Digital Signal Processing Reference
In-Depth Information
35
30
25
20
15
10
5
Forcing chirp
0
0
1
2
3
4
5
6
7
8
9
10
t (s)
FIGURE 4.5
Wigner distribution of the solution to the gliding tone problem for a critically damped case,
µ
=
ω
0
.
We point out that we have obtained the Wigner distribution from Equa-
tion 4.3 by choosing the following initial conditions:
2
W
x,x
(
−∞
3
W
x,x
(
−∞
ω)
=
∂
W
x,x
(
−∞
,
ω)
=
∂
,
ω)
=
∂
,
ω)
.
(4.11)
We have proved
15
that this choice corresponds to finding the Wigner dis-
tribution of the solution
x
W
x,x
(
−∞
,
=
0
∂
t
∂
t
2
∂
t
3
(
t
)
that has zero initial conditions at
t
=−∞
.
4.3
Approximation Method
Of course, it was fortuitous that the exact solution to Equation 4.3 was achieved,
even though one cannot write an exact solution for
x
. We were able to
achieve an exact solution because the Wigner distribution of the driving force
f
(
t
)
e
i
ω
1
t
+
i
β
t
2
2
was a delta function. We now observe that a general class of
signals that are very interesting from a physical point of view are the so-called
monocomponent signals, which are signals of the form
s
/
(
t
)
=
e
j
ϕ(
t
)
(with cer-
tain restrictions on the phase). Although the Wigner distribution of a driving
force of the form
s
(
t
)
=
e
j
ϕ(
t
)
is not in general a delta function along the instan-
(
t
)
=
)
=
ϕ
(
taneous frequency
ω
(
t
t
)
, the delta function is nonetheless a very good
i
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