Digital Signal Processing Reference
In-Depth Information
tion):
2
1
A
n
−
1
A
n
−
1
Ω
n
−
1
+
=
1
R
n
)
−
1
W
n
Ω
n
=
(α
n
·
=
W
n
·
−|
μ
n
|
A
n
−
1
A
n
−
1
·
(9.52)
with
2
10
A
n
−
1
Ω
and
1
2
n
−
1
·
Ω
1
/
+
n
−
1
1
/
2
W
n
=
− |
μ
n
|
Ω
n
−
1
=
Ω
(9.53)
1
/
2
n
−
1
a
n
, respectively complex autoregressive vector
and reflection coefficient (see Sect.
9.17.1
for more details).
In the framework of Information Geometry (
9.66
), we consider Information metric
defined as Kählerian metric where the Kähler potential is given by the Entropy of
the process
a
1
a
n
]
T
with
A
n
=[
···
and
μ
n
=
Φ(
(called Ruppeiner metric in Physics). We describe link with Rao
metric in Sect.
9.14
:
R
n
)
n
−
1
Φ(
det
R
−
1
n
2
R
n
)
=
log
(
)
−
n
log
(π
·
e
)
=
1
(
n
−
k
)
·
ln
[
1
−|
μ
k
|
]+
n
·
ln
[
π
·
e
·
P
0
]
k
=
(9.54)
Information metric is given by hessian of Entropy:
Φ
2
θ
(
n
)
=
P
0
μ
1
···
μ
n
−
1
T
∂
g
ij
≡
where
(9.55)
∂θ
(
n
)
i
∂θ
(
n
)
∗
j
n
−
1
k
{
μ
k
}
with
1
regularized Burg's reflection coefficient and
P
0
mean Power. Kählerian
metric is finally:
=
dP
0
P
0
2
n
−
1
θ
(
n
)
+
g
ij
d
2
|
d
μ
i
|
θ
(
n
)
=
ds
n
=
d
n
·
+
1
(
n
−
i
)
(9.56)
1
2
2
− |
μ
i
|
i
=
This is linked with general result on Bergman manifold and theory of homogeneous
complex manifolds. For complex manifold, where:
i
n
K
dz
1
∧···∧
dz
n
Ω
=
(
z
)
dz
1
∧···∧
dz
n
∧
(9.57)
is the given exterior differential form, the Hermitian differential form:
2
log
K
∂
(
z
)
ds
2
dz
i
dz
j
=
(9.58)
z
j
∂
z
i
∂
i
,
j
