Digital Signal Processing Reference
In-Depth Information
This equation contains no derivatives with respect to
and can hence be
solved as an ordinary differential equation in
t
. We choose as initial conditions
the following:
ω
N
0
W
x
(
0
,
ω)
=
2
,
(4.35)
ω
2
2
0
−
ω
+
4
µ
2
ω
2
W
x
∂
3
W
x
∂
∂
W
x
∂
ω)
=
∂
ω)
=
∂
(
0
,
(
0
,
(
0
,
ω)
=
0
.
(4.36)
t
t
2
t
3
2
0
This choice establishes that the system is in the condition
K
(
t
)
=
ω
for
all negative times, and that at
t
starts varying with time accord-
ing to Equation 4.32. In Figure 4.9 we show the solution to the approximate
differential equation, Equation 4.34. Notice that the computed Wigner spec-
trum
W
x
=
0
,K
(
t
)
looks exactly as we expected, in the sense that it is exactly a
time-varying bandpass function. Such solutions have not been obtained be-
fore by any means, and therefore we undertook a major simulation to ascer-
tain the accuracy of our answer. We do not describe the numerical simulations
here, but the result is shown in Figure 4.10. The agreement is excellent.
(
t,
ω)
x 10
−
3
FIGURE 4.9
Behavior of the instantaneous power spectrum obtained by solving the equation for the Wigner
spectrum.
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