Some Essentials of Data Analysis with Wavelets Slides in the wavelet - - PowerPoint PPT Presentation

Some Essentials of Data Analysis with Wavelets

Slid i h l f h i d l i Th Slides in the wavelet part of the course in data analysis at The Swedish National Graduate School of Space Technology Lecture 2: The continuous wavelet transform

Niklas Grip Department of Mathematics L leå Uni ersit of technolog Niklas Grip, Department of Mathematics, Luleå University of technology

Last update: 2009-12-10

The theme of Joh. Seb. Bach’s Goldberg variations

Source:http://www.ti6.tu-harburg.de/~rolf/Goldberg.html

Time frequency analysis Time-frequency analysis

( ): Short time Fourier transform continuous Gabor transform



( )

, , 2

( , ) ( ) ( ) ( ) ( ) , h ( ) ( ) (2 ) i (2 ) ( )

g f x f x i ft

V s x f s t g t dt s g d t ft i ft t

p

x x x

¥ ¥

• ¥
• ¥

= =

ò ò



Parseval’s relation

( )

2 ,

where ( ) g(t-x)= cos(2 ) sin(2 ) ( - ).

i ft f x

g t e ft i ft g t x

p

p p = +

A is a bounded function for which ( ) 0. wavelet t dt y y

¥

=

ò

( ) (Some extra technical contitions (MRA) must be satisfied for getting the orthonormal wavelet bases discussed in previous slides ) y y

• ¥

ò

getting the orthonormal wavelet bases discussed in previous slides.)

The : continuous wavelet transform

Parseval’s relation



( , ) ( ) ( ) ( ) ( ) ,

a b a b

s a b s t t dt s d

y

y x y x x

¥ ¥

= =

ò ò 

W

relation

( )

, ,

( , ) ( ) ( ) ( ) ( ) , 1 ( )

a b a b

t b

y

y x y x x

• ¥
• ¥
• ò

ò

( )

,

1 where ( ) .

a b

t b t a a y y =

The time-frequency

( )

localization of 1

• ( )

a b

t b t y y =

( )

, ( )

is completely described by and .

a b

a a a b y y by a d a b

Heisenberg boxes

Wavelets: STFT:

Time - frequency localization of a function g

( )

2 2 2

( ) where t ( )

g

t t g t dt t g t dt

¥ ¥

D =

• =

ò ò

( ) ( )

2 2 2

( ) ( ) ( ) h ( )

g

g g f f f df f f f df

• ¥

¥ ¥

D

ò ò ò ò

 

( )

2 2 2

( ) where ( )

g

f f g f df f f g f df

• ¥

D =

• =

ò ò



 

Formulas "borrowed" from mechanics Note

H i b t it

2

Formulas borrowed from mechanics ( , centre of mass) and probability theory ( ( ) 1 expectation Note. t f g t dt t f

¥

« = «

ò

Heisenberg uncertanity principle: the area 1

¥

theory ( ( ) 1, , expectation and , standard deviation.)

g g

g t dt t f

• ¥

= « D D «

ò



2

• 1

( ) . 4

g g

g t dt p ¥ D D ³

ò



Bearing condition monitoring Bearing condition monitoring

• Bearing failures can cause both personal

damages and economical loss

damages and economical loss.

• Often not possible to stop production to

check bearings check bearings.

• Usual monitoring techniques today

analyse analyse time domain signal or Fourier transform.

Vibration measurements with handheld device Vibration measurements with handheld device

Noise-free vibrations

CWT vibration analysis

Example plot after further analysis Example plot after further analysis

Close up:

Full:

14 16 18 8 10 12 2 4 6 2 4 6 8 10 12 14 2