Signal Based Fault Detection and Diagnosis for Rotating Electrical Machines: Issues and Solutions - Complex System Modelling and Control Through Intelligent Soft Computations - page 290

Þ ¼ P
n 2Z þ

c ð a ; b Þ ¼ c ð j ; k f ð n Þ/ j ; k ð n Þ;

ð 25 Þ

2 j 2 j k

a ¼ ; b ¼ ;

j 2 Z; k 2 Z;

, n 2 Z þ and t c the sampling

considering the simpli ed notation f ð n Þ ¼f ð n t c Þ

time, the discretization of continuous time signal f ð t Þ is considered. The inverse

transform, also called discrete synthesis, is de ned as:

X X

f ð n Þ ¼ c ð j ; k Þ/ j ; k ð n Þ: ð 26 Þ

j 2Z k 2Z

In Mallat ( 1989 ), a signal is decomposed into various scales with different time
and frequency resolutions, this algorithm is known as the multi-resolution signal
decomposition. De ning:

;

2 j = 2 2 j n

/ j ; k ð n Þ ¼ / k

;

2 j = 2 2 j n

w j ; k ð n Þ ¼ w k

2

ð j ; k Þ2Z ð 27 Þ

V j ¼

/ j ; k ; 2 Z ;

span k

;

W j ¼ span w j ; k ; k 2 Z

the wavelet function / j ; k , is the orthonormal basis of V j and the orthogonal wavelet

w j ; k , called scaling function, is the orthonormal basis of W j . In Daubechies ( 1988 )is
shown that:

V j ? W j ;

V m ¼

2

V m ; W m L ðRÞ ð 28 Þ

W m þ 1 V m þ 1 :

De ning f ð n Þ ¼ f as element of V 0 ¼ W 1 V 1 , f can be decomposed into its

components along V 1 and W 1 :

¼ þ : ð 29 Þ

f P 1 f Q 1 f

with P j the orthogonal projection onto V j and Q j the orthogonal projection onto W j .
De 1 and f ð n Þ ¼c n , it results:

f ð n Þ ¼ X k 2Z

ning j

c k / 1 ; k ð n Þþ X
k

d k w 1 ; k ð n Þ;

2Z

c k ¼ X n 2Z

h ð n 2k Þ c n ;

X n 2Z

g ð n 2k Þ c n ;
h ð n 2k Þ ¼ / 1 ; k ð n Þ; / 0 ; n ð n Þ

d k ¼

ð 30 Þ

;

:

g ð n 2k Þ ¼ w 1 ; k ð n Þ; w 0 ; n ð n Þ

2

k ; n 2 Z

: