Chapter 4. Health Reasoning for Diagnosis (Vibration based) 4 - Week 4-5... · 2019/1/4 Seoul National University -3. Chapter 4. Health Reasoning for Diagnosis (Vibration based) Continuous

Seoul National University

Chapter 4. Health Reasoning for Diagnosis (Vibration based)

Byeng D. YounSystem Health & Risk Management LaboratoryDepartment of Mechanical & Aerospace EngineeringSeoul National University

Prognostics and Health Management (PHM)


CONTENTS

2019/1/4 - 2 -

Basic Fourier Analysis1Advanced Fourier Analysis: STFT & WT2Time Synchronous Averaging (TSA)3Cepstrum4Case Study5

Seoul National University2019/1/4 - 3 -


Continuous Time Fourier Series (CTFS)Fourier Series• The basic concept of Fourier series is to express periodic signals as a summation of

sinusoidal components.

𝑥𝑥 𝑡𝑡 = 𝑥𝑥 𝑡𝑡 + 𝑇𝑇 = �𝑘𝑘=−∞

+∞

𝑎𝑎𝑘𝑘𝑒𝑒𝑗𝑗𝑘𝑘(2𝜋𝜋/𝑇𝑇)𝑡𝑡 for all 𝑡𝑡

• Calculation of 𝑎𝑎𝑘𝑘

�𝑇𝑇𝑥𝑥 𝑡𝑡 𝑒𝑒−𝑗𝑗𝑗𝑗(2𝜋𝜋/𝑇𝑇)𝑡𝑡𝑑𝑑𝑡𝑡 = �

𝑇𝑇�𝑘𝑘=−∞

+∞

𝑎𝑎𝑘𝑘𝑒𝑒𝑗𝑗𝑘𝑘(2𝜋𝜋/𝑇𝑇)𝑡𝑡𝑒𝑒−𝑗𝑗𝑗𝑗(2𝜋𝜋/𝑇𝑇)𝑡𝑡 𝑑𝑑𝑡𝑡

�𝑇𝑇𝑥𝑥 𝑡𝑡 𝑒𝑒−𝑗𝑗𝑗𝑗(2𝜋𝜋/𝑇𝑇)𝑡𝑡𝑑𝑑𝑡𝑡 = �

𝑘𝑘=−∞

+∞

𝑎𝑎𝑘𝑘𝑇𝑇𝛿𝛿[𝑘𝑘 − 𝑙𝑙] = 𝑇𝑇𝑎𝑎𝑗𝑗

– Therefore,

𝑎𝑎𝑘𝑘 =1𝑇𝑇�𝑇𝑇𝑥𝑥 𝑡𝑡 𝑒𝑒−𝑗𝑗𝑘𝑘(2𝜋𝜋/𝑇𝑇)𝑡𝑡𝑑𝑑𝑡𝑡

𝑒𝑒±𝑗𝑗𝑗𝑗 = cos 𝜃𝜃 ± 𝑗𝑗sin 𝜃𝜃cos(𝜃𝜃)=1

2𝑒𝑒+𝑗𝑗𝑗𝑗 + 𝑒𝑒−𝑗𝑗𝑗𝑗

sin(𝜃𝜃)= 12𝑗𝑗

𝑒𝑒+𝑗𝑗𝑗𝑗 − 𝑒𝑒−𝑗𝑗𝑗𝑗



Continuous Time Fourier Series (CTFS)Example

• Period: T

𝑥𝑥𝑇𝑇 𝑡𝑡 = 𝑥𝑥𝑇𝑇 𝑡𝑡 + 𝑇𝑇 = �𝑘𝑘=−∞

+∞



𝑎𝑎𝑘𝑘 =1𝑇𝑇�−𝑇𝑇/2

𝑇𝑇/2𝑥𝑥 𝑡𝑡 𝑒𝑒−𝑗𝑗𝑘𝑘(2𝜋𝜋/𝑇𝑇)𝑡𝑡𝑑𝑑𝑡𝑡 =

1𝑇𝑇�−𝑆𝑆

𝑆𝑆𝑒𝑒−𝑗𝑗𝑘𝑘(2𝜋𝜋/𝑇𝑇)𝑡𝑡𝑑𝑑𝑡𝑡 =

sin 2𝜋𝜋𝑘𝑘𝑘𝑘𝑇𝑇

𝜋𝜋𝑘𝑘=

2 sin 𝑘𝑘𝑘𝑘0𝑘𝑘𝑘𝑘𝑘𝑘0𝑇𝑇

1






1

1

Motivation• Representation of aperiodic signals• Fourier series by allowing the periodic time to tend to infinity

Example• Let 𝑥𝑥 𝑡𝑡 represent an aperiodic signal

• Periodic extension

𝑥𝑥𝑇𝑇 𝑡𝑡 = �𝑘𝑘=−∞

+∞

𝑥𝑥 𝑡𝑡 + 𝑘𝑘𝑇𝑇 , 𝑥𝑥 𝑡𝑡 = lim𝑇𝑇→∞

𝑥𝑥𝑇𝑇 (𝑡𝑡)

2019/1/4 - 5 -


Continuous Time Fourier Transform (CTFT)



Continuous Time Fourier Transform (CTFT)Fourier Transform

𝑥𝑥 𝑡𝑡 =1

2𝜋𝜋�−∞

∞𝑋𝑋(𝑗𝑗𝑘𝑘)𝑒𝑒𝑗𝑗𝑗𝑗𝑡𝑡𝑑𝑑𝑘𝑘

𝑋𝑋(𝑗𝑗𝑘𝑘) = �−∞

∞𝑥𝑥(𝑡𝑡) 𝑒𝑒−𝑗𝑗𝑗𝑗𝑡𝑡𝑑𝑑𝑡𝑡

Relationship between Fourier Transform and Laplace Transform• Laplace transform (LT): 𝑠𝑠 = 𝜎𝜎 + 𝑗𝑗𝑘𝑘 (complex function of a complex variable)

𝑋𝑋(𝑠𝑠) = �−∞

∞𝑥𝑥(𝑡𝑡) 𝑒𝑒−𝑠𝑠𝑡𝑡𝑑𝑑𝑡𝑡

• Fourier transform (FT) : 𝑠𝑠 = 𝑗𝑗𝑘𝑘 (complex function of an imaginary variable)



– If the region of convergence(ROC) of LT includes the 𝑗𝑗𝑘𝑘 axis, then the FT is equal to LT.







