Outage Performance Analysis of RSMA-Aided Semi-Grant-Free Transmission Systems

In this paper, we analyze the outage performance of a rate-splitting multiple access (RSMA)-aided semi-grant-free (SGF) transmission system, in which a grant-based user (GBU) and multiple grant-free users (GFUs) access the base station by sharing the same resource blocks. In the RSMA-aided SGF (RSMA-SGF) transmission system, the GBU and admitted GFU are respectively treated as the primary and secondary users by using the cognitive radio principle. With the aid of RSMA, the admitted GFU’s transmit power allocation, target rate allocation, and successive interference cancellation decoding order are jointly optimized to attain the maximum achievable rate for the admitted GFU, without deteriorating the GBU’s outage performance compared to orthogonal multiple access. Taking into account the extended non-outage zone achieved by rate-splitting, a closed-form expression is derived for the outage probability of the admitted GFU in the considered RSMA-SGF system. Asymptotic analysis for the admitted GFU’s outage probability is also provided. The superior outage performance and full multiuser diversity gain achieved by the RSMA-SGF transmission system are verified by the analytical and simulation results.

transmissions, which results in higher spectral efficiency than admitting only GBUs or GF users (GFUs) [5]. To opportunistically admit GFUs to resource blocks occupied by GBUs, non-orthogonal multiple access (NOMA)-aided SGF (NOMA-SGF) transmissions have been proposed, which eliminates complex handshaking processes for admitting GFUs [5], [6]. By using NOMA, user collisions caused by the GFUs' contention can be resolved by spectrum sharing with the aid of successive interference cancellation (SIC).
To avoid system performance degradation of a GBU in the NOMA-SGF transmission system, the GFU's contention must be managed appropriately taking into account the GBU's QoS requirements. Specifically, outage performance experienced by the GBU should be comparable to that of its orthogonal multiple access (OMA) counterpart [5], [6]. In [5], a distributed contention protocol was proposed to guarantee that a fixed number of GFUs are admitted. It was seen that this open-loop protocol suffered from user collisions at a rate similar to that seen in pure GF transmissions. In [6], the received GBU's signal power at the base station (BS) was used to determine an interference threshold, which was then broadcast to GFUs to facilitate distributed contentions. Considering that existing power-domain NOMA schemes rely on superposition coding and transmit power allocation, advanced SIC can be used to maintain a high transmission reliability while preventing the GBU from unnecessary awareness of the GFUs' contention [6], [7], [8]. In [9], a hybrid SIC was proposed to decode the desired signals based on their relative power levels, i.e., the admitted GFU's signal can be decoded at the first or second stage of SIC to attain the allowed maximum achievable rate, which can be used in conjunction with transmit power allocation to further improve the reliability of the NOMA-SGF transmission system [10].
When viewed in the context of underlay cognitive radio (CR), the paired NOMA users can be regarded as a primary user (PU) and a secondary user (SU), respectively [11], which signifies that GB and GF transmissions in NOMA-SGF systems can be treated as primary and secondary transmissions, respectively. Since CR-inspired NOMA (CR-NOMA) with SIC can achieve only a subset on the capacity region boundary of uplink multiple access channels (MACs), the outage performance of SU in this setting can result in decreased user fairness [12]. To avoid deterioration in the PU's outage performance, both PU and SU can be allowed to access the same BS simultaneously in CR-NOMA, and a new hybrid SIC was proposed to improve the achievable rate of the SU in such a system [13]. To achieve arbitrary points on the capacity region boundary of uplink MACs, rate-splitting multiple access (RSMA) proposed in [14] can also be implemented in a CR-inspired way, which can extend the non-outage zone for CR-NOMA [12]. In [15], the authors proposed a CR-inspired RSMA (CR-RSMA) scheme to improve the SU's outage performance by admitting both PU and SU simultaneously to access the same BS. Nevertheless, both PU and SU in the underlay CR-NOMA and CR-RSMA systems worked in a GB manner [13], [15], whereas a GFU needed to share the same time-frequency resource blocks with a GBU in the SGF transmissions through a shortened hand-shaking signaling. Moreover, the admission contention from multiple GFUs was not taken into account in CR-RSMA and the effects of the GFUs' contention on the RSMA-aided SGF (RSMA-SGF) transmission systems are still unknown.
