering of Noise only --Wavelet-SVD Denoising
0 10 20 30 40 50 60 70-1.5
-1
-0.5
0
0.5
1
1.5Wavelet Denoising then SVD
Time Index
Filtered SignalOriginal Signal
![Page 36: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/36.jpg)
36
Conclusion
This presentation gave an overview of SVD and Wavelet based Noise Filtering methods.
A Hybrid Technique, Wavelet-SVD, is proposed and its assessment is carried out.
The Hybrid Technique performs better at low SNR.
At high SNR conventional SVD performs better than the other two methods.
![Page 37: 7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649d4b5503460f94a281d7/html5/thumbnails/37.jpg)
Thanks !!!Thanks !!!