EE 478
Handout #5
Multiple User Information Theory
Thursday, October 2, 2008
Homework Set #2
Due: Thursday, October 9, 2008.
1.
Capacity with input cost.
Consider a DMC denoted by (
X
,p
(
y

x
)
,
Y
), and assume that
nonnegative cost
b
(
x
) is associated with each input symbol
x
∈ X
. The capacity of such
channel with input cost constraint
B
is given by
C
(
B
) =
max
X
:E(
b
(
X
))
≤
B
I
(
X
;
Y
)
.
(a) Show that the function
C
(
B
) is concave in
B
.
(b) The proof of the achievability is outlined in the lecture note. Complete the details of
the proof.
2.
Mutual information game.
Consider the following channel:
aA
±²
a
a
A
X
Y
Z
Throughout this problem, assume that
X
and
Z
are independent, E(
X
) = 0, E(
X
2
) =
P
,
E(
Z
) = 0, and E(
Z
2
) = 1. The channel capacity is given by
I
(
X
;
Y
) =
I
(
X
;
X
+
Z
).
Now for the game. The noise player chooses a distribution on
Z
to minimize
I
(
X
;
X
+
Z
)
,
while the signal player chooses a distribution on
X
to maximize
I
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 Spring '11
 Kelly
 Information Theory, UK, square error distortion, error distortion

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