Parametric Time-frequency Analysis (TFA)

Yang YangShanghai Jiao Tong University

August, 2015

OUTLINE

Background

Theory and methods

Applications

Vibration signals

Radar signals

Bioelectric signal

Electronic signal and

speech

Seismic data

and guided waves

Non-stationary

signals

Non-stationary signals

Fourier transform-global transform

Stationary signal Non-stationary signal

Tim

e d

om

ain

Fre

qu

ency

do

mai

n

2 4 6 8 10-1

-0.5

0

0.5

1

Time/Sec

Am

p

0 5 10 15

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Freq / Hz

Am

p

0 1 2 3 4-1

-0.5

0

0.5

1

Time/Sec

Am

p

0 10 20 30 40

0.05

0.1

0.15

0.2

0.25

Freq / Hz

Am

p

Piece-wised stationary signal

1 1.5 2 2.5 3 3.5 4

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Time/Sec

Am

p0 5 10 15 20

0.05

0.1

0.15

0.2

0.25

Freq / HzA

mp

Non-stationary signal and time-frequency analysis

Mono-component signal

1

2

d tIF t

dt

1

2

d fGD f

df

Group delay(GD)

IF of cmp1

Instantaneous frequency(IF)

IF

LGD

Controversy• UniquenessShekel、Prestley、Huang

• Physical meaningMandel、Cohen

1937：Carson & Fry1958：Ville

Multi-component signal

IF of cmp2

TFA

Non-parametric

TFA

Semi-parametric

TFA

Parametric TFA

Sparse decomposition

Global TF transform

Post-processor

Time-frequency analysis(TFA)

Background:TFA->TFR(time-frequency representation) & TFD(time-frequency distribution

Key issues of TFA:Concentration improvement and cross-term suppression

Short-time Fourier transform(STFT)

Joseph Fourier

(1768-1830)

Dennis Gabor

(1900-1979)

Alfréd Haar

(1885-1933)Jean Morlet Ingrid

Daubechies

Wigner-Ville distribution (WVD)

Eugene p Wigner

Sparse decomposition

Stephane MallatYves Meyer

Widely used TFA

TFAs

Wavelet transform

1. STFT

3.WVD

2. Continues wavelet transform

0 5 10 150

10

20

30

40

Time / Sec

Fre

q/H

z

True IF

Problem:1&2-Poor concentration, incorrect instantaneous amplitude (IA)3-Interference of cross-term4-Poor characterization of IF

4.Sparse decomposition

TFAs

Signal model

Chirplet transform (CT)

Chirplet transform

Mother chirplet

50

200

150

100

Fre

quency

/Hz

Time/Sec50 100 150 200

p1

p2

p3

50

200

150

100

Fre

quency

/Hz

Time/Sec50 100 150 200 250

p1

p2

p3

50

200

150

100

Fre

quency/H

z

Time/Sec50 100 150 200 250

p1

p2

p3

0.4

0.2

1

0.8

0.6

Norm

ali

zed e

nvelo

pe

50 100 150 200

Frequency/Hz

0.4

0.2

1

0.8

0.6

No

rmal

ized

en

vel

op

e

50 100 150 200

Frequency/Hz

0.4

0.2

1

0.8

0.6

Norm

ali

zed e

nvelo

pe

50 100 150 200

Frequency/Hz

Effect of chirplet parameter on Gaussian window function

TF

resolution

Section

(t=120sec)

STFT Wavelet transform Chirplet transform

线性调频

频移

时移

高斯窗

CT and general parametric TFA

Fre

qu

en

cy

Time

ω0

;sIF t

0t

sIF P

0tP

sIF t

Fre

qu

ency

Time

ct

θ

t

)(tIF

ω0

c c

频率旋转算子

频率平移算子

Problem：Parametric TFAs lack of general theoretical framework to implement

Contribution：General parametric TF transform and dual definition

General parameterized time-frequency transform, IEEE Transactions on Signal Processing, 62(11), pp 2751-2764, 2014