Continuous Time Fourier Transform (CTFT)Duality• To find new transform pairs


2𝜋𝜋�−∞

∞𝑋𝑋(𝑗𝑗𝑘𝑘)𝑒𝑒𝑗𝑗𝑗𝑗𝑡𝑡𝑑𝑑𝑘𝑘 , 𝑋𝑋(𝑗𝑗𝑘𝑘) = �

−∞


𝑥𝑥 𝑡𝑡 𝐹𝐹𝑇𝑇↔ 𝑋𝑋(𝑗𝑗𝑘𝑘)

𝑋𝑋 𝑡𝑡 𝐹𝐹𝑇𝑇↔ 2𝜋𝜋𝑥𝑥(𝑗𝑗(−𝑘𝑘))

• Example𝛿𝛿 𝑡𝑡 − 𝑇𝑇 𝐹𝐹𝑇𝑇

↔ 𝑒𝑒−𝑗𝑗𝑗𝑗𝑇𝑇

𝑒𝑒−𝑗𝑗𝑡𝑡𝑇𝑇 𝐹𝐹𝑇𝑇↔ 2𝜋𝜋𝛿𝛿 𝑘𝑘 + 𝑇𝑇

Relationship between CTFS and CTFT (FT of periodic function)


+∞

𝑎𝑎𝑘𝑘𝑒𝑒𝑗𝑗𝑘𝑘(2𝜋𝜋/𝑇𝑇)𝑡𝑡

𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑒𝑒𝐹𝐹 𝑇𝑇𝐹𝐹𝑎𝑎𝑇𝑇𝑠𝑠𝑇𝑇𝐹𝐹𝐹𝐹𝑇𝑇 𝑋𝑋(𝑗𝑗𝑘𝑘) = ∑𝑘𝑘=−∞+∞ 2𝜋𝜋𝑎𝑎𝑘𝑘𝛿𝛿 𝑘𝑘 − 2𝜋𝜋𝑇𝑇𝑘𝑘 , 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑒𝑒𝐹𝐹 𝑘𝑘𝑒𝑒𝐹𝐹𝐹𝐹𝑒𝑒𝑠𝑠 𝑎𝑎𝑘𝑘

In short, fourier series is for periodic signals and fourier transform is for aperiodic signals. Fourier series is used to decompose signals into basis elements (complex exponentials) while fourier transforms are used to analyze signal in another domain (e.g. from time to frequency, or vice versa).



Discrete Time Fourier Series (DTFS)Discrete Time Fourier Series• The signals in the real world are obtained and processed with the digital equipment and

software. (e.g., NI-DAQ, Matlab, Python…)

• The basic concept of Fourier series is to express signals as a summation of sinusoidal components (or basis).

𝑥𝑥[𝑇𝑇] = 𝑥𝑥[𝑇𝑇 + 𝑁𝑁] = �𝑎𝑎𝑘𝑘𝑒𝑒𝑗𝑗𝑘𝑘 2𝜋𝜋

𝑁𝑁 𝑛𝑛 for all 𝑇𝑇

• Complex exponential is periodic in N.

𝑒𝑒𝑗𝑗𝑘𝑘2𝜋𝜋𝑁𝑁 (𝑛𝑛+𝑁𝑁) = 𝑒𝑒𝑗𝑗𝑘𝑘

2𝜋𝜋𝑁𝑁 𝑛𝑛𝑒𝑒𝑗𝑗𝑘𝑘

2𝜋𝜋𝑁𝑁 𝑁𝑁 = 𝑒𝑒𝑗𝑗𝑘𝑘

2𝜋𝜋𝑁𝑁 𝑛𝑛𝑒𝑒𝑗𝑗𝑘𝑘2𝜋𝜋 = 𝑒𝑒𝑗𝑗𝑘𝑘

2𝜋𝜋𝑁𝑁 𝑛𝑛

– Therefore, Discrete Time Fourier Series are only N distinct complex exponential components

𝑥𝑥[𝑇𝑇] = 𝑥𝑥[𝑇𝑇 + 𝑁𝑁] = �𝑘𝑘=0

𝑁𝑁−1

𝑎𝑎𝑘𝑘𝑒𝑒𝑗𝑗𝑘𝑘 2𝜋𝜋



+∞




Discrete Time Fourier Series (DTFS)Discrete Time Fourier Series• Orthogonal property of periodic complex exponential components

�𝑘𝑘=0

𝑁𝑁−1

𝑒𝑒𝑗𝑗𝑘𝑘2𝜋𝜋𝑁𝑁 𝑛𝑛 = 𝑁𝑁𝛿𝛿[𝑘𝑘]


�𝑛𝑛=0

𝑁𝑁−1

𝑥𝑥 𝑇𝑇 𝑒𝑒−𝑗𝑗𝑗𝑗2𝜋𝜋𝑁𝑁 𝑛𝑛 = �

𝑛𝑛=0

𝑁𝑁−1

�𝑘𝑘=0

𝑁𝑁−1

𝑎𝑎𝑘𝑘𝑒𝑒𝑗𝑗(𝑘𝑘−𝑗𝑗) 2𝜋𝜋

𝑁𝑁 𝑛𝑛

�𝑛𝑛=0

𝑁𝑁−1

𝑥𝑥 𝑇𝑇 𝑒𝑒−𝑗𝑗𝑗𝑗2𝜋𝜋𝑁𝑁 𝑛𝑛 = �

𝑛𝑛=0

𝑁𝑁−1

𝑁𝑁𝑎𝑎𝑘𝑘𝛿𝛿[𝑘𝑘 − 𝑙𝑙] = 𝑁𝑁𝑎𝑎𝑗𝑗

– Therefore,

𝑎𝑎𝑘𝑘 =1𝑁𝑁�𝑛𝑛=0

𝑁𝑁−1

𝑥𝑥 𝑇𝑇 𝑒𝑒−𝑗𝑗𝑘𝑘2𝜋𝜋𝑁𝑁 𝑛𝑛 𝑎𝑎𝑘𝑘 =

1𝑇𝑇�𝑇𝑇𝑥𝑥 𝑡𝑡 𝑒𝑒−𝑗𝑗𝑘𝑘(2𝜋𝜋/𝑇𝑇)𝑡𝑡𝑑𝑑𝑡𝑡



Discrete Time Fourier Series (DTFS)Example

• Period: N

𝑥𝑥𝑁𝑁[𝑇𝑇] = 𝑥𝑥𝑁𝑁[𝑇𝑇 + 𝑁𝑁] = �𝑘𝑘=0

𝑁𝑁−1




𝑎𝑎𝑘𝑘 =1𝑁𝑁

�𝑛𝑛=−𝑁𝑁1

𝑁𝑁1

𝑥𝑥 𝑇𝑇 𝑒𝑒−𝑗𝑗𝑘𝑘2𝜋𝜋𝑁𝑁 𝑛𝑛 = 1 +

1𝑁𝑁�𝑛𝑛=1

𝑁𝑁1

2 cos𝑘𝑘2𝜋𝜋𝑁𝑁

𝑇𝑇 = 1 +1𝑁𝑁�𝑛𝑛=1

𝑁𝑁1

2 cos𝑘𝑘Ω0𝑇𝑇

= 1 +1𝑁𝑁�𝑛𝑛=1

𝑁𝑁1 sin(𝑘𝑘Ω0(𝑇𝑇 + 12))

