Fundamental Limits of Spectrum Sharing for NOMA-based Cooperative Relaying

Transcription

1 1 Fundamental Limits of Spectrum Sharing for NOMA-based Cooperative Relaying arxiv: v1 [cs.it] 4 Sep 18 Vaibhav Kumar, Barry Cardiff, and Mar F. Flanagan School of Electrical and Electronic Engineering University College Dublin, Belfield, Dublin 4, Ireland vaibhav.umar@ucdconnect.ie, barry.cardiff@ucd.ie, mar.flanagan@ieee.org Abstract Non-orthogonal multiple access NOMA and spectrum sharing SS are two emerging multiple access technologies for efficient spectrum utilization in the fifth-generation 5G wireless communications standard. In this paper, we present a closed-form analysis of the average achievable sum-rate and outage probability for a NOMA-based cooperative relaying system CRS in an underlay spectrum sharing scenario. We consider a pea interference constraint, where the interference inflicted by the secondary unlicensed networ on the primary-user licensed receiver PU-Rx should be less than a predetermined threshold. We show that the CRS-NOMA outperforms the CRS with conventional orthogonal multiple access OMA for large values of pea interference power at the PU-Rx. I. INTRODUCTION With the proliferation of wireless communication technologies, services and applications, one of the maor challenges for the 5G communication standard is to support large-scale heterogeneous data traffic. NOMA has recently been recognized as a promising multiple access technology for 5G wireless networs as it can accommodate several users within the same orthogonal resource bloc time, frequency and/or spreading code via multiplexing them in the power domain at the transmitter side and using successive interference cancellation SIC at the receiver to remove messages intended for other users [1]. In the case of NOMA, users with poor channel conditions have a larger share of transmission power, unlie the conventional OMA where more power is allocated to users with strong channel conditions also nown as the water-filling strategy []. Cognitive radio CR is another emerging technology intended to enhance the spectrum utilization efficiency in wireless systems via SS. There are three main SS paradigms: underlay, overlay and interweave [3]. In underlay SS, secondary-users SUs operate in a frequency band originally owned by a

2 PU such that the interference caused by the SUs on the primary networ is less than a predefined limit, often referred to as the interference temperature. Therefore, no limit is directly imposed on the power transmitted from a SU transmitter SU-Tx; it is sufficient that the interference caused at the PU receiver PU-Rx is below the threshold. In a fading channel, the secondary networ may tae advantage of this fact by opportunistically transmitting at a high power level when the interference channel between SU-Tx and PU-Rx is in a deep fade. The closed-form expression for the relation between the secondary channel capacity and the pea/average interference inflicted on the primary user for different fading distributions were quantified in [4]. The interference from the PU transmitter PU-Tx to the SU receiver SU-Rx, also termed as the primary-tosecondary interference, was not considered in [4] and hence the results derived give an upper-bound on the achievable rate for the secondary networ. The average achievable rate for the SS system in the lowpower regime considering the primary-to-secondary interference were studied in [5], for general fading channels where in addition to the interference constraint, a transmit power constraint was also imposed on the SU-Tx. Different models of spectrum sharing NOMA networs including underlay NOMA, overlay NOMA and cognitive NOMA were discussed in [6] and it was shown that cooperative relaying can improve reception reliability. As such, cooperative spectrum sharing NOMA networs have lower outage probability compared to their non-cooperative counterparts. A novel secondary NOMA relay assisted spectrum sharing scheme was proposed in [7], where first the quality-of-service QoS of the PU was guaranteed using maximal-ratio combining MRC and then the sum-rate of the SUs was maximized. Another interesting application of NOMA for spatially multiplexed transmission using a cooperative relaying system CRS- NOMA to enhance the spectral efficiency was presented in [8], where the source was able to deliver two different symbols to the destination in two-time slots with the help of a relay here Rayleigh fading was considered. The CRS-NOMA can be easily seen to be superior to the conventional OMA relaying system in which a single symbol is delivered to the destination in two time slots. A performance analysis of the CRS-NOMA in Rician fading was presented in [9]. In this paper, we present the achievable sum-rate and outage probability analysis of the CRS-NOMA in an underlay SS scenario, where the power transmitted from the SU-Tx and from the relay are constrained by placing a limit on the pea interference power received at the PU-Rx. For clarity of exposition, no other constraint on the transmit power is imposed. While in practice, the transmit power from the SU-Tx or the relay is limited by hardware capabilities and other health-related safety considerations, the rates derived in this paper serve as an upper-bound on the capacity of the CRS-NOMA based SS system under a pea interference power constraint.