Parametric TFA

TFR fusion

Vibration analysis of rotatory machines with variable speed

Dispersion analysis of guided waves

Applications

Theory and methods

Multi-component

Kernel construction and estimation

Signal decomposition

Nonlinear system identification

Rotation operator

Shift operator

General parametric TFA

Problem：Traditional TFAs need to balance the trade-offs between concentration and cross-term and between time and frequency resolution

Contribution：New parametric TFAs are constructed using different kernels and corresponding kernel parameter estimators are developed

Generalized warblet transform (GWT)

Spline chirplettransform (SCT)

Polynomial chirplettransform (PCT)

• Spline-kernelled chirplet transform for the analysis of signals with time-varying frequency and its application, IEEE Transactions on Industrial Electronics, 59(3), pp 1612-1621, 2012.

• Characterize highly oscillating frequency modulation using generalized Warblet transform, Mechanical Systems and Signal Processing, 26, pp 128-140, 2012

Parametric TFA

Parametric TFA

TFR fusion

Vibration analysis of rotatory machines with variable speed

Dispersion analysis of guided waves

Applications

Theory and methods

Multi-component

Kernel construction and estimation

Signal decomposition

Nonlinear system identification

Parameter estimation

First PCT

Second PCT

Final PCT

4

;sFCI G

P P

TFD dependent estimator (TF domain) Model-based estimator(parameter space)

Multi-component signal

How to analyze multi-component signals？

IF1

IF2f

t

Problem：Traditional TFA suffers from the poor concentration or interference of cross-term in the case of multi-component signals

Contribution：TFR fusion and signal decomposition were developed for multi-component signals

Signal decomposition

TFR manipulation

• Application of parameterized time-frequency analysis on multicomponent frequency modulated signals, IEEE Transactions on Instrumentation and Measurement, 63(12), pp 3169-3180, 2014.

• Multicomponent signal analysis based on polynomial chirplet transform, IEEE Transactions on Industrial Electronics , 60(9), pp 3948-3956, 2013

Parametric TFA

TFR fusion

Vibration analysis of rotatory machines with variable speed

Dispersion analysis of guided waves

Applications

Theory and methods

Multi-component

Kernel construction and estimation

Signal decomposition

Nonlinear system identification

Parametric TFA

High-pass 2D filter

Image processing

Rotate and filter

Connectivity labeling

RotateFilter and

recover

Multi-component signal

Application1-Hydraulic turbine

Parametric TFA

TFR fusion

Vibration analysis of rotatory machines with variable speed

Dispersion analysis of guided waves

Applications

Theory and methods

Multi-component

Kernel construction and estimation

Signal decomposition

Nonlinear system identification

Hydraulic turbine

Shut-down stageContinuous and non-stationary processRich informationMultiple non-stationary components

Components separation & parametric TFA

1X 2X

3X 4X

TFR combination Residual

• Vibration signal analysis using parameterized time-frequency method for feature extraction of varying-speed rotary machinery, Journal of Sound and Vibration，332(20), pp 350-366, 2015.

Application1-Hydraulic turbine

STFT Wavelet transform

WVD Sparse decomposition

Problem：Traditional TFA cannot characterize time-frequency pattern of multi-component non-stationary signal accurately

Contribution：The proposed multi-component signal analysis method provides better solution of analyzing such signal in TF domain

Nd:YadPulse laser

532nm

Specimen

Digital oscilloscope

2.5GS/s,100MHz

GCC-102202

mirror

GCL-010207

biconvex lens

Trigger Laser Doppler Velocimeter

0-2MHz

Frequency conversion

unit

Application2-Lamb wave analysis

0 2 4 61

2

3

4

5

6

fd/MHz.mm

Gourp

velo

city/k

m/s

S0

S1 S2

S3

A0

A1

A2

A3

Parametric TFA

TFR fusion

Vibration analysis of rotatory machines with variable speed

Dispersion analysis of guided waves

Applications

Theory and methods

Multi-component

Kernel construction and estimation

Signal decomposition

Nonlinear system identification

Dispersion curves

Structure health detection

TFR manipulation based on parametric TFA of frequency domain

Local Dispersion estimation of mode A0

Local Dispersion estimation of mode S0

Simulated Lamb wave signal

• Frequency-varying group delay estimation using frequency domain polynomial chirplet transform, Mechanical Systems and Signal Processing, 46(1), pp 146-162, 2014.

