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 Fahd University of Petroleum & Minerals Electrical Engineering Department

2 Overview  Motivation  Problem Formulation  Noise Filtering Methods  SVD(Singular Value Decomposition) based Noise Filtering  Wavelet Denoising  Hybrid Wavelet-SVD  Simulation Results  Conclusion

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.

5 Methods of Noise Filtering  Conventional Filters  LS  RLS  LAV etc... Classical Methods Modern Methods  Singular Value Decomposition (SVD)  Wavelets

6 Singular Value Decomposition(SVD)  The SVD of an m x n matrix A of rank r is defined as A=U  V T where U=[u 1... u m ], V=[v 1... v n ] 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

9 Hankel Matrix Structure  The Data Matrix Z in Hankel Structure: where N+M=T+1, N  M  The 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 Z r =U 2r  2r V 2r T =

11 Reconstruction of Noise Filtered Data  The reduced rank matrix Z r is not Hankel anymore.  We can restore the Hankel Structure by averaging the antidiagonal elements.

12 Wavelet Denoising  Besides other applications of Wavelets, they are widely used in Denoising.  Donoho proposed the formal interpretation of Denoising in  Denoising Steps  Apply Wavelet Decomposition  Threshold the Wavelet Coefficients  Use Wavelet reconstruction to obtain the estimate of the signal.

13 Wavelet Denoising In Action

Approximation and Details Before DenoisingAfter Denoising

15 Wavelet Denoising In Action (contd.) Before Denoising After Denoising

16 Wavelet Denoising Steps Wavelet Decomposition Coefficient Thresholding Reconstruction (Inverse Wavelet Transform)

17 Hybrid Wavelet-SVD based Denoising  Hybrid Techniques SVD-WaveletWavelet-SVD  Improved results are obtained at Low SNR’s. Data Wavelet Denoising SVD Filtered Data

19 Performance Comparison  Different filtering techniques are compared on the basis of Relative Mean Square Error

20 Simulation -- Test Signal  Standard Test Signal  It is a distorted voltage signal in a 3-  full wave six pulse bridge rectifier.

21 Simulation -- Test Signal Contents

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 ( )  Filtering of Noise and higher order Harmonics First 2 singular values are kept High Threshold (4-5)

RMSE vs Denoising Threshold Relative Mean Sqaure Error vs Threshold(SNR=0dB) Denoising Threshold Relative Mean Square Error 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)

26 Results---Tabular Form Filtering of Noise and Higher Harmonics (High Threshold)

Original and Noisy Signal(10dB)

Original Signal and Filtered Signal (10dB) Filtering by SVD only Time Index Original Signal Filtered 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 Filtering by SVD only Time Index Filtered Signal Original Signal

Original Signal and Filtered Signal (0dB) Filtering of Noise only --Filtering by SVD Filtering by SVD only Time Index Filtered Signal Original Signal

Original Signal and Filtered Signal (0dB) Filtering of Noise only --Wavelet Denoising Wavelet Denoising Time Index Filtered Signal Original Signal

Original Signal and Filtered Signal (0dB) Filtering of Noise only --Wavelet-SVD Denoising Wavelet Denoising then SVD Time Index Filtered Signal Original Signal

Original Signal and Filtered Signal (10dB) Filtering of Noise only --Filtering by SVD Filtering by SVD only Time Index Filtered Signal Original Signal

Original Signal and Filtered Signal (10dB) Filtering of Noise only --Wavelet Denoising Wavelet Denoising Time Index Filtered Signal Original Signal

Original Signal and Filtered Signal (10dB) Filtering of Noise only --Wavelet-SVD Denoising Wavelet Denoising then SVD Time Index Filtered Signal Original 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 !!!

38 Quality of Power & Harmonics  The Quality of Power is affected by many sources such as voltage transients, voltage sag, harmonic distortion, etc...  Harmonic Pollution is an important parameter in determining the quality of power.  To mitigate the effects of Harmonics we need the estimate of these Harmonics.

39 Estimation of Harmonics in Noise  The estimation results are greatly affected by the presence of noise.  Accurate estimation is possible only after Noise Filtering.

40 Sources of Noise  Common Sources of Noise are  Measurement Noise  Communication difficulties in telemetring these measurements to Control Centers.

41 Methods of Noise Filtering  Various methods exist for noise filtering ranging from classical Filtering methods to modern Signal Processing Tools.  Classical methods include conventional Filters, LS, RLS, LAV etc... SVD and Wavelets are among the Modern Methods for Noise Filtering.

42 Wavelet Decomposition  Continuous Wavelet Transform which correlates the signal with mother wavelet,  jk, is defined as  For denoising, detail coefficients are needed that can be computed in the discrete domain as

43  Thresholding may be of two types  Hard Thresholding -- ‘keep’ or ‘kill’  Soft Thresholding -- shrinking Values are compared with a threshold, shrinking the non-zero elements towards zero The coefficients are modified as Thresholding in Wavelet Denoising Threshold selection is very critical

44 Inverse Wavelet Transform-- Reconstruction  The modified wavelet coefficients are used along with approximations to compute Inverse Wavelet transform to reconstruct the noise free data.

45 Robustness of Singular Values  Wiley Theorem If matrix A is perturbed by E i.e. Then singular values of perturbed matrix are bounded by