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
2
Overview
MotivationProblem FormulationNoise Filtering MethodsSVD(Singular Value Decomposition) based Noise
FilteringWavelet DenoisingHybrid Wavelet-SVDSimulation ResultsConclusion
3
Motivation ...
Quality of Power
Sources of Harmonics
Harmonics deteriorate Quality of Power
Harmonics Filtering
Noise Filtering
...
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(
)()()(
5
Methods of Noise Filtering
Conventional Filters LS RLS LAV etc...
Classical Methods
Modern Methods
Singular Value Singular Value Decomposition Decomposition (SVD)(SVD)
WaveletsWavelets
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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.
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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.
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SVD based Noise Filtering Algorithm
OBTAINSAMPLES OFNOISY DATA
CONSTRUCTHANKEL MATRIX
APPLYSINGULAR
VALUEDECOMPOSITON
ESTABLISHREDUCED RANK
MATRIX
RESTORE HANKELMATRIX
TO OBTAIN NOISEFILTERED DATA
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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.
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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
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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
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.
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0 200 400 600 800 1000 1200-10
-8
-6
-4
-2
0
2
4
6
8
10
Wavelet Denoising In Action
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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
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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
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n
kjjk nnzd )()( ,
Wavelet Denoising Steps
Wavelet Decomposition
Coefficient Thresholding
)|)(|sgn( jkjknewjk ddd
Reconstruction (Inverse Wavelet
Transform)
j kkj
newjk
k
nd
nkcnZ
)(
)()()(ˆ
,
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Hybrid Wavelet-SVD based Denoising
Hybrid Techniques
SVD-Wavelet Wavelet-SVD
Improved results are obtained at Low SNR’s.
DataWavelet
DenoisingSVD Filtered
Data
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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~
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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
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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
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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)
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
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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.
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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
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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
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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
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
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
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
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
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
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
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
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
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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.
Thanks !!!Thanks !!!