A. RELATED WORK 1) RSMA: RSMA has received significant attention recently due to its capability to improve the spectral/energy efficiencies, robustness, reliability, and latency of downlink and uplink multi-user transmissions [16], [17]. In downlink RSMA, the users' messages are split into common and private streams using available channel state information at the transmitter (CSIT) [18]. By treating inter-user interference flexibly, i.e., interference can be partially decoded and partially treated as noise, downlink RSMA not only bridges space division multiple access (SDMA) and NOMA, but also achieves superior system performance, so that it is regarded as a promising enabling technology for 6G new radio (NR) [19], [20], [21], [22] (see also [16] and references therein).
However, RSMA and the corresponding applications for uplink MACs are still in their infancy. By using the rate-splitting technique, the message of each user is split into multiple sub-messages to enable non-orthogonal uplink RSMA, which provides flexible interference management, better user-fairness, and massive connectivity as compared to conventional uplink OMA and NOMA [12], [23], [24]. Being essentially different from downlink RSMA, which decodes common and private streams partially at a receiver to balance the decoding performance and complexity, the split data streams from all the users are fully decoded at the BS using SIC in uplink RSMA [23], [24], [25], [26]. In [23], the sum-rate maximization problem was investigated for uplink RSMA in which the proportional rate constraints among users were considered. To enhance the outage performance of uplink NOMA, several rate-splitting schemes were proposed in [24], [25], [26]. For single-input multiple-output (SIMO) NOMA, uplink rate-splitting was investigated to guarantee max-min user fairness in [24]. Nevertheless, an exhaustive search was needed to find the optimal SIC decoding order and optimal power allocation for uplink RSMA [23], [24]. Using the CR principle, adaptive power allocation and ratesplitting were proposed to improve the outage performance and user fairness for uplink NOMA in [12].
Recently, uplink RSMA has been applied to realize physical layer network slicing for ultra-reliable and low-latency (URLLC) and enhanced mobile broadband communications (eMMB) [27]. Also, RSMA has been used to support URLLC and eMMB in 6G NR downlink transmissions [28]. In [29], the outage performance of uplink RSMA was investigated taking into account all possible SIC decoding orders. In [30], cooperative RSMA was proposed to realize uplink user cooperation. In [31] and [32], rate-splitting schemes were designed for uplink aerial networks and satellite communications, respectively. Rate-splitting applications for cell-free machine-type communications and device-todevice fog radio access networks were investigated in [33] and [34], respectively.
2) NOMA-SGF Transmissions: To control the number of admitted GFUs for the NOMA-SGF transmission system, two contention mechanisms, namely open-loop contention and distributed contention protocols, have been proposed in [5], in which an interference temperature alike channel gain threshold was broadcast to aid the GFUs' contention. Furthermore, a dynamic interference threshold proportional to the received GBU's signal power was proposed in [7] to determine the admitted GFUs, which results in reduced GFUs' interference compared to the open-loop contention. The authors in [6] and [9] proposed CR-NOMA with hybrid SIC decoding order to enhance transmission reliability of the admitted GFU, meanwhile ensuring that the GBU experiences the same outage performance as in OMA. An ergodic rate analysis was provided for NOMA-aided SGF (NOMA-SGF) transmissions in [35]. To efficiently leverage the capability of CR-NOMA to achieve the capacity region of uplink MACs, adaptive power allocation with hybrid and fixed SIC decoding orders were proposed in [8] and [10]. Further, adaptive power allocation was proposed in [36] for the NOMA-SGF transmission system to improve the outage performance and sum rate. In [37], a downlink collided-preamble feedback was applied to facilitate massive multiple-input multiple-output assisted SGF random access. For tactile IoT networks that use NOMA-SGF transmissions, joint power allocation and sub-channel assignment were investigated in [38]. In [39], a user barring scheme was proposed to increase the average arrival rate for the NOMA-SGF transmission system taking into account multiple transmit power levels. In [40] and [41], multi-agent deep reinforcement learning (MADRL) was applied to optimize the transmit power for NOMA-SGF and NOMA-aided GF (NOMA-GF) transmissions. Moreover, MADRL was used to optimize transmit power allocation, sub-channel assignment and reflection beamforming for an intelligent reflecting surface (IRS) aided NOMA-SGF system [42].

B. MOTIVATION AND CONTRIBUTIONS
Although the NOMA-SGF transmissions can accommodate the coexistence of GB and GF transmissions, these systems achieve only a subset of the uplink MAC capacity region as noted above. In contrast, the full capacity region of uplink MACs can be achieved by using RSMA [14]. However, to maximize the sum-rate and ensure user fairness, exhaustive search was used to determine the optimal SIC decoding order and optimal power allocation for uplink RSMA, which is computationally prohibitive [23], [24]. For delay-limited transmissions, the authors in [12] proposed a CR-inspired transmit power allocation to aid rate-splitting, in which target rates of the SU's data streams were chosen heuristically.
