Lecture 2: The continuous wavelet transform

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

Music

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

Time frequency analysis

Time-frequency analysis

The cochlea

The Gabor transform

( ):

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

x x x

¥ ¥

-¥ -¥

= ò = ò ^{ Parseval’s} _relation

( )

, 2

where g

( ) t = e

g(t-x)= cos(2 p ft ) + i sin(2 p ft g t x ) ( - ).

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

¥

ò

(Some extra technical contitions (MRA) must be satisfied for

getting the orthonormal wavelet bases discussed in previous slides )

y y

-¥

ò

Continuous wavelet transform (CWT)

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

( )

,

where _{a b} ( ) 1 t b .

t a a

y = y

The time-frequency

( )

localization of

1 -

a b

( ) t b

y t = y

Wavelet TF-localization

( )

,

( )

is completely described by and .

a b

a a

a b

y y

by a a d b

Heisenberg boxes

Wavelets: STFT:

Time - frequency localization of a function g

(

)

^{( )}

^{where t}

^{( )}

g

t t g t dt t g t dt

¥ ¥

D = ò -

= ò

( )

( )

0 0

0

2 2 2

( ) ( )

( ) h ( )

g

g g

f f f df f f f df

-¥

¥ ¥

D

ò ò

ò ⁽

⁾

^

ò ^

0

( ) 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

0 0

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

0 0

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

g

g t dt t f

-¥

= «

D D «

ò



2 -

1 ( ) .

g g

4 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

Main goal

Noise-free vibrations

CWT vibration

analysis

Example plot after further analysis Example plot after further analysis

Close up:

Full:

20 40 60 80 100 120 140 160

20 40 60 80

14 16 18

8 10 12

2 4 6

2 4 6 8 10 12 14

2