3 3 II. SYSTEM MODEL Consider the spectrum sharing CRS-NOMA shown in Fig. 1, which consists of a SU-Tx S, a relay R, a SU-Rx D and a PU-Rx P. The SU-Tx S is equipped with a single transmit antenna and the PU-Rx P is equipped with a single receive antenna. The relay R is equipped with N r receive antennas and a single transmit antenna, while the SU-Rx D is equipped with N d receive antennas. It is assumed that all the nodes are operating in half-duplex mode and all the wireless lins are assumed to be independent and Rayleigh distributed. The channel coefficient between the SU-Tx and the i th receive antenna of the relay 1 i N r is denoted by h sr,i and has a mean-square value Ω sr for all i, while that between the SU-Tx and the th antenna of the SU-Rx 1 N d is denoted by h sd, and has a mean-square value Ω sd for all. The channel coefficient between the transmit antenna of the relay and the th antenna PU-Rx time slot I time slot II Relay SU-Tx SU-Rx Fig. 1: System model for CRS-NOMA with underlay spectrum sharing. of the SU-Rx is denoted by h rd, and has a mean-square value Ω rd for all. Moreover, the channel coefficient between the SU-Tx and the PU-Rx is denoted by h sp and has a mean-square value, while that between the transmit antenna of the relay and the PU-Rx is denoted by h rp and has a mean-square value Ω rp. Furthermore, it is assumed that the channels between the SU-Tx and the SU-Rx are on average weaer that those between the SU-Tx and the relay, i.e., Ω sd < Ω sr. In the CRS-NOMA scheme with spectrum sharing, the SU-Tx broadcasts a 1 P s h sp s 1 + a P s h sp s to both relay and SU-Rx, where s 1 and s are the data-bearing constellation symbols which are multiplexed in the power domain E{ s i } 1 for i 1,. P s h sp is the power transmitted from the SU-Tx and in general is a mapping from the fading coefficient h sp to the set of non-negative real numbers R + such that the instantaneous interference at the PU-Rx does not exceed a predetermined value interference temperature. Moreover, a 1 and a are power weighting coefficients satisfying the constraints a 1 +a 1 and a 1 > a. After receiving the signal from the SU-Tx, the SU-Rx decodes symbol s 1 treating the

4 4 interference from s as additional noise, while the relay first decodes symbol s 1 and then applies SIC to decode s. In the second time slot, the SU-Tx remains silent and only the relay transmits its estimate of symbol s, denoted as ŝ, to the SU-Rx with power P r h rp which in general is a mapping from the fading coefficient h rp to R + such that the instantaneous interference at the primary receiver does not exceed the predetermined threshold. In this manner, two different symbols are delivered to the secondary receiver in two time slots. In contrast to this, in the conventional OMA with underlay spectrum sharing scheme, the SU-Tx broadcasts symbol s 1 with power P s h sp in the first time slot and the relay retransmits the resulting symbol estimate ŝ 1 to the SU-Rx with power P r h rp in the second time slot. The SU-Rx then combines both the copies of symbol s 1 and in this manner only a single symbol is delivered to the SU-Rx in two time slots. III. PERFORMANCE ANALYSIS In this section we present the achievable sum-rate and outage probability analysis of the CRS-NOMA with underlay spectrum sharing under pea interference constraint. We first analyze the simple case where N r N d 1 and then we generalize these results to the case when N r 1 and N d 1. A. Scenario I: N r N d 1 In this case we denote the channel coefficients for the S D, S R, R D, S P and R P lins by h sd, h sr, h rd, h sp and h rp respectively. The signals received at the relay, the SU-Rx and the PU-Rx in the first time slot are, respectively, y sr h sr a 1 P s h sp s 1 + a P s h sp s + n sr, y sd h sd a 1 P s h sp s 1 + a P s h sp s + n sd, y sp h sp a 1 P s h sp s 1 + a P s h sp s + n sp, where n i, i {sr, sd, sp} is complex additive white Gaussian noise AWGN with zero mean and unit variance. The received instantaneous signal-to-interference-plus-noise ratio SINR at the relay for decoding symbol s 1 and the received instantaneous signal-to-noise ratio SNR for decoding symbol s assuming the symbol s 1 is decoded correctly are given by γ 1 sr λ sr a 1 P s h sp /{λ sr a P s h sp + 1} and γ sr λ sr a P s h sp respectively, where λ sr h sr. Similarly, the received instantaneous SINR at the SU-Rx for decoding of s 1 is given by γ sd λ sd a 1 P s h sp /{λ sd a P s h sp + 1}, where λ sd h sd.

