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

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1 Non-orthogonal Multile Access in Large-Scale 1 Underlay Cognitive Radio Networks Yuanwei Liu, Zhiguo Ding, Maged Elkashlan, and Jinhong Yuan Abstract In this aer, non-orthogonal multile access (NOMA is alied to large-scale underlay cognitive radio (CR networks with randomly deloyed users. In order to characterize the erformance of the considered network, arxiv: v1 [cs.it] 14 Jan 216 new closed-form exressions of the outage robability are derived using stochastic-geometry. More imortantly, by carrying out the diversity analysis, new insights are obtained under the two scenarios with different ower constraints: 1 fixed transmit ower of the rimary transmitters (PTs, and 2 transmit ower of the PTs being roortional to that of the secondary base station. For the first scenario, a diversity order of m is exerienced at the m-th ordered NOMA user. For the second scenario, there is an asymtotic error floor for the outage robability. Simulation results are rovided to verify the accuracy of the derived results. A ivotal conclusion is reached that by carefully designing target data rates and ower allocation coefficients of users, NOMA can outerform conventional orthogonal multile access in underlay CR networks. Index Terms Cognitive radio, large-scale network, non-orthogonal multile access, stochastic geometry I. INTRODUCTION Sectrum efficiency is of significant imortance and becomes one of the main design targets for future fifth generation networks. Non-orthogonal multile access (NOMA has received considerable attention because of its otential to achieve suerior sectral efficiency [1]. Particularly, different from conventional multile access (MA techniques, NOMA uses the ower domain to serve multile users at different ower levels in order to use sectrum more efficiently. A downlink NOMA and an ulink NOMA are considered in [2] and [3], resectively. The alication of multile-inut multile-outut (MIMO techniques to NOMA has been considered in [4] by using zero-forcing detection matrices. The authors in [5] investigated an ergodic caacity maximization roblem for MIMO NOMA systems. Y. Liu and M. Elkashlan are with Queen Mary University of London, London, UK ( yuanwei.liu, maged.elkashlan}@qmul.ac.uk. Z. Ding is with Lancaster University, Lancaster, UK ( z.ding@lancaster.ac.uk. J. Yuan is with the University of New South Wales, Sydney, Australia ( j.yuan@unsw.edu.au.

2 2 Another aroach to imrove sectrum efficiency is the aradigm of underlay cognitive radio (CR networks, which was roosed in [6] and has rekindled increasing interest in using sectrum more efficiently. The key idea of underlay CR networks is that each secondary user (SU is allowed to access the sectrum of the rimary users (PUs as long as the SU meets a certain interference threshold in the rimary network (PN. In [7], an underlay CR network taking into account the satial distribution of the SU relays and PUs was considered and its erformance was evaluated by using stochastic geometry tools. In [8], a new CR insired NOMA scheme has been roosed and the imact of user airing has been examined, by focusing on a simle scenario with only one rimary transmitter (PT. By introducing the aforementioned two concets, it is natural to consider the alication of NOMA in underlay CR networks using additional ower control at the secondary base station (BS to imrove the sectral efficiency. Stochastic geometry is used to model a large-scale CR network with a large number of randomly deloyed PTs and rimary receivers (PRs. We consider a ractical system design as follows: 1 All the SUs, PTs, and PRs are randomly deloyed 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 redefined ower constraint threshold to avoid interference at the PRs. New closed-form exressions of the outage robability of the NOMA users are derived to evaluate the erformance of the considered CR NOMA network. Moreover, considering two different ower constraints at the PTs, diversity order 1 analysis is carried out with roviding imortant insights: 1 When the transmit ower of the PTs is fixed, the m-th user among all ordered NOMA user exeriences a diversity order of m; and 2 When the the transmit ower of the PTs is roortional to that of the secondary BS, an asymtotic error floor exists for the outage robability. II. NETWORK MODEL We consider a large-scale underlay sectrum 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 D as its coverage. The M randomly deloyed secondary users are uniformly distributed 1 Diversity order is defined as the sloe for the outage rovability curve decreasing with the signal-to-noise-ratio (SNR. It measures the number of indeendent fading aths over which the data is received. In NOMA networks, since users channels are ordered and SIC is alied at each receiver, it is of imortance to investigate the diversity order.

