
Power Efficiency of Cooperative Transmission 77
where u>0,wehave
μ =
e
λπd
2
s,d
4
λπd
2
s,d
Γ
3
2
,
λπd
2
s,d
4
, (6.19)
which establishes the expectation of power ratio
E
[β]=
√
2e
λπd
2
s,d
4
K
λπd
2
s,d
Γ
3
2
,
λπd
2
s,d
4
, (6.20)
where
Γ(α, x)=
∞
x
e
−t
t
α−1
dt is the incomplete gamma function.
Theorem 6.4 The average power ratio of DAF cooperation relative to direct
transmission for α =2is
1
√
2K
π
4ρ
< E[β] <
1
√
2K
π
4ρ
+1
. (6.21)
Proof Let ρ := πλd
2
s,d
/4. From the definition of the incomplete gamma
function, we have
g := e
ρ
Γ
3
2
,ρ
=
∞
ρ
t
1
2
e
ρ−t
dt . (6.22)
1. Upper bound:
g =
∞
0
(ρ + s)
1
2
e
−s
ds <
∞
0
(ρ
1
2
+ s
1
2
)e
−s
ds = ρ
1
2
+Γ
3
2
.
(6.23)
2. Lower bound:
g>
∞
0
t
1
2
e
−t
dt =Γ
3
2
=
√
π
2
. (6.24)
Using the two