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Wigner-Ville Distribution: In Perspective of Fault Diagnosis

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Wigner-Ville Distribution: In Perspective of Fault Diagnosis

  1. 1. Seoul National University Wigner-Ville Distribution: In Perspective of Fault Diagnosis (Based on Time-Frequency Analysis, Cohen and Time-Frequency Toolbox for Use with Matlab, Auger) Jungho Park, Ph.D Candidate System Health & Risk Management Laboratory Department of Mechanical & Aerospace Engineering Seoul National University
  2. 2. Seoul National University2018/1/27 - 2 - Contents 4. Second class of solutions: the energy distribution 4.1. The Cohen’s class 4.1.1. The Wigner-Ville distribution 4.1.2. The Cohen’s class 4.1.3. Link with the narrow-band ambiguity function 4.1.4. Other important energy distribution 4.1.5. Conclusion Time-Frequency Toolbox For Use with MATLAB 8. The Wigner Distribution 9. General Approach and the Kernel Method 10. Characteristic Function Operator Method 11. Kernel Design for Reduced Interference 12. Some Distributions Time-Frequency Analysis, Cohen
  3. 3. Seoul National University • First class of solutions: Atomic decomposition • Fourier transform • Short-time Fourier transform • Wavelet transform 2018/1/27 - 3 - 8. The Wigner Distribution • Definition (Related to the energy of the signals) • Second class of solutions: Energy distribution • Wigner Distribution • Choi-Williams distribution • Zhao-Atlas-Marks 𝑊 𝑥 𝑡, 𝜈 = ' 𝑥 𝑡 + 𝜏 2 𝑥∗ (𝑡 − 𝜏 2 )𝑒012345 𝑑𝜏 78 08 𝑋 𝜈 = ' 𝑥 𝑡 𝑒01234: 𝑑𝑡 78 08 𝐹 𝑥 𝑡, 𝜈; ℎ = ' 𝑥 𝑢 ℎ∗(𝑢 − 𝑡)𝑒01234: 𝑑𝑢 78 08 𝑇 𝑥 𝑡, 𝑎; Ψ = ' 𝑥 𝑠 Ψ:,C ∗ (𝑠)𝑑𝑠 78 08 𝑊 𝑥 𝑡, 𝜈 = ' 𝑥 𝑡 + 𝜏 2 𝑥∗(𝑡 − 𝜏 2 )𝑒012345 𝑑𝜏 78 08 𝑃 𝐶𝑊 𝑡, 𝜔 = 1 4𝜋J/2 ' ' 1 𝜏2/𝜎 exp[− (𝑢 − 𝑡)2 4𝜏2/𝜎 − 𝑗𝜏𝜔] ×𝑠∗ 𝑢 − 𝜏/2 ℎ 𝑢 + 𝜏/2 𝑑𝑢𝑑𝜏 𝑍𝐴𝑀 𝑥 𝑡, 𝑣 = ' ℎ(𝜏) ' 𝑥 𝑠 + 𝜏 2 𝑥∗ (𝑡 − 𝜏 2 ) 𝑑𝑠 :7 5 /2 :0 5 /2 𝑒012345 𝑑𝜏 78 08
