network information theory chap5 课后题答案
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Solutions to Selected Problems in Chapter 5 1
SOLUTIONS TO SELECTED PROBLEMS IN CHAPTER 5
.. Verify that the characterizations of the capacity region for the BS-BC and the
AWGN-BC in (??) and (??), respectively, are convex.
.. Show that the converse for the degraded DM-BC can be proved with the auxiliary
random variable identication U
i
=(M
2
, Y
i−1
2
)or U
i
=(M
2
, Y
i−1
1
, Y
i−1
2
).
.. Prove the converse for the capacity region of the AWGN-BC by starting from the
single-letter characterization
R
1
≤I(X; Y
1
|U ),
R
2
≤I(U ; Y
2
)
for some F(u, x)with E(X
2
)≤P.
.. Show that if a DM-BC is degraded, then it is also less noisy.
.. Show that if a DM-BC is less noisy, then it is also more capable.
.. Given a DM-BC p(y
1
, y
2
|x), let D(p(x)) = I(X; Y
1
)−I(X; Y
2
), where p(x)is the
pmf of X. Show that if D(p(x))is convex in p(x), then Y
1
is less noisy than Y
2
.
.. Prove the classication of the BSC–BEC broadcast channel in Example ??.
.. Prove the converse for the capacity region of the less noisy DM-BC p(y
1
, y
2
|x).
(Hint: Follow the steps of the converse for the more capable DM-BC.)
.. Show that the t wo characterizations of the capacity region for the more capable
DM-BC in Section ?? are equal.
.. Another simple outer bound. Consider the DM-BC with three messages.
(a) Show that a rate triple (R
0
, R
1
, R
2
)is in the capacity region if
R
0
+R
1
≤I(X; Y
1
) ,
R
0
+R
2
≤I(X; Y
2
)
for some p(x).
(b) Show that this outer bound on the capacity region is tighter than the simple
outer bound in Figure ??. (Hint: Consider two antisymmetric Z-channels.)
(c) S et R
1
= R
2
= 0 in the outer bound in part (a) to show that the common-
message capacity is
C
0
=max
p(x)
min{I(X; Y
1
) , I(X; Y
2
)}.
Argue that C
0
is in general strictly smaller than min{C
1
, C
2
}.
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