Analysis of Non-repetitive Signals

8/13/2019 Analysis of Non-repetitive Signals

http://slidepdf.com/reader/full/analysis-of-non-repetitive-signals 1/29

An Introduction to the Analysis ofNon-Repetitive Signals Using

Joint Time-FrequencyDistributions and Wavelets

Paul Wright

Applied Electrical and Magnetics Group, NPL.



Outline

• Non-Repetitive Waveforms

• Windowing and the STFT

• Wigner-Ville Distribution• Joint Time Frequency Distributions

• Wavelets• Polynomial Demoduation



Fourier Transform of a Burst Signal

H1 and aburst of H2

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 16 32 48 64 80 96

Fourier Frequency

M a g n i t u d e

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Sample Number

A m p l i t u

d e

2/5 3/5

(H1 and H2 are the same amplitude)

FT

∫−∞

∞−

−

= dt et s f S t f j π 2)()(



Analysing Non-repetitivewaveforms

Break the waveform up into adjacent sectionsBreak the waveform up into adjacent sections

Do a DFT on each sectionDo a DFT on each section – – Now have T & F InformationNow have T & F Information

Each Section Should contain an integer number of cyclesEach Section Should contain an integer number of cycles

Sometime called a Short Time Fourier Transform (STFT)Sometime called a Short Time Fourier Transform (STFT)

∫

−∞

∞−

−

= dt et s f S t f j π 2)()(

No longerNo longer

reliable!reliable!



Short Time FT Using 8 Windows

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Sample Number

A m p l i t u d

2/5 3/5



Time Frequency Distributions

TimeTimeFrequencyFrequency

MagnitudeMagnitude

Percentage of Full Scale Magnitude

0 40 60 80 10020



STFT Using 8 Windows

2/5

3/5



STFT Using 16 Windows

2/5

3/5



Number of

Windows

Fundamental

Cycles/window

Frequency

Resolution

2nd

Harmonic Error

Zero 32 1/32 -55.5 %

4 8 1/8 -37.1 %

8 4 1/4 -11.1 %

16 2 1/2 -8.0 %

32 1 1 -2.2 %

STFT Results withVarious Windows



The Wigner-VilleDistribution

Auto-correlation function of signal s(t) :

Wigner Distribution of signal s(t) :



Analysis of a Burst

16 Window STFT

Wigner Ville Distribution



FT ν

FT ω

FT τ

FT t

IFT t

IFT ω

IFT ν

IFT τ

W(t,ω)Wigner Domain

R(t,τ )

Auto-Correlation

Domain (Temporal)

R(υ ,ω)

Auto-Correlation

Domain (Spectral)

A(υ ,τ )

Ambiguity Domain



Frequency Shift, ν

T i m e L

a g , τ

0

0

Ambiguity Domain



Frequency Shift, ν

Time Lag, τ

M a g

n i t u d e

Kernel Filter Response

Design Filter Kernels in the Ambiguity Domain

Choi-Williams Filter



T i m e L a g , τ

0

Frequency Shift, ν

0

Filtered Ambiguity

Function



Choi-WilliamsDistribution

2/5

3/5



Wavelets

)],(),([)( t t f ncorrelatioF =

∫∞

∞−

= dt t t f F ),().()( ω ψ ω

Fourier TransformFourier Transform

)](),([)( t t f ncorrelatiot W abab =

∫∞

∞−

= dt t t f t W abab

)().()( ψ

Wavelet TransformWavelet Transform



Translation and Dilation

)2(2)( 2 k t t j

jk

j

−= ψ ψ

Where the integer j is the dilation parameter and integer k is used for translation

Harr Wavelet

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.5 1

Translation (k) = 1

Dilation (j) = 1

Harr Wavelet

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.5 1

Translation (k) = 1

Dilation (j) = 2

Harr Wavelet

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.5 1

Translation (k) = 2

Dilation (j) = 2

Harr Wavelet

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.5 1

Translation (k) = 3

Dilation (j) = 2

Harr Wavelet

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.5 1

Translation (k) = 3

Dilation (j) = 3

Harr Wavelet

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.5 1

Translation (k) = 4

Dilation (j) = 3



Wavelet Families

1 0 1 2

1

0

1

φ x( )

ψ x( )

x

Haar

2 0 2 42

1

0

1

2

φ x( )

ψ x( )

x

Daubechies 4

2 0 2 41

0

1

2

φ x( )

ψ x( )

x

Symmlet 20Battle-Lemarie 6

2 0 2 42

1

0

1

2

φ x( )

ψ x( )

x



Vibration Analysis Using Wavelets

D 1 1

D 1 0

D 9

D 8

D 7

D 6

D 5

D 4

D 3

S a m p le N u m b e r

F r

e q u e n c y

A m p l i t u d e

( w i t h a r b . o

f f s e t s )



WaveletData/Image

Compression

Reject all wavelet coefficientsbelow a given Threshold,then invert what’s left….

Signal

75% data compression in this

example.

Compressed Data Signal



Perform DWT, retain only

highest 10 % of coefficients

Reconstructed Image –

10:1 compressionPerform DWT, retain only

highest 1 % of coefficients

Reconstructed Image –

100:1 compression

Image Compression

Original Image



JPEG Compression Wavelet Compression

Image Compression

W l t D t /I F i



Fine Spatial Texture Image

Coarse Spatial Spectral Image

Fused Image

Wavelet Data/Image Fusion

(NB this is illustrative!)



Smooth Modulation

Made of H1,H2 and H3

But…

A H i M d l ti



A Harmonic ModulationModel

∑∑ ==

+++=

N

n

n

N

n

n t nqqt n p pt x1

0

1

0 )(sin)(cos)( ω ω

A signal x(t) with N non-modulated harmonics:

For modulated harmonics, use modulation functions

m(t) for the p ’s & q ’s:

pn(t) = an,0 m<0> + an,1 m<1> + … an,K m<K>

qn(t) = bn,0 m<0> + bn,1 m<1> + … bn,K m<K>

(if m <k> = t k then the scheme is a polynomial modulation model)

De De - - modulate using Method of Least Squares modulate using Method of Least Squares



When m(t) = t , the modulators are polynomials,

Highly accurate demodulation for smooth modulation functions.

Polynomial modulation functionsPolynomial modulation functions

Using Wavelet Basis Functions



Using Wavelet Basis Functions

For signals with discontinuities, the m(t) functions can be wavelets…

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

0 100 200 300 400 500

Sample Number

A m p l i t u d e

Test Signal: Modulated H1 and H2

H1

H2

L.S. D4 Wavelet demodulation

High Accuracy Demodulation – Handles the discontinuity!

S



Summary

• Short Time Fourier Transforms

• Time Resolution – Frequency Resolution Trade Off

• Wigner Distributions

• Ambiguity Domain Filtering• Wavelets

• Time Series Modulation Methods

Documents

Analysis of Non-repetitive Signals