sin 𝑘𝑘Ω02where Ω0 =

2𝜋𝜋𝑁𝑁

𝑎𝑎𝑘𝑘 =1𝑇𝑇�−𝑇𝑇/2

𝑇𝑇/2𝑥𝑥 𝑡𝑡 𝑒𝑒−𝑗𝑗𝑘𝑘(2𝜋𝜋/𝑇𝑇)𝑡𝑡𝑑𝑑𝑡𝑡 =

2 sin𝑘𝑘𝑘𝑘0𝑘𝑘𝑘𝑘𝑘𝑘0𝑇𝑇



Discrete Time Fourier Transform (DTFT)Motivation• To generalize to aperiodic signals• To consider aperiodic signal as periodic with infinite period

Example• Let 𝑥𝑥[𝑇𝑇] represent an aperiodic signal

• Periodic extension

𝑥𝑥𝑁𝑁[𝑇𝑇] = �𝑘𝑘=−∞

+∞

𝑥𝑥 𝑇𝑇 + 𝑘𝑘𝑁𝑁 , 𝑥𝑥[𝑇𝑇] = lim𝑁𝑁→∞

𝑥𝑥𝑁𝑁 [𝑇𝑇]


+∞

𝑥𝑥 𝑡𝑡 + 𝑘𝑘𝑇𝑇 , 𝑥𝑥 𝑡𝑡 = lim𝑇𝑇→∞

𝑥𝑥𝑇𝑇 (𝑡𝑡)



Discrete Time Fourier Transform (DTFT)Discrete time Fourier Transform

𝑥𝑥 𝑇𝑇 =1

2𝜋𝜋�2𝜋𝜋𝑋𝑋(𝑒𝑒𝑗𝑗Ω)𝑒𝑒𝑗𝑗Ω𝑛𝑛𝑑𝑑Ω

𝑋𝑋(𝑒𝑒𝑗𝑗Ω) = �𝑛𝑛=−∞

∞

𝑥𝑥 𝑇𝑇 𝑒𝑒−𝑗𝑗Ω𝑛𝑛

Relationship between Fourier Transform and z Transform• Z transform (complex function of a complex variable)

𝑋𝑋(𝑧𝑧) = �𝑛𝑛=−∞

∞

𝑥𝑥 𝑇𝑇 𝑧𝑧−𝑛𝑛

• Discrete time Fourier transform (complex function of a unit circle in complex domain)


∞


– If the region of convergence(ROC) of z transform includes the unit circle, then the Discrete time Fourier transform is equal to z transform.


2𝜋𝜋�−∞

∞𝑋𝑋(𝑗𝑗𝑘𝑘)𝑒𝑒𝑗𝑗𝑗𝑗𝑡𝑡𝑑𝑑𝑘𝑘



𝑥𝑥[𝑇𝑇] = 𝑥𝑥[𝑇𝑇 + 𝑁𝑁] = �𝑘𝑘=0

𝑁𝑁−1


𝑁𝑁 𝑛𝑛



Relations among Fourier RepresentationsContinuous time Fourier Series


+∞

𝑎𝑎𝑘𝑘𝑒𝑒𝑗𝑗𝑘𝑘(2𝜋𝜋/𝑇𝑇)𝑡𝑡 , 𝑎𝑎𝑘𝑘 =1𝑇𝑇�𝑇𝑇𝑥𝑥 𝑡𝑡 𝑒𝑒−𝑗𝑗𝑘𝑘(2𝜋𝜋/𝑇𝑇)𝑡𝑡𝑑𝑑𝑡𝑡

Continuous time Fourier Transform


2𝜋𝜋�−∞

∞𝑋𝑋(𝑗𝑗𝑘𝑘)𝑒𝑒𝑗𝑗𝑗𝑗𝑡𝑡𝑑𝑑𝑘𝑘 , 𝑋𝑋(𝑗𝑗𝑘𝑘) = �

−∞


Discrete time Fourier Series

𝑥𝑥 𝑇𝑇 = 𝑥𝑥 𝑇𝑇 + 𝑁𝑁 = �𝑘𝑘=0

𝑁𝑁−1


𝑁𝑁 𝑛𝑛 , 𝑎𝑎𝑘𝑘 =1𝑁𝑁�𝑛𝑛=0

𝑁𝑁−1

𝑥𝑥 𝑇𝑇 𝑒𝑒−𝑗𝑗𝑘𝑘2𝜋𝜋𝑁𝑁 𝑛𝑛

Discrete time Fourier Transform

𝑥𝑥 𝑇𝑇 =1

2𝜋𝜋�2𝜋𝜋𝑋𝑋(𝑒𝑒𝑗𝑗Ω)𝑒𝑒𝑗𝑗Ω𝑛𝑛𝑑𝑑Ω , 𝑋𝑋(𝑒𝑒𝑗𝑗Ω) = �

𝑛𝑛=−∞

∞




Relations among Fourier RepresentationsSampling• Continuous time signal to discrete time signal

