Non-orthogonal Multiple Access in Large-Scale Underlay Cognitive Radio Networks

0.21 MB·

by  Yuanwei Liu

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Non-orthogonal Multiple Access in Large-Scale
Underlay Cognitive Radio Networks
Yuanwei Liu, Zhiguo Ding, Maged Elkashlan, and Jinhong Yuan
Abstract
In this paper, non-orthogonalmultiple access (NOMA) is applied to large-scale underlaycognitive radio (CR)
networks with randomly deployed users. In order to characterize the performance of the considered network,
6 new closed-form expressions of the outage probability are derived using stochastic-geometry. More importantly,
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0 by carrying out the diversity analysis, new insights are obtained under the two scenarios with different power
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constraints: 1) fixed transmit power of the primary transmitters (PTs), and 2) transmit power of the PTs being
n
a
J proportionaltothatofthesecondarybasestation.Forthefirstscenario,a diversityorderofmisexperiencedatthe

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1 m-th ordered NOMA user. For the second scenario, there is an asymptotic error floor for the outage probability.

] Simulationresultsareprovidedtoverifytheaccuracyofthederivedresults.Apivotalconclusionisreachedthatby
T
I carefullydesigningtargetdataratesandpowerallocationcoefficientsofusers,NOMAcanoutperformconventional
.
s
c
orthogonalmultiple access in underlay CR networks.
[

