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284 13 Compressive Data Gathering
Substitute Eq. (13.22) into Eq. (13.20), and sum up the interferences from all annu-
luses, we have:
P
f
=
∞
∑
i=1
P
f
(A
i
) <
∞
∑
i=1
2πP
0
(1 + δ)(i + 1)
((1 + δ)ri)
α
=
2πP
0
r
α
(1 + δ)
α−1
∞
∑
i=1
1
i
α−1
+
1
i
α
=
2πP
0
(ζ (α −1) + ζ (α))
r
α
(1 + δ)
α−1
, (13.23)
where ζ (·) is the Riemann Zeta function. When α > 2, ζ (α) <
π
2
6
, and ζ (α −1)
converges to a constant. Denote c
2
= ζ (α) + ζ(α −1). Then, when r = r
0
and δ =
δ
0
>
α−1
q
2πβ c
2
1−βr
α
N
0
/P
0
−1, Eq. (13.23) can be written into:
P
f
<
P
0
r
α
0
β
−N
0
. (13.24)
Substitute Eq. (13.19 and Eq. (13.24) into Eq. (13.18), we obtain SINR
j
> β . This
proves that a feasible scheduling under protocol model with r =