# Achievable Rate of NOMA-Based Cooperative Spectrum-Sharing CRN over Nakagami-\(m\) Channels
## 2 Related Work
In addition to the half-duplex transmission, the statistical performance characterization of NOMA-based CRNs with full-duplex transmission also received substantial attention [6; 18; 19; 20]. In [18], the full-duplex mode of transmission is used at a double antenna relay to support multiple cell-edge users and to achieve a high throughput. To overcome interferences in the system, the source-to-relay and relay-to-destination transmission phases are carried out in licensed and unlicensed bands, respectively. Moreover, in [18], the performance of the system is characterized in terms of outage probabilities and ergodic sum rates. In [19], the performance of a NOMA system is improved by transforming one of the user (with a good channel condition) into a full duplex relay. For such a system, the authors in [19] also derived a simple achievable rate region. A full-duplex based device-to-device (D2D) cooperative NOMA transmission strategy was proposed in [6], where strong NOMA users support weak NOMA users. Therein, an adaptive multiple access method was also investigated in [6], where depending on the residual self interference level, the system switches between a cooperative NOMA and conventional NOMA and OMA schemes. This dynamic multiple access method yields the best outage performance. The outage probability expression of the system was also derived in [6]. A downlink NOMA system with cooperative relaying was studied in [20], where the users near the base-station act as full-duplex relays for users that are far from the base station. Closed-form expressions for outage probability and ergodic sum rate were also presented in [20]. Most existing studies on the performance characterization of half-duplex NOMA-based CRNs assume Rayleigh fading channels. A cooperative spectrum-sharing protocol based on NOMA to protect the quality of service of cell-center primary users, in which the secondary transmitter works with the primary to attain full rates via cooperative multiplexing, was considered in [21]. They derived closed-form expressions for the average bit error rate and evaluated the performance of primary users. A cooperative NOMA-based spectrum-sharing system was introduced in [7], where the primary network shares the spectrum with the secondary network. In exchange, the secondary network transmits the messages of the primary and secondary networks using NOMA. The proposed scheme is useful when the secondary transmitter has better channel conditions relative to the primary receiver but lacks the radio spectrum. To obtain better insights of the system, it is useful to extend the analysis of the system to the ergodic achievable rate with arbitrary values of fading parameter \(m\). In this study, we derive ergodic achievable rate expressions for NOMA-based cooperative spectrum-sharing CRNs over Nakagami\(-m\) fading channels. For our performance analyses, we use the same system model and channel model as those in [7], where the authors derived outage probability expressions for both primary and secondary users. In contrast to [7], here, we provide the ergodic capacity analysis for primary and secondary users in the NOMA-based cooperative spectrum-sharing CRN. The derived analytical achievable rate expressions presented in this paper provide some insights with respect to how the system's performance is affected under different fading severity and NOMA power levels. The contributions of this study are summarized below:  
* For the cooperative NOMA-based spectrum-sharing system, assuming the Nakagami\(-m\) fading channel model, we derive the exact closed form expressions of the ergodic sum rate for both primary and secondary users. * The ergodic sum rate expressions derived in this paper are comprehensive as they are valid for any integer or non-integer values of the Nakagami-\(m\) fading severity parameter. The performance analysis presented in this study assists in predicting the performance of NOMA-based CRNs for various propagation conditions, such as line-of-sight (LoS) or non-line-of-sight (NLoS) conditions.