Signal Based Fault Detection and Diagnosis for Rotating Electrical Machines: Issues and Solutions - Complex System Modelling and Control Through Intelligent Soft Computations - page 289

4.1.2 Wavelet Transform

The Wavelet Transform (WT) is de ned as the integral of the signal f ð t Þ multiplied

by scaled, shifted version of basic wavelet function / ð t Þ , that is a real valued

function whose Fourier transform satis es the admissibility criteria stated in Li et al.

( 1999 ). Then the wavelet transformation c ð; Þ of a signal f ð t Þ is de ned as:

Þ ¼ R R

a dt

p /

t b

c ð a ; b f ð t Þ

1

ð 20 Þ

2 R þ f

a 0 g

b 2 R;

where a is the so-called scaling parameter, b is the time localization parameter. Both
a and b can be continuous or discrete variables. Multiplying each coef cient by an

appropriately scaled and shifted wavelet it yields the constituent wavelets of the
original signal. For signals of

finite energy, continuous wavelets synthesis provides

the reconstruction formula:

Z Z da

a 2 db

1
K /

t b

f ð t Þ ¼ c ð a ; b Þ/ ð 21 Þ

a

R þ

R

where:

Z þ1

j/ðnÞj

2

K / ¼ d n ð 22 Þ

jnj

1

/

denotes a (Wavelet speci c) normalization parameter in which is the Fourier

/

transform of . Mother wavelets must satisfy the following properties:

þ1 Z þ1 þ1

2 dt

j/ð t Þj dt

\ 1;

j/ð t Þj ¼ 1 ; /ð t Þ dt ¼ 0 : ð 23 Þ

1 1 1

To avoid intractable computations when operating at every scale of the Con-
tinuous WT (CWT), scales and positions can be chosen on a power of two, i.e.
dyadic scales and positions. The Discrete WT (DWT) analysis is more ef

cient and
accurate, as reported in Li et al. ( 1999 ) and Daubechies ( 1988 ). In this scheme a and
b are given by:

a 0 ; b 0 a 0 k

2

a ¼ b ¼ ; ð j ; k Þ2Z ; Z : ¼ f 0 ; 1 ; 2 ; g: ð 24 Þ

The variables a 0 and b 0 are fixed constants that are set, as in Daubechies ( 1988 ),

to: a 0 ¼ 2 and b 0 ¼ 1. The discrete wavelet analysis can be described mathemat-

ically as: