EE 376B/Stat 376B
Handout #15
Information Theory
Tuesday, May 16, 2006
Prof. T. Cover
Solutions to Homework Set #5
1.
Gaussian multiple access.
A group of
m
users, each with power
P
, is using a Gaussian multiple access channel
at capacity, so that
m
X
i
=1
R
i
=
C
mP
N
¶
,
(1)
where
C
(
x
) =
1
2
log(1 +
x
) and
N
is the receiver noise power.
A new user of power
P
0
wishes to join in.
(a) At what rate can he send without disturbing the other users?
(b) What should his power
P
0
be so that the new users rate is equal to the combined
communication rate
C
(
mP/N
) of all the other users?
Solution: Gaussian multiple access.
(a) If the new user is not to disturb other users, his message should be decodable at
first. Therefore,
R
=
C
P
0
mP
+
N
¶
.
(b) We need
C
P
0
mP
+
N
¶
=
C
mP
N
¶
,
or equivalently,
P
0
=
mP
(
mP
+
N
)
N
.
2.
Capacity of multiple access channels.
Find the capacity region for each of the following multiple access channels:
(a) Additive modulo 2 multiple access access channel.
X
1
∈ {
0
,
1
}
, X
2
∈ {
0
,
1
}
, Y
=
X
1
⊕
X
2
.
(b) Multiplicative multiple access channel.
X
1
∈ {
1
,
1
}
, X
2
∈ {
1
,
1
}
, Y
=
X
1
·
X
2
.
1
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Solution: Capacity of multiple access channels.
(a) Additive modulo 2 multiple access channel.
Quite clearly we cannot send at a total rate of more than 1 bit, since
H
(
Y
)
≤
1.
We can achieve a rate of 1 bit from sender 1 by setting
X
2
= 0, and similarly
we can send 1 bit/transmission from sender 2.
By simple time sharing we can
achieve the entire capacity region which is shown in Figure 1.

6
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
R
1
R
2
0
C
2
= 1
C
1
= 1
Figure 1: Capacity region of additive modulo 2 MAC.
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 Spring '05
 TomCover
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