3 3 within the disc which is the user zone for NOMA. The secondary BS communicates with all SUs within the disc by alying the NOMA transmission rotocol. It is worthy ointing out that the ower 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 lane. The satial toology of all the PTs and PRs are modeled using homogeneous oisson oint rocesses (PPPs, denoted by Φ b and Φ l with density λ b and λ l, resectively. All channels are assumed to be quasi-static Rayleigh fading where the channel coefficients are constant for each transmission block but vary indeendently between different blocks. According to underlay CR, the transmit ower P t at the secondary BS is constrained as follows: I P t = min max g l 2,P s, (1 l Φ l where I is the maximum ermissible interference ower at the PRs, P s is maximum transmission ower at the secondary BS, g l 2 = ĝ l 2 L(d l is the overall channel gain from the secondary BS to PRs l. Here, ĝ l is small-scale fading with ĝ l CN(,1, L(d l = 1 1+d α l is large-scale ath loss, d l is the distance between the secondary BS and the PRs, and α is the ath loss exonent. A bounded ath loss model is used to ensure the ath 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 m = h m an P t x n +n m, (2 n=1 where n m is the additive white Gaussian noise (AWGN at the m-th user with variance σ 2, a n is the ower allocation coefficient for the n-th SU with M n=1 a n = 1, x n is the information for the n-th user, and h m is the channel coefficient between the m-th user and the secondary BS. For the SUs, they also observe the interferences of the randomly deloyed 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, which means that there is no interference from PTs allowed inside this zone [1]. We assume d 1 in

4 4 this aer. The interference links from the PTs to the SUs are dominated by the ath loss and is given by I B = b Φ b L(d b, (3 where L(d b = 1/(1+d α b is the large-scale ath loss and d b is the distance from the PTs to the SUs. Without loss of generality, all the channels of SUs are assumed to follow the order as h 1 2 h 2 2 h M 2. The ower allocation coefficients are assumed to follow the order as a 1 a 2 a M. According to the NOMA rincile, 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. Secifically, the m-th user first decodes the messages of all the (m users, and then successively subtracts these messages to obtain its own information. Therefore, the received signal-to-interference-lus-noise ratio (SINR for the m-th user to decode the information of the j-th user is given by } h m 2 γ t a j γ m,j =, (4 M h m 2 γ t a i +ρ b I B +1 i=j+1 ρ where γ t = min,ρ max g l 2 s, ρ = I,ρ σ 2 s = Ps,ρ σ 2 b = P B σ 2, and P B is the transmit ower of the PTs, h m 2 l Φ l 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 M,M = h M 2 γ ta M ρ b I B +1. III. OUTAGE PROBABILITY In this section, we rovide exact analysis of the considered networks in terms of outage robability. 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 m = hm 2 γ t ρ b I B +1. Based on (4, the cumulative distribution function (CDF of X m is given by } h m 2 γ t F Xm (ε = Pr ρ b I B +1 < ε. (5 ( We denote ε j = τ j / a j τ M j i=j+1 a i for j < M, τ j = 2 R j 1, R j is the target data rate for the j-th user, ε M = τ M /a M, and ε max m exressed as follows: = maxε 1,ε 2,...,ε m }. The outage robability at the m-th user can be P m = PrX m < ε max m } = F Xm (ε max m, (6

5 5 where the condition a j τ j M i=j+1 a i > should be satisfied due to alying NOMA, otherwise the outage robability will always be one [2]. We need calculate the CDF of X m conditioned on I B and γ t. Rewrite (5 as follows: F Xm I B,γ t (ε = F hm 2 ( (ρb I B +1ε γ t, (7 where F hm 2 is the CDF of h m. Based on order statistics [12] and alying binomial exansion, the CDF of the ordered channels has a relationshi with the unordered channels as follows: M m F 2(y = ψ ( M m ( ( m+ hm m F m+ h 2 (y, (8 = where y = (ρ bi B +1ε M! γ t, ψ m =, and (M m!(m! h 2 = arbitrary SU. Here, ĥ 2 L(d is the unordered channel gain of an ĥ is the small-scale fading coefficient with ĥ CN(,1, L(d = 1 1+d α is the largescale ath loss, and d is a random variable reresenting the distance from the secondary BS to an arbitrary SU. Then using the assumtion of homogenous PPP and alying the olar coordinates, we exress F h 2(y as follows: F 2(y = 2 RD ( 1 e (1+r α y rdr. (9 h RD 2 Note that it is challenging to obtain an insightful exression for the unordered CDF. As such, we aly the Gaussian-Chebyshev quadrature [13] to find an aroximation for (9 as F 2(y N b h n e cny, (1 n= where N is a comlexity-accuracy tradeoff arameter, b n = ω N 1 φ 2 n (φ n +1, b = N b n, c n = 1+ ( R D2 (φ n +1 α, ωn = π N, and φ n = cos ( 2n 2N π. Substituting (1 into (8 and alying the multinomial theorem, we obtain M m F hm 2 (y = ψ ( M m ( ( ( m+ N m m+ q + +q N = q + +q N =m+ n= b qn n e n=1 n=y. (11