5 5 In the next time slot, the relay transmits the decoded symbol ŝ to the SU-Rx with power P r h rp. The received signals at the SU-Rx and the PU-Rx are, respectively, y rd h rd P r h rp ŝ + n rd, y rp h rp P r h rp ŝ + n rp, where n, {rd, rp} is complex AWGN with zero mean and unit variance. The received SNR at the SU-Rx for decoding the symbol s is given by γ rd λ rd P r h rp, where λ rd h rd. Since the symbol s 1 should be decoded correctly at the SU-Rx as well as at the relay for SIC, while satisfying the interference constraint at the PU-Rx, the average achievable rate for symbol s 1 is given by c.f. [8, eqn. 8], [4, eqn. 1, eqn. ] C s1 max.5 log 1 + min{γ sr 1, γ sd } g 1 h sp g h sr g 3 h sd d h sp d h sr d h sd, P sh sp h sp h sr h sd 1 s.t. P s h sp Q, where Q is the pea interference power that the PU-Rx can tolerate from the secondary networ, and where g 1 h sp, g h sr and g 3 h sd denote the probability density functions PDFs of h sp, h sr and h sd respectively. Assuming no other limitation on the power transmitted from the SU-Tx, the optimal transmit power P s h sp which maximizes 1 the achievable rate is given by Q/. Hence, the average achievable rate for symbol s 1 is given by C s1.5 log 1 + h sp h sr h sd [.5 log 1+Qxf X xdx where X min{λ sr, λ sd }/. min{λ sr,λ sd} Qa 1 min{λ sr,λ sd} g 1 h sp g h sr g 3 h sd d h sp d h sr d h sd Qa + 1 ] log 1+Qa xf X xdx, 3 Theorem 1. The closed-form expression for the average achievable rate for the symbol s 1 is obtained as where φ 1/Ω sr + 1/Ω sd. Q φ C s1.5 Q log Q φ a Q log aq a Q φ φ, 4 1 Here we refer to maximization of the achievable sum-rate for a given pair of power allocation coefficients a 1 and a. The achievable sum-rate can be further maximized by optimizing the power allocation coefficients a 1 and a for a given optimal transmit power level P s h sp.

6 6 Proof : See Appendix A. Similarly, the average achievable rate for symbol s is given by c.f. [8, eqn. 9], [4, eqn. 1, eqn. ] C s max.5 log 1+min{γ sr, γ rd } g 1 h sp g h sr g 4 h rp g 5 h rd P sh sp h sp h rp h sr h rd P rh rp d h sp d h sr d h rp d h rd, 5 s.t. P s h sp Q, λ rp P r h rp Q, 6 7 where g 4 h rp and g 5 h rd denote the PDFs of h rp and h rd respectively. Assuming no other limitation on the power transmitted from the SU-Tx and the relay, the optimal transmit power levels P s h sp and P r h rp which maximize the achievable rate are given by Q/ and Q/λ rp, respectively. Therefore, the average achievable rate for symbol s is given by { λsr a C s.5 log 1+min, λ } rd Q g 1 h sp g h sr g 4 h rp g 5 h rd λ rp h sp.5 h rp h sr h rd log 1+Qxf Y xdx 1 log eq where Y min {λ sr a /, λ rd /λ rp }. 1 F Y x 1 + Qx d h sp d h sr d h rp d h rd dx, 8 Theorem. The closed-form expression for the average achievable rate for the symbol s is given by.5 a Ω rd Ω sr Q C s Ω rd a Ω rp Ω sr Ω rd Q Ω rp a Ω sr Q [ a Ω rp Ω sr Ω rp log Ω rd Proof : See Appendix B. + a Ω rp Ω sr Q log Ωrd Q Ω rp ] a Ω sr Q Ω rd Q log. 9 Using 4 and 9, the average achievable sum-rate for the CRS-NOMA is given as C sum C s1 + C s. 1 For the case of CRS-OMA, the signals received at the relay, the SU-Rx and the PU-Rx in the first time slot are, respectively, y sr,oma h sr Ps h sr s 1 + n sr, y sd,oma h sd Ps h sd s 1 + n sd, y sp,oma h sp P s h sp s 1 + n sp.

