Digital Signal Processing Reference
In-Depth Information
With this notation we can write in a compact form a general partial derivative
∂
|
α
|
D
α
=
x
α
m
.
(4.108)
x
α
1
1
x
α
2
2
∂
∂
···
∂
The entire class of linear partial differential equations of order
N
with arbitrary
varying coefficients can then be written as
D
α
u
a
α
(
)
(
)
=
(
)
x
x
f
x
,
(4.109)
|
α
|≤
N
where
f
is the general coefficient.
To write the equation for the Wigner distribution, we first need to extend
its definition to the case of
m
-dimensional fields,
(
x
)
is the forcing term, and
a
α
(
x
)
Z
u, u
(
x, p
)
1
2
m
u
∗
(
=
x
1
−
1
/
2
τ
1
,
...
,x
m
−
1
/
2
τ
)
u
(
x
1
+
1
/
2
τ
1
,
...
,x
m
+
1
/
2
τ
)
m
m
π
e
−
i
τ
1
p
1
−
i
τ
2
p
2
+···−
i
τ
m
p
m
d
×
τ
1
d
τ
···
d
τ
m
,
(4.110)
2
where the momentum
p
is given by
p
=
(
p
1
,p
2
,
...
,p
m
).
(4.111)
We now define the following operators:
A
=
(
A
1
,
...
,A
m
)
,
B
=
(
B
1
,
...
,B
m
)
,
(4.112)
E =
(
E
1
,
...
,
E
m
)
,
F =
(
F
1
,
...
,
F
m
)
,
(4.113)
where
∂
∂
∂
∂
1
2
1
2
=
x
r
−
=
x
r
+
A
r
ip
r
,
r
ip
r
,
(4.114)
1
2
i
∂
1
2
i
∂
∂
E
=
x
r
+
,
F
=
x
r
−
,
(4.115)
r
r
∂
p
r
p
r
for any integer
r
,m
. Using these definitions one can write the equation
for the Wigner distribution
Z
associated with the general partial differential
equation, Equation 4.109,
|
α
|≤
=
1
,
...
|
β
|≤
a
α
(
E
)
A
α
B
β
Z
u, u
a
β
(
F
)
(
x, p
)
=
Z
f, f
(
x, p
)
,
(4.116)
N
N
where the meaning of the powers of the operators
A, B
is the following:
A
α
=
A
α
1
A
α
2
A
α
m
···
.
(4.117)
2
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