  4. 4. Seoul National University • Property (Refer to Cohen to check the proof) 1. Real value • The calculated values are real (It can be proved by the fact that the distribution and its complex conjugate are same.) 2018/1/27 - 4 - • Definition (Related to the energy of the signals) 8. The Wigner Distribution 𝑊 𝑥 𝑡, 𝜈 = ' 𝑥 𝑡 + 𝜏 2 𝑥∗ (𝑡 − 𝜏 2 )𝑒012345 𝑑𝜏 78 08 𝑊∗ 𝑡, 𝜔 = 1 2𝜋 ' 𝑠 𝑡 + 𝜏 2 𝑠∗ (𝑡 − 𝜏 2 )𝑒15Z 𝑑𝜏 = − [ 23 ∫ 𝑠 𝑡 + 5 2 𝑠∗ (𝑡 − 5 2 )𝑒015Z 𝑑𝜏 08 8 = [ 23 ∫ 𝑠 𝑡 + 5 2 𝑠∗ (𝑡 − 5 2 )𝑒015Z 𝑑𝜏 8 08 = 𝑊(𝑡, 𝜔)
  5. 5. Seoul National University 𝐸 = ' ' 𝑊 𝑡, 𝜔 𝑑𝜔𝑑𝑡 = ' 𝑠(𝑡) 2 𝑑𝜏 = 1 • Property (Refer to Cohen to check the proof) 2. Marginality • The energy spectral density 𝑺(𝝎) 𝟐 and the instantaneous power 𝒔(𝒕) 𝟐 can be obtained by marginal distribution of the Wigner distribution 2018/1/27 - 5 - • Definition (Related to the energy of the signals) Wigner distribution is energy distribution 8. The Wigner Distribution 𝑊 𝑥 𝑡, 𝜈 = ' 𝑥 𝑡 + 𝜏 2 𝑥∗ (𝑡 − 𝜏 2 )𝑒012345 𝑑𝜏 78 08 𝑃 𝑡 = ' 𝑊 𝑡, 𝜔 𝑑𝜔 = 1 2𝜋 ' ' 𝑠∗ 𝑡 − 𝜏 2 𝑠 𝑡 + 𝜏 2 𝑒015Z 𝑑𝜏𝑑𝜔 = ∫ 𝑠∗ 𝑡 − 5 2 𝑠 𝑡 + 5 2 𝛿(𝜏)𝑑𝜏 = 𝑠(𝑡) 2
  6. 6. Seoul National University • Property (Refer to Cohen to check the proof) 3. Non-positivity • The distribution could have negative values (Contradictory to the concept of energy density) 2018/1/27 - 6 - • Definition (Related to the energy of the signals) 8. The Wigner Distribution 𝑊 𝑥 𝑡, 𝜈 = ' 𝑥 𝑡 + 𝜏 2 𝑥∗ (𝑡 − 𝜏 2 )𝑒012345 𝑑𝜏 78 08 (Figure from Cohen)
  7. 7. Seoul National University2018/1/27 - 7 - • How the negative values are treated in the literature Normal 50% fault 100% fault Staszewski, Wieslaw J., Keith Worden, and Geof R. Tomlinson. "Time–frequency analysis in gearbox fault detection using the Wigner–Ville distribution and pattern recognition." Mechanical systems and signal processing 11.5 (1997): 673-692. 327 cited “The negative values of the distribution were set to zero to avoid difficulties with the physical interpretation.” Baydar, Naim, and Andrew Ball. "A comparative study of acoustic and vibration signals in detection of gear failures using Wigner–Ville distribution." Mechanical systems and signal processing 15.6 (2001): 1091-1107. 272 cited Normal 25% fault 50% fault “To overcome this problem and reduce the presence of interference components, a smoothed version of the WVD (SPWVD) is used.” 8. The Wigner Distribution