7 7 In the second time slot, the relay transmits its estimate of s 1, denoted by ŝ 1 to the SU-Rx. The signals received at the SU-Rx and the PU-Rx in the second time slot are respectively, y rd,oma h rd Pr h rd ŝ 1 + n rd, y rp,oma h rp P r h rp ŝ 1 + n rp. Following the same pea interference constraint as in the case of NOMA, the average achievable rate for the CRS-OMA system is given as C OMA.5E Z [log 1 + QZ], 11 { } where Z min λsr, λsd + λrd λ rp and E W [ ] denotes the expectation with respect to the random variable W. Outage probability for CRS-NOMA: We define O 1 as the outage event for symbol s 1, i.e., the event where either the relay or the SU-Rx fails to decode s 1 successfully. Hence the outage probability for symbol s 1 is given by [ 1 PrO 1 PrC s1 <R 1 Pr log 1+ a ] 1QX <R 1 PrX < Θ 1 a QX + 1 φθ φ Θ 1, Θ1 φ 1 + φ x dx using where C s1 is the instantaneous achievable rate for symbol s 1, R 1 is the target data rate for symbol s 1, ɛ 1 R1 1 and Θ 1 1 ɛ 1 Qa 1 ɛ 1a. The integration above is solved using [1, eqn ]. The system design must ensure that a 1 > ɛ 1 a, otherwise the outage probability for symbol s 1 will always be 1 as noted in [11]. Next, we define O as the outage event for symbol s. This outage event can be decomposed as the union of the following disoint events: i symbol s 1 cannot be successfully decoded at the relay; ii symbol s 1 is successfully decoded at the relay, but symbol s cannot be successfully decoded at the relay; and iii both symbols are successfully decoded at the relay, but symbol s cannot be successfully decoded at the SU-Rx. Therefore, the outage probability for the symbol s may be expressed as Pr PrO Pr λsr <Θ 1 +Pr Θ 1, λ sr λsr <Θ + Pr Θ, λ rd < ɛ ; if Θ 1 < Θ λ rp Q λsr <Θ 1 + Pr Θ 1, λ rd < ɛ ; otherwise λ rp Q ɛ ɛ F λsr λ Q λspθf rd, 13 λrp Q λsr λsr F λsr λspθ+f λ rd λrp where R is the target data rate for symbol s, ɛ R 1, Θ ɛ a Q and Θ maxθ 1, Θ. Here we assume that the SU-Rx applies MRC on the two copies of s 1.

8 8 B. Scenario II: N r 1, N d 1 In this subsection, we generalize the results obtained in the previous subsection for the case when N r 1, N d 1 and selection combining SC, i.e., selection of the antenna with highest instantaneous SNR, is used for reception at both relay and SU-Rx. The received instantaneous SINR at the relay for decoding symbol s 1 and the instantaneous SNR for decoding symbol s assuming the symbol s 1 is decoded correctly are given by γ 1 sr,sc δ sra 1 P s h sp /{δ sr a P s h sp + 1} and γ sr,sc δ sra P s h sp, respectively, where i argmax 1 i h sr,i and δ sr h sr,i. Similarly, the received instantaneous SINR at the SU-Rx for decoding symbol s 1 is given by γ sd,sc δ sd a 1 P s h sp /{δ sd a P s h sp +1}, where argmax 1 h sd, and δ sd h sd,. In the next time slot, the received instantaneous SNR at the SU-Rx for decoding symbol s is given by γ rd,sc δ rd P r h rp, where argmax 1 h rd, and δ rd h rd,. Following similar arguments as in the previous subsection, the average achievable rate for symbol s 1 using SC is given by C s1,sc.5 where X min{δ sr, δ sd }/. log 1 + Qxf X x dx.5 log 1 + Qa xf X x dx, 14 Theorem 3. The closed form expression for the average achievable rate for symbol s 1 using SC is obtained as C s1,sc where ξ, /Ω sr + /Ω sd. Proof : See Appendix C. 1 + Q log Q ξ, Q ξ, Similarly, the average achievable rate for symbol s using SC is given by C s,sc.5 where Y min {δ sr a /, δ rd /λ rp }. a Q log aq a Q ξ, ξ,, 15 log 1 + Qx f Y x dx, 16 Theorem 4. The closed-form expression for the average achievable rate for symbol s using SC is obtained as a C s,sc.5q 1 1 Ω sr log aω srq + a Ω sr Q N r Ωrd log ΩrdQ Ω rp Ω 1 rd Q Ω rp Ω rd log Ωrp QΩ rd Ω rd a Ω rp Ω sr QΩ rd Ω rp a Ω rpω sr log aqω sr Ω rd a Ω rp Ω sr a QΩ sr

