Outage Performance of A Unified Non-Orthogonal
Multiple Access Framework
Xinwei Yue∗, Zhijin Qin† , Yuanwei Liu‡ , Xiaoming Dai§ , and Yue Chen‡
∗

Beihang University, Beijing, China
Lancaster University, Lancaster, UK
‡ Queen Mary University of London, London, UK
University of Science and Technology Beijing, Beijing, China
†

arXiv:1801.08181v1 [] 24 Jan 2018

§

Abstract—In this paper, a unified framework of non-orthogonal
multiple access (NOMA) networks is proposed, which can be
applied to code-domain NOMA (CD-NOMA) and power-domain
NOMA (PD-NOMA). Since the detection of NOMA users mainly
depend on efficient successive interference cancellation (SIC)
schemes, both imperfect SIC (ipSIC) and perfect SIC (pSIC)
are taken into considered. To characterize the performance of
this unified framework, the exact and asymptotic expressions
of outage probabilities as well as delay-limited throughput for
CD/PD-NOMA with ipSIC/pSIC are derived. Based on the
asymptotic analysis, the diversity orders of CD/PD-NOMA are
provided. It is confirmed that due to the impact of residual
interference (RI), the outage probability of the n-th user with
ipSIC for CD/PD-NOMA converges to an error floor in the
high signal-to-noise ratio (SNR) region. Numerical simulations
demonstrate that the outage behavior of CD-NOMA is superior
to that of PD-NOMA.

I. I NTRODUCTION
To enhance spectrum efficiency and massive connectivity,
non-orthogonal multiple access (NOMA) [1, 2] has been identified as one of the key technologies for the fifth generation
(5G) networks. The pivotal feature of NOMA is its capability
of sharing the same physical resource element (RE), where
multiple users’ signals are linearly superposed over different
power levels by using the superposition coding scheme. To
get the desired signal, multi-user detection algorithm [3], i.e.,
successive interference cancellation (SIC) or message passing
algorithm (MPA) is carried out at the receiver.
More particularly, based on spreading signature of multiple
access (MA), NOMA schemes can be divided into two categories: power-domain NOMA (PD-NOMA) and code-power
NOMA (CD-NOMA). In [4], two evaluation metrics of PDNOMA networks including outage probability and ergodic rate
have been proposed, where the outage behaviors of users and
ergodic rate have been discussed by applying stochastic geometry. From a practical perspective, the authors in [5] studied
the performance of PD-NOMA for the two-user case with
imperfect channel state information (CSI), where the closedform expressions of outage probability were derived. When
NOMA users have similar channel conditions, the authors of
[6] proposed a PD-NOMA based multicast-unicast scheme
and verified that the spectral efficiency of PD-NOMA based
multicast-unicast scheme is higher than that of orthogonal
multiple access (OMA) based one.

As a further advance, CD-NOMA is viewed as a special
extension of PD-NOMA, in which the data streams of multiple
users are directly mapped into multiple REs (or subcarriers
K) through the sparse matrix/codebook or low density spread
sequence. Actually, CD-NOMA mainly include sparse code
multiple access (SCMA), pattern division multiple access
(PDMA), multi-user sharing access (MUSA), etc. In [7], the
authors proposed a sub-optimal design approach to design the
sparse codebook of SCMA. On the condition of the fixed
sparse pattern matrix, the authors of [8] evaluated the link
level performance of PDMA and confirmed that PDMA can
achieve the higher spectrum efficiency than OMA. In [9],
MUSA is capable of adopting a grant-free scheme to support
Internet of Things (IoT) scenario. However, up to now, there is
no work investigating the performance of the unified NOMA
framework.
Driven by this, we investigate the outage performance of
the unified NOMA framework by invoking stochastic geometry. Since the detection of NOMA users mainly depend on
efficient SIC schemes, both imperfect SIC (ipSIC) and perfect
SIC (pSIC) are taken into considered. We derive the exact
expressions of outage probability for a pair of NOMA users
(i.e., the n-th user and m-th user) in the unified framework. To
obtain deep insights, we further derive the asymptotic outage
probability of two users and attain the corresponding diversity
orders. Due to the impact of residual interference (RI), the
outage behavior of the n-th user with ipSIC for CD/PDNOMA (CD-NOMA and PD-NOMA) converges to an error
floor. Furthermore, we confirm that the outage behavior of CDNOMA is superior to that of PD-NOMA. Additionally, we
analyze system throughput for CD/PD-NOMA in the delaylimited transmission.
II. N ETWORK M ODEL
A. Network Descriptions
Consider a unified NOMA downlink transmission scenario,
where a base station (BS) transmits the information to M
randomly users. The BS directly maps the data streams
of multiple users into K subcarriers or REs by utilizing
one sparse spreading matrix GK×M , in which there are a
few number of non-zero entries within it and satisfies the
relationship M > K. For simplicity, we assume that the
BS and NOMA users are equipped with a single antenna,

