Digital Signal Processing Reference
In-Depth Information
Therefore,
G
u
x
,x
,t
G
u
1
2
1
2
τ
x, x
,p,p
,t
G
W
(
)
=
−
(4.133)
π
x
x
,t
e
+
i
τ
p
e
−
i
τ
p
d
1
2
τ
τ
−
τ
dx
.
×
+
,
τ
d
(4.134)
This is a general result that always holds for any field equation.
However, for the cases we are considering it is easier to solve the problem
in momentum space
e
−
iap
2
t
φ(
)
=
φ(
)
p, t
p,
0
,
(4.135)
e
−
Dp
2
t
U
U
(
p, t
)
=
(
p,
0
)
,
(4.136)
and substituting into the definition of the Wigner distribution we have
2
p
,
0
U
p
,
0
e
−
i
θ
x
d
e
−
2
Dtp
2
1
2
1
2
θ
1
2
θ
2
e
−
Dt
θ
/
W
u
(
x, p, t
)
=
+
−
θ
,
π
(4.137)
φ
∗
p
,
0
p
,
0
e
−
i
θ
x
d
1
2
1
2
θ
1
2
θ
e
2
ia
θ
pt
W
ψ
(
x, p, t
)
=
+
φ
−
θ.
(4.138)
π
But from the definition of the Wigner distribution,
U
∗
p
,
0
U
x
,
0
1
2
θ
1
2
θ
e
i
θ
x
dx,
+
−
=
W
u
(
x, p,
0
)
(4.139)
φ
∗
p
,
0
x
,
0
1
2
θ
1
2
θ
e
i
θ
x
dx,
+
φ
−
=
W
ψ
(
x, p,
0
)
(4.140)
and therefore
e
−
2
Dp
2
t
e
−
D
θ
1
2
2
t
2
e
i
θ(
x
−
x
)
W
u
(
/
x
,p,
0
dx
d
W
u
(
x, p, t
)
=
)
θ
,
(4.141)
π
e
i
2
a
θ
pt
e
i
θ(
x
−
x
)
W
ψ
(
1
2
x
,p,
0
dx
d
W
ψ
(
x, p, t
)
=
)
θ.
(4.142)
π
In both cases the
θ
integration can be done to yield
exp
W
u
(
e
−
2
Dp
2
t
x
)
2
1
−
(
x
−
x
,p,
0
dx
,
W
u
(
x, p, t
)
=
√
2
)
(4.143)
2
Dt
π
Dt
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