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

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

Yuanwei Liu, Zhiguo Ding, Maged Elkashlan, and Jinhong Yuan Y. Liu and M. Elkashlan are with Queen Mary University of London, London, UK (email: {yuanwei.liu, maged.elkashlan}@qmul.ac.uk). Z. Ding is with Lancaster University, Lancaster, UK (e-mail: z.ding@lancaster.ac.uk). J. Yuan is with the University of New South Wales, Sydney, Australia (e-mail: j.yuan@unsw.edu.au).
###### Abstract

In this paper, non-orthogonal multiple access (NOMA) is applied to large-scale underlay cognitive radio (CR) networks with randomly deployed users. In order to characterize the performance of the considered network, new closed-form expressions of the outage probability are derived using stochastic-geometry. More importantly, by carrying out the diversity analysis, new insights are obtained under the two scenarios with different power constraints: 1) fixed transmit power of the primary transmitters (PTs), and 2) transmit power of the PTs being proportional to that of the secondary base station. For the first scenario, a diversity order of is experienced at the -th ordered NOMA user. For the second scenario, there is an asymptotic error floor for the outage probability. Simulation results are provided to verify the accuracy of the derived results. A pivotal conclusion is reached that by carefully designing target data rates and power allocation coefficients of users, NOMA can outperform conventional orthogonal multiple access in underlay CR networks. {keywords} Cognitive radio, large-scale network, non-orthogonal multiple access, stochastic geometry

## I Introduction

Spectrum efficiency is of significant importance and becomes one of the main design targets for future fifth generation networks. Non-orthogonal multiple access (NOMA) has received considerable attention because of its potential to achieve superior spectral efficiency [1]. Particularly, different from conventional multiple access (MA) techniques, NOMA uses the power domain to serve multiple users at different power 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 using zero-forcing detection matrices. The authors in [5] investigated an ergodic capacity maximization problem for MIMO NOMA systems.

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 order111Diversity 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. analysis is carried out with providing important insights: 1) When the transmit power of the PTs is fixed, the -th user among all ordered NOMA user experiences a diversity order of ; 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 with radius as its coverage. The randomly deployed secondary users are uniformly distributed 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 and with density and , respectively. All channels are assumed to be quasi-static Rayleigh fading where the channel coefficients are constant for each transmission block but vary independently between different blocks.

According to underlay CR, the transmit power at the secondary BS is constrained as follows:

 Pt=min⎧⎪ ⎪⎨⎪ ⎪⎩Ipmaxℓ∈Φℓ|gℓ|2,Ps⎫⎪ ⎪⎬⎪ ⎪⎭, (1)

where is the maximum permissible interference power at the PRs, is maximum transmission power at the secondary BS, is the overall channel gain from the secondary BS to PRs . Here, is small-scale fading with , is large-scale path loss, is the distance between the secondary BS and the PRs, and 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 -th secondary user is given by

 ym=hmM∑n=1√anPtxn+nm, (2)

where is the additive white Gaussian noise (AWGN) at the -th user with variance , is the power allocation coefficient for the -th SU with , is the information for the -th user, and is the channel coefficient between the -th user and the secondary BS.

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 to each secondary NOMA user with radius of , which means that there is no interference from PTs allowed inside this zone [10]. We assume in this paper. The interference links from the PTs to the SUs are dominated by the path loss and is given by

 IB=∑b∈ΦbL(db), (3)

where is the large-scale path loss and 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 . The power allocation coefficients are assumed to follow the order as . According to the NOMA principle, successive interference cancelation (SIC) is carried out at the receivers [11]. It is assumed that . In this case, the -th user can decode the message of the -th user and treats the message for the -th user as interference. Specifically, the -th user first decodes the messages of all the users, and then successively subtracts these messages to obtain its own information. Therefore, the received signal-to-interference-plus-noise ratio (SINR) for the -th user to decode the information of the -th user is given by

 γm,j=|hm|2γtaj|hm|2γtM∑i=j+1ai+ρbIB+1, (4)

where , , and is the transmit power of the PTs, is the overall ordered channel gain from the secondary BS to the -th SU. For the case , it indicates the -th user decodes the message of itself. Note that the SINR for the -th SU is .

## 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 -th user can not detect any of the -th user’s message, where due to the SIC. Denote . Based on (4), the cumulative distribution function (CDF) of is given by

 FXm(ε)=Pr{|hm|2γtρbIB+1<ε}. (5)

We denote for , , is the target data rate for the -th user, , and . The outage probability at the -th user can be expressed as follows:

 Pm=Pr{Xm<εmaxm}=FXm(εmaxm), (6)

where the condition should be satisfied due to applying NOMA, otherwise the outage probability will always be one [2].

