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A Rate Splitting Strategy for Uplink CR-NOMA Systems

In uplink non-orthogonal multiple access (NOMA) channels, the existing cooperative successive interference cancellation (SIC) and power control (PC) schemes lack the capability of achieving the full capacity region, which restricts the outage performance of uplink NOMA users. For the uplink cognitive radio (CR) inspired NOMA system, we propose a new rate splitting (RS) strategy to maximize the achievable rate of the secondary user without deteriorating the primary user's outage performance. Based on the interference threshold and its own channel gain, the secondary user adaptively conducts RS, transmit power allocation and SIC, which utilizes the transmit power efficiently. Closed-form expression of the outage probability is derived for the secondary user. Numerical results show that the proposed RS scheme achieves the best outage performance for the secondary user among the existing cooperative SIC and PC schemes.


I. INTRODUCTION
In power-domain NOMA systems, multiple users' signals are multiplexed on the same time/frequency resource block, which results in inter-user interference (IUI) at the receiver side. Due to the powerful capability of eliminating IUI and simple structure, successive interference cancellation (SIC) has been utilized to detect the multiplexing signals in NOMA systems [1]. Nevertheless, the decoding performance of a NOMA receiver strongly depends on how the SIC decoding order and the associated power control (PC) are optimized [2]- [7].
Under the quality of service (QoS) oriented criterion, the users with more stringent QoS requirements are decoded first, while the others are decoded in the remaining stages of the SIC [2]- [4]. Under the channel state information (CSI)based criterion, the users with better CSI are firstly decoded before decoding the users with the worse CSI [1], [5]- [7]. When a fixed SIC decoding order is applied in the uplink NOMA systems, the users decoded in the primary stages of the SIC are inevitably affected by the IUI, which may result in severe outage probability error floor. Fortunately, the recently proposed QoS-guaranteed SIC (QoS-SIC) scheme in [2] and [3] avoids the outage probability error floor to some extent by adopting a hybrid decoding order, which takes into account both the CSI and QoS requirements for calculating the users' priorities in SIC. Furthermore, a new hybrid SIC (NH-SIC) scheme was proposed in [4], which introduced the PC method to increase the achievable rate.
However, the existing SIC and PC schemes cannot fully achieve the capacity region of the uplink multiple access channels (MACs) compared to the rate splitting (RS) [8], [9], hence only suboptimal outage performance can be achieved by the QoS-SIC and NH-SIC schemes. In this paper, we propose an RS strategy for the uplink cognitive radio (CR) inspired NOMA (CR-NOMA) system. Without deteriorating the outage performance of the primary user compared to that in orthogonal multiple access (OMA), the secondary user splits its signal into two streams and allocates the transmit power efficiently to each stream to maximize the achievable rate. Benefiting from advantage of the RS for achieving the full capacity region of MACs, our proposed RS scheme attains the best outage performance for the secondary user. To clarify the advantage of the proposed RS scheme, we carry out a comprehensive analysis and provide detailed comparisons over the existing SIC and PC schemes [3], [4].