9 9 Proof : See Appendix D. Using 15 and 17, the average achievable rate for the CRS-NOMA using SC is given by C sum,sc C s1,sc + C s,sc. 18 With some algebraic manipulations, it can be shown that for N r N d 1, 18 reduces to 1. For the case of CRS-OMA with SC, the average achievable rate is given as C OMA,SC.5E Z [log 1 + QZ], 19 where Z min{ δsr, δsd + δrd λ rp }. Outage probability for CRS-NOMA with SC: Similar to the previous subsection, we define O 1 as the outage event for symbol s 1. Hence the outage probability for symbol s 1 using SC is given by PrO 1 Pr C s1,sc < R 1 PrX < Θ 1 Θ1 f X xdx ξ, Θ ξ, Θ 1, where C s1,sc is the instantaneous achievable rate for symbol s 1 in the CRS-NOMA using SC. The integration above is solved using 7 and [1, eqn ]. Next, we define O as the outage event for symbol s using SC, similar to the previous subsection. Hence, the outage probability for symbol s is given by PrO F δsr δ λspθ+f rd λrp ɛ F δsr δ Q λspθf rd λrp ɛ Q. 1 IV. RESULTS AND DISCUSSION In this section, we present the analytical and simulation 3 results for the average achievable rate and outage probability for the spectrum sharing based cooperative relaying system. We consider a scenario where Ω sd 1, Ω sr Ω rd 1 and Ω rp 5.5. For all the NOMA systems, we consider a.1, R 1 R 1 bps/hz. Fig. a shows a comparison of the average achievable rate for the SS based CRS with N r N d 1. It is clear from the figure that for small values of Q, the SS based CRS-OMA gives higher achievable rate, but for higher values of Q, the CRS-NOMA based SS system outperforms its OMA counterpart and achieves higher spectral efficiency. In Fig. b, the average achievable rate for SS based CRS with selection combining is shown for different values of N r and N d. It is interesting to note that with an increase in the number of receive 3 We do not realize the actual scenario for simulation, but rather generate the random variables and then evaluate the average achievable rate.

10 a Simple case: N r N d 1. b General case: N r 1, N d 1. Fig. : Comparison of average achievable rate for CRS based SS a Symbol s 1. b Symbol s. Fig. 3: Outage probability for CRS-NOMA with SS. antennas at the relay and at the SU-Rx, the threshold value Q at which the CRS-NOMA outperforms its OMA counterpart becomes lower. Also, the agreement between the analytical and simulation results in Fig. confirms the correctness of our sum-rate analysis. Fig. 3a and Fig. 3b show the outage probability for symbol s 1 and s respectively against Q and as expected, the outage probability decreases as the number of receive antennas is increased at the relay and at the SU-Rx.

11 11 V. CONCLUSION In this paper, we provided a comprehensive achievable sum-rate and outage probability analysis of a NOMA based cooperative relaying system with spectrum sharing considering a pea interference power constraint. We considered the scenario where the relay and the secondary receiver are equipped with multiple receive antennas and where both apply selection combining to combine the received signal. It was shown that for higher values of pea interference power Q, the spectrum sharing system based on CRS-NOMA outperforms the spectrum sharing system based on conventional CRS-OMA, achieving higher spectral efficiency. Our results indicate that significant capacity gains can be achieved when the interference channel between the secondary transmitter/relay to primary receiver is in deep fade. ACKNOWLEDGMENT This publication has emanated from research conducted with the financial support of Science Foundation Ireland SFI and is co-funded under the European Regional Development Fund under Grant Number 13/RC/77. APPENDIX A PROOF OF THEOREM 1 Since h i, i {sr, sd, rd, sp, rp} is Rayleigh distributed, the PDF and the cumulative distribution function CDF of λ i h i are, respectively, given by f λi x 1 x x exp, F λi x 1 exp. Ω i Therefore, the PDF of min{λ sr, λ sd } is given as 4 Ω i f min{λsr,λ sd}x f λsr x[1 F λsd x]+ f λsd x[1 F λsr x] φ exp φx, Ω i where φ 1/Ω sr +1/Ω sd. Using a transformation of random variables, the PDF of X min{λ sr, λ sd }/ is therefore given as f X x yf min{λsr,λ sd}yxf λsp y, dy φ y exp [ φx+ 1 ] y dy φ 1 + φ x. 4 Given two random variables U and V with PDFs f Ux and f Vx respectively, and CDFs F Ux and F Vx respectively, the PDF of W min{u, V} is given by f Wx f Ux[1 F Vx] + f V[1 F Ux] and the CDF of W is given by F Wx F Ux + F Vx F UxF Vx.