Application2-Guided wave analysis

FGWT+TFR fusionFPCT

D-STFTWVD

Wavelet transformSTFT

A0

A1

S0

Nonlinear (time-varying) system1. varying restoring forces 2. varying natural frequencies

Application3-System identification

Parametric TFA

TFR fusion

Vibration analysis of rotatory machines with variable speed

Dispersion analysis of guided waves

Applications

Theory and methods

Multi-component

Kernel construction and estimation

Signal decomposition

Nonlinear system identification

System identification

Dynamic system modeling

System control

3 3

3 3

0.05 0.8 0

0.05 5.4 0.5 0.5 0.5 0

STFT PCT Real value

1.3638 0.9963 1

1.4856 1.0057 1

4.0777 0.7851 0.8

-0.3430 0.8525 1

4.9860 5.3695 5.4

-2.9287 0.4985 0.5

0.4498 0.4918 0.5

4.2297 0.5228 0.5

1. Extract TF features using parametric TFA(IF &IA);2. Estimate mode shape; 3. reconstruct backbone; 4. Estimate nonlinear stiffness and coupling coef.

Backbone and coupling

Application3-System identification

TFR of

TFR of

1. Yang Y., Dong X.J., Peng Z.K., Zhang W. M., Meng G., Vibration signal analysis using parameterized time-frequency

method for feature extraction of varying-speed rotary machinery, Journal of Sound and Vibration，332(20), pp

350-366, 2015.

2. Yang Y., Dong X.J., Zhang W.M., Peng Z.K., Meng G., Component Extraction for Non-stationary Multi-component

Signal Using Parameterized De-chirping and Band-pass Filter, IEEE Signal Processing Letters, 2015.

3. Yang Y., Peng Z.K., Dong X.J., Zhang W.M., General parameterized time-frequency transform, IEEE Transactions on

Signal Processing, 62(11), pp 2751-2764, 2014.

4. Yang Y., Peng Z.K., Dong X.J., Zhang W.M., Application of parameterized time-frequency analysis on

multicomponent frequency modulated signals, IEEE Transactions on Instrumentation and Measurement, 63(12),

pp 3169-3180, 2014.

5. Yang Y., Zhang W.M., Peng Z.K., Meng G., Multicomponent signal analysis based on polynomial chirplet transform,

IEEE Transactions on Industrial Electronics, , 60(9), pp 3948-3956, 2013.

6. Yang Y., Peng Z.K., Zhang W.M., Meng G., Spline-kernelled chirplet transform for the analysis of signals with time-

varying frequency and its application, IEEE Transactions on Industrial Electronics, 59(3), pp 1612-1621, 2012.

7. Yang Y., Peng Z.K., Zhang W.M., Meng G., Frequency-varying group delay estimation using frequency domain

polynomial chirplet transform, Mechanical Systems and Signal Processing, 46(1), pp 146-162, 2014.

8. Yang Y., Peng Z.K., Zhang W.M., Meng G., Characterize highly oscillating frequency modulation using generalized

Warblet transform, Mechanical Systems and Signal Processing, 26, pp 128-140, 2012.

9. Yang Y., Liao Y.X., Meng G., Zhang W.M., A hybrid feature selection scheme for unsupervised learning and its

application in bearing fault diagnosis, Expert Systems with Applications, 38(9), pp 11311-11320, 2011.

Publications

Thanks!