1 Index Terms
v
3
Cognitive radio, large-scale network, non-orthogonalmultiple access, stochastic geometry
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6
I. INTRODUCTION
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0
1. Spectrum efficiency is of significant importance and becomes one of the main design targets for future
0
6 fifth generation networks. Non-orthogonal multiple access (NOMA) has received considerable attention
1
:
v because of its potential to achieve superior spectral efficiency [1]. Particularly, different from conventional
i
X
multipleaccess (MA) techniques, NOMA uses the power domain to serve multipleusers at different power
r
a
levels in order to use spectrum more efficiently. A downlink NOMA and an uplink NOMA are considered
in [2] and [3], respectively. The application of multiple-input multiple-output (MIMO) techniques to
NOMA has been considered in[4] by usingzero-forcing detection matrices.The authorsin [5] investigated
an ergodic capacity maximization problem for MIMO NOMA systems.
Y.LiuandM.ElkashlanarewithQueenMaryUniversityof London, London, UK(email:{yuanwei.liu, maged.elkashlan}@qmul.ac.uk).
Z. Ding is with Lancaster University, Lancaster, UK (e-mail: [email protected]).
J. Yuan is with the University of New South Wales, Sydney, Australia (e-mail: [email protected]).
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Another approach to improve spectrum efficiency is the paradigm of underlay cognitive radio (CR)
networks, which was proposed in [6] and has rekindled increasing interest in using spectrum more
efficiently. The key idea of underlay CR networks is that each secondary user (SU) is allowed to access
the spectrum of the primary users (PUs) as long as the SU meets a certain interference threshold in the
primary network (PN). In [7], an underlay CR network taking into account the spatial distribution of the
SU relays and PUs was considered and its performance was evaluated by using stochastic geometry tools.
In [8], a new CR inspired NOMA scheme has been proposed and the impact of user pairing has been
examined, by focusing on a simple scenario with only one primary transmitter (PT).
By introducing the aforementioned two concepts, it is natural to consider the application of NOMA in
underlay CR networks using additional power control at the secondary base station (BS) to improve the
spectral efficiency. Stochastic geometry is used to model a large-scale CR network with a large number of
randomly deployed PTs and primary receivers (PRs). We consider a practical system design as follows:
1) All the SUs, PTs, and PRs are randomly deployed based on the considered stochastic geometry model;
2) Each SU suffers interference from other NOMA SUs as well as the PTs; and 3) The secondary BS
must satisfy a predefined power constraint threshold to avoid interference at the PRs. New closed-form
expressions of the outage probability of the NOMA users are derived to evaluate the performance of
the considered CR NOMA network. Moreover, considering two different power constraints at the PTs,
diversity order1 analysis is carried out with providing important insights: 1) When the transmit power of
the PTs is fixed, the m-th user among all ordered NOMA user experiences a diversity order of m; and 2)
When the the transmit power of the PTs is proportional to that of the secondary BS, an asymptotic error
floor exists for the outage probability.
II. NETWORK MODEL
We consider a large-scale underlay spectrum sharing scenario consisting of the PN and the secondary
network (SN). In the SN, we consider that a secondary BS is located at the origin of a disc, denoted by
D with radius R as its coverage. The M randomly deployed secondary users are uniformly distributed
D
1Diversity order is defined as the slope for the outage provability curve decreasing with the signal-to-noise-ratio (SNR). It measures the
number of independent fading paths over which the data is received. In NOMA networks, since users’ channels are ordered and SIC is
applied at each receiver, it is of importance to investigate the diversity order.
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within the disc which is the user zone for NOMA. The secondary BS communicates with all SUs within
the disc by applying the NOMA transmission protocol. It is worthy pointing out that the power of the
secondary transmitter is constrained in order to limit the interference at the PRs. In the PN, we consider
a random number of PTs and PRs distributed in an infinite two dimensional plane. The spatial topology
of all the PTs and PRs are modeled using homogeneous poisson point processes (PPPs), denoted by Φ
b
and Φ with density λ and λ , respectively. All channels are assumed to be quasi-static Rayleigh fading
ℓ b ℓ
where the channel coefficients are constant for each transmission block but vary independently between
different blocks.
According to underlay CR, the transmit power P at the secondary BS is constrained as follows:
t
I
p
P = min ,P , (1)
t max|g |2 s
ℓ
ℓ∈Φℓ 
where I is the maximum permissible interference power at the PRs, P is maximum transmission power
p   s
at the secondary BS, |g |2 = |gˆ |2L(d ) is the overall channel gain from the secondary BS to PRs ℓ. Here,
ℓ ℓ ℓ
gˆ is small-scale fading with gˆ ∼ CN(0,1), L(d ) = 1 is large-scale path loss, d is the distance
ℓ ℓ ℓ 1+dα ℓ
ℓ
between the secondary BS and the PRs, and α is the path loss exponent. A bounded path loss model is
used to ensure the path loss is always larger than one even for small distances [2,9].
According to NOMA, the BS sends a combination of messages to all NOMA users, and the observation
at the m-th secondary user is given by