We need calculate the CDF of conditioned on and . Rewrite (5) as follows:

 FXm|IB,γt(ε)=F|hm|2((ρbIB+1)εγt), (7)

where is the CDF of . Based on order statistics [12] and applying binomial expansion, the CDF of the ordered channels has a relationship with the unordered channels as follows:

 F|hm|2(y)=ψmM−m∑p=0(M−mp)(−1)pm+p(F∣∣~h∣∣2(y))m+p, (8)

where , , and is the unordered channel gain of an arbitrary SU. Here, is the small-scale fading coefficient with , is the large-scale path loss, and is a random variable representing the distance from the secondary BS to an arbitrary SU.

Then using the assumption of homogenous PPP and applying the polar coordinates, we express as follows:

 F∣∣~h∣∣2(y)=2R2D∫RD0(1−e−(1+rα)y)rdr. (9)

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

 F∣∣~h∣∣2(y)≈N∑n=0bne−cny, (10)

where is a complexity-accuracy tradeoff parameter, , , , , and .

Substituting (10) into (8) and applying the multinomial theorem, we obtain

 F|hm|2(y)=ψmM−m∑p=0(M−mp)(−1)pm+p∑q0+⋯+qN=m+p(m+pq0+⋯+qN)(N∏n=0bqnn)e−N∑n=0qncny. (11)

where . Based on (11), the CDF of can be expressed as follows:

 FXm(εj)= = ψmM−m∑p=0(M−mp)(−1)pm+p∑q0+⋯+qN=m+p(m+pq0+⋯+qN)(N∏n=0bqnn) ×∫∞0e−εjzN∑n=0qncn∫∞0e−xρbεjzN∑n=0qncnfIB(x)dxQ2fγt(z)dzQ1, (12)

where is the PDF of . We express in (III) as follows:

 Q2=∫∞0e−xρbεjzN∑n=0qncnfIB(x)dx=EΦb⎧⎪⎨⎪⎩e−xρbεjzN∑n=0qncn⎫⎪⎬⎪⎭=LIB(xρbεjzN∑n=0qncn). (13)

In this case, the Laplace transformation of the interferences from the PT can be expressed as[10]

 LIB(s) =exp⎛⎜ ⎜ ⎜ ⎜⎝−λbπ⎡⎢ ⎢ ⎢ ⎢⎣(e−sd−α0−1)d20+sδ∫sd−α00t−δe−tdtΘ⎤⎥ ⎥ ⎥ ⎥⎦⎞⎟ ⎟ ⎟ ⎟⎠, (14)

where and is the lower incomplete Gamma function.

To obtain an insightful expression, we use Gaussian-Chebyshev quadrature to approximate the lower incomplete Gamma function in (III), can be expressed as follows:

 Θ≈s1−δL∑l=1βle−tlsd−α0, (15)

where is a complexity-accuracy tradeoff parameter, , , , and . Substituting (15) into (III), we approximate the Laplace transformation as follows:

 LIB(s)≈ e−λbπ((e−sd−α0−1)d20+sL∑l=1βle−tlsd−α0). (16)

Substituting (16) into (III), is given by

 Q2=e−λbπ⎛⎜ ⎜ ⎜ ⎜⎝⎛⎜ ⎜ ⎜ ⎜⎝e−ρbεjd−α0zN∑n=0qncn−1⎞⎟ ⎟ ⎟ ⎟⎠d20+ρbεjzN∑n=0qncnL∑l=1βle−tlρbεjzdα0N∑n=0qncn⎞⎟ ⎟ ⎟ ⎟⎠. (17)

The following theorem provides the PDF of .

###### 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−ρpxDirac(x−ρs)+(ρpx+δ)aℓxδ−1e−aℓxδe−ρpx−ρpxU(ρs−x), (18)

where , is the unit step function, and is the impulse function. {proof} See Appendix A.

