Wigner-Ville Distribution: In Perspective of Fault Diagnosis

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 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

• First class of solutions: Atomic
decomposition
• Fourier transform
• Short-time Fourier transform
• Wavelet transform
2018/1/27 - 3 -
• 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

• 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 -
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)

𝐸 = ' ' 𝑊 𝑡, 𝜔 𝑑𝜔𝑑𝑡 = ' 𝑠(𝑡) 2 𝑑𝜏 = 1
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 -
Wigner distribution is energy 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

3. Non-positivity
• The distribution could have negative values (Contradictory to the concept of energy
density)
2018/1/27 - 6 -
𝑊 𝑥 𝑡, 𝜈 = ' 𝑥 𝑡 +
𝜏
2
𝑥∗
(𝑡 −
𝜏
2
)𝑒012345
𝑑𝜏
78
08
(Figure from Cohen)

• 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.”

4. Global average
2018/1/27 - 8 -
ß Global average
(due to marginal property)
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

5. Local average
• Instantaneous frequency and group delay can be derived from local averages of the
Wigner distribution
2018/1/27 - 9 -
ß Local average
𝜑 : phase
𝜓 : spectral phase
Instantaneous frequency Group delay
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)

6. Time and Frequency shift
2018/1/27 - 10 -
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)

7. Cross-term (Interference)
• For multi-component signals, cross-terms come out due to quadratic calculation
2018/1/27 - 11 -
Cross-terms
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)

• 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)

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)
• Two difference views on cross-terms
𝑊 𝑥 𝑡, 𝜈 = ' 𝑥 𝑡 +
𝜏
2
𝑥∗
(𝑡 −
𝜏
2
)𝑒012345
𝑑𝜏
78
08

• 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)
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

• 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)
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

• Smoothed-pseudo Wigner-Ville distribution (SPWVD): To solve cross-term problems
(figure from Auger)
WVD PWVD SPWVD
Smoothing in freq. Smoothing in time

• 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)

• Some Distributions and Their Kernels
(Table from Cohen)

• 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
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
'𝐶 𝑡, 𝜔 𝑑𝜔 =
1
2𝜋
' ' ' 𝛿(𝜏)𝑠∗
𝑢 −
𝜏
2
𝑠 𝑢 +
𝜏
2
𝜙 𝜃, 𝜏 𝑒1}(z0:)
=
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

• Basic properties related to the kernel
• Time and frequency shift
• Scaling invariance
• Local average
• Global average
• …
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
=
1
4𝜋2
' ' ' 𝜙 𝜃, 𝜏 𝑠∗
𝑢 −
𝜏
2
𝑠 𝑢 +
𝜏
2
𝑒01}:015(Z0Zn)71}(z7:n)
=
1
4𝜋2
' ' ' 𝜙 𝜃, 𝜏 𝑠∗
𝑢 −
𝜏
2
𝑠 𝑢 +
𝜏
2
𝑒01}(:0:n)015(Z0Zn)71}z
= 𝐶 𝑡 − 𝑡o, 𝜔 − 𝜔o

• 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

• 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

• 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
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

• Choi-Williams method
WD
C-W
Spectrogram
Figure from Cohen

• Born-Jordan Distribution: Reduced interference
• Zhao-Atlas-Marks Distribution: Reduced interference by placing cross-terms
under the self-terms
𝜙 𝜃, 𝜏 =
sin(𝑎𝜃𝜏)
𝑎𝜃𝜏
𝜙š›œ 𝜃, 𝜏 = 𝑔 𝜏 𝜏
sin(𝑎𝜃𝜏)
𝑎𝜃𝜏
Figure from Cohen

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)

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)

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

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Wigner-Ville Distribution: In Perspective of Fault Diagnosis

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