Digital Signal Processing Reference
In-Depth Information
This is the equation of motion for
Z
does not
appear in the equation, we can integrate it out to obtain an equation for the
standard Wigner distribution,
W
(
x, p, t,
ω).
However, since
ω
(
x, p, t
)
,
2
W
∂
∂
W
∂
c
∂
W
∂
D
2
∂
2
Dp
2
W,
t
+
x
=
x
2
−
(4.126)
which is Equation 4.48 of the text.
Appendix 4.6: Green's Function for the Wigner Distribution
First, we obtain Green's function for the Wigner distribution in terms of
Green's function for the field. Suppose we propagate the field by
x, x
,t
x
,
0
dx
,
u
(
x, t
)
=
G
u
(
)
u
(
)
(4.127)
and the Wigner distribution by (done by Cohen
8
and Moyal,
12
who did it for
the Schr odinger equation)
x, x
,p,p
,t
x
,p
,
0
dx
dp
.
W
(
x, p, t
)
=
G
W
(
)
W
(
)
(4.128)
Substituting Equation 4.127 into the
definition of the classic Wigner distribution, Equation 4.43, we obtain
We want to express
G
W
in terms of
G
u
.
G
u
x
,x
,t
G
u
x
,x
,t
1
2
1
2
τ
1
2
τ
W
u
(
x, p, t
)
=
−
+
π
u
∗
(
x
,
0
x
,
0
e
−
i
τ
p
d
dx
dx
.
×
)
u
(
)
τ
(4.129)
But from the definition of the Wigner distribution, we have
e
−
i
(
x
+
x
)
p
dp,
u
∗
(
x
,t
x
,t
x
+
x
)/
)
u
(
)
=
W
((
2
,p,t
)
(4.130)
and inserting this into Equation 4.129 we obtain
G
u
x
,x
,t
G
u
x
x
,t
1
2
1
2
τ
1
2
τ
τ
−
W
u
(
x, p, t
)
=
−
+
,
(4.131)
π
(τ
/
2
,p
,
0
e
+
i
τ
p
e
−
i
τ
p
d
τ
dp
dx
.
×
W
)
τ
d
(4.132)
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