Digital Signal Processing Reference
In-Depth Information
This means that in general Equation 4.26 can be written as
b
4
∂
)
−
1
W
x
4
b
3
∂
3
b
2
∂
2
b
1
∂
∂
2
t
4
+
t
3
+
t
2
+
t
+
(
|
(ω)
|
(
ω)
=
(
ω).
H
t,
W
F
t,
(4.28)
∂
∂
∂
Now, it is known that the Wigner spectrum of white noise is
(
ω)
=
.
W
F
t,
N
0
(4.29)
By substituting Equation 4.29 into the equation for the Wigner, Equation 4.28,
we readily find that the solution is constant with respect to time, and in
particular
2
(4.30)
We note that we have solved Equation 4.28 putting the initial conditions at
t
W
x
(
t,
ω)
=
N
0
|
H
(ω)
|
.
. Because we are considering a stable system, at any finite time
t
we will assume that any initial condition has come to a steady-state solution
and hence use the term
stationary solution
. Equation 4.30 is the same as the
classical power spectrum but we emphasize that the Wigner spectrum is a
two-dimensional quantity and what we have shown is that it does not change
in time. A simple explanation is that, considering Equation 4.18 as a filtering
problem, the system filters the input noise
F
=−∞
(
t
)
with a bandpass filter to
produce an output signal
x
. But because both the input random noise and
the filter/system are stationary in time, then the output is also stationary.
(
t
)
4.4.1
Harmonic Oscillator with Time-Dependent Coefficients
We now consider the considerably more difficult problem
d
2
x
dt
2
+
dx
dt
+
2
µ
K
(
t
)
x
=
F
(
t
)
(4.31)
with
2
0
K
(
t
)
=
ω
+
t,
1
.
(4.32)
The equation for the Wigner spectrum is
A
t
+
)
B
t
+
)
W
x
2
µ
A
t
+
K
(
E
2
µ
B
t
+
K
(
F
(
t,
ω)
=
W
F
(
t,
ω)
,
(4.33)
t
t
(
)
and if we take into account the fact that
K
t
is slowly varying, we can ap-
proximate this equation by
1
∂
4
3
2
16
∂
t
4
+
2
∂
1
2
(
2
2
t
3
+
µ
+
K
(
t
)
+
ω
)
∂
∂
∂
t
2
2
W
x
=
)
∂
∂
2
2
2
2
+
2
µ(
K
(
t
)
+
ω
t
+
(
K
(
t
)
−
ω
)
+
4
µ
ω
W
f
(4.34)
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