  8. 8. Seoul National University • Property (Refer to Cohen to check the proof) 4. Global average 2018/1/27 - 8 - • Definition (Related to the energy of the signals) ß Global average (due to marginal property) 8. The Wigner Distribution 𝑊 𝑥 𝑡, 𝜈 = ' 𝑥 𝑡 + 𝜏 2 𝑥∗ (𝑡 − 𝜏 2 )𝑒012345 𝑑𝜏 78 08 < 𝑔[ 𝑡 + 𝑔2 𝜔 >= ' ' 𝑔[ 𝑡 + 𝑔2 𝜔 𝑊 𝑡, 𝜔 𝑑𝜔𝑑𝑡 = ∫ 𝑔[ 𝑡 𝑠(𝑡) 2 𝑑𝑡 + ∫ 𝑔2 𝜔 𝑆(𝜔) 2 𝑑𝜔 < 𝑔 𝑡, 𝜔 >= ' ' 𝑔 𝑡, 𝜔 𝑊 𝑡, 𝜔 𝑑𝜔𝑑𝑡
  9. 9. Seoul National University • Property (Refer to Cohen to check the proof) 5. Local average • Instantaneous frequency and group delay can be derived from local averages of the Wigner distribution 2018/1/27 - 9 - • Definition (Related to the energy of the signals) ß Local average 𝜑 : phase 𝜓 : spectral phase Instantaneous frequency Group delay 8. The Wigner Distribution 𝑊 𝑥 𝑡, 𝜈 = ' 𝑥 𝑡 + 𝜏 2 𝑥∗ (𝑡 − 𝜏 2 )𝑒012345 𝑑𝜏 78 08 < 𝜔 >:= 1 𝑠(𝑡) 2 ' 𝜔𝑊 𝑡, 𝜔 𝑑𝜔 < 𝑡 >Z= 1 𝑆(𝜔) 2 ' 𝑡𝑊 𝑡, 𝜔 𝑑𝑡 𝑡 ; < 𝜔 >:= 𝜑′(𝑡) ; < 𝑡 >Z= −𝜓′(𝜔)
  10. 10. Seoul National University • Property (Refer to Cohen to check the proof) 6. Time and Frequency shift 2018/1/27 - 10 - • Definition (Related to the energy of the signals) 8. The Wigner Distribution 𝑊 𝑥 𝑡, 𝜈 = ' 𝑥 𝑡 + 𝜏 2 𝑥∗ (𝑡 − 𝜏 2 )𝑒012345 𝑑𝜏 78 08 if 𝑠 𝑡 → 𝑒1Zn: 𝑠 𝑡 − 𝑡o then 𝑊 𝑡, 𝜔 → 𝑊(𝑡 − 𝑡o,𝜔 − 𝜔o) 𝑊st 𝑡, 𝜔 = 1 2𝜋 ' 𝑒01Zn :05/2 𝑠∗ (𝑡 − 𝑡o − 𝜏 2 ) ×𝑒1Zn :75/2 𝑠(𝑡 − 𝑡o + 5 2 )𝑒015Z 𝑑𝜏 = [ 23 ∫ 𝑠∗ (𝑡 − 𝑡o − 5 2 )𝑠(𝑡 − 𝑡o + 5 2 ) 𝑒015(Z0Zn) 𝑑𝜏 = 𝑊(𝑡 − 𝑡o, 𝜔 − 𝜔o)
  11. 11. Seoul National University • Property (Refer to Cohen to check the proof) 7. Cross-term (Interference) • For multi-component signals, cross-terms come out due to quadratic calculation 2018/1/27 - 11 - • Definition (Related to the energy of the signals) Cross-terms 8. The Wigner Distribution 𝑊 𝑥 𝑡, 𝜈 = ' 𝑥 𝑡 + 𝜏 2 𝑥∗ (𝑡 − 𝜏 2 )𝑒012345 𝑑𝜏 78 08 𝑠 𝑡 =𝑠1 𝑡 +𝑠2 𝑡 𝑊 𝑡, 𝜔 = 𝑊11 𝑡, 𝜔 + 𝑊22 𝑡, 𝜔 + 𝑊12 𝑡, 𝜔 + 𝑊21 𝑡, 𝜔 where 𝑊12 𝑡, 𝜔 = ' 𝑠[ ∗ 𝑡 − 𝜏 2 𝑠2(𝑡 + 𝜏 2 )𝑒015Z 𝑑𝜏 𝑊 𝑡, 𝜔 = 𝑊11 𝑡, 𝜔 + 𝑊22 𝑡, 𝜔 + 2Re {𝑊12 𝑡, 𝜔 } (Figure from Auger)
