LI et al.: RESIDUAL TRANSCEIVER HARDWARE IMPAIRMENTS ON COOPERATIVE NOMA NETWORKS 683
E(x)=ˆr
2
i
Γ(μ
i
+2/α)/(μ
2/α
i
i
Γ(μ
i
)),whereˆr
i
is defined as
the ˆr
i
=
α
i
E(R
α
i
)-root mean of the amplitude of random
variable.
Using order statistics [36], the PDF and CDF of the ordered
variable, ˜ρ
sd
n
,aregivenby
f
˜ρ
sd
n
(x)
=Ξ
N−n
k=0
(−1)
k
N − n
k
f
ρ
sd
n
(x)
F
ρ
sd
n
(x)
n+k−1
,
(4)
F
˜ρ
sd
n
(x)
=Ξ
N−n
k=0
N − n
k
(−1)
k
n + k
F
ρ
sd
n
(x)
n+k
, (5)
where Ξ=
N!
(n−1)!(N−n)!
.
Without loss of generality, we assumed that the estimated
channel gain between S and D
n
are sorted as |ˆg
sd
1
|
2
≤
|ˆg
sd
1
|
2
≤···≤|ˆg
sd
N
|
2
[5].
A. Non-Cooperative NOMA
According to NOMA protocol, S sends
N
n=1
√
a
n
P
s
x
n
to
D
n
,wherex
n
is the message for the n-th user with E(|x
n
|
2
)=
1, P
s
is the average transmit power of S, a
n
is the power
allocation coefficient for ensuring the fairness among users
with a
1
>a
2
> ···>a
N
,and
N
n=1
a
n
=1. Thus, referring
to channel estimation error m ode in [37] and distortion noise
model in [23], the received signal at D
n
is expressed as
y
sd
n
=(ˆg
sd
n
+e
sd
n
)
N
n=1
a
n
P
s
x
n
+ η
sd
n
+ n
sd
n
, (6)
where η
sd
n
∼CN
0,κ
2
sd
n
P
s
is the aggregated distortion
noise from transceiver; κ
sd
n
represents the level of hardware
impairment at transceivers, which can be measured in practice
based on the error vector magnitude (EVM) [38]; n
sd
n
∼
CN (0,N
sd
n
) represents the additive white complex Gaussian
noise (AWGN). In addition, we define γ = P
s
/N
sd
n
.
According to NOMA pro tocol, SIC is carried out at users.
Considering imperfect SIC, the received signal-to-interference-
plus-noise ratio (SINR) for D
n
to decode D
j
’s message x
j
,
j ≤ n, can be expressed as
γ
sd,j→n
=
a
j
γ ˜ρ
sd
n
γ
˜
Δ
j
+Δ
j
+κ
2
sd
n
˜ρ
sd
n
+d
˜α
sd
n
σ
2
e
sd
n
γ
1+κ
2
sd
n
+1
,
(7)
where
˜
Δ
j
=
j−1
˜p=1
ξ
˜p
a
˜p
, 0 <ξ
˜p
< 1 indicates imperfect SIC,
ξ
˜p
=0and ξ
˜p
=1denote perfect SIC and no SIC, respec-
tively; Δ
j
=
N
p=j+1
a
p
, ˜α is the path loss parameter. If x
j
is detected under the condition of imperfect SIC, the r eceived
SINR at D
n
to detect its own message is given by
γ
sd
n
=
a
n
γ ˜ρ
sd
n
γ
˜
Δ
n
+Δ
n
+κ
2
sd
n
˜ρ
sd
n
+d
˜α
sd
n
σ
2
e
sd
n
γ
1+κ
2
sd
n
+1
,
(8)
where
˜
Δ
n
=
n−1
˜p=1
ξ
˜p
a
˜p
, Δ
n
=
N
q=n+1
a
q
.
B. Cooperative NOMA
For cooperative NOMA, the entire transmission process is
accomplished in two time slots.
The First Phase: S transmits the superposed signal to R
and D
n
according to the NOMA protocol. Thus, the received
signal at R and D
n
are g iven as
y
˜
i
=(ˆg
˜
i
+ e
˜
i
)
N
n=1
a
n
P
s
x
n
+ η
˜
i
+ n
˜
i
,
˜
i = sr, sd
n
,
(9)
where η
˜
i
∼CN
0,κ
2
˜
i
P
s
denote the distortion noise from
transceivers, n
˜
i
∼CN(0,N
˜
i
) is the AWGN noise.
At the receivers, the received SINR at D
n
to decode D
j
’s
message x
j
, j ≤ n under imperfect SIC, is expressed as
γ
sd,j→n
=
a
j
γ ˜ρ
sd
n
γ
˜
Δ
j
+Δ
j
+κ
2
sd
n
˜ρ
sd
n
+d
˜α
sd
n
σ
2
e
sd
n
γ
1+ κ
2
sd
n
+1
,
(10)
For the case of imperfect SIC, after D
j
’s message x
j
is
detected, the received SINR at D
n
to detect its own message
x
n
is given by
γ
sd
n
=
a
n
γ ˜ρ
sd
n
γ
˜
Δ
n
+Δ
n
+κ
2
sd
n
˜ρ
sd
n
+d
˜α
sd
n
σ
2
e
sd
n
γ
1+κ
2
sd
n
+1
,
(11)
The Second Phase: The relay amplifies and forwards the
received signal to the intended users, then the received signal
at D
n
is obtained as
y
rd
n
=(ˆg
rd
n
+ e
rd
n
)(Gy
sr
+ η
rd
n
)+n
rd
n
, (12)
where G
Δ
=
P
r
P
s
(1+κ
2
sr
)|ˆg
sr
|
2
+P
s
(1+κ
2
sr
)σ
2
e
sr
+N
sr
is the ampli-
fication factor, η
rd
n
∼CN
0,κ
2
rd
n
P
r
is th e aggregated
distortion noise from R and D
n
, n
rd
n
∼CN(0,N
rd
n
)
represents AWGN noise, P
r
is the transmit power at R.For
convenience, we have the following definitions: λ
sr
= P
s
/N
sr
and λ
rd
n
= P
r
/N
rd
n
as the SNR at S and R, respectively.
We further assume λ
sr
= c
1
γ, λ
rd
n
= c
2
γ,wherec
1
and c
2
are constants.
Here, the SIC can be invoked by D
n
for detecting D
j
. Thus,
the SINR for D
n
to decode D
j
’s message under the condition
of imperfect SIC can be given by, (13), as shown at the bottom
of the next page, where
d = κ
2
sr
+ κ
2
rd
n
+ κ
2
sr
κ
2
rd
n
,φ
1,n
= d
˜α
rd
n
c
2
γσ
2
e
rd
n
(1 + d)+κ
2
sr
+1
,
φ
2,n
= d
˜α
sr
c
1
γσ
2
e
sr
(1 + d)+κ
2
rd
n
+1
,
φ
3,n
= d
˜α
sr
d
˜α
rd
n
×
c
1
c
2
γ
2
σ
2
e
sr
σ
2
e
rd
n
(1 + d)
+ c
1
γσ
2
e
sr
1+κ
2
sr
+ c
2
γσ
2
e
rd
n
1+κ
2
rd
n
+1
.
Considering imperfect SIC, when the message x
j
is decoded
from the superposed message, the SINR for D
n
to decode its
own message is obtained as, (14), as shown at the bottom of
the n ext page.
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