EE 670 : Homework #4 Solution
Prof. Uf Tureli, Stevens Institute of Technology
email/tel/fax: utureli@stevenstech.edu, 201.216.5603/8246
Note: Cover and Thomas, Q. 8.6,8.12,9.1,9.3
1. Question:
Using two channels at once.
Consider two discrete memoryless channels
(
X
1
, p
(
y
1

x
1
)
,
Y
1
) and (
X
2
, p
(
y
2

x
2
)
,
Y
2
) with capacities
C
1
, and
C
2
respectively.
A new
channel (
X
1
×X
2
, p
(
Y
1

x
1
)
×
p
(
y
2

x
2
)
,
Y
1
times
Y
2
) is formed in which
x
1
∈ X
1
and
x
2
∈ X
2
are simultaneously sent, resulting in
y
1
, y
2
. Find the capacity of this channel.
Solution:
Suppose we are given two channels, (
X
1
, p
(
y
1

x
1
)
,
Y
1
) and (
X
2
, p
(
y
2

x
2
)
,
Y
2
)
which we can use simultaneously.
We can de±ne the product channel as the channel,
(
X
1
× X
2
, p
(
y
1
, y
2

x
1
, x
2
) =
p
(
y
1

x
1
)
p
(
y
2

x
2
)
,
Y
1
× Y
2
) To ±nd the capacity of the product
channel, we must ±nd the distribution of
p
(
x
1
, x
2
) on the input alphabet
X
1
× X
2
that
maximizes