International Journal of Antennas and Propagation
Each CUE is associated with the nearest BS. erefore, the
probability density function (PDF) of the distance
𝐵𝐶
can
be derived as
𝑑
𝐵𝐶
() = exp(−
𝐵
2
)2
𝐵
according to
the null probability of a D Poisson process []. Each DUE
transmitter and its DUE receiver have a xed distance of
𝐷𝐷
. Meanwhile, Rayleigh fading is assumed for the BS-CUE,
DUE-CUE, BS-DUE, and DUE-DUE links. We consider the
interference limited regime; that is, noise power is negligible
comparedtotheinterferencepower[]. In the following,
we will characterize the performance of beamforming and
interference cancellation strategies.
3. Beamforming Strategy
With multiple antennas, the BS chooses to increase its own
cellular user signal power by performing beamforming.
𝐶0
and
𝐷0
are the received signal at the typical CUE (
𝐶0
)and
typical DUE receiver (
𝐷0
), respectively,
𝐶0
=
𝑛:Γ
𝐵𝑛
∈𝜙
𝐵
𝐵
−𝑎/2
𝐵𝐶,𝑛
q
0𝑛
u
𝑛
𝐵𝑛
+
𝑛:Γ
𝐷𝑛
∈𝜙
𝐷
𝐷
−𝑎/2
𝐷𝐶,𝑛
0𝑛
𝐷𝑛
,
𝐷0
=
𝑛:Γ
𝐷𝑛
∈𝜙
𝐷
𝐷
−𝑎/2
𝐷𝐷,𝑛
0𝑛
𝐷𝑛
+
𝑛:Γ
𝐵𝑛
∈𝜙
𝐵
𝐵
−𝑎/2
𝐵𝐷,𝑛
g
0𝑛
u
𝑛
𝐵𝑛
,
()
where q
0𝑛
∈ C
1×𝑁
is the channel between BS in
𝐵
and
𝐶0
.
0𝑛
is the channel between
𝐶0
and DUE transmitter
in
𝐷
.
0𝑛
is the channel between
𝐷0
and DUE transmitter
in
𝐷
. g
0𝑛
∈ C
1×𝑁
is the channel between
𝐷0
and BS in
𝐵
.u
𝑛
∈ C
𝑁×1
isthebeamformerusedbytheth transmitter.
𝐵𝑛
and
𝐷𝑛
are the data signals transmitted from BS and
DUE, respectively.
𝐵𝑛
and
𝐷𝑛
are ∼CN(0,1).esignalto
interference ratio (SIR) in
𝐷0
with beamforming strategy is
SIR
𝐷
BF
=
𝐷
−𝑎
𝐷𝐷
00
2
𝐷
BF
,
()
where
𝐷
BF
denotes the total interference,
𝐷
BF,1
denotes the
interference from DUEs, and
𝐷
BF,2
denotes the interference
from BSs.
𝐷
BF,1
and
𝐷
BF,2
aresimpletransformof
𝐷
BF,1
and
𝐷
BF,2
by normalizing the desired signal power, respectively,
𝐷
BF
=
𝑛:Γ
𝐷𝑛
∈𝜙
𝐷
\{Γ
𝐷0
}
𝐷
−𝑎
𝐷𝐷,𝑛
0𝑛
2
𝐼
𝐷
BF,1
+
𝑛:Γ
𝐵𝑛
∈𝜙
𝐵
𝐵
−𝑎
𝐵𝐷,𝑛
g
0𝑛
u
𝑛
2
𝐼
𝐷
BF,2
,
𝐷
BF,1
=
𝑛:Γ
𝐷𝑛
∈𝜙
𝐷
\{Γ
𝐷0
}
−𝑎
𝐷𝐷,𝑛
0𝑛
2
−𝑎
𝐷𝐷
,
𝐷
BF,2
=
𝑛:Γ
𝐵𝑛
∈𝜙
𝐵
−𝑎
𝐵𝐷,𝑛
g
0𝑛
u
𝑛
2
−𝑎
𝐷𝐷
𝐷
/
𝐵
.
()
e signal to interference ratio (SIR) in
𝐶0
with beamform-
ingstrategyis
SIR
𝐶
BF
=
𝐵
−𝑎
𝐵𝐶
q
00
u
0
2
𝐶
BF
,
()
where
𝐶
BF
denotes the total interference at
𝐶0
and
𝐶
BF,1
denotes the interference from BSs.
𝐶
BF,2
denotes the inter-
ference from DUEs and
𝐶
BF,1
and
𝐶
BF,2
aresimpletransform
of
𝐶
BF,1
and
𝐶
BF,2
by normalizing the desired signal power,
respectively,
𝐶
BF
=
𝑛:Γ
𝐵𝑛
∈𝜙
𝐵
\{Γ
𝐵0
}
𝐵
−𝑎
𝐵𝐶,𝑛
q
0𝑛
u
𝑛
2
𝐼
𝐶
BF,1
+
𝑛:Γ
𝐷𝑛
∈𝜙
𝐷
𝐷
−𝑎
𝐷𝐶,𝑛
0𝑛
2
𝐼
𝐶
BF,2
,
𝐶
BF,1
=
𝑛:Γ
𝐵𝑛
∈𝜙
𝐵
\{Γ
𝐵0
}
−𝑎
𝐵𝐶,𝑛
q
0𝑛
u
𝑛
2
−𝑎
𝐵𝐶
,
𝐶
BF,2
=
𝑛:Γ
𝐷𝑛
∈𝜙
𝐷
−𝑎
𝐷𝐶,𝑛
0𝑛
2
−𝑎
𝐵𝐶
𝐷
/
𝐵
.
()
According to the BF criterion via receiving power maxi-
mization, the precoding vector u
𝑛
should align with the same
direction as the channel itself u
𝑛
= q
𝑜𝑛
/q
𝑜𝑛
2
; then the signal
power at the
𝐶0
is |q
00
u
0
|
2
∼
2
(2)and the interference
power at the
𝐶0
from other BS is |q
0𝑛
u
𝑛
|
2
∼
2
(2)as [].
Similarly, |g
0𝑛
u
0
|
2
∼
2
(2).Because
0𝑛
∼CN(0,1),wehave
|
0𝑛
|
2
∼
2
(2).
A performance metric of interest in this study is the
success transmission probability of BS
Suc.𝐶
(
𝐶
)with respect
to a predened SIR threshold
𝐶
, similarly, the success
transmission probability of DD network
Suc.𝐷
(
𝐷
)with
respect to a predened SIR threshold
𝐷
.eDDand
cellular network success probability are given in eorem .
eorem 1. For the cellular network underlay with D2D com-
munication, the cellular BSs are congured with antennas for
beamforming strategy. e cellular and D2D success probability
are given by
𝑆𝑢𝑐.𝐷
𝐵𝐹
𝐷
=exp −
𝑑1
2
𝐷
, ()
𝑆𝑢𝑐.𝐶
𝐵𝐹
𝐶
=
𝐵
0
+
𝑁−1
𝑘=1
𝑘
𝑗=1
(
−1
)
𝑘
!
𝐵
𝑘
𝑗
Γ+1
0
𝑗+1
,
()