Substituting (18) and (17) into (III), we express as follows:

 Q1= e−aℓρδse−ρpρs−εjN∑n=0qncnρs−λbπ⎛⎜ ⎜ ⎜⎝⎛⎜ ⎜ ⎜⎝e−ρbεjρsdα0N∑n=0qncn−1⎞⎟ ⎟ ⎟⎠d20+ρbεjρsN∑n=0qncnL∑l=1βle−tlρbεjρsdα0N∑n=0qncn⎞⎟ ⎟ ⎟⎠ +∫ρs0aℓ(ρpz+δ)zδ−1e−aℓzδe−ρpz−ρp+εjN∑n=0qncnzQ2dzΨ. (19)

We notice that it is very challenging to solve the integral in (III), therefore, we apply the Gaussian-Chebyshev quadrature to approximate the integral as follows:

 Ψ≈K∑k=1ηke−ρp+εjN∑n=0qncnρssk−λbπ⎛⎜ ⎜ ⎜⎝⎛⎜ ⎜ ⎜⎝e−ρbεjρsskdα0N∑n=0qncn−1⎞⎟ ⎟ ⎟⎠d20+ρbεjρsskN∑n=0qncnL∑l=1βle−tlρbεjρsskdα0N∑n=0qncn⎞⎟ ⎟ ⎟⎠, (20)

where is a complexity-accuracy tradeoff parameter, , , , and .

Substituting (III) and (20) into (III) and applying , based on (6), we obtain the closed-form expression of the outage probability at the -th user as follows:

 Pm=ψmM−m∑p=0(M−mp)(−1)pm+p∑q0+⋯+qN=m+p(m+pq0+⋯+qN)(N∏n=0bqnn) ×⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣e−aℓρδse−ρpρs−εmaxN∑n=0qncnρs−λbπ⎛⎜ ⎜ ⎜⎝⎛⎜ ⎜ ⎜⎝e−ρbεmaxρsdα0N∑n=0qncn−1⎞⎟ ⎟ ⎟⎠d20+ρbεmaxρsN∑n=0qncnL∑l=1βle−tlρbεmaxρsdα0N∑n=0qncn⎞⎟ ⎟ ⎟⎠ +K∑k=1ηke−ρp+εmaxN∑n=0qncnρssk−λbπ⎛⎜ ⎜ ⎜⎝⎛⎜ ⎜ ⎜⎝e−ρbεmaxρsskdα0N∑n=0qncn−1⎞⎟ ⎟ ⎟⎠d20+ρbεmaxρsskN∑n=0qncnL∑l=1βle−tlρbεmaxρsskdα0N∑n=0qncn⎞⎟ ⎟ ⎟⎠⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦. (21)

## Iv Diversity Analysis

Based on the analytical results for the outage probability in (III), we aim to provide asymptotic diversity analysis for the ordered NOMA users. The diversity order of the user’s outage probability is defined as

 d=−limρs→∞logPm(ρs)logρs. (22)

### Iv-a Fixed Transmit Power at Primary Transmitters

In this case, we examine the diversity with the fixed transmit SNR at the PTs (), while the transmit SNR of secondary BS () and the maximum permissible interference constraint at the PRs () go to the infinity. Particularly, we assume is proportional to , i.e. , where is a positive scaling factor. 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 , similar to (8), the ordered CDF has the relationship with unordered CDF as

 F∞Xm|IB,γt∗(y∗)=ψmM−m∑p=0(M−mp)(−1)pm+p(F∞∣∣~h∣∣2(y∗))m+p, (23)

where . When , we observe that . In order to investigate an insightful expression to obtain the diversity order, we use Gaussian-Chebyshev quadrature and to approximate (9) as

 F∞∣∣~h∣∣2(y∗)≈N∑n=1χny∗, (24)

where . Substituting (24) into (23), since , we obtain

 F∞Xm|IB,γt∗(εj)=ξ((ρbIB+1)εjρsγt∗)m+o(((ρbIB+1)εjρsγt∗)m), (25)

where . Based on (6), (11), and (25), the asymptotic outage probability is given by

 P∞mF≈1ρms∫∞0∫∞0ξ((ρbx+1)εmaxz)mfIB(x)fγt∗(z)dxdzC, (26)

where the PDF of . Since is a constant independent of , (26) can be expressed as follows:

 P∞mF=1ρmsC+o(ρs−m), (27)

Substituting (27) into (22), we obtain the diversity order of this case is . 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 -th user, since the interferences from all the other users are canceled, it obtains a diversity of .