On the other hand, the recently proposed CR-RSMA can improve the SU's outage performance [15] and reduce the task-offloading latency for mobile edge computing [43] by optimizing the transmit power and target rate allocations. Motivated by the capacity region achieved by SGF transmissions, we apply the CR-RSMA that adopts RSMA and CR principles to assist SGF transmissions, with the aim of improving the system performance under the uplink RSMA framework. In the considered RSMA-SGF transmission system, rate-splitting is conducted at the admitted GFU, while the optimal transmit power and target rate allocations at the admitted GFU and SIC decoding order at the BS are jointly optimized using the CR principle, which improves the admitted GFU's outage performance significantly.
The main contributions of this paper are summarized as follows: • For the RSMA-SGF system, we show that the nonoutage zone is extended significantly compared to the NOMA-SGF scheme by exploiting the full capacity region. Due to the extended non-outage zone, additional target rate pairs containing the higher target rates for the admitted GFU are supported by the RSMA-SGF scheme. Therefore, the outage performance of the admitted GFU is significantly improved when RSMA-SGF scheme is applied. • We derive exact expressions for the outage probability of the admitted GFU and its approximation in the high signal-to-noise ratio (SNR) region. These analytical results reveal that the full multiuser diversity gain is achieved by the RSMA-SGF transmission system. Moreover, benefiting from the obtained maximum achievable rate for the admitted GFU, the multiuser diversity can be achieved in a wider target rate regions than that of the NOMA-SGF transmission system. • Various computer simulation results are presented to verify the accuracy of the derived analytical results and corresponding high SNR approximations. The superior outage performance of the admitted GFU achieved by the RSMA-SGF scheme is verified by these simulation results. The impacts of the target rate and number of GFUs on the outage performance of the admitted GFU are revealed. The remainder of this paper is organized as follows: Section II presents the system model and the RSMA-SGF scheme, respectively; In Section III, the outage performance of the admitted GFU achieved by the RSMA-SGF transmission system is analyzed and the high SNR approximation for the outage probability is derived; In Section IV, simulation results are presented for corroborating the superior outage performance of the RSMA-SGF transmission system, and Section V summarizes this work.

A. SYSTEM MODEL
We assume that the considered system consists of multiple GBUs and multiple GFUs. To prevent GF transmissions from generating too much interference to GBUs, all the GFUs and GBUs are divided into multiple groups, each of which contains K GFUs and one GBU. In each group, we assume that only one out of K GFUs is paired with the GBU for simultaneous uplink transmissions, whereas multiple GFUs from different groups can be simultaneously admitted through wireless resource allocation among the groups.
Without loss of generality, we consider SGF transmissions within a single group, in which the GBU and the kth GFU are denoted by U 0 and U k ∈ {U 1 , U 2 , . . . , U K }, respectively. The channel coefficients from U 0 and U k to the BS are denoted by h 0 and h k (k = 1, 2, . . . , K), 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 quasi-static fading, i.e., the channel coefficients remain constant during a single transmission block and can vary from one transmission block to another independently. Moreover, the channel gains can be ordered as where the subscripts denote the ordered indices. It should be noted that the above ordering information is unavailable to the K GFUs and the BS. In the considered system, we assume that the K GFUs have the knowledge of their own channel state information (CSI) and the admitted GFU's CSI is not required to be known at the BS prior to the SGF transmissions. In addition, the BS has acquired the information of the GBU's CSI and transmit power. To compare the system performance of the RSMA-SGF scheme with the benchmarks, we also assume perfect SIC at the BS receiver as those in [6], [13]. Without loss of generality, we assume that the th GFU U is admitted to transmit (1 ≤ ≤ K). For each block of transmissions, U can split its message signalx into two partsx ,1 andx ,2 . Corresponding to the message signals x ,x ,1 , andx ,2 , the generated transmit signals at U are denoted by x , x ,1 , and x ,2 , respectively. In each transmission block, U 0 and U simultaneously transmit their signals to the BS. Then, the received signal at the BS can be written as: (2) where P 0 and P s denote the transmit power of U 0 and U , respectively, x 0 is the transmit signal of U 0 , α is the transmit power allocation factor at U satisfying 0 ≤ α ≤ 1, and w is additive white Gaussian noise (AWGN) at the BS with zero mean and unit variance. We assume that each transmit signalx ∈ {x 0 , x , x ,1 , x ,2 } is coded by an independent Gaussian code book and satisfies E{|x| 2 } = 1, where E{·} is the expectation operator.