A. System Model
We consider an uplink CR-NOMA system consisting of a primary user U 0 , a secondary user U 1 , and a base-station (BS). Each node is equipped with a single antenna. The channel coefficients from U 0 and U 1 to the BS are denoted by h 0 and h 1 , respectively, which are modeled as independent and identically distributed (i.i.d.) circular symmetric complex Gaussian random variables with zero mean and unit variance. We assume that the channels follow a statistical fading. In addition, we assume the same channel state information (CSI) acquisition procedure as that in [4] and take perfect SIC for comparing the system performance of the proposed RS scheme with the benchmarks [3], [4].
Required by the stringent QoS requirements, U 0 transmits at a fixed data rate. U 1 is allowed to share the same resource block with U 0 only when the U 1 's transmission does not deteriorate the U 0 's outage performance compared to the counterpart of U 0 in OMA. In other words, U 0 experiences the same outage performance as in OMA for delay-limited transmissions.
To begin with, we introduce P i , g i , and ε i for user U i (i = 0, 1) to represent its transmit power, channel gain, and target rate related metric, respectively, where g i = |h i | 2 and ε i = 2R i − 1 withR i denoting the target rate of U i . Prior to each transmission block, the BS sends an interference threshold to U 1 to guarantee the QoS requirements of U 0 . Considering that U 0 can achieve the same system performance as in OMA, the interference threshold is given by With respect to τ , the target rateR 0 is achievable for U 0 when log 2 (1 + P 0 g 0 ) ≥R 0 ; Otherwise, U 0 experiences an outage event due to log 2 (1 + P 0 g 0 ) <R 0 .
To enhance the transmission reliability, U 1 applies RS for each block of transmission. Specifically, U 1 splits its signal x 1 into two parts x 1,1 and x 1,2 for transmission [8], [9]. At the end of each transmission block, the received signal at the BS can be expressed as where x 0 is the transmit signal of U 0 , n is the Gaussian background noise modeled with a normalized power, and α is the power allocation factor satisfying 0 ≤ α ≤ 1. We assume that all the signals, {x 0 , x 1 , x 11 , x 12 }, are independently coded with Gaussian code book and each signal has a unit power in expectation. The signal model in (2) is also called as rate splitting multiple access (RSMA) [8]. In this paper we use the RS as in (2) to improve the transmission robustness of the secondary user in the considered CR-NOMA system. According to [8], the SIC decoding order x 11 → x 0 → x 12 is applied at the BS receiver to achieve the benefit of RSMA. Consequently, the received SNR/SINRs for decoding x 11 , x 0 , and x 12 can be expressed as γ 11 = αP1g1 (1−α)P1g1+1 , and γ 12 = (1 − α)P 1 g 1 ,, respectively. For the primary and secondary users, the achievable rates are given by R 0 = log 2 (1 + γ 0 ) and R 1 = R 11 + R 12 , respectively, with R 11 = log 2 (1 + γ 11 ) and R 12 = log 2 (1 + γ 12 ).

B. RS Strategy
The RS strategy is designed to obtain the maximum allowed achievable rate R 1 for U 1 , meanwhile keeping U 0 attaining the same outage performance as in OMA. To begin with, we assume that x 11 has been detected correctly in the first SIC stage using the decoding order x 11 → x 0 → x 12 . Then, the remaining SIC is to detect x 0 and x 12 , sequentially. When τ = 0, the U 0 's transmission is always in outage and the detection of x 12 will fail in the remaining SIC with decoding order x 0 → x 12 . Nevertheless, when τ > 0 and γ 12 ≤ τ , it can be shown not only R 0 ≥R 0 , but also the achievable rate R 1 = R 11 + R 12 for U 1 . To maximize the achievable rate R 1 , the RS strategy is designed to maximize R 12 = log 2 (1 + γ 12 ) considering that the detection of x 12 is interference-free in SIC.
Depending on the different values of P 1 g 1 and τ , the RS strategy is designed as follows: 1) Case I: τ > 0 and P 1 g 1 ≤ τ . In order to maximize R 1 for this case, it is needed to allocate all the transmit power P 1 to transmit x 12 , such that R 12 = log 2 (1 + γ 12 ) is maximized. Consequently, the RS strategy sets α = 0 and x 12 = x 1 . Since x 11 is not transmitted, the decoding order x 11 → x 0 → x 12 degrades to x 0 → x 12 (or equivalently x 0 → x 1 ), so that the achievable rate for U 1 becomes ( 2) Case II: τ > 0 and P 1 g 1 > τ . In this case, to maximize the achievable rate R 1 , R 12 = log 2 (1 + γ 12 ) needs to be firstly maximized subject to the constraint of log 2 (1+γ 12 ) ≤ log 2 (1+τ ), i.e., the transmission of x 12 cannot decrease the outage performance of U 0 for its transmission of x 0 . Obviously, the allowed maximum R 12 = log 2 (1 + τ ) is achieved by setting log 2 (1 + γ 12 ) = log 2 (1 + τ ), which results in the optimal power allocation factor α = 1 − τ P1g1 for transmitting x 12 with the power (1−α)P 1 . Meanwhile, the remaining transmit power, αP 1 , is allocated to transmit x 11 , which yields the achievable rate R 11 = log 2 1 + P1g1−τ P0g0+τ +1 . In this case, the achievable rate for the U 1 's transmission is given by To ensure that U 0 achieves the same outage performance as in OMA, it requires the successful detection of x 11 before detecting x 0 . Nevertheless, x 11 cannot be detected correctly if R (II) <R 1 . Therefore, in the proposed RS strategy, U 1 is allowed to transmit only if R (II) ≥R 1 . Otherwise, U 1 keeps silent.
3) Case III: τ = 0. In this case, the U 0 's transmission always encounters outage. To efficiently utilize transmit power, U 1 does not allocate any part of P 1 to transmit x 12 which cannot be detected correctly due to failure detection of x 0 . To maximize the achievable rate R 1 in this case, the RS strategy sets α = 1 and x 11 = x 1 . Consequently, the decoding order x 11 → x 0 → x 12 degrades to x 11 → x 0 (or equivalently x 1 → x 0 ), so that the achievable rate for U 1 becomes Obviously, the difference between the proposed RS scheme and the QoS-SIC and NH-SIC schemes [3], [4] is that the QoS-SIC and NH-SIC schemes can attain the achievable rate in the form of R (I) and R (III) , whereas only the proposed RS scheme can attain the achievable rate R (II) in Case II.