### Iv-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 () is proportional to the transmit SNR of secondary BS (). Particularly, we assume , where is a positive scaling factor. We still assume is proportional to . Applying , and to (III), we obtain the asymptotic outage probability of the -th user in this case as follows:

 P∞mP≈ ψmM−m∑p=0(M−mp)(−1)pm+p∑q0+⋯+qN=m+p(m+pq0+⋯+qN)(N∏n=0bqnn) ×⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣e−a∞ℓe−κ−λbπ⎛⎜ ⎜ ⎜⎝⎛⎜ ⎜ ⎜⎝e−νεmaxdα0N∑n=0qncn−1⎞⎟ ⎟ ⎟⎠d20+νεmaxN∑n=0qncnL∑l=1βle−tlνεmaxdα0N∑n=0qncn⎞⎟ ⎟ ⎟⎠ +K∑k=1η∞ke−κsk−λbπ⎛⎜ ⎜ ⎜⎝⎛⎜ ⎜ ⎜⎝e−νεmaxskdα0N∑n=0qncn−1⎞⎟ ⎟ ⎟⎠d20+νεmaxskN∑n=0qncnL∑l=1βle−tlνεmaxskdα0N∑n=0qncn⎞⎟ ⎟ ⎟⎠⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦. (28)

where and .

It is observed that is a constant independent of . Substituting (IV-B) into (22), we find that 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 m. The Gaussian-Chebyshev parameters are chosen with , , and . Monte Carlo simulation results are marked as “” to verify our derivation.

Fig. 1 plots the outage probability of the -th user for the first scenario when is fixed and is proportional to . In Fig. 1(a), the power allocation coefficients are , and . The target data rate for each user is assumed to be all the same as bit per channel use (BPCU). The dashed and solid curves are obtained from the analytical results derived in (III). Several observations can be drawn as follows: 1) Reducing the coverage of the secondary users zone 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 and . The target rate is and BPCU. The performance of a conventional OMA is also shown in the figure as a benchmark for comparison. It can be observed that for different values of the path loss, NOMA can achieve a lower outage probability than the conventional OMA.

Fig. 2 plots the outage probability of the -th user for the second scenario when both and are proportional to . The power allocation coefficients are and . The target rates are BPCU. The dashed and solid curves are obtained from the analytical results derived in (III). One observation is that error floors exist in both Figs. 2(a) and 2(b), which verifies the asymptotic results in (IV-B). Another observation is that user two () outperforms user one (). The reason is that for user two, by applying 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 and decrease, which is due to less interference from PTs and the relaxed interference power constraint at the PRs. It is also worth noting that with these system parameters, NOMA outperforms OMA for user one while OMA outperforms NOMA for user two, which indicates the importance of selecting appropriate power allocation coefficients and target data rates for NOMA. In Fig. 2(b), it is observed that the error floors become smaller as decreases. This is due to the fact that smaller means a lower transmit power of PTs, which in turn reduces the interference at SUs.

## Vi Conclusions

In this paper, we have studied non-orthogonal multiple access (NOMA) in large-scale underlay cognitive radio networks with randomly deployed users. Stochastic geometry tools were used to evaluate the outage performance of the considered network. New closed-form expressions were derived for the outage probability. Diversity order of NOMA users has been analyzed in two situations based on the derived outage probability. An important future direction is to optimize the power allocation coefficients to further improve the performance gap between NOMA and conventional MA in CR networks.

## Appendix A: Proof of Theorem 1

The CDF of is given by

 Fγt(x) =Pr⎧⎪ ⎪⎨⎪ ⎪⎩min⎧⎪ ⎪⎨⎪ ⎪⎩ρpmaxℓ∈Φℓ|gℓ|2,ρs⎫⎪ ⎪⎬⎪ ⎪⎭≤x⎫⎪ ⎪⎬⎪ ⎪⎭ =Pr{maxℓ∈Φℓ|gℓ|2≥max{ρpx,ρpρs}}+Pr{maxℓ∈Φℓ|gℓ|2≤ρpρs,ρs≤x} =1−U(ρs−x)Pr{maxℓ∈Φℓ|gℓ|2≥ρpx}Ω. (A.1)

Denote , we express as follows:

 ¯Ω =Pr{maxℓ∈Φℓ|gℓ|2≤ρpx}=EΦℓ⎧⎨⎩∏ℓ∈ΦℓPr{|^gℓ|2≤(1+dαℓ)ρpx}⎫⎬⎭ =EΦℓ⎧⎨⎩∏ℓ∈ΦℓF|^gℓ|2((1+dαℓ)ρpx)⎫⎬⎭. (A.2)

Applying the generating function, we rewrite (Appendix A: Proof of Theorem 1) as follows:

 ¯Ω= exp⎡⎢ ⎢⎣−λℓ∫R2(1−F|^gℓ|2((1+dαℓ)μ))rdr⎤⎥ ⎥⎦=exp[−2πλℓe−μ∫∞0re−μrαdr]. (A.3)

Applying [14, Eq. (3.326.2)], we obtain

 Ω=1−¯Ω=1−e−e−μδπλℓΓ(δ)<