  12. 12. Seoul National University2018/1/27 - 12 - • Definition (Related to the energy of the signals) 8. The Wigner Distribution • Property (Refer to Cohen to check the proof) 7. Cross-term (Interference) • For multi-component signals, cross-terms come out due to quadratic calculation 𝑊 𝑥 𝑡, 𝜈 = ' 𝑥 𝑡 + 𝜏 2 𝑥∗ (𝑡 − 𝜏 2 )𝑒012345 𝑑𝜏 78 08 Cross-terms 𝑠 𝑡 =𝑠1 𝑡 +𝑠2 𝑡 𝑊 𝑡, 𝜔 = 𝑊11 𝑡, 𝜔 + 𝑊22 𝑡, 𝜔 + 𝑊12 𝑡, 𝜔 + 𝑊21 𝑡, 𝜔 where 𝑊12 𝑡, 𝜔 = ' 𝑠[ ∗ 𝑡 − 𝜏 2 𝑠2(𝑡 + 𝜏 2 )𝑒015Z 𝑑𝜏 𝑊 𝑡, 𝜔 = 𝑊11 𝑡, 𝜔 + 𝑊22 𝑡, 𝜔 + 2Re {𝑊12 𝑡, 𝜔 } (Figure from Cohen)
  13. 13. Seoul National University2018/1/27 - 13 - • Definition (Related to the energy of the signals) ü First let us make clear that it is not generally true that the cross terms produce undesirable effects. ~~~ In fact, since any signal can be broken up into a sum of parts in an arbitrary way, the cross terms can be neither bad nor good since they are not uniquely defined; they are different for different decompositions. The Wigner distribution does not know about cross terms, since the breaking up of a signal into parts is not unique. (P.126, Cohen) ü However, the localization and amplitude of these additional terms often make the use and interpretation of the representation difficult, or even impossible when the signal contains a large number of “elementary components”. Since these interference terms distribute the real part of the scalar product in the time-frequency plane, they distribute negative values when the scalar product is negative. (P. 148-149, Auger) 8. The Wigner Distribution • Property (Refer to Cohen to check the proof) 7. Cross-term (Interference) • Two difference views on cross-terms 𝑊 𝑥 𝑡, 𝜈 = ' 𝑥 𝑡 + 𝜏 2 𝑥∗ (𝑡 − 𝜏 2 )𝑒012345 𝑑𝜏 78 08
  14. 14. Seoul National University • Property • Instantaneous frequency and group delay can be derived by local average. • The outputs could have negative values, which is counter-intuitive. • Suffers from the fact that confusing artifacts could be achieved for multicomponent signals (Cross-terms) 2018/1/27 - 14 - • Comparison between the Wigner distribution and the spectrogram Wigner distribution Spectrogram • Property • Instantaneous frequency and group delay can only be approximated. • The outputs always have positive values. • The multi-component could not be effectively resolved. (Window size) 8. The Wigner Distribution 𝑊 𝑥 𝑡, 𝜈 = ' 𝑥 𝑡 + 𝜏 2 𝑥∗(𝑡 − 𝜏 2 )𝑒012345 𝑑𝜏 78 08 𝐹 𝑥 𝑡, 𝜈; ℎ = ' 𝑥 𝑢 ℎ∗(𝑢 − 𝑡)𝑒01234z 𝑑𝑢 78 08