III. OUTAGE ANALYSIS FOR THE PROPOSED RS SCHEME
In the proposed RS scheme, the primary user experiences the same outage performance as in OMA and the outage probability experienced by the secondary user can be expressed as follows: where P are provided in the following theorems.
Theorem 1: Corresponding to the RS scheme's operation of Case II, the probability that U 1 is in outage is given by with η 0 ε0 P0 and Proof: See Appendix A. Although the operation of the proposed RS scheme in Case II is different to those of the QoS-SIC and NH-SIC schemes [3], [4], their operations in Cases I and III are the same. As a result, closed-form expressions for the P (I) out and P (III) out can be retrieved from [3] and [4] as in the following theorem.
Theorem 2: Corresponding to the RS scheme's operations in Cases I and III, probabilities that U 1 is in outage are given by respectively, where η 1 ε1 P1 . Corollary 1: The outage probability experienced by the secondary user in the proposed RS scheme is given by Proof: By combing P out , and P (III) out , the outage probability is obtained as (11).
Remark 1: In the high SNR region, i.e., P 0 = P 1 → ∞, the probability terms can be approximated as P , which results in an approximation for the outage probability as P out ≈ η 1 .
Remark 2: Recalling the relationship between the upper and lower bounds on the channel gain g 1 for deriving P out,II as in Appendix A, only the constraint η 0 < g 0 < η 0 (1 + ε 1 ) is required without additional restricting on the values of ε 0 and ε 1 , whereas the QoS-SIC scheme of [3] needs ε 0 ε 1 < 1 to avoid the error floor for the outage probability. Consequently, the proposed scheme can achieve the diversity gain of 1 for all the feasible values of ε 0 and ε 1 as indicated by P out ≈ η 1 .

IV. COMPARISON BETWEEN THE PROPOSED SCHEME AND BENCHMARK SCHEMES
In this section, we compare the system performance of our proposed RS scheme with those of the QoS-SIC and NH-SIC schemes [3], [4], respectively.
For the considered three schemes, the operations in Cases I and III are same resulting in the same probabilities P (I) out and P (III) out for U 1 . The difference among the three schemes only lies on the operation for the Case II. For Case II, the QoS-SIC and NH-SIC schemes attain the achievable ratesR (II) = log 2 1 + P1g1 P0g0+1 andR (II) = max log 2 1 + P1g1 P0g0+1 , log 2 (1 + τ ) , respectively [3], [4]. For the QoS-SIC and NH-SIC schemes, the probabilities that U 1 is in outage for Case II can be evaluated as respectively [3], [4]. Furthermore,P (II) out in (12) only exists for ε 0 ε 1 < 1. When ε 0 ε 1 ≥ 1,P For the considered schemes working for Case II, the conditional outage probability for U 1 can be expressed as P (II),con out, 1+P1η0 . Remark 3: The proposed RS scheme can achieve a lower outage probability than that of the NH-SIC scheme due to ∆P (II) out ≥ 0. Furthermore, as P 0 = P 1 → ∞, we have ∆P (II) out → 0. As a result, the proposed scheme and NH-SIC scheme achieve the same outage probability of U 1 in the high SNR region.
Remark 4: An advantage of the proposed RS scheme is that it always attains a higher achievable rate than the QoS-SIC and NH-SIC schemes, which can be representatively demonstrated by the following examples.