M
y = h a P x +n , (2)
m m n t n m
n=1
Xp
where n is the additive white Gaussian noise (AWGN) at the m-th user with variance σ2, a is the
m n
power allocation coefficient for the n-th SU with M a = 1, x is the information for the n-th user,
n=1 n n
and h is the channel coefficient between the m-thPuser and the secondary BS.
m
For the SUs, they also observe the interferences of the randomly deployed PTs in the PN. Usually, when
the PTs are close to the secondary NOMA users, they will cause significant interference. To overcome
this issue, we introduce an interference guard zone D to each secondary NOMA user with radius of d ,
0 0
which means that there is no interference from PTs allowed inside this zone [10]. We assume d ≥ 1 in
0
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this paper. The interference links from the PTs to the SUs are dominated by the path loss and is given by
I = L(d ), (3)
B b
bX∈Φb
where L(d ) = 1/(1+dα) is the large-scale path loss and d is the distance from the PTs to the SUs.
b b b
Without loss of generality, all the channels of SUs are assumed to follow the order as |h |2 ≤ |h |2 ≤
1 2
··· ≤ |h |2. The power allocation coefficients are assumed to follow the order as a ≥ a ≥ ··· ≥ a .
M 1 2 M
According to the NOMA principle, successive interference cancelation (SIC) is carried out at the receivers
[11]. It is assumed that 1 ≤ j ≤ m < i. In this case, the m-th user can decode the message of the j-th
user and treats the message for the i-th user as interference. Specifically, the m-th user first decodes
the messages of all the (m−1) users, and then successively subtracts these messages to obtain its own
information. Therefore, the received signal-to-interference-plus-noise ratio (SINR) for the m-th user to
decode the information of the j-th user is given by
|h |2γ a
m t j
γ = , (4)
m,j
M
|h |2γ a +ρ I +1
m t i b B
i=j+1
P
where γ = min ρp ,ρ , ρ = Ip,ρ = Ps,ρ = PB, and P is the transmitpower of the PTs, |h |2
t (ℓm∈aΦxℓ|gℓ|2 s) p σ2 s σ2 b σ2 B m
is the overall ordered channel gain from the secondary BS to the m-th SU. For the case m = j, it indicates
the m-th user decodes the message of itself. Note that the SINR for the M-th SU is γ = |hM|2γtaM.
M,M ρbIB+1
III. OUTAGE PROBABILITY
In this section, we provide exact analysis of the considered networks in terms of outage probability. In
NOMA, an outage occurs if the m-th user can not detect any of the j-th user’s message, where j ≤ m
due to the SIC. Denote X = |hm|2γt. Based on (4), the cumulative distribution function (CDF) of X is
m ρbIB+1 m
given by
|h |2γ
m t
F (ε) = Pr < ε . (5)
Xm ρ I +1
( b B )
We denote ε = τ / a −τ M a for j < M, τ = 2Rj − 1, R is the target data rate for the
j j j j i=j+1 i j j
(cid:16) (cid:17)
j-th user, ε = τ /a , and εPmax = max{ε ,ε ,...,ε }. The outage probability at the m-th user can be
M M M m 1 2 m
expressed as follows:
P = Pr{X < εmax} = F (εmax), (6)
m m m Xm m
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where the condition a − τ M a > 0 should be satisfied due to applying NOMA, otherwise the
j j i=j+1 i
outage probability will alwayPs be one [2].
We need calculate the CDF of X conditioned on I and γ . Rewrite (5) as follows:
m B t
(ρ I +1)ε
b B
F (ε) = F , (7)
Xm|IB,γt |hm|2 γ
(cid:18) t (cid:19)
where F is the CDF of h . Based on order statistics [12] and applying binomial expansion, the CDF
|hm|2 m
of the ordered channels has a relationship with the unordered channels as follows:
M−m p m+p
M −m (−1)
F (y) = ψ F (y) , (8)
|hm|2 m p m+p |h˜|2
p=0 (cid:18) (cid:19) (cid:18) (cid:19)
X
2 2
where y = (ρbIB+1)ε , ψ = M! , and h˜ = hˆ L(d) is the unordered channel gain of an
γt m (M−m)!(m−1)!
(cid:12) (cid:12) (cid:12) (cid:12)
arbitrary SU. Here, hˆ is the small-scale fading co(cid:12)effi(cid:12)cient(cid:12)w(cid:12)ith hˆ ∼ CN(0,1), L(d) = 1 is the large-
(cid:12) (cid:12) (cid:12) (cid:12) 1+dα
scale path loss, and d is a random variable representing the distance from the secondary BS to an arbitrary
SU.
Then using the assumptionof homogenous PPP and applying the polar coordinates, we express F (y)
|˜h|2
as follows:
2 RD
F (y) = 1−e−(1+rα)y rdr. (9)
|˜h|2 R2
D Z0
(cid:0) (cid:1)
Note that it is challenging to obtain an insightful expression for the unordered CDF. As such, we apply
the Gaussian-Chebyshev quadrature [13] to find an approximation for (9) as
N
F (y) ≈ b e−cny, (10)
|˜h|2 n
n=0
X
N
where N is a complexity-accuracy tradeoff parameter, b = −ω 1−φ2 (φ +1), b = − b , c =
n N n n 0 n n
n=1
1+ RD (φ +1) α, ω = π, and φ = cos 2n−1π . p P
2 n N N n 2N
Su(cid:0)bstituting (10(cid:1)) into (8) and applying the(cid:0)multin(cid:1)omial theorem, we obtain
M−m p N N
F|hm|2 (y) = ψm Xp=0 (cid:18)M −p m(cid:19)m(−+1)p q0+···+XqN=m+p(cid:18)q0 +m··+·+p qN(cid:19) nY=0bqnn!e−nP=0qncny. (11)
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where m+p = (m+p)!. Based on (11), the CDF of X can be expressed as follows:
q0+···+qN q0!···qN! m
(cid:0) (cid:1) ∞ ∞ (ρ x+1)ε
b j
F (ε ) = F f (x)f (z)dxdz
Xm j |hm|2 z IB γt
Z0 Z0 (cid:18) (cid:19)
M−m p N
M −m (−1) m+p
=ψ bqn
m p m+p q +···+q n
Xp=0 (cid:18) (cid:19) q0+···+XqN=m+p(cid:18) 0 N(cid:19) nY=0 !
× ∞e−εzj nPN=0qncn ∞e−xρzbεj nPN=0qncnfIB (x)dxfγt (z)dz, (12)
Z0 Z0
Q2
| Q1 {z }
where fγt is the PDF of γt|. We express Q2 in (12) a{szfollows: }
Q2 = Z0∞e−xρbzεj nPN=0qncnfIB (x)dx = EΦb(e−xρzbεj nPN=0qncn) = LIB xρzbεj nN=0qncn!. (13)
X
In this case, the Laplace transformation of the interferences from the PT can be expressed as [10]
LIB (s) = exp −λbπ e−sd−0α −1 d20 +sδγ 1−δ,sd−0α
(cid:16) h(cid:16) (cid:17) i(cid:17)
(cid:0) (cid:1)
sd−α
= exp−λbπ e−sd−0α −1 d20 +sδ 0 t−δe−tdt, (14)
Z0
(cid:16) (cid:17)
  