6 6 where ( m+ q + +q N = (m+! F Xm (ε j =. Based on (11, the CDF of X q! q N! m can be exressed as follows: f IB (xf γt (zdxdz M m =ψ m = ( (ρb x+1ε j F hm 2 z ( M m ( e εj z m+ q + +q N =m+ q nc n n= e xρbεj z ( m+ q + +q N ( N n= nc n n=q f IB (xdxf γt (zdz, (12 }} } } Q 1 where f γt is the PDF of γ t. We exress Q 2 in (12 as follows: Q 2 = e xρbεj z n= Q 2 f IB (xdx = E Φb e xρ b ε j z } n= = L IB ( xρ b ε j z b qn n q n c n. (13 In this case, the Lalace transformation of the interferences from the PT can be exressed as [1] where δ = 2 α ( L IB (s = ex λ b π = ex λ ( bπ [(e sd α d 2 +sδ γ ( ] 1 δ,sd α e sd α sd α d 2 +s δ and γ( is the lower incomlete Gamma function. } } Θ n= t δ e t dt, (14 To obtain an insightful exression, we use Gaussian-Chebyshev quadrature to aroximate the lower incomlete Gamma function in (14, Θ can be exressed as follows: Θ s 1 δ L l=1 β l e t lsd α, (15 where L is a comlexity-accuracy tradeoff arameter, β l = 1 2 d2 α ω L 1 θ 2 l t l δ, t l = 1 2 (θ l +1, ω L = π L, and θ l = cos ( 2l π. Substituting (15 into (14, we aroximate the Lalace transformation as follows: 2L L IB (s e λ bπ Substituting (16 into (12, Q 2 is given by Q 2 = e λ b π e ρbεjd α z n= The following theorem rovides the PDF of γ t. ( ( e sd α d 2 +s L β l e t l sd α l=1 d 2 +ρ b ε j z L n= l=1 t l ρ b ε j zd β l e α. (16 n=. (17

7 7 Theorem 1: Consider the use of the comosite channel model with Rayleigh fading and ath loss, the PDF of the effective ower of the secondary BS is given by f γt (x = e a lρ δ se ρ x Dirac(x ρ s + ( ρ x +δ a l x δ e a lx δ e ρ x ρ x U (ρs x, (18 where a l = δπλ lγ(δ, U ( is the unit ste function, and Dirac( is the imulse function. ρ δ Proof: See Aendix A. Substituting (18 and (17 into (12, we exress Q 1 as follows: ε j a l ρ δ s e ρ ρs n= ρs Q 1 =e ρs ρ b ε j ρsd λ b π e α n= d 2 +ρ b ε j ρs L n= l=1 t l ρ b ε j ρsd β l e α n= ( ρ ρ+ε + a l z +δ z δ e a lz δ e ρ j z n= z Q 2 dz. (19 }} Ψ We notice that it is very challenging to solve the integral Ψ in (19, therefore, we aly the Gaussian- Chebyshev quadrature to aroximate the integral as follows: Ψ K k=1 η k e ρ+ε j ρ b ε j n= ρss λ ρss b π e k d α n= k d 2 + ρ b ε j ρss k L n= l=1 t l ρ b ε j ρss β l e k d α n=, (2 where K is a comlexity-accuracy tradeoff arameter, ω K = π, ϕ K k = cos ( 2k π, s 2K k = 1 (ϕ 2 k +1, and ( ρ η k = ω K 2 1 ϕ 2 ρ k ρ ss k +δ a l ρ δ s sδ k e a lρ δ ss δ k e ρss k. Substituting (19 and (2 into (12 and alying ε max ε j, based on (6, we obtain the closed-form exression of the outage robability at the m-th user as follows: M m ( M m ( ( ( m+ N P m = ψ m m+ q = q + +q N =m+ + +q N n= e + K k=1 a l ρ δ s e ρ ρs η k e εmax n= ρ b εmax ρsd λ ρs b π e α n= ρ+εmax n= ρ b εmax ρss λ ρss b π e k d α n= k d 2 +ρ b εmax ρs d 2 +ρ b εmax ρss k b qn n L n= l=1 L n= l=1 t l ρ b εmax ρsd β l e α t l ρ b εmax ρss β l e k d α n= n=. (21

