
162 Energy-Efficient Cooperative Wireless Communication and Networks
d
s,d
d
s,r
d´
s,r
d´
r,d
d
r,d
d
r´
r
s
FIGURE 11.2 The position of a mapping relay.
From (7.6), we define f (a, b)=p
∗
+q
∗
.Since
∂f(a,b)
∂a
> 0 and
∂f(a,b)
∂b
> 0,
we can obtain
p
∗
+ q
∗
= f (a, b) >f(a
,b) >f(a
,b
)=p
∗
+ q
∗
,which
completes the proof.
2
Theorem 11.1 For path loss α =2, the best relay location that minimizes
β
∗
for the optimal DAF cooperation is at the destination.
Proof From Result 11.1, we can find that the relay location that minimizes
β
∗
for the optimal DAF cooperation is surely on the line sd, namely d
s,r
+
d
r,d
= d
s,d
. Bringing this result into (11.2), we can obtain the ratio for α =2:
β
∗