network information theory chap6 课后题答案
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Solutions to Selected Problems in Chapter 6 1
SOLUTIONS TO SELECTED PROBLEMS IN CHAPTER 6
.. Provide an achievability proof for the treat-interference-as-noise inner bound (??).
.. Prove Sato’s outer bound (??).
.. Prove Lemma ??.
Solution: First note that the strong interference condition implies that
I(X
1
; Y
1
|X
2
, U) ≤ I(X
1
; Y
2
|X
2
, U)for all p(u)p(x
1
|u)p(x
2
|u). (.)
We will prove the lemma by induction on n.
If n = 1, the lemma is true following immediately from the strong interference
condition.
Suppose it is true when n = k, i.e. I(X
k
1
; Y
k
1
|X
k
2
)≤ I(X
k
1
; Y
k
2
|X
k
2
) , then
I(X
k+1
1
; Y
k+1
1
|X
k+1
2
)= I(X
k
1
; Y
k
1
|X
k+1
2
)+I(X
k
1
; Y
1,k+1
|Y
k
1
, X
k+1
2
)+I(X
1,k+1
; Y
k+1
1
|X
k
1
, X
k+1
2
)
= I(X
k
1
; Y
k
1
|X
k
2
)+0 +I(X
1,k+1
; Y
1,k+1
|X
2,k+1
, X
k
1
, X
k
2
, Y
k
1
)
(a)
≤ I(X
k
1
; Y
k
2
|X
k
2
)+I(X
1,k+1
; Y
2,k+1
|X
k
1
, X
k+1
2
, Y
k
1
)
= I(X
k+1
1
; Y
k+1
2
|X
k+1
2
) ,
where (a)comes from induction hypothesis and taking U =(X
k
1
, X
k
2
, Y
k
1
)in ().
.. Successive cancellation decoding. Consider a DM-IC p(y
1
, y
2
|x
1
, x
2
). Assume the
same codebook generation and encoding in Section ??. Consider the successive
cancellation decoding scheme, where receiver rst decodes M
2
and then decodes
its own message M
1
. Likewise receiver rst decodes M
1
and then decodes M
2
.
(a) Find the achievable rate region.
(b) Show that this region is always contained in the inner bound (??) in Section ??.
Solution:
(a) e decoding scheme is equivalent to successive decoding for two DM-MAC
p(y
1
|x
1
, x
2
)and p(y
2
|x
1
, x
2
)sharing the same inputs X
1
and X
2
. e rate con-
straints are
[ From the rst MAC: ]
R
1
≤ I(X
1
; Y
1
|X
2
, Q),
R
2
≤ I(X
2
; Y
1
|Q)
[From the second MAC: ]
R
1
≤ I(X
1
; Y
2
|Q),
R
2
≤ I(X
2
; Y
1
|X
1
, Q)
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