  15. 15. Seoul National University2018/1/27 - 15 - • Smoothed-pseudo Wigner-Ville distribution (SPWVD): To solve cross-term problems WVD: PWVD: SPWVD: (Smoothing in frequency-domain) (Smoothing both in time- and frequency-domain) 8. The Wigner Distribution 𝑊 𝑥 𝑡, 𝜈 = ' 𝑥 𝑡 + 𝜏 2 𝑥∗ (𝑡 − 𝜏 2 )𝑒012345 𝑑𝜏 78 08 𝑃𝑊 𝑥 𝑡, 𝜈 = ' ℎ(𝜏)𝑥 𝑡 + 𝜏 2 𝑥∗ (𝑡 − 𝜏 2 )𝑒012345 𝑑𝜏 78 08 𝑆𝑃𝑊 𝑥 𝑡, 𝜈 = ' ℎ(𝜏) ' 𝑔(𝑠 − 𝑡)𝑥 𝑠 + 𝜏 2 𝑥∗ (𝑠 − 𝜏 2 )𝑒012345 𝑑𝜏 78 08 78 08
  16. 16. Seoul National University2018/1/27 - 16 - • Smoothed-pseudo Wigner-Ville distribution (SPWVD): To solve cross-term problems (figure from Auger) WVD PWVD SPWVD Smoothing in freq. Smoothing in time 8. The Wigner Distribution
  17. 17. Seoul National University2018/1/27 - 17 - • Definition (Cohen) (Auger) Kernel function Parameterization function • Types of kernels • Product kernel: General case • Separable kernel 9. General Approach and the Kernel Method (The Cohen’s class) 𝐶 𝑡, 𝜔 = 1 4𝜋2 ' ' ' 𝑠∗ 𝑢 − 𝜏 2 𝑠 𝑢 + 𝜏 2 𝜙 𝜃, 𝜏 𝑒01}:015Z71}z 𝑑𝑢𝑑𝜏𝑑𝜃 𝐶~ 𝑡, 𝜐; 𝑓 = ' ' ' 𝑒123• s0: 𝑓(𝜉, 𝜏)𝑥 𝑠 + 𝜏 2 𝑥∗(𝑠 − 𝜏 2 )𝑒012345 𝑑𝜉𝑑𝑠𝑑𝜏 78 08 𝜙(𝜃, 𝜏) = 𝜙ƒ„ 𝜃𝜏 = 𝜙(𝜃𝜏) 𝜙 𝜃, 𝜏 = 𝜙[(𝜃)𝜙[(𝜏)
  18. 18. Seoul National University2018/1/27 - 18 - • Some Distributions and Their Kernels (Table from Cohen) 9. General Approach and the Kernel Method (The Cohen’s class)
  19. 19. Seoul National University2018/1/27 - 19 - • Basic properties related to the kernel • Marginals: Instantaneous Energy / Energy Density Spectrum Basic form Integrating wrt frequency For the integration to be instantaneous power ( )For frequency marginal For total energy 9. General Approach and the Kernel Method (The Cohen’s class) 𝐸 = ' ' 𝑊 𝑡, 𝜔 𝑑𝜔𝑑𝑡 = ' 𝑠(𝑡) 2 𝑑𝜏 𝑃 𝑡 = ' 𝑊 𝑡, 𝜔 𝑑𝜔 = 𝑠(𝑡) 2 𝐶 𝑡, 𝜔 = 1 4𝜋2 ' ' ' 𝑠∗ 𝑢 − 𝜏 2 𝑠 𝑢 + 𝜏 2 𝜙 𝜃, 𝜏 𝑒01}:015Z71}z 𝑑𝑢𝑑𝜏𝑑𝜃 '𝐶 𝑡, 𝜔 𝑑𝜔 = 1 2𝜋 ' ' ' 𝛿(𝜏)𝑠∗ 𝑢 − 𝜏 2 𝑠 𝑢 + 𝜏 2 𝜙 𝜃, 𝜏 𝑒1}(z0:) 𝑑𝑢𝑑𝜏𝑑𝜃 = 1 2𝜋 ' '𝜙 𝜃, 0 𝑠(𝑢) 2 𝑒1}(z0:) 𝑑𝜃𝑑𝑢 1 2𝜋 '𝜙 𝜃, 0 𝑒1}(z0:) 𝑑𝜃 = 𝛿(𝑡 − 𝑢) 𝜙 𝜃, 0 =1 𝜙 0, 𝜏 =1 𝜙 0,0 =1