  Θ 
  
where δ = 2 and γ(·) is the lower incomplete Gamma function.| {z }
α
To obtain an insightful expression, we use Gaussian-Chebyshev quadrature to approximate the lower
incomplete Gamma function in (14), Θ can be expressed as follows:
L
Θ ≈ s1−δ βle−tlsd−0α, (15)
l=1
X
where Lisacomplexity-accuracytradeoffparameter, β = 1d2−αω 1−θ2t −δ , t = 1 (θ +1), ω = π,
l 2 0 L l l l 2 l L L
and θ = cos 2l−1π . Substituting (15) into (14), we approximate tphe Laplace transformation as follows:
l 2L
(cid:0) (cid:1) −λbπ e−sd−0α−1 d20+s L βle−tlsd−0α
LIB (s) ≈e (cid:18) (cid:19) lP=1 !. (16)
Substituting (16) into (12), Q is given by
2
−λbπe−ρbεjzd−0αnPN=0qncn−1d20+ρbzεj N qncn L βle−tlzρdbα0εj nPN=0qncn
n=0 l=1
Q2 = e   P P . (17)
  
The following theorem provides the PDF of γ .
t
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Theorem 1: Consider the use of the composite channel model with Rayleigh fading and path loss, the
PDF of the effective power of the secondary BS is given by
fγt (x) = e−aℓρδse−ρxpDirac(x−ρs)+ ρxp +δ aℓxδ−1e−aℓxδe−ρxp−ρxpU (ρs −x), (18)
(cid:16) (cid:17)
where a = δπλℓΓ(δ), U (·) is the unit step function, and Dirac(·) is the impulse function.
ℓ ρδ
p
Proof: See Appendix A.
Substituting (18) and (17) into (12), we express Q as follows:
1
−aℓρδse−ρρps−εjnPN=ρ0sqncn−λbπe−ρρsbdεα0j nPN=0qncn−1d20+ρρbsεj N qncn L βle−tρlsρbdεα0j nPN=0qncn
n=0 l=1
Q1 =e   P P 
  
N
ρs ρ ρp ρp+εj P qncn
+ aℓ p +δ zδ−1e−aℓzδe− z − nz=0 Q2dz. (19)
z
Z0 (cid:16) (cid:17)
Ψ
We notice that it is very challenging to solve the integral Ψ in (19), therefore, we apply the Gaussian-
| {z }
Chebyshev quadrature to approximate the integral as follows:
K −ρp+εjρnsPN=sk0qncn−λbπe−ρsρsbkεdjα0 nPN=0qncn−1d20+ρρsbsεkj nN=0qncnl=L1βle−ρtslsρkbεdjα0 nPN=0qncn
Ψ ≈ ηke   P P , (20)
  
k=1
X
where K is a complexity-accuracy tradeoff parameter, ω = π, ϕ = cos 2k−1π , s = 1 (ϕ +1), and
K K k 2K k 2 k
ρp
ηk = ω2K 1−ϕ2k ρρsspk +δ aℓρδsskδ−1e−aℓρδssδke−ρssk. (cid:0) (cid:1)
(cid:16) (cid:17)
Substiptuting (19) and (20) into (12) and applying ε → ε , based on (6), we obtain the closed-form
max j
expression of the outage probability at the m-th user as follows:
M−m p N
M −m (−1) m+p
P = ψ bqn
m m p m+p q +···+q n
Xp=0 (cid:18) (cid:19) q0+···+XqN=m+p(cid:18) 0 N(cid:19) nY=0 !
−aℓρδse−ρρps−εmaxnρPN=s0qncn−λbπe−ρbρεsmdα0axnPN=0qncn−1d20+ρbερmsax N qncn L βle−tlρρbsεdmα0ax nPN=0qncn
×e   nP=0 lP=1 
  



