웨이브렛 변환을 이용한 시스템 식별에 관한 연구

백옥진; 한정우; 강성주; 정창복; Wook Jin Baek; Jeong Woo Han; Sung Ju Kang; Chang Bock Chung

HWAHAK KONGHAK, Vol.39, No.2, 163-175, April, 2001

Export Citation

웨이브렛 변환을 이용한 시스템 식별에 관한 연구

A Study on System Identification Using Wavelet Transformation

백옥진, 한정우, 강성주 ^†, 정창복 ¹

전남대학교 공과대학 화학공학과, 광주 500-757
¹전남대학교 응용화학공학부, 광주 500-757

Wook Jin Baek, Jeong Woo Han, Sung Ju Kang^†, and Chang Bock Chung¹

Department of Chemical Engineering, Chonnam National University, Kwangju 500-757, Korea
¹Faculty of Applied Chemical Engineering, Chonnam National University, Kwangju 500-757, Korea

E-mail:

초록

후리에(Fourier) 변환의 단점을 극복하고자 개발된 웨이브렛(wavelet) 변환은 여러 연구분야(de-noising, 자료압축, 편미분 방정식의 해석 등)에서 다양한 용도로 활용되고 있다. 그 중 de-noising은 웨이브렛 변환의 대표적인 응용분야 중의 하나로서 최근 활발한 연구가 진행되고 있다. De-noising 효과는 사용하는 shrinkage 함수와 이에 적용되는 threshold 값을 선택하는 방법에 달려있다. 본 연구에서는 서로 다른 특성을 갖는 신호와 잡음의 크기에 따라 여러 threshold 알고리듬을 적용시킨 결과를 서로 비교하고, 그 응용으로 de-noising의 여러 알고리듬을 이용하여 잡음을 제거한 신호와 그렇지 않은 신호를 시스템 식별(system identification)에 적용시켜 그 성능을 서로 비교하였다.

The wavelet transformation, which was developed in order to overcome the defects of traditional Fourier transformation, is applied to many fields of study in various ways-for example, de-noising, data compression and mathematic applications such as solving partial differential equations, etc. De-noising is one of the main application areas of the wavelet transformation and has been studied by many researchers. The effect of de-noising depends upon the shrinkage function and the method of choosing the threshold value for the function. The objective of this work is to analyze the results of applying various threshold algorithms according to characteristics for signals and noise level. By applying the de-noising to the system identification, we compared the performances of signals which went through the de-noising process with those of signals without de-noising.

Keywords:Wavelet;De-noising;Shrinkage;Threshold;System Identification

[References]

Graps A, IEEE Computational Sci. Eng., 2(2), 50 (1995)
Donoho DL, "Wavelet Shrinkage and W.V.D.: A Ten-minute Tour," Technical report 416, Dep. of Statistics, Stanford University (1993)
Donoho DL, Johnstone IM, J. Amer. Statist. Assoc., 90(432), 1200 (1995)
Donoho DL, Johnstone IM, Biometrika, 81(3), 425 (1994)
Johnston IM, Kerkyacharian G, Picard D, J. Roy. Statist. Soc., B(57), 301 (1995)
Johnston IM, Silverman BW, J. Roy. Statist. Soc., B(59), 319 (1997)
Bruce A, Gao HY, "WaveShrink: Shrinkage Functions and Thresholds," Technical report, StatSci Division, MathSoft, Inc., 1700 Westlake Ave. N, Seattle, WA98109.9891 (1995)
Gao HY, "Wavelet Shrinkage Denoising Using the Non-Negative Garrote," The Wavelet Digest, 6(3) (1997)
Gao HY, Bruce AG, Statistica Sinica, 7(4), 855 (1997)
Gao HY, "Threshold Selection in WaveShrink," Technical report, StatSci Division, MathSoft, Inc., 1700 Westlake Ave. N, Seattle, WA98109 (1997)
Gao HY, "Wavelet Shrinkage Smoothing for Heteroscedastic Data," The Wavelet Digest, 6(3) (1997)
Palavajjhala S, Motard RL, Joseph B, AIChE J., 42(3), 777 (1996)
Nikolaou M, Vuthandam P, AIChE J., 44(1), 141 (1998)
Carrier JF, Stephanopoulos G, AIChE J., 44(2), 341 (1998)
Flehmig F, Von Watzdorf R, Marquardt W, Comput. Chem. Eng., 22(S), 491 (1998)
Akansu AN, Haddad RA, "Multiresolution Signal Decomposition: Transform, Subbands, and Wavelets," Academic Press (1992)
Chui CK, "An Introduction to Wavelets," Academic Press (1992)
Strang G, Nguyen T, "Wavelets and Filter Banks," Wellesley-Cambridge Press (1996)
Nason GP, "Wavelet Regression by Cross-validation," Technical Report 447, Dep. of Statistics, Stanford University (1994)
Weyrich N, Warhola GT, In S.P. Singh, editor, Approximation Theory, Wavelets and Applications, NATO ASI Series C(454), 523 (1995)
Jansen M, Malfait M, Bultheel A, Signal Processing, 56(1), 33 (1997)
Jansen M, Uytterhoeven G, Bultheel A, "Image De-noising by Integer Wavelet Transforms and Generalized Cross Validation," Technical Report TW264, Dep. of Computer Science, K.U. Leuven (1997)
Jansen M, Bultheel A, "Multiple Wavelet Threshold Estimation by Generalized Cross Validation for Data with Correlated Noise," Technical Report TW250, Dep. of Computer Science, K.U. Leuven (1997)
Press WH, Teukolsky SA, Vetterling WT, Flannery BP, "Numerical Recipes in C, the Art of Scientific Computing," 2nd edn, Cambridge: Cambridge University Press (1992)
Zhu YC, Backx T, "Identification of Multivariable Industrial Processes for Simulation, Diagnosis and Control," Springer-Verlag (1993)
Ljung L, "System Identification: Theory for the User," Prentice-Hall, Englewood Cliffs, N.J (1987)
Kang SJ, "System Identification with Modified Finite Impulse Response Models: Comparison with Other Identification Methods," Ph.D. Dissertation, University of Missouri-Columbia, MO, USA (1994)
Ljung L, "System Identification Toolbox User's Guide," The MathWorks, Inc. (1997)