8 8 IV. DIVERSITY ANALYSIS Based on the analytical results for the outage robability in (21, we aim to rovide asymtotic diversity analysis for the ordered NOMA users. The diversity order of the user s outage robability is defined as logp m (ρ s d = lim. (22 ρ s logρ s A. Fixed Transmit Power at Primary Transmitters In this case, we examine the diversity with the fixed transmit SNR at the PTs (ρ b, while the transmit SNR of secondary BS (ρ s and the maximum ermissible interference constraint at the PRs (ρ go to the infinity. Particularly, we assume ρ is roortional to ρ s, i.e. ρ = κρ s, where κ is a ositive scaling factor. This assumtion alies 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 t = γt ρ s = min κ,1 max g l 2 l Φ l similar to (8, the ordered CDF has the relationshi with unordered CDF as M m ( M m ( ( m+ FX m I B,γ t (y = ψ m F m+ h 2 (y, (23 = where y = (ρ bi B +1ε j ρ sγ t. When ρ s, we observe that y. In order to investigate an insightful exression to obtain the diversity order, we use Gaussian-Chebyshev quadrature and 1 e y y to aroximate (9 as F h 2 (y χ n y, (24 where χ n = ω N 1 φ 2 n (φ n +1c n. Substituting (24 into (23, since y, we obtain ( m (( m F (ε (ρb I B +1ε j (ρb I B +1ε j Xm I B,γ t j = ξ +o, (25 ρ s γ t ρ s γ t where ξ = ( m ψ m χ n n=1 m n=1. Based on (6, (11, and (25, the asymtotic outage robability is given by Pm F 1 ( m (ρb x+1ε max ξ f ρ m IB s z (xf γt (zdxdz, (26 } } C where f γt the PDF of γ t. Since C is a constant indeendent of ρ s, (26 can be exressed as follows: P m F = 1 ρ m s C +o ( ρ s m, (27,

9 9 Substituting (27 into (22, we obtain the diversity order of this case is m. This can be exlained as follows. Note that SIC is alied at the ordered SUs. For the first user with the oorest channel gain, no interference cancelation is oerated at the receiver, therefore its diversity gain is one. While for the m-th user, since the interferences from all the other (m users are canceled, it obtains a diversity of m. B. Transmit Power of Primary Transmitters Proortional to that of Secondary Ones In this case, we examine the diversity with the transmit SNR at the PTs (ρ b is roortional to the transmit SNR of secondary BS (ρ s. Particularly, we assume ρ b = νρ s, where ν is a ositive scaling factor. We still assume ρ is roortional to ρ s. Alying ρ s, ρ = κρ s and ρ b = νρ s to (21, we obtain the asymtotic outage robability of the m-th user in this case as follows: M m ( M m ( ( ( N Pm m+ P ψ m b qn n m+ q = q + +q N =m+ + +q N n= e + K k=1 a νεmax l e κ d λ b π e α n= κ ηk e νεmax s λ s b π e k d α n= k d 2 +νεmax d 2 +νεmax s k L n= l=1 L n= l=1 where a l = δπλ lγ(δ and η κ δ k = ( ω K 2 1 ϕ 2 κ k s k +δ a κ l sδ k e a l sδ k e sk. t l νεmax d β l e α t l νεmax s β l e k d α n= n=. (28 It is observed that P m P is a constant indeendent of ρ s. Substituting (28 into (22, we find that asymtotically there is an error floor for the outage robability of SUs. V. NUMERICAL RESULTS In this section, numerical results are resented to verify the accuracy of the analysis as well as to obtain more imortant 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 arameters are chosen with N = 5, K = 1, and L = 1. Monte Carlo simulation results are marked as to verify our derivation. Fig. 1 lots the outage robability of the m-th user for the first scenario when ρ b is fixed and ρ is roortional to ρ s. In Fig. 1(a, the ower allocation coefficients are a 1 =.5, a 2 =.4 and a 3 =.1. The target data rate for each user is assumed to be all the same as R 1 = R 2 = R 3 =.1 bit er channel use (BPCU. The dashed and solid curves are obtained from the analytical results derived in (21. Several

