Digital Signal Processing Reference
In-Depth Information
provided the integral exists. The joint probability Pr(
d
,
y
) can be factored as
Pr(
yjd
) Pr(
d
), revealing its dependence on the channel transition probability function
Pr(
yjd
).
A closed-form expression for the maximized mutual information is often elusive,
but can be obtained in some special cases. Perhaps the most common of these is a
linear model for which
y ¼Hdþb:
If the input and output sequences are continuous amplitude processes and the noise is
Gaussian and white, so that E(
bb
T
)
¼ s
2
I
, and the input is normalized to unit power,
so that
E
(
d
i
)
¼
1, the capacity follows the log-determinant formula [45, 46]
H
T
H
s
2
1
2
log
2
det
Iþ
Capacity
¼
:
(3
:
6)
This capacity bound applies to our setting once we recognize that the channel model
from (3.1) may be written in matrix form as
2
3
2
3
2
3
y
1
y
2
y
3
.
h
0
0
0
0
0
b
1
b
2
b
3
.
4
5
4
5
4
5
h
1
h
0
00
0
2
4
3
5
d
1
d
2
d
3
.
h
2
h
1
h
0
0
0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
¼
.
þ
.
.
h
L
1
h
0
y
L
1
y
L
.
h
L
0
b
L
1
b
L
.
.
0
h
L
h
L
1
h
0
d
N
.
.
.
.
.
.
|
{z
}
d
.
.
.
.
.
.
.
.
y
LþN
1
b
LþN
1
0
000
h
L
|
{z
}
y
|
{z
}
b
|
{z
}
H
in which the matrix
H
, of dimensions (
LþN
2
1)
N, is a convolution matrix. Its
grammian
H
T
H
is thus a symmetric Toeplitz matrix built from the channel autocorre-
lation sequence
2
4
3
5
r
0
r
1
r
N
1
.
.
.
.
r
1
r
0
with
r
k
¼
X
j
H
T
H¼
,
h
j
h
jþk
:
.
.
.
.
.
.
.
r
1
r
N
1
r
1
r
0
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