respectively. Assuming that the BS is located at the center
of circular cluster denoted as D, with radius RD and the
M NOMA users are uniformly distributed within circular
cluster [10]. To facilitate analysis, we assumed that M users
are divided into M/2 orthogonal pairs, in which the distant
user and the nearby user can be distinguished based on their
disparate channel conditions. Furthermore, each pair of users
is randomly selected to carry out the NOMA protocol [4, 11].
A bound pass model is employed to model the channel coefficients in networks from the BS to users. Meanwhile, these
wireless links are disturbed by additive white Gaussian noise
(AWGN) with mean power N0 . Without loss of generality, the
effective channel gains between the BS and users are sorted as
2
2
2
2
khM k2 > · · · > khn k2 > · · · > khm k2 > · · · > kh1 k2 [12,
13] with the assistance of order statistics. In this paper, we
focus on the m-th user paired with the n-th user for NOMA
transmission.

2

γn =

ρ kdiag (hn ) gn k2 an
2

̟ρ khI k2 + 1

Regarding the unified NOMA downlink transmission scenario, the BS transmits the superposed signals to multiple
users, where the data stream of each user spreads over one
column of sparse matrix. Hence the observation at the ϕ-th
user over K subcarriers yϕ = [yϕ1 yϕ2 · · · yϕK ]T is given by
p
p
yϕ =diag (hϕ ) (gn Ps an xn + gm Ps am xm ) + nϕ , (1)

where ϕ ∈ (n, m), xn and xm are the normalized unity
power signals for the n-th and m-th users, respectively, i.e,
E{x2n } = E{x2m } = 1. We assume the fixed power allocation coefficients satisfy the condition that am > an with
am + an = 1, which is for fairness considerations. Ps denotes
the normalized transmission power at BS. The sparse indicator
vector of the ϕ-th user is denoted by gϕ = [gϕ1 gϕ2 · · · gϕK ]T ,
which is one column of GK×M . More specifically, gϕk is
the subcarrier index, where gϕk = 1 indicates the signals
are mapped into the corresponding RE, otherwise, gϕk = 0.
Let hϕ = [h̃ϕ1 h̃ϕ2 · · · h̃ϕK ]T denotes the channel vector
between√the BS and ϕ-th user occupying K subcarriers with
ηh
h̃ϕk = √1+dϕkα , where hϕk ∼ CN (0, 1) is the Rayleigh fading
channel gain between the BS and ϕ-th user occupying the
k-th subcarrier, η is a frequency dependent factor, α is the
path loss exponent and d is the distance from BS to ϕ-th user.
nϕ ∼ CN (0, N0 IK ) denotes the AWGN.
To maximize the output SNRs and diversity orders, we
employ the maximal ratio combiner (MRC) ∗at the ϕ-th user
(diag(h )g )
over K subcarriers. Let uϕ = kdiag(hϕϕ )gϕϕ k , and then the
received signal at the ϕ-th user can be written as
p
p
ỹϕ = uϕ diag (hϕ ) (gn Ps an xn + gm Ps am xm ) + uϕ nϕ .
(2)
Based on aforementioned assumptions, the signal-plusinterference-to-noise ratio (SINR) at the n-th user to detect
the m-th user’s signal xm is given by
2