10 1 1 1 Outage robability Analytical RD=1 m Analytical RD=12 m Simulation m=1 m=2 m=3 Outage robability NOMA m=1 NOMA m=2 OMA m=1 OMA m=2 α=3 α= SNR (db SNR (db (a For different user zone, with λ b = 1 3, λ l = 1 3, κ = 1, α = 4, ρ b = 2 db, and M = 3. (b For different α, with λ b = 1 3, λ l = 1 3, κ = 1, R D = 5 m, ρ b = 2 db, and M = 2. Fig. 1: Outage robability of the m-th user versus ρ s of the first scenario. Outage robability OMA m=1 OMA m=2 NOMA m=1 NOMA m=2 Simulation Analytical error floor λb=λι=1 3 λb=λι= SNR (db Outage robability ν =1.,.5,.1 m=1 m=2 Analytical error floor ν =1.,.5, SNR (db (a For different density of PTs and PRs, with α = 4, κ = 1, ν = 1, R D = 1 m, ρ b = νρ s, and M = 2. (b For different ν, with α = 4, λ b = 1 4, λ l = 1 4, κ =.5, R D = 1 m, ρ b = νρ s, and M = 2. Fig. 2: Outage robability of the m-th user versus ρ s of the second scenario. observations can be drawn as follows: 1 Reducing the coverage of the secondary users zone D can achieve a lower outage robability because of a smaller ath loss. 2 The ordered users with different channel conditions have different decreasing sloe because of different diversity orders, which verifies the derivation of (26. In Fig. 1(b, the ower allocation coefficients are a 1 =.8 and a 2 =.2. The target rate is R 1 = 1 and R 2 = 3 BPCU. The erformance of a conventional OMA is also shown in the figure as a benchmark for comarison. It can be observed that for different values of the ath loss, NOMA can

11 11 achieve a lower outage robability than the conventional OMA. Fig. 2 lots the outage robability of the m-th user for the second scenario when both ρ b and ρ are roortional to ρ s. The ower allocation coefficients are a 1 =.8 and a 2 =.2. The target rates are R 1 = R 2 =.1 BPCU. The dashed and solid curves are obtained from the analytical results derived in (21. One observation is that error floors exist in both Figs. 2(a and 2(b, which verifies the asymtotic results in (28. Another observation is that user two (m = 2 outerforms user one (m = 1. The reason is that for user two, by alying SIC, the interference from user one is canceled. While for user one, the interference from user two still exists. In Fig. 2(a, it is shown that the error floor become smaller when λ b and λ l decrease, which is due to less interference from PTs and the relaxed interference ower constraint at the PRs. It is also worth noting that with these system arameters, NOMA outerforms OMA for user one while OMA outerforms NOMA for user two, which indicates the imortance of selecting aroriate ower allocation coefficients and target data rates for NOMA. In Fig. 2(b, it is observed that the error floors become smaller as ν decreases. This is due to the fact that smaller ν means a lower transmit ower of PTs, which in turn reduces the interference at SUs. VI. CONCLUSIONS In this aer, we have studied non-orthogonal multile access (NOMA in large-scale underlay cognitive radio networks with randomly deloyed users. Stochastic geometry tools were used to evaluate the outage erformance of the considered network. New closed-form exressions were derived for the outage robability. Diversity order of NOMA users has been analyzed in two situations based on the derived outage robability. An imortant future direction is to otimize the ower allocation coefficients to further imrove the erformance ga between NOMA and conventional MA in CR networks. APPENDIX A: PROOF OF THEOREM 1 The CDF of γ t is given by F γt (x = Pr min ρ max g l 2,ρ s x l Φ l = Pr max g l 2 ρ max l Φ l x, ρ }} +Pr max g l 2 ρ },ρ s x ρ s l Φ l ρ s = 1 U (ρ s xpr max g l 2 ρ }. (A.1 l Φ l x }} Ω

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