12 1 The integral above is solved using [1, eqn ]. Using, the first integral in 3 can be solved as ln1 + Qxdx Q log I 1 log eφ 1 + φ x. 3 Q φ Q φ The integration above is solved using [1, eqn ]. Similarly, the second integral in 3 can be solved as ln1 + a Qxdx a Q log aq I log eφ 1 + φ x φ. 4 a Q φ Hence, using 3, 3 and 4, the closed-form expression for the average achievable rate for symbol s 1 in CRS-NOMA reduces to 4; this completes the proof. APPENDIX B PROOF OF THEOREM Using a transformation of random variables, we have f λsra x 1 x f λsr 1 x exp. a a a Ω sr a Ω sr Hence, and f λsra / x 1 a Ω sr yf λsra yxf λsp y dy F λsra / x [ x y exp + 1 ] y dy a Ω sr x f λsra / t dt The integration above is solved using [1, eqn ]. Similarly, Therefore, we have f λrd/λ rp x Ω rd Ω rp Ω rd + Ω rp x, F λ rd/λ rp x x a Ω sr + x. 1 F Y x1 F λsra / x F λrd/λ rp x+f λrd/λ rp xf λsra / x Using 8 and 5, the average achievable rate for symbol s is given as C s lim Λ Λ Ω rp x Ω rd + Ω rp x. a Ω sr a Ω sr + x, using [1, ] a Ω sr Ω rd a Ω sr + xω rd + Ω rp x. 5.5 log e Q a Ω sr Ω rd dx a Ω sr + xω rd + Ω rp x1 + Qx. 6 Solving the integral above using partial fractions, 6 reduces to 9; this completes the proof.

13 13 APPENDIX C PROOF OF THEOREM 3 The CDF and PDF of δ sr are, respectively, F δsr x exp 1 1 f δsr x 1 1 exp Ω sr x Ω sr x The CDF and PDF of δ sd can be obtained by replacing N r by N d and Ω sr by Ω sd in the corresponding equations above. The PDF of min{δ sr, δ sd } is given by f min{δsr,δ sd}x Ω sr,. ξ, exp ξ, x, where ξ, /Ω sr + /Ω sd. The PDF of X min{δ sr, δ sd }/ is therefore given by 1 + ξ [, f X x yf min{δsr,δ sd}yxf λsp ydy y exp ξ, x ξ, 11 ξ, x+ 1 ] y dy. 7 The integral above is solved using [1, eqn ]. Now, the first integral in 14 can be solved as 1 + ξ, I 3 log e ln1+qxf X xdx ln1+qx ξ, x+ 1 dx ln Q Q ξ, log Q. 8 ξ, The integration above is solved using [1, eqn ]. Similarly, the second integral in 14 can be solved as I 4 log e ln1 + Qa xf X x dx a Q a Q ξ, log a Q ξ,. 9 Using 14, 8 and 9, the closed-form expression for the average achievable rate for symbol s 1 in the CRS-NOMA with SC reduces to 15; this completes the proof. APPENDIX D PROOF OF THEOREM 4 Using a transformation of random variables, the PDF of δ sr a is given as f x δsr a 1 1 x f δsra x exp. a a Ω sr a Ω sr 1