𝑥𝑥 𝑇𝑇 = 𝑥𝑥(𝑇𝑇𝑇𝑇)– Sampling frequency 1/T

• Signal generation with an impulse train

𝑥𝑥𝑝𝑝 𝑡𝑡 = 𝑥𝑥 𝑡𝑡 𝑝𝑝 𝑡𝑡 where 𝑝𝑝 𝑡𝑡 = �𝑘𝑘=−∞

+∞

𝛿𝛿 𝑡𝑡 − 𝑘𝑘𝑇𝑇

• Fourier transform of an impulse train

– (Fourier Series) 𝑎𝑎𝑘𝑘 = 1𝑇𝑇 ∫−𝑇𝑇/2

𝑇𝑇/2 𝑝𝑝 𝑡𝑡 𝑒𝑒−𝑗𝑗𝑘𝑘(2𝜋𝜋/𝑇𝑇)𝑡𝑡𝑑𝑑𝑡𝑡 = 1𝑇𝑇

– (Fourier Transform) 𝑃𝑃 𝑗𝑗𝑘𝑘 = ∑𝑘𝑘=−∞+∞ 2𝜋𝜋𝑎𝑎𝑘𝑘𝛿𝛿 𝑘𝑘 − 2𝜋𝜋𝑇𝑇𝑘𝑘 = ∑𝑘𝑘=−∞+∞ 2𝜋𝜋

𝑇𝑇𝛿𝛿 𝑘𝑘 − 2𝜋𝜋

𝑇𝑇𝑘𝑘

1

0 𝑇𝑇 2𝑇𝑇−2𝑇𝑇 −𝑇𝑇… …

……



Relations among Fourier RepresentationsRelationship between CTFT and DTFT

• Continuous time Fourier Transform (CTFT) of 𝑥𝑥𝑝𝑝 𝑡𝑡

𝑋𝑋𝑝𝑝 𝑗𝑗𝜔𝜔 = �−∞

∞𝑥𝑥𝑝𝑝 𝑡𝑡 𝑒𝑒−𝑗𝑗𝜔𝜔𝑡𝑡𝑑𝑑𝑡𝑡 = �

−∞

∞𝑥𝑥 𝑡𝑡 �

𝑛𝑛=−∞

+∞

𝛿𝛿 𝑡𝑡 − 𝑇𝑇𝑇𝑇 𝑒𝑒−𝑗𝑗𝜔𝜔𝑡𝑡𝑑𝑑𝑡𝑡

= �−∞

∞�𝑛𝑛=−∞

+∞

𝑥𝑥[𝑇𝑇] 𝛿𝛿 𝑡𝑡 − 𝑇𝑇𝑇𝑇 𝑒𝑒−𝑗𝑗𝜔𝜔𝑡𝑡𝑑𝑑𝑡𝑡 = �𝑛𝑛=−∞

+∞

𝑥𝑥[𝑇𝑇]𝑒𝑒−𝑗𝑗𝜔𝜔𝑇𝑇𝑛𝑛

• Discrete time Fourier Transform (DTFT) of 𝑥𝑥[𝑇𝑇]


∞


– Therefore,𝑋𝑋𝑝𝑝 𝑗𝑗𝜔𝜔 = 𝑋𝑋 𝑒𝑒𝑗𝑗Ω |Ω = 𝜔𝜔T

Ω = 𝜔𝜔T



Relations among Fourier RepresentationsRelationship between CTFS and CTFT (FT of periodic function)

• Assume 𝑥𝑥 𝑡𝑡 is aperiodic signal in finite duration. (CTFT)



• Convolution with an impulse train → Periodic signal (Periodic extension) (CTFS)


+∞

𝑥𝑥 𝑡𝑡 − 𝑘𝑘𝑇𝑇 = 𝑥𝑥 𝑡𝑡 ∗ 𝑝𝑝 𝑡𝑡 𝑘𝑘𝑤𝑒𝑒𝐹𝐹𝑒𝑒 𝑝𝑝 𝑡𝑡 = �𝑘𝑘=−∞

+∞

𝛿𝛿 𝑡𝑡 − 𝑘𝑘𝑇𝑇

𝑋𝑋𝑇𝑇 𝑗𝑗𝑘𝑘 = 𝑋𝑋 𝑗𝑗𝑘𝑘 𝑃𝑃 𝑗𝑗𝑘𝑘 = 𝑋𝑋 𝑗𝑗𝑘𝑘 �𝑘𝑘=−∞

+∞2𝜋𝜋𝑇𝑇𝛿𝛿 𝑘𝑘 −

2𝜋𝜋𝑇𝑇𝑘𝑘

– Sampling in frequency domain

• Therefore, CTFS of periodic signal is equal to sampling of CTFT of aperiodic signal.

𝑋𝑋𝑇𝑇 𝑗𝑗𝑘𝑘 = 𝑋𝑋 𝑗𝑗𝑘𝑘 𝑃𝑃 𝑗𝑗𝑘𝑘= 𝑋𝑋 𝑗𝑗𝑘𝑘 ∑𝑘𝑘=−∞+∞ 2𝜋𝜋


𝑇𝑇𝑘𝑘


Example

• 𝑥𝑥 𝑡𝑡 is aperiodic signal → Fourier Transform

𝑋𝑋 𝑗𝑗𝑘𝑘 = �−∞

∞𝑥𝑥 𝑡𝑡 𝑒𝑒−𝑗𝑗𝑗𝑗𝑡𝑡𝑑𝑑𝑡𝑡 =

sin(𝑘𝑘/2)𝑘𝑘/2

• How about periodic signal with period 4?– Periodic Extension

2019/1/4 - 17 -


Relations among Fourier Representations

𝑥𝑥 𝑡𝑡 = �1 (−12≤ 𝑡𝑡 ≤

12

)

0 (𝐹𝐹𝑡𝑡𝑤𝑒𝑒𝐹𝐹𝑘𝑘𝐹𝐹𝑠𝑠𝑒𝑒)

𝑥𝑥(𝑡𝑡)

𝑡𝑡

1

−12 +

12

𝑎𝑎𝑘𝑘 =1𝑇𝑇�−𝑆𝑆

𝑆𝑆𝑒𝑒−𝑗𝑗𝑘𝑘(2𝜋𝜋/𝑇𝑇)𝑡𝑡𝑑𝑑𝑡𝑡 =

sin 2𝜋𝜋𝑘𝑘𝑘𝑘𝑇𝑇

𝜋𝜋𝑘𝑘=

2 sin𝑘𝑘𝑘𝑘0𝑘𝑘𝑘𝑘𝑘𝑘0𝑇𝑇


Example (cont.)• 𝑥𝑥4 𝑡𝑡 = 𝑥𝑥4 𝑡𝑡 + 4 is periodic signal.• Calculation of Fourier Series from Fourier Transform of 𝑥𝑥(𝑡𝑡)

𝑋𝑋4 𝑗𝑗𝑘𝑘 = 𝑋𝑋 𝑗𝑗𝑘𝑘 𝑃𝑃 𝑗𝑗𝑘𝑘 = �𝑘𝑘=−∞

+∞sin(𝑘𝑘/2)𝑘𝑘/2

2𝜋𝜋4𝛿𝛿 𝑘𝑘 −

2𝜋𝜋4𝑘𝑘

𝑋𝑋4 𝑗𝑗𝑘𝑘 = �𝑘𝑘=−∞

+∞

2𝜋𝜋𝑎𝑎𝑘𝑘𝛿𝛿 𝑘𝑘 −2𝜋𝜋4𝑘𝑘

– Therefore,

𝑎𝑎𝑘𝑘 =14

sin(𝑘𝑘/2)𝑘𝑘/2

|w =2𝜋𝜋

4𝑘𝑘 =

sin𝜋𝜋𝑘𝑘4𝜋𝜋𝑘𝑘

2019/1/4 - 18 -



𝑥𝑥4(𝑡𝑡)