V. SIMULATION RESULTS
In this section, we present the simulation results to clarify the outage performance achieved by the proposed RS scheme and verify the accuracy of the developed analytical expressions. For comparison purposes, the CSI-based SIC (CSI-SIC) scheme is also considered [1], in which the stronger user's signal is decoded before the weaker user's signal. In Fig. 1, the outage probability achieved by the proposed RS scheme is compared with the existing schemes for various transmit power settings. Particularly, we assume P 0 = P 1 in Fig. 1(a) and P 0 = P1 10 in Fig. 1(b), respectively. The curves in Figs. 1(a) and 1(b) verify the accuracy of the analytical result in Theorem 1 and the approximation expression for P out . Furthermore, we can see that the proposed scheme achieves the lowest outage performance among the four schemes. Compared to the CSI-SIC and QoS-SIC schemes that cause the error floors on the outage probability, the proposed RS scheme achieves the outage probability monotonically decreasing with an increasing transmit SNR. In addition, the proposed RS scheme achieves a lower outage probability than that of the NH-SIC scheme mostly in the middle SNR region. In the high SNR region, the proposed scheme and the NH-SIC scheme almost achieve the same outage probability due to ∆P (II) out → 0. Similarly to the NH-SIC scheme, the proposed scheme avoids the error floor without restricting the values of ε 0 and ε 1 .
In Fig. 2, we investigate the conditional outage probability of U 1 achieved by the proposed RS scheme under the Case II CSI condition. Based on the analytical results in Section III, we know that the outage performance difference among the existing schemes only relies on the operations for Case II. For the numerical results in Fig. 2(a), we set P 0 = P1 10 . In Fig. 2(a), the proposed RS scheme achieves the lowest conditional outage probability. For the QoS-SIC scheme, the conditional outage probability achieved by it has an error floor in the middle and high SNR regions. However, the conditional outage probability values achieved by both the NH-SIC scheme and the proposed RS scheme monotonically decrease with an increasing of the transmit SNR. The impact of the target rate on the probability that U 1 is in outage under the Case II CSI condition is investigated in Fig. 2(b), where we set a fixed target rateR 0 = 1 BPCU in addition to P 0 = 15 dB and P 1 = 20 dB. The results in Fig. 2(b) show that the conditional outage probability achieved by all the considered schemes monotonically increases with an increasing of the target rateR 1 . Furthermore, the CSI-SIC scheme achieves the highest conditional outage probability among the four schemes, while the conditional outage probability achieved by the QoS-SIC scheme is lower than that of the CSI-SIC scheme when ε 0 ε 1 < 1. Among the four schemes, the proposed RS scheme achieves the lowest conditional outage probability in the considered target rate region. In Fig. 3, we evaluate the average throughput achieved by U 1 to verify the superior performance of the proposed RS scheme. It is clearly illustrating that our RS scheme achieves the highest average throughput among the existing ones. Compared to the CSI-SIC, QoS-SIC, and NH-SIC schemes, the average throughput achieved by the RS scheme is much larger in the middle and high SNR regions, which verifies that the proposed RS scheme exploits the transmit power more efficiently than the CSI-SIC, QoS-SIC and NH-SIC schemes.

VI. CONCLUSIONS
In this paper, we have proposed an RS scheme to enhance the outage performance of the secondary user in the uplink CR-NOMA system without deteriorating the outage performance of the primary user compared to its counterpart in OMA. The proposed RS scheme can effectively allocate the transmit power to its split signal streams to improve the achievable rate in a CR-inspired way. The exact and approximated expressions for the outage probability of the secondary user have been derived and the numerical results have been provided, which clarify the superior outage performance of the proposed RS scheme over the existing SIC schemes.

APPENDIX: A PROOF OF THEOREM 1
The probability term P (II) out can be rewritten as P (II) out = Pr g 0 > η 0 , g 1 > where η 0 ε0 P0 , E{·} stands for the expectation operation, and S is defined by S Pr Since that g 1 is positive in practice, the inequality (1+ε0)(1+ε1)−(1+P0g0) P1 > 0 holds true for (15), which results in the constraint g 0 < η 0 (1 + ε 1 ) + ε1 P0 for the expectation in (15). For the expression of S in (16), the upper bound on g 1 should be larger than the lower bound on g 1 , which results in the following hidden constraint g 0 < η 0 (1 + ε 1 ).
Consequently, the expectation for S in (15) should be taken over η 0 < g 0 < η 0 (1 + ε 1 ) and P After some integration manipulations, a closed-form expression for P out,II can be obtained, which completes the proof.