Rate-Splitting Multiple Access-Based Cognitive Radio Network With ipSIC and CEEs

In this paper, we study the outage and ergodic rate performance of a rate-splitting multiple access (RSMA)-based cognitive radio (CR) system, where the secondary transmitter aims to communicate with two RSMA users. Imperfect successive interference cancellation (ipSIC) and channel estimation errors (CEEs) are considered in the proposed analysis. Analytical expressions for the outage probability (OP) and the ergodic rate (ER) are calculated. For a deeper understanding, the asymptotic behavior of OP and ER at high signal-to-noise ratio (SNR) regimes are carried out. Illustrative simulation results are presented and reveal that: i) The OP decreases and the ER gradually increases with the increase of SNR, and eventually approaching a constant; ii) ipSIC and CEEs have a negative impact on the considered system; iii) The outage and ER performance of RSMA-based CR network outperforms CR-NOMA network due to its flexible control of interference; iv) The OP first decreases and then increases with the power allocated to the common message, and there exists an optimal power allocation factor to ensure the reliable performance.


I. INTRODUCTION
Recently, the ever increasing demands for high data rates and the rapid development of intelligent terminals have led to the search for the Xuesong Gao and Xingwang Li are with the School of Physics and Electronic Information Engineering, Henan Polytechnic University, Jiaozuo 454003, China (e-mail: gaoxuesong@home.hpu.edu.cn;lixingwangbupt@gmail.com).
Congzheng Han is with the Middle Atmosphere and Global environment Observation (LAGEO) Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China (e-mail: c.han@mail.iap.ac.cn).
Ming Zeng is with the Department of Electrical and Computer Engineering, Laval University, Quebec, QC G1V 0A6, Canada (e-mail: ming.zeng@gel.ulaval.ca).
Hongwu Liu is with the School of Information Science and Electrical Engineering, Shandong Jiaotong University, Jinan 250357, China (e-mail: liuhongwu@sdjtu.edu.cn).
Arumugam Nallanathan is with the School of Electronic Engineering and Computer Science, Queen Mary University of London, E1 4NS London, U.K. (e-mail: a.nallanathan@qmul.ac.uk).
Digital Object Identifier 10.1109/TVT.2023.3305960higher efficient spectrum and massive connectivity schemes [1], [2].As two promising candidate techniques of the future wireless communication networks, rate-splitting multiple access (RSMA) and cognitive radio (CR) have been widely studied since it have the capability to support massive connectivity and high spectral efficiency [3], [4].RSMA has aroused a heated attention as a powerful multiple access technique [5].The dominant feature of RSMA is that the message required by the each user is divided into a common part and a private part at the transmitter.The common parts of all users are combined into a common message for each user to decode, while the private parts are independently encoded as private message that can only be recovered by its own user [6], [7].RSMA decodes partial interference and treats the rest as noise at the receiver [8].Therefore, it enables flexible interference control and is considered as a bridge between non-orthogonal multiple access (NOMA) and space-division multiple access (SDMA) [9], [10].
On a parallel avenue, cognitive radio (CR) is known as a potential technique that can effectively improve spectrum utilization [11].The CR network consists of a primary network and a secondary network.It provides an opportunity for the secondary network to share the licensed spectrum with the primary network.In general, underlay, overlay and interweave are the three typical modes in CR network according to the different spectrum access paradigms.Among these, underlay CR is more appealing since it needs less collaborative overhead and low implementation complexity [12].In underlay network, the secondary network can access the spectrum of the primary network for communication as long as its interference power to the primary network is within a certain power constraint [13].
RSMA combines the advantages of NOMA and SDMA, achieving massive connection, flexibly interference control and data transmission, while avoiding the complex receiver design problem in NOMA.Furthermore, CR compensates for the shortcomings of static spectrum allocation by dynamically accessing idle spectrum through spectrum monitoring.Therefore, applying rate splitting (RS) strategy to CR network will further improve the system performance and alleviate the problem of spectrum efficiency.In [14], the authors proposed an underlay multiple-input single-output network based on the downlink RSMA, and achieved the goal of minimizing the total transmit power by optimizing the precode vectors and common rate variables.The authors in [15] introduced RSMA into the downlink multi-antenna multicarrier CR network and maximized the ergodic mutual information of secondary users under the consideration of imperfect channel state information (CSI) at transmitter.However, the above literature did not take imperfect successive interference cancellation (ipSIC) into account and were all focused on optimization rather than performance analysis.Liu et al. in [16] introduced RS strategy into an uplink CR-NOMA system, and proved the superiority of the RS for considered system's reliability by analyzing the outage probability (OP) of users.
Currently, most works on the combination of CR and RSMA focus on optimizing network models to achieve better performance, with few analysis.Liu et al. in [16] proposed an uplink RS-based CR system with Rayleigh fading channels.There also exist many differences between uplink and downlink of the networks, and thus, the analysis for uplink may not be applied to downlink.To the best of our knowledge, there has not been much research on the RMSA-underlay CR yet.Although both amplify-and-forward (AF) and underlay CR can improve the performance of primary users, these are two different schemes.Underlay CR communicates in two networks, and AF protocol has the drawback of noise amplification in the process of signal amplification.
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Motivated by these reasons, the underlay CR is selected.In addition, there was no work to introduce Nakagami-m fading channels into RSMA-based underlay CR networks and the current studies do not consider the impact of ipSIC and the channel estimation errors (CEEs).
To fill this gap, RS strategy is adopted in the downlink underlay CR network in this paper.Secondary transmitter communicates with the secondary receivers without affecting the quality of service (QoS) of the primary network.The main contributions of this paper are as follows: 1) We analyze the OP and ER of the RSMA-based CR networks based on Nakagami-m fading channels under CEEs and ipSIC; 2) We discuss in depth the asymptotic performance of OP and ER at high signal-to-noise ratio (SNR) regions to shed light into the considered network; 3) We validate the advantage of the proposed RSMA-based CR networks by comparing with CR-NOMA networks.

