
17-11Nonlinear System Modeling with Invariance to Fourier Transform for Fault Diagnosis
and the corresponding Fourier transform is
T T
j t j t
T
T
j t e dt t e dt( ) ( ) ( )
ω ψ ψ
ω ω
= =
−∞
+∞
− −
−
∫ ∫
(1
7
.26
)
Using
a ge
neralization
of Pa
rseval’s
th
eorem,
th
e
av
erage
po
wer
of a re
al-valued
po
wer
sig
nal
ψ
T
(t) is
co
mputed
f
rom
t
he
F
ourier
t
ransform
a
s
P
T
t dt
P
T
j j
T
T
T
T
T T
ψ
ψ
ψ
π
ω ω
=
= −
→∞
−
→∞
−∞
∞
∫
∫
lim | ( )|
lim | ( ) ( )
1
2
1
4
2
or
Ψ Ψ
dd
P
T
j d
T
ω
π
ω ω
ψ
or
=
→∞
−∞
∞
∫
lim | ( )|
1
4
2
Ψ
(1
7
.27
)
e power spectral density of the signal is computed from the Fourier transform as
S
j
T
T
T
ψ
ω
ω
( ) lim
| ( )|
=
→∞
Ψ
2
2
(17.28)
17.4.3 Power Spectrum of Stochastic Signals
e power spectral density is ...