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the existence of a fault in a system under monitoring [
17
,
222
,
223
]. Thus to conclude
if a dynamical system has been subjected to fault of its components, information
coming from the power spectrum of its outputs can be processed. To find the spectral
density of a signal .t/ with the use of its Fourier transform .j!/, the following
definition is used:
E
D
R
C1
1
2
R
C1
.t/.
R
C1
1
. .t//
2
dt
D
1
.j!/e
j!t
d!/
dt
i.e.
2
R
C1
1
(12.29)
1
E D
.j!/.j!/d!
1
Taking that .t/ is a real signal it holds that .j!/ D
.j!/ which is the
signal's complex conjugate. Using this in Eq. (
12.29
) one obtains
2
R
C1
1
.j!/
.j!/d! or
E
D
2
R
C1
1
(12.30)
1
2
d!
E
D
1
j.j!/j
1
This means that the energy of the signal is equal to
2
times the integral over
frequency of the square of the magnitude of the signal's Fourier transform. This
is
Parseval's theorem
. The integrated term j.j!/j
2
is the energy density per unit
of frequency and has units of magnitude squared per Hertz.
12.8.2
Power Spectrum of the Signal Using
the Gauss-Hermite Expansion
As shown in Eqs. (
12.11
) and (
12.17
) the Gauss-Hermite basis functions satisfy the
orthogonality property, i.e. for these functions it holds
Z
C1
1 if m D k
0 if m¤k
m
.x/
k
.x/
dx
D
1
Therefore, using the definition of the signal's energy one has
E D
R
C1
1
. .t//
2
dt
D
R
C1
1
Œ
P
kD1
c
k
k
.t/
2
dt
(12.31)
and exploiting the orthogonality property one obtains
E D
P
kD1
c
k
(12.32)
Therefore the square of the coefficients c
k
provides an indication of the distribution
of the signal's energy to the associated basis functions. One could arrive at the
same results using the Fourier transformed description of the signal and Parseval's
theorem. It has been shown that the Gauss-Hermite basis functions remain invariant
under the Fourier transform subject only to a change of scale. Denoting by .j!/