II. SYSTEM MODEL
We consider a downlink RSMA-based underlay CR system, which consists of one secondary transmitter ST, one far user SR f , one near user SR n and one primary user PR.The following assumptions are considered: i) All nodes are equipped with a single antenna; ii) All channels are subject to the independent non-identically Nakagami-m fading channels 1 .
As [17], linear minimum mean square error (LMMSE) algorithm is adopted since the perfect CSI is impossible to acquire.Thus, the channel coefficient can be denoted by where ĥi represents the estimated channel coefficient and e i ∼ CN (0, σ 2 e i ) is the CEE.In addition, secondary communication is allowed if the PR does not receive harmful interference from the STblue.Accordingly, the following restriction should be met at the ST: P s = min(P max , , where P max is the maximum transmit power of ST, and P I represents the interference of ST on PR.
According to the RSMA protocol, the signal transmitted by the ST can be written as where α c and α p,1 , α p,2 are the power allocation factors of common message and private messages of the SR f and SR n , respectively, satisfying α c + 2 k=1 α p,k = 1.The received message at SR i (i = f, n) can be expressed as where n i ∼ CN (0, N i ) denotes the complex additive white Gaussian noise (AWGN).The signal-interference-plus-noise ratio (SINR) of decoding the common message x c and private message x p,k for SR i is given by 1 Nakagami-m fading channel is a general fading channel.It becomes Gaussian fading channel, Rayleigh fading channel and Rician fading channel when 2˜K+1 , respectively, where K is the Rician factor.All the Gaussian fading channel, Rayleigh fading channel and Rician fading channel are its special cases.

III. PERFORMANCE ANALYSIS
This section evaluates the outage and ER performance of the RSMAbased underlay CR network by calculating the analytcal expressions for OP and ER.Furthermore, we analyze the asymptotic behavior of OP and ER at high SNR regimes for the users to have a comprehensive sight of the proposed system.

A. Outage Performance Analysis 1) Outage Probability of SR i : When both the common message
x c and private message x p,k can be successfully decoded by SR i , the outage behavior will not occur.Therefore, the OP of SR i can be denoted as where γ c th and γ p,i th are the target rate of decoding x c and x p,k of SR i .Theorem 1: For Nakagami-m fading channels, the OP of SR f is represented as where the integer m i represents the fading severity parameter, Ω i denotes the average power and its value is positive, ), , Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Pmax ), and γ s = Pmax N 1 .Proof: See Appendix A. Corollary 1: At high SNRs, the asymptotic expression of OP for SR f is given by Next, the optimal value of α c are derived to achieve the optimal reliability of the SR f by performing a series of calculations on (8).We set th .It is obvious that A > 0 and C < 0, so the values of Δ 1 and Δ 2 decrease and increase with a c , respectively.Accordingly, P SR f out,γ→∞ will reach its optimal value when Δ 1 =Δ 2 .Therefore, the optimal power allocation coefficient for the common messages Similarly, the optimal value for the SR n can be computed using the same method.
Similarly, the following Theorem gives the analytical OP expression of SR n .
Theorem 2: For Nakagami-m fading channels, the analytical expression for the OP of SR n is represented as where , and .
Corollary 2: The asymptotic OP expression for SR n at high SNRs can be written as For a deeper exploration of the RSMA-based underlay CR network, we analyze the diversity orders of SR f and SR n .The diversity order is represented as Corollary 3: The diversity orders of SR f , SR n can be represented as Remark 1: From Corollaries 1 and 2, we can observe that the OP of SR f and SR n decreases with P I , the reliability continues to increase, and eventually saturates to a constant.This implies that the OPs have error floors, resulting in the diversity orders to be 0. In addition, there is an optimal value for α c to ensure the most reliable performance for the considered system.