−12

+12

−4 +4

1

𝑋𝑋𝑇𝑇 𝑗𝑗𝑘𝑘 = 𝑋𝑋 𝑗𝑗𝑘𝑘 𝑃𝑃 𝑗𝑗𝑘𝑘= 𝑋𝑋 𝑗𝑗𝑘𝑘 ∑𝑘𝑘=−∞+∞ 2𝜋𝜋


𝑇𝑇𝑘𝑘


Example (cont.)• Relationship in frequency domain

2019/1/4 - 19 -



𝑋𝑋 𝑗𝑗𝑘𝑘

𝑋𝑋4 𝑗𝑗𝑘𝑘

𝑎𝑎𝑘𝑘

𝑘𝑘

𝑘𝑘

𝑘𝑘

1

𝜋𝜋/2

1/4


Example (cont.)• Sampling in time domain

– Continuous time signals to Discrete time signals

– Sampling period 𝑇𝑇 = 12

, 𝑥𝑥𝑠𝑠 𝑇𝑇 = 𝑥𝑥 12𝑇𝑇

• DTFT of 𝑥𝑥𝑠𝑠 𝑇𝑇

𝑋𝑋𝑠𝑠 𝑒𝑒𝑗𝑗Ω = �𝑛𝑛=−∞

∞

𝑥𝑥𝑠𝑠 𝑇𝑇 𝑒𝑒−𝑗𝑗Ω𝑛𝑛 = 1 + 2 cosΩ

2019/1/4 - 20 -



𝑥𝑥(𝑡𝑡)

𝑡𝑡

1

−12 +

12

𝑥𝑥𝑠𝑠 𝑇𝑇

𝑇𝑇0

1

1 2 …






Example (cont.)• Calculation of CTFT from DTFT

𝑥𝑥𝑝𝑝 𝑡𝑡 = 𝑥𝑥 𝑡𝑡 𝑝𝑝 𝑡𝑡 𝑘𝑘𝑤𝑒𝑒𝐹𝐹𝑒𝑒 𝑝𝑝 𝑡𝑡 = �𝑘𝑘=−∞

+∞

𝛿𝛿 𝑡𝑡 −12𝑘𝑘

𝑋𝑋𝑝𝑝 𝑗𝑗𝑘𝑘 = 𝑋𝑋 𝑗𝑗𝑘𝑘 ∗ �𝑘𝑘=−∞

+∞

4𝜋𝜋𝛿𝛿 𝑘𝑘 − 4𝜋𝜋𝑘𝑘

𝑋𝑋𝑝𝑝 𝑗𝑗𝑘𝑘 = �−∞

∞𝑥𝑥𝑝𝑝 𝑡𝑡 𝑒𝑒−𝑗𝑗𝑗𝑗𝑡𝑡𝑑𝑑𝑡𝑡 = �

𝑛𝑛=−∞

+∞

𝑥𝑥[𝑇𝑇]𝑒𝑒−𝑗𝑗12𝑗𝑗𝑛𝑛 = 1 + 2 cos

12𝑘𝑘

𝑋𝑋𝑝𝑝 𝑗𝑗𝑘𝑘 = 𝑋𝑋 𝑒𝑒𝑗𝑗Ω |Ω =1

2𝑘𝑘

2019/1/4 - 21 -



𝑥𝑥𝑠𝑠 𝑇𝑇

𝑇𝑇0

1

1 2 …

𝑥𝑥𝑝𝑝 𝑡𝑡

𝑡𝑡0

1

1/2


Example (cont.)• Relationship in frequency domain

2019/1/4 - 22 -



𝑋𝑋 𝑗𝑗𝑘𝑘

𝑘𝑘

𝑘𝑘

1

Ω

3 𝑋𝑋 𝑒𝑒𝑗𝑗Ω

4𝜋𝜋 𝑋𝑋𝑝𝑝 𝑗𝑗𝑘𝑘

−4𝜋𝜋 4𝜋𝜋

−2𝜋𝜋 2𝜋𝜋




Continuous time Fourier Series

(CTFS)

Continuous time Fourier transform

(CTFT)

Discrete time Fourier Series

(DTFS)

Discrete time Fourier transform

(DTFT)

Period 𝑇𝑇 → ∞

Periodic Extension(Sampling in frequency)

Samplingin time

Samplingin time

Period 𝑇𝑇 → ∞

Periodic Extension(Sampling in frequency)


Motivation• FFT(Fast Fourier Transform) does not provide time information in frequency domain.

Therefore, it is impossible to tell when a particular event has taken place.• Signals contain numerous non-stationary or transitory characteristics: drift, trends, abrupt

changes, and beginnings and ends of events.

• In 1946, Gabor adapted Fourier Transform to analyze only a small section of the signal at a time.

2019/1/4 - 24 -


Short-time Fourier Transform (STFT)



Short-time Fourier Transform (STFT)Concept• The basic idea of STFT is to break up the signal into small time segments and apply

Fourier analyse at each time segment to determine the existing frequencies. • The total of such spectra indicates how the spectrum is varying with time.

Steps of STFTStep 1) Choose a window function of finite lengthStep 2) Compute the FT of the truncated signal by the window functionStep 3) Incrementally slide the window to the rightStep 4) Go to step 2 until window reaches the end of the signal

Time Domain

Time (s)Frequency (Hz)

Short-time Fourier Transform



Short-time Fourier Transform (STFT)Definition• STFT of 𝑇𝑇(𝑡𝑡): computed for each window centred at 𝑡𝑡.

• One can define an energy density, called the spectrogram:

𝑘𝑘 𝑡𝑡,𝑘𝑘 = �𝑇𝑇 τ 𝑔𝑔 τ − 𝑡𝑡 𝑒𝑒_𝑗𝑗𝑘𝑘𝑡𝑡𝑑𝑑𝑑𝑑τ

𝑔𝑔 τ ---Window function. It is real, symmetric and normalize with 𝑔𝑔 τ = 1. It istranslated by 𝑡𝑡 and modulated by frequency 𝑘𝑘;𝑇𝑇(τ) --- Signal to be analyzed;𝑡𝑡 --- Time parameter;𝑘𝑘 --- Frequency parameter;

𝑃𝑃𝑘𝑘 𝑡𝑡,𝑘𝑘 = |𝑘𝑘 𝑡𝑡,𝑘𝑘 |2 = |�𝑇𝑇 τ 𝑔𝑔 τ − 𝑡𝑡 𝑒𝑒_𝑗𝑗𝑘𝑘𝑡𝑡𝑑𝑑𝑑𝑑τ |2

It measures the energy in the time-frequency neighborhood of 𝑡𝑡,𝑘𝑘 .


