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the Fourier transformed signal of .t/ and by
k
.j!/ the Fourier transform of the
k-th Gauss-Hermite basis function one obtains
.j!/D
P
kD1
c
k
k
.j!/
(12.33)
and the energy of the signal is computed as
2
R
C1
1
2
d!
(12.34)
E
D
1
j.j!/j
Substituting Eqs. (
12.33
)into(
12.34
) one obtains
2
R
C1
1
j
P
kD1
c
k
k
.j!/j
1
2
d!
(12.35)
E
D
and using the invariance of the Gauss-Hermite basis functions under Fourier
transform one gets
2
R
C1
1
j
P
kD1
c
k
˛
1
2
k
.˛
1
j!/j
1
2
d!
(12.36)
E
D
while performing the change of variable !
1
D ˛
1
! it holds that
2
R
C1
1
j
P
kD1
c
k
˛
2
k
.j!
1
/j
1
2
d!
1
(12.37)
E
D
Next, by exploiting the orthogonality property of the Gauss-Hermite basis functions
one gets that the signal's energy is proportional to the sum of the squares of the
coefficients c
k
which are associated with the Gauss-Hermite basis functions, i.e. a
relation of the form
E
D
P
kD1
c
k
(12.38)
12.8.3
Detection of Changes in the Spectral Content
of the System's Output
The thermal model of a system, i.e. variations of the temperature output signal, has
been identified considering the previously analyzed neural network with Gauss-
Hermite basis functions. As shown in Figs.
12.26
and
12.27
, thanks to the multi-
frequency characteristics of the Gauss-Hermite basis functions, such a neural model
can capture with increased accuracy spikes and abrupt changes in the temperature
profile [
164
,
167
,
169
,
221
]. The RMSE (Root Mean Square Error) of training the
Gauss-Hermite neural model was of the order of 4 10
3
.
The update of the output layer weights of the neural network is given by a
gradient equation (LMS-type) of the form
w
i
.k C 1/ D
w
i
.k/ e.k/
T
.k/
(12.39)