+K ηke−ρp+εmaρxssnkPN=0qncn−λbπe−ρρbsεsmkdaα0x nPN=0qncn−1d20+ρbρεsmskax nPN=0qncnlP=L1βle−tlρρsbsεkmdα0axnPN=0qncn. (21)
  
Xk=1 



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IV. DIVERSITY ANALYSIS
Based on the analytical results for the outage probability in (21), we aim to provide asymptoticdiversity
analysis for the ordered NOMA users. The diversity order of the user’s outage probability is defined as
logP (ρ )
m s
d = − lim . (22)
ρs→∞ logρs
A. Fixed Transmit Power at Primary Transmitters
In this case, we examine the diversity with the fixed transmit SNR at the PTs (ρ ), while the transmit
b
SNR of secondary BS (ρ ) and the maximum permissible interference constraint at the PRs (ρ ) go to the
s p
infinity. Particularly, we assume ρ is proportional to ρ , i.e. ρ = κρ , where κ is a positive scaling factor.
p s p s
This assumption applies to the scenario where the PRs can tolerate a large amount of interference from the
secondary BS and the target data rate is relatively small in the PN. Denote γ = γt = min κ ,1 ,
t∗ ρs (ℓm∈aΦxℓ|gℓ|2 )
similar to (8), the ordered CDF has the relationship with unordered CDF as
M−m p m+p
M −m (−1)
F∞ (y∗) = ψ F∞ (y∗) , (23)
Xm|IB,γt∗ m p m+p |h˜|2
p=0 (cid:18) (cid:19) (cid:18) (cid:19)
X
where y∗ = (ρbIB+1)εj. When ρ → ∞, we observe that y∗ → 0. In order to investigate an insightful
ρsγt∗ s
expression to obtain the diversity order, we use Gaussian-Chebyshev quadrature and 1 − e−y∗ ≈ y∗ to
approximate (9) as
N
F∞ (y∗) ≈ χ y∗, (24)
|h˜|2 n
n=1
X
where χ = ω 1−φ2 (φ +1)c . Substituting (24) into (23), since y∗ → 0, we obtain
n N n n n
p (ρ I +1)ε m (ρ I +1)ε m
F∞ (ε ) = ξ b B j +o b B j , (25)
j
Xm|IB,γt∗ (cid:18) ρsγt∗ (cid:19) (cid:18)(cid:18) ρsγt∗ (cid:19) (cid:19)
m
N
ψm χn
where ξ = n=1 ! . Based on (6), (11), and (25), the asymptotic outage probability is given by
Pm
1 ∞ ∞ (ρ x+1)ε m
P∞ ≈ ξ b max f (x)f (z)dxdz, (26)
mF ρm z IB γt∗
s Z0 Z0 (cid:18) (cid:19)
C
where f the PDF of γ . Since C is a constant independent of ρ , (26) can be expressed as follows:
γt∗ t∗ | {z s }
1
P∞ = C +o ρ −m , (27)
mF ρm s
s
(cid:0) (cid:1)
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Substituting (27) into (22), we obtain the diversity order of this case is m. This can be explained as
follows. Note that SIC is applied at the ordered SUs. For the first user with the poorest channel gain, no
interference cancelation is operated at the receiver, therefore its diversity gain is one. While for the m-th
user, since the interferences from all the other (m−1) users are canceled, it obtains a diversity of m.
B. Transmit Power of Primary Transmitters Proportional to that of Secondary Ones
In this case, we examine the diversity with the transmit SNR at the PTs (ρ ) is proportional to the
b
transmit SNR of secondary BS (ρ ). Particularly, we assume ρ = νρ , where ν is a positive scaling
s b s
factor. We still assume ρ is proportional to ρ . Applying ρ → ∞, ρ = κρ and ρ = νρ to (21), we
p s s p s b s
obtain the asymptotic outage probability of the m-th user in this case as follows:
M−m p N
M −m (−1) m+p
P∞ ≈ψ bqn
mP m p m+p q +···+q n
Xp=0 (cid:18) (cid:19) q0+···+XqN=m+p(cid:18) 0 N(cid:19) nY=0 !
−a∞ℓ e−κ−λbπe−νεdmα0ax nPN=0qncn−1d20+νεmax N qncn L βle−tlνdεα0max nPN=0qncn
×e   nP=0 lP=1 
  



