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

Some Essentials of Data Analysis with Wavelets Slides in the wavelet part of the course in data analysis at The Slid i h l f h i d l i Th 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 ): Parseval’s ¥ ¥ ò ò   = = x x x V s x f ( , ) s t g ( ) ( ) t dt s ( ) g ( ) d , g f x , f x , relation -¥ -¥ p p i i 2 2 ft ft = ( ( p + p ) ) where h g ( ) ( ) t t e g(t-x)= cos(2 ( ) (2 ft ft ) ) i i sin(2 i (2 ft ft ) ) g t ( - ( t x ). ) f x ,

¥ ò ò y y y y = A wavelet is a bounded function for which ( ) ( ) t dt 0. -¥ (Some extra technical contitions (MRA) must be satisfied for getting the orthonormal wavelet bases discussed in previous slides.) getting the orthonormal wavelet bases discussed in previous slides ) Continuous wavelet transform (CWT) Parseval’s The continuous wavelet transform : relation relation ¥ ¥  ò ò ò ò  W = y y = x y x y x x x x s a b ( , ) ( , ) s t ( ) ( ) ( ) ( ) t dt s ( ) ( ) ( ) ( ) d , , y y a b a b , a b a b , -¥ -¥ ( ( ) ) - 1 1 t t b b y = y where ( ) ( ) t . a b , a a

The time-frequency localization of ( ( ) ) 1 t - b y y = y y , ( ) ( ) t a b a b a a Wavelet TF-localization is completely described by and . by a a a d b b

Heisenberg boxes Wavelets: STFT:

Time - frequency localization of a function g ¥ ¥ 2 ò ò 2 ò ò 2 ( ( ) ) D = - = t t g t g ( ) ( ) dt where t t g t g ( ) ( ) dt g g 0 0 0 0 TF-localization -¥ 0 ¥ ¥ 2 2 2 2 2 2 ò ò ( ( ) ) ò ò D D = -   =   f f f f g f ( ) ( ) f df df where h f f f f g f ( ) ( ) f df df  0 0 g -¥ 0 Note Note. Formulas borrowed from mechanics Formulas "borrowed" from mechanics H i Heisenberg uncertanity b t it « ( , t f centre of mass) and probability principle: the area 0 0 ¥ ¥ 2 ò ò = = « « theory ( theory ( g t g t ( ) ( ) dt dt 1 1, , t t f f expectation expectation 1 1 2 0 0 ò D D ³ g t ( ) dt .  g -¥ g p ¥ 4 D D « and , standard deviation.) -  g g

Bearing condition monitoring Bearing condition monitoring •Bearing failures can cause both personal damages and economical loss damages and economical loss. Bearing condition monitoring •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 Full: Close up: 18 80 60 16 40 20 14 20 40 60 80 100 120 140 160 12 10 8 6 4 2 2 2 4 6 8 10 12 14