Some Essentials of Data Analysis with WaveletsSlid i h l f h i d l i ThSlides 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 technologNiklas 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 analysisTime-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 tp

x x x¥ ¥

-¥ -¥= =ò ò Parseval’s

relation( )2

,where ( ) g(t-x)= cos(2 ) sin(2 ) ( - ).i ftf xg t e ft i ft g t xp p p= +

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

=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’srelation

( , ) ( ) ( ) ( ) ( ) ,a b a bs a b s t t dt s dy 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 bt b

ta a

y y=

The time-frequency

( )localization of

1 -( )a b

t bty y=

Wavelet TF-localization

( ), ( )

is completely described

by and .

a b a a

a b

y y

by a da b

Heisenberg boxes

Wavelets: STFT:

Time - frequency localization of a function g

( )2 2 20 0( ) where t ( )g t t g t dt t g t dt

¥ ¥

D = - =ò òTF-localization

( )

( )

0 00

2 2 2

( ) ( )

( ) h ( )

g g g

f f f df f f f df

-¥¥ ¥

D

ò ò

ò ò ( )2 2 20 0

0

( ) where ( )g f f g f df f f g f df-¥

D = - =ò ò

Formulas "borrowed" from mechanicsNote H i b t it0 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 0theory ( ( ) 1, , expectation

and , standard deviation.)g g

g t dt t f-¥

= «

D D «

ò

2

-

1( ) .

4g gg t dt

p¥

D D ³ ò

Bearing condition monitoringBearing condition monitoring•Bearing failures can cause both personal damages and economical loss

Bearing condition monitoring

damages and economical loss.•Often not possible to stop production to check bearingscheck bearings.

•Usual monitoring techniques today analyseanalysetime domain signal or Fourier transform.

Vibration measurements with handheld deviceVibration measurements with handheld device

Main goal

Noise-free vibrations

CWT vibration analysis

Example plot after further analysisExample plot after further analysisClose 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