Digital Signal Processing Reference
In-Depth Information
toward this approach.
4
,
5
,
8
,
12
Equations of motion have been written for both
the Wigner distribution and other distributions.
In contrast, in signal analysis, the Wigner distribution has not been used
in such a manner. We describe methods that allow one to write a dynamical
equation for the Wigner distribution that corresponds to the solution of an
ordinary or partial differential equation. We have found many advantages in
this approach, both from the point of view of insight and also in devising new
methods of solution and approximation to the original equation. As is well
known, a differential equation where the dependent variable is time can be
converted into an equivalent equation in the frequency domain. Our approach
converts the differential equation in the combined time-frequency domain. In
this chapter we present these ideas and methods and we do so using concrete
examples. While we give the general approach, we emphasize and develop the
ideas for the harmonic oscillator, with constant and time varying coefficients,
and also for the deterministic and random case. In the appendices we give
the general results that can be applied to arbitrary differential equations.
4.2
Harmonic Oscillator with Chirp Driving Force
Consider the harmonic oscillator with a deterministic driving force,
d
2
x
(
)
(
)
t
dx
t
2
+
2
µ
+
ω
0
x
=
f
(
t
)
,
(4.1)
dt
2
dt
where
x
is the driving
term. If we want to study the time-frequency properties we could solve this
equation and substitute the answer into the Wigner distribution,
(
t
)
is the state variable (e.g., position, current) and
f
(
t
)
x
∗
t
x
t
e
−
i
τω
d
1
2
1
2
τ
1
2
τ
W
x,x
(
t,
ω)
=
−
+
τ.
(4.2)
π
We want to write the equation of motion for the Wigner distribution and solve
that directly. As mentioned above we have devised a general procedure to do
that which is reviewed in Appendix 4.2. For the harmonic oscillator it is
a
4
∂
a
0
W
x,x
(
4
3
2
a
3
∂
a
2
∂
a
1
∂
∂
t
4
+
t
3
+
t
2
+
t
+
t,
ω)
=
W
f, f
(
t,
ω)
,
(4.3)
∂
∂
∂
where
=
ω
2
2
2
0
2
2
,
a
0
−
ω
+
4
µ
ω
(4.4)
µ
ω
2
,
2
0
a
1
=
2
+
ω
(4.5)
2
ω
2
,
1
0
2
a
2
=
+
ω
+
2
µ
(4.6)
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