Digital Signal Processing Reference
In-Depth Information
⎡
⎣
⎤
⎦
P
(
y
1
|
x
1
)
P
(
y
1
|
x
1
) ...
P
(
y
1
|
x
1
)
y
1
|
y
1
|
y
1
|
P
(
x
1
)
P
(
x
1
) ...
P
(
x
1
)
[
P
(
Y
|
X
)
]
=
(8.17)
.
.
.
...
P
(
y
1
|
x
1
)
P
(
y
1
|
x
1
) ...
P
(
y
1
|
x
1
)
The matrix [
P
] is called the
channel matrix.
Since each input to the channel results in some output, each row of the channel
matrix must sum to unity. This means that
(
Y
|
X
)
n
P
y
j
|
x
i
=
1,
∀
i
(8.18)
j
=
1
Now, if the input probability P(X) is represented by the row matrix, then we have
[
P
(
X
)
]
=
[
P
(
x
1
)
P
(
x
2
) ...
P
(
x
m
)
]
(8.19)
Also, the output probability P(Y) is represented by the row matrix as follows:
=
P
y
n
)
[
P
(
Y
)
]
(
y
1
)
P
(
y
2
) ...
P
(
(8.20)
then
[
P
]
=
[
P
][
P
Y
|
X
]
(
Y
)
(
X
)
(
)
(8.21)
Now, P(X) is represented as a diagonal matrix, then
[
P
(
X
,
Y
)
]
=
[
P
(
X
)
][
P
(
Y
|
X
)
]
(8.22)
The matrix P(X, Y) is the
joint probability matrix.
8.6 Special Channels
There are some special channels also which are neither discrete nor exactly
continuous. These channels are having their own channel matrices.
8.6.1 Lossless Channel
In lossless channels, no source information is lost due to transmission [
2
,
5
]. This
channel descriptor or channel matrix holds only one non-zero element each column,
as shown in the Eq. (
8.23
) and Fig.
8.4
.
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