14 14 The PDF of δ sr a / is given by f δsra / x yf δsra yxf λsp y dy a Ω sr 1 x + 1. a Ω sr 1 1 a Ω sr [ x y exp + 1 ] y dy a Ω sr The integration above is solved using [1, ]. Using [1, eqn ], the CDF of δ sr a / is given by Similarly, F δsra x x f δsr/λ rp x F δrd/λ rp x f δsra t dt Ω rd + Ω rp Therefore, for Y min {δ sr a /, δ rd /λ rp }, we have 1 F Y x 1 F δsr a x F δ rd λsp λrp Ω rp x Ω rd + Ω rp x. + F δsr a xf δ rd x λsp λrp 1 1 x a Ω sr + x x a Ω sr + x. x + 1, Ω rd Ω rp 1 1 Ω rp x Ω rd + Ω rp x Using 16, the average achievable rate for symbol s using SC is given as C s,sc.5 Now we define the integral I 5 as 1 x I Qx a Ω sr + x Qx Λ dx lim Λ log 1 + Qxf Y x dx.5 log eq dx Qx a Ω sr dx a Ω sr + x1 + Qx 1 + Ω rp x a Ω sr + xω rd +Ω rp x. 3 1 F Y x 1 + Qx a Ω sr dx a Ω sr + x The integration above is solved using partial fractions. Similarly, 1 Ω rp x 1 I 6 dx 1 + Qx Ω rd + Ω rp x 1 + Qx dx Ω rd ln Ω rd Q Ω rp dx. 31 dx a Ω sr ln aω srq 1 + Qx. 3 a Ω sr Q Ωrd Q Ω rp, 33

15 15 and I 7 1 Ω rp x dx 1 + Qx a Ω sr + xω rd + Ω rp x [ dx Ω rd Qx a Ω rp Ω sr Ω rd Ω rd + Ω rp x + a Ω rpω ] sr Ω rd a Ω rp Ω sr a Ω sr + x dx Ω 1 + Qx + rd ln Ωrp Ω rdq a Ω rd a Ω rp Ω sr Ω rd Q Ω rp + Ω rpω sr ln aω srq Ω rd a Ω rp Ω sr a Ω sr Q. Moreover, we also have dx Qx Using 31 35, the closed-form expression for the average achievable rate of symbol s in CRS-NOMA using SC reduces to 17; this completes the proof. REFERENCES [1] Y. Liu, Z. Qin, M. Elashlan, Z. Ding, A. Nallanathan, and L. Hanzo, Nonorthogonal multiple access for 5g and beyond, Proc. of the IEEE, vol. 15, no. 1, pp , Dec 17. [] Z. Ding, Y. Liu, J. Choi, Q. Sun, M. Elashlan, C. L. I, and H. V. Poor, Application of non-orthogonal multiple access in LTE and 5G networs, IEEE Commun. Mag., vol. 55, no., pp , February 17. [3] A. Goldsmith, S. A. Jafar, I. Maric, and S. Srinivasa, Breaing spectrum gridloc with cognitive radios: An information theoretic perspective, Proc. of the IEEE, vol. 97, no. 5, pp , May 9. [4] A. Ghasemi and E. S. Sousa, Fundamental limits of spectrum-sharing in fading environments, IEEE Trans. Wireless Commun., vol. 6, no., pp , Feb 7. [5] L. Sboui, Z. Rezi, and M. S. Alouini, Achievable rate of spectrum sharing cognitive radio systems over fading channels at low-power regime, IEEE Trans. Wireless Commun., vol. 13, no. 11, pp , Nov 14. [6] L. Lv, J. Chen, Q. Ni, Z. Ding, and H. Jiang, Cognitive non-orthogonal multiple access with cooperative relaying: A new wireless frontier for 5G spectrum sharing, IEEE Commun. Mag., vol. 56, no. 4, pp , Apr 18. [7] B. Chen, Y. Chen, Y. Chen, Y. Cao, N. Zhao, and Z. Ding, A novel spectrum sharing scheme assisted by secondary NOMA relay, IEEE Wireless Commun. Lett., to appear. [8] J. B. Kim and I. H. Lee, Capacity analysis of cooperative relaying systems using non-orthogonal multiple access, IEEE Commun. Lett., vol. 19, no. 11, pp , Nov 15. [9] R. Jiao, L. Dai, J. Zhang, R. MacKenzie, and M. Hao, On the performance of NOMA-based cooperative relaying systems over Rician fading channels, IEEE Trans. Veh. Technol., vol. 66, no. 1, pp , Dec 17. [1] A. Jeffrey and D. Zwillinger, Table of Integrals, Series, and Products, 7th ed. Elsevier Science, 7. [11] Z. Ding, H. Dai, and H. V. Poor, Relay selection for cooperative NOMA, IEEE Wireless Commun. Lett., vol. 5, no. 4, pp , Aug 16.