ρ kdiag (hn ) gm k2 am
ρ kdiag (hn ) gn k22 an + 1

,

(3)

,

(4)

where ̟= 0 and ̟= 1 denote the pSIC and ipSIC operations,
respectively. Note that hI = [hI1 hI2 · · · hIK ]T denotes the RI
channel vector at K subcarriers with hIk ∼ CN (0, ΩI ).
The SINR of the m-th NOMA user to decode the information of itself can be expressed as
γm =

ρ kdiag (hm ) gm k22 am
2

ρ kdiag (hm ) gn k2 an + 1

III. P ERFORMANCE

B. Signal Model

γn→m =

Ps
denotes the transmit SNR. For the sake of
where ρ = N
0
simplicity, assuming that gm and gn have the same column
weights for GK×M .
By applying SIC [14], the SINR of the n-th user, who needs
to decode the information of itself is given by

.

(5)

EVALUATION

In this section, the outage probability for a pair of NOMA
users is selected as a metric to evaluate the performance of
the unified downlink NOMA networks.
A. The outage probability of the m-th user
The outage event of the m-th user is that the m-th user cannot detect its own information. Hence the outage probability
of the m-th user for CD-NOMA can be expressed as
Pm,CD = Pr (γm < εm ) ,

(6)

where εm = 2Rm − 1 and Rm is the target rate of the m-th
user. The following theorem provides the outage probability
of the m-th user.
Theorem 1. The outage probability of the m-th user for CDNOMA is given by
M−m
X M − m (−1)p
Pm,CD ≈φm
m+p
p
p=0
" U
i !#m+p
K−1 
X
τ cu X 1
τ
c
u
,
×
bu 1 − e− η
i! η
u=1
i=0
(7)
εm
with am
>
ε m an ,
ρ(am −εm an )

(M−m)!
M−m
M!
φm
=
,
,
=
(M−m)!(m−1)!
p
p!(M−m−p)!
p
α
RD
π
2
bu = 2U 1 − θu (θu + 1), cu =1 +
2 (θu + 1) ,
π
and
U
is
a
parameter
to ensure a
θu = cos 2u−1
2U

where

τ

=

complexity-accuracy tradeoff.
Proof. See Appendix A.

Corollary 1. For the special case with K = 1, the outage
probability of the m-th user for PD-NOMA is given by
" U
#
M−m

 m+p
X M − m (−1)p X
τ cu
Pm,P D ≈ φm
.
bu 1 − e− η
p
m
+
p
p=0
u=1
(8)

B. The outage probability of the n-th user
As stated in [4, 15], the outage for the n-th user can happen
in the following two cases : 1) The n-th user cannot decode
the message of the m-th user; and 2) The n-th user can decode
the message of the m-th user, then carries out SIC operations,
but cannot decode the information of itself. Hence the outage
probability of the n-th user can be expressed as
Pn,CD = Pr {γn→m ≤ εm }
+ Pr {γn→m > εm , γn ≤ εn } ,

Corollary 3. The asymptotic outage probability of the m-th
user for CD-NOMA is given by
∞
Pm,CD

" U
#m
X b u  τ cu  K
M!
.
≈
(M − m)!m! u=1 K! η

(9)

where εn = 2Rn − 1 with Rn being the target rate at the n-th
user to detect the m-th user. The following theorem provides
the outage probability of the n-th user with ipSIC for CDNOMA.
Theorem 2. The outage probability of the n-th user with ipSIC
εn
n
, ϑ = ̟ε
for CD-NOMA is given by (10), where β = ρa
an and
n
̟ = 1.
Proof. See Appendix B.
Substituting ̟ = 0 into (10), the outage probability of the
n-th user with pSIC for CD-NOMA is given by
" U
M−n
X M − n (−1)p X
pSIC
Pn,CD ≈φn
bu
p
n + p u=1
p=0
!#n+p
K−1
X 1  βcu i
− βcηu
.
(11)
× 1−e
i! η
i=0
Corollary 2. For the special case with K = 1, the outage
probability of the n-th user with ipSIC for PD-NOMA is given
by