+K ηk∞e−sκk−λbπe−νsεkmdaα0xnPN=0qncn−1d20+νεsmkax nPN=0qncnlP=L1βle−tlsνkεdmα0ax nPN=0qncn. (28)
  
Xk=1 


where a∞ = δπλℓΓ(δ) and η∞ = ωK 1−ϕ2 κ +δ a∞sδ−1e−a∞ℓ sδke−sκk. 
ℓ κδ k 2 k sk ℓ k
(cid:16) (cid:17)
It is observed that P∞ is a copnstant independent of ρ . Substituting (28) into (22), we find that
mP s
asymptotically there is an error floor for the outage probability of SUs.
V. NUMERICAL RESULTS
In this section, numerical results are presented to verify the accuracy of the analysis as well as to obtain
more important insights for NOMA in large-scale CR networks. In the considered network, the radius of
the guard zone is assumed to be d = 2 m. The Gaussian-Chebyshev parameters are chosen with N = 5,
0
K = 10, and L = 10. Monte Carlo simulation results are marked as “•” to verify our derivation.
Fig. 1 plots the outage probability of the m-th user for the first scenario when ρ is fixed and ρ is
b p
proportional to ρ . In Fig. 1(a), the power allocation coefficients are a = 0.5, a = 0.4 and a = 0.1.
s 1 2 3
The target data rate for each user is assumed to be all the same as R = R = R = 0.1 bit per channel
1 2 3
use (BPCU). The dashed and solid curves are obtained from the analytical results derived in (21). Several
10
100 100
10−1
bility 10−2 m=1 bility 10−1
a a
b b
pro pro10−2 α=3
e 10−3 m=2 e
g g
a a NOMA m=1
Out Analytical RD=10 m m=3 Out10−3 NOMA m=2
10−4 ASinmaulylatitcioaln RD=12 m OOMMAA mm==12 α=4
10−4
0 10 20 30 40 50 0 10 20 30 40 50
SNR (dB) SNR (dB)
(a) For different user zone, with λ = 10−3, λ = 10−3, κ = 1, (b) For different α, with λ = 10−3, λ = 10−3, κ= 1, R =
b ℓ b ℓ D
α=4, ρ =20 dB, and M =3. 5 m, ρ =20 dB, and M =2.
b b
Fig. 1: Outage probability of the m-th user versus ρ of the first scenario.
s
100
100
λb=λι=10−3 m=1
m=2
Analytical error floor
y
abilit10−1 bility 10−1 ν =1.0, ν = 01..50,,
Outage prob10−2 NOOOMMMAAA mm m==12=1 Outage proba10−2 00..51, 0.1
NOMA m=2
Simulation λb=λι=10−4
Analytical error floor
10−3 10−3
0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70
SNR (dB) SNR (dB)
(a) For different density of PTs and PRs, with α = 4, κ = 1, (b) Fordifferentν,withα=4,λ =10−4,λ =10−4,κ=0.5,
b ℓ
ν =1, R =10 m, ρ =νρ , and M =2. R =10 m, ρ =νρ , and M =2.
D b s D b s
Fig. 2: Outage probability of the m-th user versus ρ of the second scenario.
s
observations can be drawn as follows: 1) Reducing the coverage of the secondary users zone D can
achieve a lower outage probability because of a smaller path loss. 2) The ordered users with different
channel conditions have different decreasing slope because of different diversity orders, which verifies the
derivation of (26). In Fig. 1(b), the power allocation coefficients are a = 0.8 and a = 0.2. The target
1 2
rate is R = 1 and R = 3 BPCU. The performance of a conventional OMA is also shown in the figure
1 2
as a benchmark for comparison. It can be observed that for different values of the path loss, NOMA can