Uncertainty Principle• Since ||𝑔𝑔(τ)||= ʃ|𝑔𝑔(τ-t)|2dτ=1, we can interpret |𝑔𝑔(τ)|2 as probability distribution.

• The size of the box σλσ𝑡𝑡 is independent. In other words, STFT has the same resolution across the time-frequency plane.

• For good time resolution, the window must be narrow in time domain. But, also, for good frequency resolution, the bandwidth should be small. From uncertainty principle, we cannot get good resolutions both in time and frequency at the same time.

2019/1/4 - 27 -


Short-time Fourier Transform (STFT)

σ𝑡𝑡2 = � τ − 𝑡𝑡 2|𝑔𝑔(τ)|2𝑑𝑑τ

The bandwidth of window function:

where 𝐺𝐺(λ) is Fourier Transform of 𝑔𝑔(τ)

σλ2 =1

2𝜋𝜋� λ − 𝑘𝑘 2|𝐺𝐺(λ)|2𝑑𝑑λ

The time-width of window function:

The uncertainty principle for the window is:

σλσ𝑡𝑡 ≥ 1/4π



Short-time Fourier Transform (STFT)Window Function• Window should be narrow enough to ensure that the portion of the signal falling within

the window is stationary.• However, extremely narrow windows do not offer good localization in the frequency

domain.

Wide window: Good frequency resolution, poor time resolution.Narrow window: Good time resolution, poor frequency resolution.

𝑔𝑔(t)=1: STFT turns into FTProviding excellent frequency localization, but no time localization.

𝑔𝑔(t)= δ(t)*: STFT becomes time domain.Providing excellent time localization but no frequency localization.

δ(t)*: Dirac delta function



Short-time Fourier Transform (STFT)Window Function: Example• Different time-frequency resolutions for the same signal due to different window sizes

were used in generating the STFT.

50Hz [0~0.2]s

75Hz [0.2~0.4]s

100Hz [0.4~0.6]s

125Hz [0.6~1.0]s

window length = 101

window length = 181window length = 151



Wavelet Transform (WT)Motivation• STFT (Short-time Fourier Transform) can transfer the signals from time domain to time-

frequency domain. But, the window function and resolution is fixed all the time. So it could not satisfy the demand of non-stationary signals.

• Narrower windows are more appropriate at high frequencies, but wider windows are more appropriate at low frequencies.

• In 1980s, Grossman and Morlet proposed the wavelet transform (WT) to deal with the problem.

Time Domain

Time (s)Frequency (Hz)

Short-time Fourier Transform



Wavelet Transform (WT)Concept• A wavelet is a waveform of effectively limited duration that has an average value of zero.

• Basis functions of WT are the wavelets which can be obtained using scaling and translation of a scaling function and wavelet function:

𝜓𝜓𝑠𝑠, 𝜏𝜏𝑡𝑡 =

1𝑠𝑠𝜓𝜓(𝑡𝑡 − 𝜏𝜏𝑠𝑠 )

𝑠𝑠---scale. Large s means long wavelength𝜏𝜏---translation

Signal

Wavelet



Wavelet Transform (WT)Definition• Wavelet Transform of 𝑇𝑇(𝑡𝑡):

𝛾𝛾 𝑠𝑠, 𝜏𝜏 = �𝑇𝑇 𝑡𝑡 𝜓𝜓𝑠𝑠, 𝜏𝜏𝑡𝑡 𝑑𝑑𝑡𝑡

𝜓𝜓𝑠𝑠, 𝜏𝜏𝑡𝑡 =

1𝑠𝑠𝜓𝜓(𝑡𝑡 − 𝜏𝜏𝑠𝑠

)

where:

𝑇𝑇(𝑡𝑡) --- time series𝛾𝛾(𝑠𝑠,𝜏𝜏)--- coefficients of wavelets with scales, 𝑠𝑠 and translation, 𝜏𝜏𝜓𝜓 --- mother wavelet



Wavelet Transform (WT)Wavelets• Commonly used wavelet functions:

Admissibility condition on wavelets• Admissibility constant

where Ψ 𝜔𝜔 is Fourier transform of mother wavelet, 𝜓𝜓• Sufficient decay for well localized function

where Ψ 𝜔𝜔 is Fourier transform of mother wavelet, 𝜓𝜓

𝐶𝐶𝜓𝜓 = �−∞

∞ Ψ 𝜔𝜔𝜔𝜔

𝑑𝑑𝜔𝜔

Ψ 0 = �−∞

∞𝜓𝜓 𝑡𝑡 𝑑𝑑𝑡𝑡 = 0

WT has the property for the effective localization in both time and frequency domain contaray to the Fourier transform, which decomposes the signal in terms of sines and cosines, i.e., infinite duration waves.



Summary: STFT & WTShort-time Fourier Transform (STFT)• STFT is a kind of linear transform. It provides time information and frequency

information at the same time. • But the window function and window size in STFT method is fixed. The resolutions in

time and frequency can not be changed. It could not offer good localizations both in time and frequency.

Wavelet Transform (WT)• The window function and window size in WT are flexible. The resolutions in time and

frequency can be changed. It could offer good localizations both in time and frequency.• It is a little difficult to choose a suitable wavelet basis function.

STFT WT


Objectives• To isolate vibration from random noise and various sources of vibrationMethod• To resample and interpolate the vibration signal equal sample size per rotation• To divide signal into several groups length corresponding to one rotation• To average them reduced Gaussian noise

2019/1/4 - 35 -


Time synchronous average technique (TSA)

1st cycle 2nd cycle Nth cycle

TSA signal

Resa

mpl

ing

…

Divi

ding

Aver

agin

g

Noise

Noise

1st cycle 2nd cycle Nth cycle

…


ResamplingDefinition• Resampling means combining interpolation and decimation to change the sampling rate

by a rational factor.– Practically, it is usually used to interface two systems having different sampling rates.– For example, in audio signal, professional audio equipment and consumer’s have

different sampling rate. Therefore, it needs to change sampling rate.

Resampling for non-stationary signals• In the view of rotational system, there are lots of cases that angular velocity has some

fluctuation or is non-stationary. – Of course, there are lots of techniques using time-frequency analysis (E.g., STFT,

Wigner-Ville, Wavelet, …).– However, Time-Frequency analysis have some restriction (e.g., Uncertainty principle)

and difficulty. So synchronous sampling is used to make signal stationary in angular domain.