M−n 
p
φn X M − n (−1)
ipSIC
Pn,P
≈
D
p
ΩI p=0
n+p
"
#
Z ∞
U

 n+p
X
cu (ϑy+β)
− Ωy
bu 1 − e− η
×
e I
dy.
0

where P ∞ (ρ) denotes the asymptotic outage probability at
high SNR region.

u=1

(12)

Substituting ̟ = 0 into (12), the outage probability of the
n-th user with pSIC for PD-NOMA is given by
" U
M−n
X M − n (−1)p X
pSIC
Pn,P D ≈φn
bu
p
n + p u=1
p=0
in+p

βcu
.
(13)
× 1 − e− η

Proof. By definition, Θ1 = 1 − e

d = − lim

ρ→∞

∞

log (P (ρ))
,
log ρ

(14)

K−1
X
i=0

|


i
1 τ cu
. Applyi! η
{z
}
Θ2

ing power series expansion, the summation
term Θ2 can be
i
∞
τ cu
P
1 τ cu
η
rewritten as Θ2 = e
−
. Substituting Θ2 into
i!
η
i=K

Θ1 , when x → 0, Θ1 withthe approximation
of e−x ≈ 1 − x
K
τ cu
1
is formulated as Θ1 ≈ K!
. Furthermore, substituting
η
Θ1 into (7) and taking the first term (p = 0), we obtain (15).
The proof is completed.

For the special case with K = 1, the asymptotic outage
probability of the m-th user for PD-NOMA is given by
∞
Pm,P
D

" U
#m
X  τ cu 
M!
.
bu
≈
(M − m)!m! u=1
η

(16)

Remark 1. Upon substituting (15) and (16) into (14), the
diversity orders of the m-th user for CD-NOMA and PDNOMA are mK and m, respectively.
Corollary 4. The asymptotic outage probability of the n-th
user with ipSIC for CD-NOMA is given by
M−n
X M − n (−1)p Z ∞
φn
y K−1
n
+
p
p
(K − 1) ΩK
0
I p=0
" U
!#n+p


K−1
i
X
yϑcu X 1
yϑcu
− y
dy.
bu 1 − e− η
×e ΩI
i!
η
u=1
i=0
(17)

ipSIC,∞
Pn,CD
≈

Substituting ̟ = 0 into (17), the asymptotic outage probability of the n-th user with pSIC for CD-NOMA is given
by

C. Diversity Order Analysis

To obtain deep insights, diversity order analysis is present,
which highlights the slope of the curves for outage probabilities varying with the SNRs. The definition of diversity order
is given by

− τ cηu

(15)

pSIC,∞
Pn,CD

" U
#n
X bu  βcu K
M!
.
≈
(M − n)!n! u=1 K! η

(18)

Remark 2. Upon substituting (17) and (18) into (14), the
diversity orders of the n-th user with ipSIC/pSIC for CDNOMA are zero and nK, respectively.

ipSIC
Pn,CD

" U
M−n
X M − n (−1)p Z ∞
X
y
φn
K−1 − ΩI
≈
bu
y
e
p
n+p 0
(K − 1) ΩK
I p=0
u=1

Corollary 5. For the special case with K = 1, the asymptotic
outage probability of the n-th user with ipSIC for PD-NOMA
is given by

M−n 
p
φn X M − n (−1)
ipSIC,∞
Pn,P
≈
D
ΩI p=0
n+p
p
#
"
Z ∞
U

 n+p
X
yϑcu
− Ωy
×
e I
dy.
bu 1 − e− η
0

u=1

(19)

Substituting ̟ = 0 into (19), the asymptotic outage probability of the n-th user with pSIC for PD-NOMA is given
by
" U
#n
X  τ cu 
M!
pSIC,∞
Pn,P D ≈
.
(20)
bu
(M − n)!n! u=0
η