At the BS, the decoding order x ,1 → x 0 → x ,2 is adopted in SIC to recover the signal x ,1 , x 0 , and x ,2 , sequentially. According to [15], the SIC decoding order x ,1 → x 0 → x ,2 ensures that U obtains the allowed maximum achievable rate. Also, the full capacity region boundary of MACs can be achieved by applying the SIC decoding order x ,1 → x 0 → x ,2 [14]. Then, the received signal-to-interferenceplus-noise ratio (SINR) (or signal-to-noise ratio (SNR)) for decoding x ,1 , x 0 , and x ,2 can be expressed as follows: and For the admitted U 0 and U , the achievable rates are given by R 0 = log 2 (1+γ 0 ) and R = R ,1 +R ,2 , respectively, with R ,1 = log 2 (1 + γ ,1 ) and R ,2 = log 2 (1 + γ ,2 ) representing the achievable rates to transmit x ,1 and x ,2 , respectively. Remark 1: By simply interchanging x ,1 and x ,2 , all possible SIC decoding orders at the BS can be categorized into three types as Using the logarithmic product law, we can readily prove that applying x ,1 → x ,2 → x 0 is equal to applying x → x 0 in the sense of attaining the achievable rates R and R 0 . In such a case, rate-splitting is unnecessary, so does the decoding order x 0 → x ,1 → x ,2 . As it will be seen in the next subsection, NOMA with the SIC decoding orders x → x 0 and x 0 → x cannot approach the full capacity region boundary of MACs. Thus, the decoding order x ,1 → x 0 → x ,2 is used in the RSMA-SGF transmission system with respect to its capability to achieve the full capacity region boundary.

B. RSMA-SGF TRANSMISSIONS
For the considered the RSMA-SGF transmission system, to guarantee that the GBU U 0 achieves the same outage performance as in OMA, a CR analogous interference threshold is broadcast to the K GFUs to aid user contention. Let R 0 ,R s ,R s,1 , andR s,2 denote the target rates to transmit x 0 , x , x ,1 , and x ,2 , respectively. In addition, we definê is the target rate allocation factor. With respect to the SIC decoding order x ,1 → x 0 → x ,2 , the signal x 0 can be decoded correctly only when the following constraints are satisfied, i.e., Under the constraints in (6), the GBU achieves the same outage performance as in OMA when Pr(R ,1 ≥R s,1 ) = 1 and By substituting (4) into (7), the equality in (7) holds under the condition of The equation (8) indicates that the interference power caused by the transmission of √ (1 − α)P s h x ,2 cannot surpassτ to ensure the correct decoding of x 0 at the second stage of SIC. By assuming that Pr(R ,1 ≥R s,1 ) = 1 is guaranteed in the RSMA-SGF transmission system, as it will be explained later in this subsection, the interference threshold to be broadcasted by the BS is determined as [6]: The admission procedure of the RSMA-SGF scheme is presented as follows: • The BS broadcasts the pilot signals to assist the users to estimate CSI. • U 0 feeds back its CSI and P 0 to the BS. • The BS calculates τ according to (9) and broadcasts it to the K GFUs. • Each GFU calculates the achievable rate using its own CSI and the corresponding optimal α * and β * , which will be provided in the later part of this subsection. • Through the distributed contention, the GFU that obtains the maximum achievable rate is admitted by the BS. With respect to the capability of uplink RSMA to achieve the full capacity region boundary of MACs, the design goal of the RSMA-SGF scheme is to maximize the achievable rate for the admitted GFU meanwhile guaranteeing that the GBU achieves the same outage performance as in OMA by jointly optimizing the transmit power allocation and target rate allocation. The optimization problem can be formulated as: In (10), C1 is the QoS constraint of U 0 , C2 is the constraint of the transmit power allocation, and C3 is the constraint of the target rate allocation. Considering all the possible QoS realizations of U 0 and all the possible CSI realizations, the received power levels of P 0 |h 0 | 2 and P s |h K | 2 can be classified into three cases, namely, Case I: and Case III: τ = 0. As an illustration, three examples of the received power levels corresponding to Cases I, II, and III are respectively depicted in Fig. 1. In the following, the optimal α * and β * of the RSMA-SGF scheme are jointly designed for Cases I, II, and III, respectively.
