Digital Signal Processing Reference
In-Depth Information
x 10
−
3
−
0.5
−
1
−
50
FIGURE 4.14
Wigner distribution of Green's function for the diffusion equation, for
t
=
10.
4.5.2
Wigner Distribution of Green's Function
It is of interest to calculate the Wigner distribution of Green's function for
each case. For the Schr odinger case
iat
exp
,
x
)
2
1
−
(
x
−
x, x
,t,t
)
=
G
ψ
(
√
4
(4.57)
4
iat
π
the Wigner distribution is (time is assumed to be positive)
1
2
x
−
W
G
ψ
(
x, p, t
)
=
π
δ(
x
−
2
apt
).
(4.58)
For the diffusion equation where
exp
,
x
)
2
1
−
(
x
−
x, x
,t
√
4
G
u
(
)
=
(4.59)
4
Dt
π
Dt
the Wigner distribution is
exp
2
Dtp
2
x
)
2
1
−
(
x
−
W
G
u
(
x, p, t
)
=
√
8
−
.
(4.60)
2
Dt
π
3
Dt
Thus we see the Wigner distribution shows a significant physical difference
between the two Green's functions although superficially the original wave
Search WWH ::
Custom Search