Remark 3. Upon substituting (19) and (20) into (14), the
diversity orders of the n-th user with ipSIC/pSIC for PDNOMA are zero and n, respectively.
D. Throughput Analysis

In this subsection, the system throughput of the unified
NOMA framework is characterized in delay-limited transmission mode. In this mode, the BS transmits information at
a constant rate R, which is subject to the effect of outage
probability. Hence the system throughput of CD/PD-NOMA
with ipSIC/pSIC is given by


ψ
Rm ,
(21)
Rφψ = (1 − Pm,φ ) Rn + 1 − Pn,φ

where ψ ∈ (ipSIC, pSIC), φ ∈ (CD, P D). Pm,CD and
ipSIC
pSIC
Pm,P D are given by (7) and (8), respectively. Pn,CD
, Pn,CD
,
ipSIC
pSIC
Pn,P D and Pn,P D are given by (10) and (11), (12) and (13),
respectively.
IV. N UMERICAL R ESULTS

In this section, simulation results are presented to verify
the analytical results derived in the above sections. In this
unified framework considered, we assume the power allocation
coefficients of a pair of users are am = 0.8 and an = 0.2,
respectively. The target rates are set to be Rn = Rm = 0.01
BPCU, where BPCU is short for bit per channel use. Setting
the pathloss exponent to be α = 2 and the system carrier
frequency is equal to 1 GHz. The complexity-vs-accuracy
tradeoff parameter is set to be N = 15. Without loss of generality, the OMA is selected to be a benchmark for comparison
purposes. Note that NOMA users with low target data rate
can be applied to the IoT scenarios, which require low energy
consumption, small packet service and so on.

1−e

− cu (ϑy+β)
η

K−1
X
i=0


i !#n+p
1 (ϑy + β) cu
dy.
i!
η
(10)

Fig. 1 plots the outage probability of a pair of NOMA
users (the m-th and n-th user) versus the transmit SNR with
pSIC/pSIC, where K = 2. The exact analytical curves for
the outage probability of the m-th user is plotted according
to (7). Furthermore, the exact analytical curves for the outage
probability of the n-th user with ipSIC/pSIC are plotted based
on (10) and (11), respectively. Obviously, the exact outage
probability curves match perfectly with the Monte Carlo
simulations results. We observed that the outage behavior of
conventional OMA is inferior to the n-th user with pSIC and
superior to the m-th user. This is due to the fact that NOMA
is capable of providing better fairness since multiple users are
served simultaneously, which is the same as the conclusions
in [4, 16].
Additionally, as can be seen from Fig. 1, the dashed curves
represent the asymptotic COP of the m-th user and n-th user
with pSIC, which can be obtained by numerically evaluating
(15) and (18). One can observe that the asymptotic outage
probabilities are approximated to the analytical results in the
high SNR region. The dotted curves represent the asymptotic
outage probabilities of the n-th user with ipSIC, which are
calculated from (17), respectively. It is shown that the outage
performance of the n-th user converges to an error floor and
obtain zero diversity order. Due to the influence of RI, the
outage behavior of the n-th user with ipSIC is inferior to
OMA. This is because that the RI signal from imperfect
cancellation operation is the dominant impact factor. With
the value of RI increasing from −30 dB to −20 dB, the
outage behavior of the n-th user is becoming more worse and
deteriorating. Hence the design of effective multiuser receiver
algorithm is significant to improve the performance of NOMA
networks in practical scenario.
Fig. 2 plots the outage probability versus SNR with the
different number of subcarriers (i.e., K = 3 and K = 1).
For the special case with K = 1, the unified framework of
NOMA becomes PD-NOMA. The exact outage probability
curve of the m-th user for PD-NOMA is plotted according
to (8). The exact outage probability curves of the n-th user
with ipSIC/pSIC are given by Monte Carlo simulations and
precisely match with the analytical expressions, which have
been derived in (12) and (13), respectively. The asymptotic
outage probabilities of this pair of users for PD-NOMA are
also approximated with the analytical results in the high SNR
region. We observe that CD-NOMA has a more steep slope
and can provide better outage performance than PD-NOMA.
This is due to the fact that CD-NOMA is capable of achieving
the higher diversity orders.
Fig. 3 plots the outage probabilities versus SNR for different
user target rates. We observe that with increasing target rates,

