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

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.

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.

8

SVD based Noise Filtering Algorithm

OBTAINSAMPLES OFNOISY DATA

CONSTRUCTHANKEL MATRIX

APPLYSINGULAR

VALUEDECOMPOSITON

ESTABLISHREDUCED RANK

MATRIX

RESTORE HANKELMATRIX

TO OBTAIN NOISEFILTERED DATA

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.

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

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

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.

13

0 200 400 600 800 1000 1200-10

-8

-6

-4

-2

0

2

4

6

8

10

Wavelet Denoising In Action

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

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

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

,

17

Hybrid Wavelet-SVD based Denoising

Hybrid Techniques

SVD-Wavelet Wavelet-SVD

Improved results are obtained at Low SNR’s.

DataWavelet

DenoisingSVD Filtered

Data

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~

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

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

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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

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.

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

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

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

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

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.

Thanks !!!Thanks !!!