
144 Energy-Efficient Cooperative Wireless Communication and Networks
Proof Laneman [84] proposed an approximation for (10.6) but with the as-
sumption that each node has the same normalized power. We extend this
result to the s cenario where the normalized power of each node
P
C
i
, ∀i ∈
{s}∪K
s
can be different. Take one item from (10.7) and let δ =2
2R
− 1,
ρ
i
s = P
C
i
;thenwehave
lim
s→∞
s · Pr[P
C
i
|a
i,d
|
2
<δ] = lim
s→∞
s · Pr[ρ
i
s|a
i,d
|
2
<δ]
= lim
s→∞
s · Pr
1
|a
i,d
|
2
<
δ
ρ
i
s
2
= lim
s→∞
s ·
δ
ρ
i
s
d
α
i,d
=
δ
ρ
i
d
α
i,d
.
(10.8)
By applying Theorem 10.10.1 in Laneman [84], the result (10.8) being
utilized
K times, yields the approximation
out
= Pr
P
C
s
|a
s,d
|
2
+
i∈K
s
P
C
i
|a
i,d
|
2
< 2
2R
− 1
≈
(2
2R