0

0

10

10

−1

−1

10
2

−2

E{||hI||2}= −30, −25 , −20 (dB)

10

−3

10

−4

10

−5

10

−6

10

Sim
OMA
Error floor
Asymptotic
Exact − User m
Exact − User n − pSIC
Exact − User n − ipSIC

0

10

20

Outage Probability

Outage Probability

10

R = R =1

−2

Rn = Rm = 0.01 BPCU

−3

10

2

E{||hI||2}= −30 (dB)

−4

10

Sim
Error floor
Asymptotic
Exact − User m
Exact − User n − pSIC
Exact − User n − ipSIC

10

−6

40

10

50

0

10

20

SNR (dB)

30

40

50

SNR (dB)

Fig. 1: Outage probability versus the transmit SNR, with K =
2, M = 3, n = 2, m = 1, RD = 2 m, Rn = Rm = 0.01 BPCU.

Fig. 3: Outage probability versus the transmit SNR, with M =
3, n = 2, m = 1, K = 2.

0

0.02

10

2

K=1

System Throughput (BPCU)

0.018

−2

Outage Probability

m

BPCU

10

−5

30

n

Rn = Rm = 0.1 BPCU

10

K=3
2

E{||hI||2}= −30 (dB)
−4

10

−6

10

0

Sim
OMA
Error floor
Asymptotic
Exact − User m
Exact − User n − pSIC
Exact − User n − ipSIC

10

20

E{||hI||2}= −30, −25 , −20 (dB)

0.016
0.014
0.012
0.01

CD−NOMA
(K = 2)

0.008
PD−NOMA
(K = 1)

0.006
0.004

NOMA − pSIC
NOMA − ipSIC

0.002
30

40

50

SNR (dB)

0
0

10

20

30

40

50

SNR (dB)

Fig. 2: Outage probability versus the transmit SNR, with M =
3, n = 2, m = 1, Rn = Rm = 0.01 BPCU.

Fig. 4: System throughput versus the transmit SNR, with M =
3, n = 2, m = 1, K = 2.

the lower outage probabilities are achieved. This is due to
the fact that the achievable rates are directly related to the
target SNRs. It is beneficial to detect the superposed signals
for the selected user pairing with smaller target SNRs. It
is worth pointing out that the impact of practical scenario
parameter frequency dependent factor η has been taken into
account in the unified NOMA framework. Furthermore, the
incorrect choice of Rn and Rm will lead to the improper
outage behavior for the unified framework.
Fig. 4 plots the system throughput versus SNR in the delaylimited transmission mode. The solid black curves represent
throughput of CD/PD-NOMA with ipSIC/pSIC, which can
be obtained from (21). The dash-dotted blue curves represent
throughput of CD-NOMA and PD-NOMA with ipSIC for the
different values of RI. We observe that CD-NOMA attains a
higher throughput compared to PD-NOMA, since CD-NOMA
has the smaller outage probabilities. This is due to that CDNOMA is capable of attaining the larger diversity order than

that of PD-NOMA. Another observation is that increasing the
values of RI from −30 dB to −20 dB will reduce the system
throughput in high SNR region. This is because that CD/PDNOMA converge to the error floors in the high SNR region.
V. C ONCLUSIONS
This paper has investigated the outage performance of a
unified NOMA framework insightfully by invoking stochastic
geometry. The exact expressions for outage probability of a
pair of users with ipSIC/pSIC have been derived. It has been
observed that the diversity orders of the m-th user for CD/PDNOMA are mK and m, respectively. However, due to the
influence of RI, the diversity orders achieved by the n-th
user with ipSIC are zeros for CD/PD-NOMA. On the basis
of analytical results, we observed that the outage behaviors
of CD-NOMA is superior to that of PD-NOMA. Additionally,
the system throughput of CD/PD-NOMA with ipSIC/pSIC has
been discussed in the delay-limited transmission mode.