Synchronous sampling(resampling)

Principle• Using encoder signal, we can estimate the average angular velocity at each cycle. • From that, original signal in time domain can be converted to resampled signal in

angular domain.• It is easier to understand system characteristics because non-stationary behaviors of the

system can be minimized.

FFT


Resampling


Example


Resampling

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7Frequency analysis

Frequency

|Y(f)

|

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5Order analysis

Order

|Y(f)

|

Time [sec]

0

0

0

0

𝜽𝜽𝟒𝟒

𝜽𝜽𝟏𝟏

𝜽𝜽𝟑𝟑

𝒕𝒕𝟐𝟐 𝒕𝒕𝟑𝟑𝒕𝒕𝟏𝟏 𝒕𝒕𝟒𝟒

𝜽𝜽𝟐𝟐

Cum

ulat

ive

Deg

ree

0 100 200 300-1

-0.5

0

0.5

1

Sample Number

Ampl

itude

1 2 3 4 …

Sample Points

Vibr

atio

n

Rotation of the gearRaw Signal (𝑇𝑇𝑠𝑠: 𝑐𝑐𝐹𝐹𝑇𝑇𝑠𝑠𝑡𝑡.)

Time [sec]

0

0

0

0

𝜽𝜽𝟒𝟒∗

𝜽𝜽𝟏𝟏∗

𝜽𝜽𝟑𝟑∗

𝒕𝒕𝟐𝟐∗ 𝒕𝒕𝟑𝟑∗𝒕𝒕𝟏𝟏∗ 𝒕𝒕𝟒𝟒∗

𝜽𝜽𝟐𝟐∗

Cum

ulat

ive

Deg

ree

0 50 100 150 200 250-1

-0.5

0

0.5

1

Rotation of the gear1 2 3 4 …

Sample Points

Vibr

atio

n

After resampling


Basic principle of interpolation filter• Necessity

– We set new time points for getting resampled signals.– Practically, we can’t get every point matched with defined point (by resampling).

Therefore, we need interpolation.

• Application as filter– The process of interpolation is fitting a continuous function (e.g., linear, spline, …) to

the discrete points in the digital signal.


Interpolation


Basic principle of interpolation filter• Calculation

– The interpolation function is shifted so that its peak is aligned with the position of the resampled point.

– Then, the value of the resampled point is equal to the sum of the values of the original function scaled by the corresponding values of the interpolation function.


Interpolation


Types of interpolation function

Nearest neighbor

Linear

Cubic B-spline

Interpolation function Magnitude ofFourier transform

Logarithm of magnitude ofFourier transform


Interpolation


Types of interpolation function

• Properties– As learned before section, we can also regard interpolation function as filtering

system. – Generally, it is preferred to choose a cubic spline method considering the property of

frequency domain on each interpolation function.


Interpolation

High resolutioncubic spline(𝜶𝜶 = −𝟏𝟏)

High resolutioncubic spline(𝜶𝜶 = −𝟎𝟎.𝟓𝟓)

Interpolation function Magnitude ofFourier transform

Logarithm of magnitude ofFourier transform


Averaging

• When the signals are resampled using a trigger, we can average the synchronized signals (Synchronous averaging)

• The magnitudes of noises are reduced as the number of averaging gets larger.

1st average

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-5

0

5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-5

0

5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-5

0

5

10th average

100th average




CepstrumDefinition• In 1963, Bogert, Healy, and Tukey proposed the concept of “Cepstrum [/ˈkɛpstrʌəm/,

/ˈsɛpstrʌəm/)]” by paraphrasing Spectrum , and cepstrum can be defined as the ‘Fourier transform of the logarithm of the Fourier transform’.

• The complex cepstrum 𝐶𝐶𝑐𝑐 𝜏𝜏 of time domain signals 𝑇𝑇 𝑡𝑡 can be formulated as

where ℱ[�] is the Fourier transform, ℱ−1[�] is the inverse Fourier transform, and 𝜏𝜏 is referred to as “Quefrency” by paraphrasing “Frequency”.(The forward and inverse Fourier transforms give the same results while being a difference in scaling factor)

• As the real signals indicate signals with zero phase, the real cepstrum can be defined as

ℱ 𝑇𝑇 𝑡𝑡 = 𝐴𝐴 𝑇𝑇 + 𝑗𝑗𝜙𝜙 𝑇𝑇

where 𝐴𝐴 𝑇𝑇 and 𝜙𝜙 𝑇𝑇 are amplitude and phase of the spectrum[2].

𝐶𝐶𝑐𝑐 𝜏𝜏 = ℱ−1{log ℱ(𝑇𝑇(𝑡𝑡)) }

𝐶𝐶𝑟𝑟 𝜏𝜏 = �ℱ−1 log 𝐴𝐴 𝑇𝑇 + 𝑗𝑗𝜙𝜙 𝑇𝑇𝜙𝜙 𝑓𝑓 =0

= ℱ−1 log 𝐴𝐴 𝑇𝑇



CepstrumTerminology• Commonly used terms in cepstral domain are as follows:

– Quefrency from Frequency– Rahmonics from Harmonics– Lifter from Filter

Log magnitude of Fourier transform

Frequency [kHz]

Fundamental “Period” = 80Hz

Quefrency [ms]

Cepstral domain

First rahmonic peak at 12.5

– Gamnitude from Magnitude– Saphe from Phase



CepstrumApplication of Cepstrum: (1) Speech Analysis• The human speech (=𝑣𝑣(𝑡𝑡)) can be modelled as the convolution of vocal tract (𝑎𝑎(𝑡𝑡)) and

glottal excitation (𝑤(𝑡𝑡)).• Therefore, the cepstrum can be used for speech analysis to separate convoluted signal

mixtures. • The convolutive mixtures are deconvoluted in the cepstral domain as an additive form.

𝑣𝑣 𝑡𝑡 = 𝑎𝑎 𝑡𝑡 ∗ 𝑤(𝑡𝑡)

𝑉𝑉 𝑇𝑇 = 𝐴𝐴 𝑇𝑇 � 𝐻𝐻 𝑇𝑇

log 𝑉𝑉 𝑇𝑇 = log 𝐴𝐴 𝑇𝑇 + log 𝐻𝐻 𝑇𝑇

ℱ−1 log 𝑉𝑉 𝑇𝑇 = ℱ−1 log 𝐴𝐴 𝑇𝑇

Fourier transform

Logarithm

Inverse Fourier transform

Vocal tract+ℱ−1 log 𝐻𝐻 𝑇𝑇 Glottal excitation

𝑣𝑣𝑡𝑡

𝑉𝑉𝑇𝑇

log𝑉𝑉𝑇𝑇

ℱ−1

log𝑉𝑉𝑇𝑇



CepstrumApplication of Cepstrum: (2) Machine diagnostics• Faults of mechanical components (e.g., motor and gear) are often characterized by side-

bands in the frequency domain signals. • As the sidebands are usually smaller than main harmonics, the logarithmic scale of the

cepstrum could provide a better way to visualize them.

