Signal Based Fault Detection and Diagnosis for Rotating Electrical Machines: Issues and Solutions - Complex System Modelling and Control Through Intelligent Soft Computations - page 291

where the terms g and h are high-pass and low-pass filter coef cients derived from

the bases w and / . Considering a dataset of N ð n ¼ 1 ; ...; N Þ samples, and intro-

ducing a vector notation, c k and d k can be rewrite as Daubechies ( 1988 ):

c 1 Hc 0

¼ ;

ð 31 Þ

d 1 Gc 0 ¼ ;

with

2 3

h ð 0 Þ h ð 1 Þ h ð N Þ

4 5 ;

h ð 2 Þ h ð 1 Þ h ð N 2 Þ

H ¼ ð 32 Þ

. . .

h ð 2k Þ h ð 1 2k Þ h ð N 2k Þ

2 3

g ð 0 Þ g ð 1 Þ g ð N Þ

4 5 :

g ð 2 Þ g ð 1 Þ g ð N 2 Þ

G ¼ ð 33 Þ

. . .

g ð 2k Þ g ð 1 2k Þ g ð N 2k Þ

The procedure can be iterated obtaining:

c j ¼ Hc j 1

;

ð 34 Þ

d j ¼ Gd j 1 :

Then:

c j ¼ H j c 0 ;

ð 35 Þ

d j ¼ G j d 0 ;

where H j is obtained by applying the H filter j times, and G j is obtained by applying

the H filter once. Hence any signal may be decomposed

into its contributions in different regions of the time-frequency space by projection
on the corresponding wavelet basis function. The lowest frequency content of the
signal is represented on a set of scaling functions. The number of wavelet and
scaling function coef

filter j 1 times and the G

cients decreases dyadically at coarser scales due to the dyadic
discretization of the dilation and translation parameters. The algorithms for com-
puting the wavelet decomposition are based on representing the projection of the
signal on the corresponding basis function as a filtering operation (Mallat 1989 ).

Convolution with the filter H represents projection on the scaling function, and

convolution with the filter G represents projection on a wavelet. Thus, the signal

f is decomposed at different scales, the detail scale matrices and approximation

scale matrices. De

ð n Þ

ning L the decomposition levels, the approximation scale A L