A PPENDIX A: P ROOF

OF

T HEOREM 1

The proof starts by assuming gm and gn have the same col2
umn weights for GK×M . That is to say that kdiag (hm ) gm k2
2
and kdiag (hm ) gn k2 follow the same distribution. Hence
based on (5), the expression for outage probability of the m-th
user is rewritten as


εm
∆
(A.1)
=τ ,
Pm,CD = Pr Zm <
ρ (am − εm an )
where Zm = kdiag
observed that Y =

(hm ) gm k22
K
P

=

gmk h̃mk

η
1+dα
2

K
P

εm
where J1 = FZn (τ ), τ = ρ(am −ε
with am > εm an and
m an )
̟ = 1. Noting that YI is also subjective to a Gamma distribution with the parameters of (K, ΩI ) and the corresponding
PDF fYI is give by

f YI

−

y

y K−1 e ΩI
(y) =
.
(K − 1) !ΩK
I

(B.2)

After some mathematical manipulations, J2 is calculated as
J2 =Pr (τ < Zn < ϑYI + β)

2

gmk h̃mk . It is

k=1

is subject to a Gamma

k=1

distribution with the parameters of (K, 1). The corresponding
K−1
P yi
CDF of Y is given by FY (y) = 1 − e−y
i! .
i=0

In addition, on the basis of order statistics [12], the CDF
of the sorted channel gains between the BS and users over
K subcarriers has a specific relationship with the unsorted
channels, which can be expressed as follows:
M−m
im+p
X M − m (−1)p h
F ∼ (z)
, (A.2)
FZm (z) = φm
m+p Z
p
p=0

=

Z
|

0

∞

fYI (y) FZn (ϑy + β) dy −FZn (τ ) ,
{z
}

(B.3)

J3

n
and ϑ = ̟ε
where β =
an . Similar to the proving process
of (A.1), based on (B.2), J3 can be given by
M−n
X M − n (−1)p Z ∞
φn
− y
J3 ≈
y K−1 e ΩI
K
p
n+p 0
(K − 1) ΩI p=0
" U
!#n+p
K−1
X
X 1  (ϑy + β) cu i
− cu (ϑy+β)
η
×
bu 1 − e
dy.
i!
η
u=1
i=0
(B.4)

εn
ρan

where F ∼ (z) denotes the CDF of unsorted channels for the
Z
m-th user. Due to the assumption of homogeneous PPPs [10] Substituting (B.4) and (B.3) into (B.1), we can obtain (10) and
for randomly users and applying polar coordinate conversion, complete the proof.
the CDF F ∼ (z) is given by
R EFERENCES
Z
"
#
[1]
Y.
Cai,
Z.
Qin,
F.
Cui,
G.
Y. Li, and J. A. McCann, “Modulation and
i
Z RD
K−1 
α
z (1+rα ) X 1
z
(1
+
r
)
2
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Z
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(A.3)
Obviously, it is difficult to obtain effective insights from the
above integral. We employ the Gaussian-Chebyshev quadrature to provide an approximation of (A.3) and rewrite it as
follows:
!
U
K−1
X
X 1  zcu i
− zcηu
F ∼ (z) ≈
bu 1 − e
.
(A.4)
Z
i! η
u=1
i=0
Substituting (A.4) and (A.2) into (A.1), we can obtain (7) and
complete the proof.
A PPENDIX B: P ROOF

OF

kdiag (hn ) gn k22

T HEOREM 2

Denote Zn =
= kdiag (hn ) gm k22 and
2
YI = khI k2 , respectively. Substituting (3) and (4) into (9),
ipSIC
the COP of Pn,CD
can be expressed as


ρZn am
ipSIC
< εm
Pn,CD = Pr
ρZn an + 1
|
{z
}
J1


ρan Zn
ρZn am
(B.1)
> εm ,
< εn ,
+ Pr
ρZn an + 1
̟ρYI + 1
|
{z
}
J2

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