CepstrumSpectrum



Case Study: Motor-driven gearbox in a pulverizer (1/3)

[8]

Problem description• Pulverizer: A grinding system of coil for boiler in steam generating power cycle• A motor driven gearbox is connected to the grinding system.• Exposed to external noise and disturbance due to harsh operating conditions• Need to find a fault feature which is robust to external noises and disturbance.



Technique–FFT• FFT was performed to observe spectral behaviors of acceleration signals• Used acceleration signals measured from a high speed shaft.

(Normal idle, Normal loading, Failure 1, and Failure 2).• From the FFT results, we could observe periodic spectral behaviors in the failure

cases.Spectrum

0 200 400 600 800 1000 12000

0.10.2

Amp.

Normal: Idle

0 200 400 600 800 1000 12000

0.10.2

Amp.

Normal: Loading

0 200 400 600 800 1000 12000

0.10.2

Amp.

Failure 1

0 200 400 600 800 1000 1200

Freq. (Hz)

00.10.2

Amp.

Failure 2

Acc. data at the high speed shaft




Technique–Cepstrum• The cepstrum technique was used to quantify periodic spectral behaviors.• The quefrency values at the rotating speed of high speed shaft increased for the failure

cases.

Log spectrum Cepstrum

0 0.1 0.2 0.3 0.4 0.50

0.5

Amp.

Normal: Idle

0 0.1 0.2 0.3 0.4 0.50

0.5

Amp.

Normal: Loading

0 0.1 0.2 0.3 0.4 0.50

0.5Am

p.

Failure 1

0 0.1 0.2 0.3 0.4 0.5

Quef. (s)

0

0.5

Amp.

Failure 2

0.079

0.096

0.167

0.286

0 200 400 600 800 1000 1200-15-10

-50

Log(

Amp.

)

Normal: Idle

0 200 400 600 800 1000 1200-15-10

-50

Log(

Amp.

)

Normal: Loading

0 200 400 600 800 1000 1200-15-10

-50

Log(

Amp.

)

Failure 1

0 200 400 600 800 1000 1200

Freq. (Hz)

-15-10

-50

Log(

Amp.

)

Failure 2

Health Data




Case Study: Planetary gearbox in an excavator (1/3) Problem description• Excavator: A grinding system of coil for boiler in steam generating power cycle• A swing device in the excavator suffers from severe loading conditions.• Lots of noise exists from other components like a motor and a pump.• Failure mode: Surface damage in the sun gear at the 2nd stage.



Case Study: Planetary gearbox in an excavator (2/3) Technique–STFT• STFT was performed to observe spectral behaviors of each vibration source

along the time.• Lots of vibration sources were observed along with gear vibration signals.

646.9

431

215.6192.6

385.2

577.8

770.3

PumpOn

Gear rotatestart

CW

1st GMF

2nd GMF

Vibration source identification for a pump

Vibration source identification for a gear

644

322

Gear rotating freq.

Swing motor rotating freq.

Pump rotating freq.

Freq

uenc

y(k

Hz)

Time (Mins)



Case Study: Planetary gearbox in an excavator (3/3) Technique–AR-MED filter• AR-MED filter was used to remove noises from other vibration sources.• Filter parameters were optimized based on the gear rotating frequency.• AR-MED filter could maximize the fault features in the fault signals.

Level 3 (D 2.5mm)

Level 2 (D 1.0mm)

Level 1 (D 0.5mm)

FaultNormal

AR-MED filter


THANK YOUFOR LISTENING

2019/1/4 - 54 -


Reference

[1] Kreyszig, E., Advanced Engineering Mathematics, WILEY, 2011.[2] Kim, N., et al., Prognostics and Health Management of Engineering Systems, Springer,

2017.[3] Oppenheim,A.V., et al., Signal & Systems, Prentice-Hall, 1997.[4] Shin, K., et al., Fundamentals of signal processing for sound and vibration engineers,

Wiley, 2008.[5] Randall,R.B., Vibration-based Condition Monitoring, Wiley, 2011.[6] https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-003-

signals-and-systems-fall-2011/lecture-notes/[7] D. Lee Fugal. (2010). Conceptual Wavelets In Digital Signal Processing. Space &

Signals Technologies LLC.[8] Mehala, N., & Dahiya, R. (2008, December). A comparative study of FFT, STFT and

wavelet techniques for induction machine fault diagnostic analysis. In Proceedings ofthe 7th WSEAS International Conference on Computational Intelligence, Man-Machine Systems and Cybernetics, Cairo, Egypt (Vol. 2931).

[9] Debnath, L. (2012). Wavelet transforms and time-frequency signal analysis. SpringerScience & Business Media.

[10] B.P. Bogert, M.J.R. Healy, and J.W. Tukey, “The quefrency alanysis of time series forechoes: Cepstrum, pseudo-autocovariance, cross-cepstrum, and saphe cracking,” in TimeSeries Analysis, M. Rosenblatt, Ed., 1963, ch.15,pp. 209–243.


Reference

[11] Randall, Robert B. "A history of cepstrum analysis and its application to mechanical problems." Mechanical Systems and Signal Processing 97 (2017): 3-19.

[12] Norton, Michael Peter, and Denis G. Karczub. Fundamentals of noise and vibration analysis for engineers. Cambridge university press, 2003.

[13] Oppenheim, Alan V., and Ronald W. Schafer. "From frequency to quefrency: A history of the cepstrum." IEEE signal processing Magazine 21.5 (2004): 95-106.

[14] iitg.vlab.co.in,. (2011). Cepstral Analysis of Speech. Retrieved 2 February 2018, from iitg.vlab.co.in/?sub=59&brch=164&sim=615&cnt=1

[15] Liang, Bo, S. D. Iwnicki, and Yunshi Zhao. "Application of power spectrum, cepstrum, higher order spectrum and neural network analyses for induction motor fault diagnosis." Mechanical Systems and Signal Processing 39.1-2 (2013): 342-360.

[16] Park, Jungho, et al. " Failure prediction of a motor-driven gearbox in a pulverizerunder external noise and disturbance." SMART STRUCTURES AND SYSTEMS 22.2 (2018): 185-192

[17] 미분기 Pulverizer, from http://blog.naver.com/yunggu7410/220591847988

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