B. Ergodic Rate Analysis
The analytical expression for the ER of SR i can be represented as Unfortunately, it is very difficult, if not impossible, to obtain exact expressions for the ERs of SR f and SR n .To this end, we seek to obtain the approximations of the ER of the two users in the following Theorem.
Theorem 3: For Nakagami-m fading channels, the analytic expression for the ER of SR f is approximately denoted by where . Proof: See Appendix B. Corollary 4: At high SNRs, the asymptotic expression for ER of SR f can be written as where ) and ).
Similar to SR f , the following Theorem gives the ER of SR n .Theorem 4: For Nakagami-m fading channels, the analytical expression for the ER of SR n is approximately written by where Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.where e SRn ).Remark 2: From Corollaries 4 and 5, it can be observed that the ER increases with P I and finally tends to be a fix constant, indicating that there exists a ceiling for ER.The ER of SR f and SR n increases with α c , but decreases with the coefficients of ipSIC and CEEs.Therefore, a higher value of α c is preferred and more accurate hardware devices should be selected to reduce the coefficients of ipSIC and CEEs.

IV. NUMERICAL RESULTS
The Fig. 1 presents the OP of the far user and near user versus P I , respectively.We can clearly observe that OP decreases with P I , while the reliability of the system increases with P I , which can be deduced from ( 7) and (9).OP gradually reaches a fixed constant when P I tends to infinity, resulting in an error floor.This is due to the flexible decoding method of RSMA.As the OP reaches saturation, the ability to increase the reliable performance of the system by adding P I is no longer significant.In addition, it also find that the reliable of CR-RSMA system outperforms CR-NOMA system because of the flexible management of interference.Fig. 2 plots the effect of power allocation coefficient α c on the reliable performance under the Nakagami-m fading channels and Rayleigh fading channels.It is clear that OP first decreases and then increases with α c , showing a critical point.There exits an optimal α c to minimize the OPs, and the system performance will deteriorate if α c deviates from the optimal value.The larger α c becomes, the larger corresponding power of common message and the smaller corresponding power of private message.The common message link is basically in an outage state if α c is extremely small, vice versa.The change in the value of α c causes the OP to generate the state shown in the Fig. 2. Furthermore, we consider the effect of different P I values on user OP.It is not difficult to see that when the P I gets larger, the performance improves better.In addition, we can observed that the OPs of SR f and SR n are smaller and the reliability is higher under the Nakagami-m fading channels.
Under ideal and nonideal conditions, the curves of ER are depicted versus P I in Fig. 3.The experiments are established on 10 5 times.We set m SP = 2, m SR f = 8, m SRn = 7 Ω SP = 2, Ω SR f = 3 and Ω SRn = 0.3.It can be clearly seen that with the increase of P I , the ER performance of the RSMA-based underlay CR system improves, which can be inferred from ( 14) and ( 16).The ER tends to be a fixed constant when P I reaches infinity.As can be seen from the Fig. 3, the CEEs and ipSIC negatively affect the ER performance of the considered system.By comparing RSMA-based CR network with CR-NOMA, we find that RSMA system outperforms NOMA system for ER.

V. CONCLUSION
In this paper, we analyzed the outage and ER performance of the proposed RSMA-based CR systems.To obtain a better view of the system performance, the approximate behavior of OP and ER at high SNR were investigated.The simulation results showed that the OP decreases and the ER increases gradually with the transmit SNR of ST, and tended to be a fixed constant.Moreover, ipSIC and non-ideal CSI had a negative impact on the considered network.Finally, we found that the performance of RSMA-based CR networks was better than that of CR-NOMA networks.
The solution process of outage performance for SR n is similar to SR f , so it is omitted due the space limited.
Substituting (4) and ( 5) into (13), the ER of SR f can be expressed as

Manuscript received 1
December 2022; revised 20 May 2023 and 8 July 2023; accepted 12 August 2023.Date of publication 30 August 2023; date of current version 17 January 2024.This work was supported in part by Henan Province Key R&D and Promotion Special Project under Grant 212102210166, in part by Henan Scientific and Technological Research Project under Grant 232102211073, in part by the Key Laboratory of Cognitive Radio and Information Processing, Ministry of Education (Guilin University of Electronic Technology) under Grant CRKL220203, and in part by the Key Laboratory of Middle Atmosphere and Global Environment Observation (LAGEO) Institute of Atmospheric Physics, Chinese Academy of Sciences under Grant LAGEO-2022-02.The review of this article was coordinated by Prof. Francesco Verde.(Corresponding author: Xingwang Li.)

Fig. 1 .
Fig. 1.The OP versus the P I for ideal and non-ideal conditions.
analysis of this work are corroborated by the Monte Carlo simulations.All experiments are established on 10 6 times.Unless stated explicitly, we have the following settings: The noise power are N 0 = N 1 = 1, The maximum transmit power of ST is P max = 1; The target rates are γ c th = γ p,f th = γ p,n th = 0.1; The power allocation coefficients for common and private message are α c = 0.6, α p,1 = 0.25 and α p,2 = 0.15; The fading severity parameters are m SP = 6, m SR f = 4; m SRn = 2; The average power are Ω SP = 4, Ω SR f = 3, Ω SRn = 4; The channel estimation errors are σ 2 e SR f = σ 2 e SRn = 0.08; The ipSIC is ζ = 0.01.