  20. 20. Seoul National University2018/1/27 - 20 - • Basic properties related to the kernel • Time and frequency shift • Scaling invariance • Local average • Global average • … 9. General Approach and the Kernel Method (The Cohen’s class) 𝐶st 𝑡, 𝜔 = 1 4𝜋2 ' ' ' 𝑒01Zn(z0 5 2 0:n) 𝑒1Zn(z7 5 2 0:n) × 𝑠∗ 𝑢 − 5 2 − 𝑡o 𝑠 𝑢 + 5 2 − 𝑡o 𝜙 𝜃, 𝜏 𝑒01}:015Z71}z 𝑑𝑢𝑑𝜏𝑑𝜃 = 1 4𝜋2 ' ' ' 𝜙 𝜃, 𝜏 𝑠∗ 𝑢 − 𝜏 2 𝑠 𝑢 + 𝜏 2 𝑒01}:015(Z0Zn)71}(z7:n) 𝑑𝑢𝑑𝜏𝑑𝜃 = 1 4𝜋2 ' ' ' 𝜙 𝜃, 𝜏 𝑠∗ 𝑢 − 𝜏 2 𝑠 𝑢 + 𝜏 2 𝑒01}(:0:n)015(Z0Zn)71}z 𝑑𝑢𝑑𝜏𝑑𝜃 = 𝐶 𝑡 − 𝑡o, 𝜔 − 𝜔o
  21. 21. Seoul National University2018/1/27 - 21 - • Objective: To maintain the good properties of the Wigner distribution 11. Kernel Design for Reduced Interference where *Weak finite support *Strong finite support For product kernel, 𝜙(𝜃, 𝜏) = 𝜙ƒ„ 𝜃𝜏 = 𝜙(𝜃𝜏) (Table from Cohen) ℎ 𝑡 = 1 2𝜋 '𝜙 𝑥 𝑒1~: 𝑑𝑥 ; 𝜙 𝜃𝜏 = 'ℎ 𝑡 𝑒01}5: 𝑑𝑡 𝑃 𝑡, 𝜔 = 0 for 𝑡 outside 𝑡[, 𝑡2 if 𝑠 𝑡 is zero outside 𝑡[, 𝑡2 𝑃 𝑡, 𝜔 = 0 for 𝜔 outside 𝜔[, 𝜔2 if 𝑆 𝜔 is zero outside 𝜔[, 𝜔2 𝑃 𝑡, 𝜔 = 0 if 𝑠 𝑡 = 0 for a particular time 𝑃 𝑡, 𝜔 = 0 if 𝑆 𝜔 = 0 for a particular frequency
  22. 22. Seoul National University2018/1/27 - 22 - • Choi-Williams method • Properties • Product kernel • Both marginal are satisfied (The energy spectral density 𝑺(𝝎) 𝟐 and the instantaneous power 𝒔(𝒕) 𝟐 can be obtained) • Distribution 12. Some distributions *H.I. Choi: Faculty of the Global School Of Media at the Soongsil University *W.J. Williams: Faculty of the Department of Electrical Engineering and Computer Science at the University of Michigan (For frequency marginal) (For time marginal) Kernel function ! ", $ = 1 4() * ** +∗ - − / 2 + - + / 2 2 3, / 45678569:;67< =-=/=3 𝜙 𝜃, 𝜏 = 𝑒0}‘5‘/’ 𝜙 0, 𝜏 = 1 𝜙 𝜃, 0 = 1 𝑃“” 𝑡, 𝜔 = 1 4𝜋J/2 ' ' 1 𝜏2 /𝜎 exp (𝑢 − 𝑡)2 4𝜏2/𝜎 − 𝑗𝜏𝜔 × 𝑠∗ 𝑢 − 5 2 𝑠 𝑢 + 5 2 𝑑𝑢𝑑𝜏
  23. 23. Seoul National University2018/1/27 - 23 - • Choi-Williams method: Examples • For the sum of two sine waves ( ), the distribution will be calculated as where à The distribution would have a large peak at 𝝎 = 𝝎 𝟏 7𝝎 𝟐 𝟐 for large 𝝈 12. Some distributions Wigner distribution C-W with a large 𝝈 C-W with a small 𝝈 *C-W becomes WD for 𝜎 → ∞ 𝜙 𝜃, 𝜏 = 𝑒0}‘5‘/’ 𝑠 𝑡 = 𝐴[ 𝑒1Z˜: + 𝐴2 𝑒1Z‘: 𝐶“” 𝑡, 𝜔 = 𝐴[ 2 𝛿 𝜔 − 𝜔[ + 𝐴2 2 𝛿 𝜔 − 𝜔2 + 2𝐴[ 𝐴2 cos[ 𝜔2 − 𝜔[ 𝑡]𝜂(𝜔, 𝜔[, 𝜔2, 𝜎) 𝜂 𝜔, 𝜔[, 𝜔2, 𝜎 = 1 4𝜋 𝜔[ − 𝜔2 2/𝜎 exp 𝜔 − 1 2 𝜔[ + 𝜔2 2 4𝜋 𝜔[ − 𝜔2 2/𝜎 Figure from Cohen
