Digital Signal Processing Reference
In-Depth Information
A
n
−
1
0
A
(
−
)
n
A
n
=
+
μ
n
·
−
1
(11.13)
1
where
V
(
−
)
=
JV
∗
and
J
is the antidiagonal unit matrix,
μ
n
is reflection coefficient
defined in the unit disk with
|
μ
k
|
<
1
(
k
=
1
,
···
,
n
)
and computed by Regu-
larized Burg algorithm. Actually,
are uniquely determined by
the covariance matrix
R
n
once the radar data is given. Thus, the covariance matrix
of the radar data can be parameterized by the regularized reflection coefficients
(
μ
k
(
k
=
1
,
···
,
n
)
P
0
,μ
1
,
···
,μ
n
−
1
)
where
P
0
=
c
0
.
11.4.2 Kahler Metric on Reflection Coefficients
A seminal paper of Erich Kahler has introduced natural extension of Riemannian
geometry to Complex Manifold during 1930s of last century. Considering the
manifold
ζ
n
consisting of all the Toeplitz Hermitian Positive Definite Matrices
of dimension
n
, for any
R
n
θ
(
n
)
∈
ζ
n
, its coordinates can be expressed by
=
=[
θ
(
n
)
1
,θ
(
n
)
2
,
···
,θ
(
n
)
T
. Here we introduce kahler metric
on this manifold as the Hessian of Entropy [
10
]:
T
[
P
0
,μ
1
,
···
,μ
n
−
1
]
]
n
Φ
2
∂
Φ(
g
ij
=
with
R
)
=−
ln
(
|
R
|
)
−
n
ln
(π
e
)
(11.14)
∂
H
i
∂
H
j
Considering the block structure of
R
n
in According to the formula
|
G
|=|
a
|·|
B
−
, the entropy of the complex autoregressive process can
be expressed by the reflection coefficients as follows:
aV
+
WB
a
−
1
WV
+
|
if
G
=
n
−
1
ln
1
2
Φ(
R
n
)
=
1
(
n
−
k
)
·
−|
μ
k
|
+
n
ln
(π
eP
0
)
(11.15)
k
=
For complex autoregressive models, in the framework of Affine information geom-
etry, Kahler metric is defined as:
n
dP
0
P
0
2
n
n
−
1
2
|
d
μ
i
|
ds
n
=
g
i j
dz
i
d z
j
2
=
+
1
(
n
−
i
)
(11.16)
(
−|
μ
i
|
2
)
2
1
d
i
,
j
=
1
i
=
By integration then, the distance between two complex autoregressive models
G
A
=
[
P
0
,μ
1
,
···
,μ
n
T
P
0
,μ
1
,
···
,μ
n
T
1
]
and
G
B
=[
1
]
can be expressed by the
−
−
kahler metric as