[Related Articles]

[1] Graps A, IEEE Computational Sci. Eng., 2(2), 50 (1995)

[2] Donoho DL, "Wavelet Shrinkage and W.V.D.: A Ten-minute Tour," Technical report 416, Dep. of Statistics, Stanford University (1993)

[3] Donoho DL, Johnstone IM, J. Amer. Statist. Assoc., 90(432), 1200 (1995)

[4] Donoho DL, Johnstone IM, Biometrika, 81(3), 425 (1994)

[5] Johnston IM, Kerkyacharian G, Picard D, J. Roy. Statist. Soc., B(57), 301 (1995)

[6] Johnston IM, Silverman BW, J. Roy. Statist. Soc., B(59), 319 (1997)

[7] Bruce A, Gao HY, "WaveShrink: Shrinkage Functions and Thresholds," Technical report, StatSci Division, MathSoft, Inc., 1700 Westlake Ave. N, Seattle, WA98109.9891 (1995)

[8] Gao HY, "Wavelet Shrinkage Denoising Using the Non-Negative Garrote," The Wavelet Digest, 6(3) (1997)

[9] Gao HY, Bruce AG, Statistica Sinica, 7(4), 855 (1997)

[10] Gao HY, "Threshold Selection in WaveShrink," Technical report, StatSci Division, MathSoft, Inc., 1700 Westlake Ave. N, Seattle, WA98109 (1997)

[11] Gao HY, "Wavelet Shrinkage Smoothing for Heteroscedastic Data," The Wavelet Digest, 6(3) (1997)

[12] Palavajjhala S, Motard RL, Joseph B, AIChE J., 42(3), 777 (1996)

[13] Nikolaou M, Vuthandam P, AIChE J., 44(1), 141 (1998)

[14] Carrier JF, Stephanopoulos G, AIChE J., 44(2), 341 (1998)

[15] Flehmig F, Von Watzdorf R, Marquardt W, Comput. Chem. Eng., 22(S), 491 (1998)

[16] Akansu AN, Haddad RA, "Multiresolution Signal Decomposition: Transform, Subbands, and Wavelets," Academic Press (1992)

[17] Chui CK, "An Introduction to Wavelets," Academic Press (1992)

[18] Strang G, Nguyen T, "Wavelets and Filter Banks," Wellesley-Cambridge Press (1996)

[19] Nason GP, "Wavelet Regression by Cross-validation," Technical Report 447, Dep. of Statistics, Stanford University (1994)

[20] Weyrich N, Warhola GT, In S.P. Singh, editor, Approximation Theory, Wavelets and Applications, NATO ASI Series C(454), 523 (1995)

[21] Jansen M, Malfait M, Bultheel A, Signal Processing, 56(1), 33 (1997)

[22] Jansen M, Uytterhoeven G, Bultheel A, "Image De-noising by Integer Wavelet Transforms and Generalized Cross Validation," Technical Report TW264, Dep. of Computer Science, K.U. Leuven (1997)

[23] Jansen M, Bultheel A, "Multiple Wavelet Threshold Estimation by Generalized Cross Validation for Data with Correlated Noise," Technical Report TW250, Dep. of Computer Science, K.U. Leuven (1997)

[24] Press WH, Teukolsky SA, Vetterling WT, Flannery BP, "Numerical Recipes in C, the Art of Scientific Computing," 2nd edn, Cambridge: Cambridge University Press (1992)

[25] Zhu YC, Backx T, "Identification of Multivariable Industrial Processes for Simulation, Diagnosis and Control," Springer-Verlag (1993)

[26] Ljung L, "System Identification: Theory for the User," Prentice-Hall, Englewood Cliffs, N.J (1987)

[27] Kang SJ, "System Identification with Modified Finite Impulse Response Models: Comparison with Other Identification Methods," Ph.D. Dissertation, University of Missouri-Columbia, MO, USA (1994)

[28] Ljung L, "System Identification Toolbox User's Guide," The MathWorks, Inc. (1997)