  24. 24. Seoul National University2018/1/27 - 24 - • Choi-Williams method 12. Some distributions WD C-W Spectrogram Figure from Cohen
  25. 25. Seoul National University2018/1/27 - 25 - • Born-Jordan Distribution: Reduced interference • Zhao-Atlas-Marks Distribution: Reduced interference by placing cross-terms under the self-terms 12. Some distributions 𝜙 𝜃, 𝜏 = sin(𝑎𝜃𝜏) 𝑎𝜃𝜏 𝜙š›œ 𝜃, 𝜏 = 𝑔 𝜏 𝜏 sin(𝑎𝜃𝜏) 𝑎𝜃𝜏 Figure from Cohen
  26. 26. Seoul National University2018/1/27 - 26 - Literature review Feng, Zhipeng, Ming Liang, and Fulei Chu. "Recent advances in time–frequency analysis methods for machinery fault diagnosis: A review with application examples." Mechanical Systems and Signal Processing 38.1 (2013): 165-205. 283 cited • Linear time–frequency representation STFT WT Signal: 𝑥 𝑡 = sin 2𝜋𝑓 ¡¢£ 𝑡 + 2 cos 2𝜋𝑓¤¥¦¦¡£¦ 𝑡 + 153.6 cos 2𝜋𝑓«¬ 𝑡 + 𝑛(𝑡)
  27. 27. Seoul National University2018/1/27 - 27 - Literature review Feng, Zhipeng, Ming Liang, and Fulei Chu. "Recent advances in time–frequency analysis methods for machinery fault diagnosis: A review with application examples." Mechanical Systems and Signal Processing 38.1 (2013): 165-205. 283 cited • Bilinear time–frequency distribution WVD SPWVD C-H Signal: 𝑥 𝑡 = sin 2𝜋𝑓 ¡¢£ 𝑡 + 2 cos 2𝜋𝑓¤¥¦¦¡£¦ 𝑡 + 153.6 cos 2𝜋𝑓«¬ 𝑡 + 𝑛(𝑡)
  28. 28. Seoul National University2018/1/27 - 28 - Literature review • Basic principles of gear fault diagnosis à Based on side-band detection *Feng, Zhipeng, and Ming Liang. "Fault diagnosis of wind turbine planetary gearbox under nonstationary conditions via adaptive optimal kernel time–frequency analysis." Renewable Energy 66 (2014): 468-477. 56 cited * * The interference terms from WVD would make it difficult to diagnose the fault in the system
  29. 29. Seoul National University THANK YOU FOR LISTENING 2018/1/27 - 29 -

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