In this case, the maximum interference level caused by admitting an arbitrary GFU is not greater than the interference threshold τ . When U is admitted, the maximum achievable rate is given by R = R ,2 = log 2 (1 + γ ,2 ) considering that x ,2 is interference-freely decoded at the last stage of SIC. In other words, U will allocate all of P s to transmit x by setting x ,2 = x . Consequently, U K , which has the greatest channel gain among all the GFUs, is admitted to attain the maximum achievable rate R K = R K,2 = log 2 (1 + γ K,2 ). In this case, the optimal transmit power and target rate allocation factors are respectively given by where (·) * denotes the optimal solution for the corresponding parameter.
Since only Consequently, the achievable rate of the admitted GFU can be expressed as Case II: 0 < τ < P s |h 1 | 2 or P s |h k | 2 < τ < P s |h k+1 | 2 with k = 1, 2, . . . , K − 1. In this case, it can be seen that τ > 0. Then, a GFU U whose channel gain is larger than τ will be admitted, where k + 1 ≤ ≤ K. Due toτ > 0, the GBU's signal x 0 can be correctly decoded at the second stage of SIC (x ,1 → x 0 → x ,2 ) subject to the constraints R ,1 >R ,1 and R ,2 ≤ log 2 (1 +τ ). Thus, the admitted GFU's achievable rate R = R ,1 + R ,2 can be maximized by first maximizing R ,2 = log 2 (1 + γ ,2 ), which is obtained as R ,2 = log 2 (1+τ ) by setting γ ,2 =τ . The corresponding optimal transmit power allocation factor is given by By setting the target rateR ,2 = log 2 (1 +τ ) to transmit x ,2 , the optimal target rate allocation factor is given by Furthermore, R ,1 can be maximized by setting = K considering that U K has the greatest channel gain, i.e., the achievable rate for the GFU is maximized by admitting U K . Accordingly, the optimal transmit power and target rate allocation factors are respectively given by α * (K) and β * , and the achievable rate in this case can be written as where R (II) and R (II) Since the received SINR/SNR γ 0 (α * (K)) and γ K,2 (α * (K)) provide the necessary conditions for the correct decoding of x 0 and x K,2 in SIC processing x K,1 → x 0 → x K,2 , only the failure decoding of x K,1 can result in error propagation in SIC. To avoid this error propagation, R (II) K,1 ≥R s,1 is required to decode x K,1 correctly or equivalently R (II) K ≥R s ; Otherwise, both U 0 and U K encounter outage. Therefore, in the RSMA-SGF scheme, U K is permitted to transmit only when R (II) K,1 ≥R s,1 ; Otherwise, U K keeps silence while U 0 is transmitting alone to the BS.
Case III: τ = 0. In this case, it can be seen thatτ < 0, so that x 0 cannot be correctly decoded due to a weak channel gain |h 0 | 2 . To avoid error propagation in SIC processing x ,1 → x 0 → x ,2 caused by the failure decoding of x 0 , the admitted GFU is chosen not to transmit x ,2 rather to transmit x by setting x ,1 = x and using the whole P s . Therefore, the optimal transmit power and target rate allocation factors are given by α * = 1 and β * = 1.
Since U K has the greatest channel gain among the GFUs, it is admitted to attain the maximum achievable rate as and the corresponding SIC decoding order x K,1 → x 0 → x K,2 degenerates to x K → x 0 . Remark 2: Due to the channel gain ordering |h 1 | 2 ≤ |h 2 | 2 ≤ · · · ≤ |h K | 2 , the maximum achievable rates in Cases I, II, and III for the admitted GFU are always attained by admitting U K , no matter which decoding order is applied in SIC. Since the channel coefficients of all the K GFUs follow i.i.d. complex Gaussian distribution, all the GFUs have the equal probability to have the greatest channel gain, or equivalently, to be admitted. Thus, the admission probabilities for all the K GFUs are equal in the RSMA-SGF transmission system.
Remark 3: In Cases I and III, the NOMA-SGF scheme also admits U K and apply the SIC decoding order x 0 → x K and x K → x 0 , respectively, to obtain the maximum achievable rates [6]. Nevertheless, in Case II, the NOMA-SGF scheme admits U K (or U k ) subject to P s |h 1 | 2 > τ (or P s |h k | 2 < τ < P s |h k+1 | 2 ). For the NOMA-SGF scheme, the achievable rate of the admitted GFU in Case II is given by [6] R (II) i.e., the RSMA-SGF scheme always achieves a larger achievable rate for the admitted GFU than that of the NOMA-SGF scheme in Case II.
Remark 4: In Case II, U K only transmits its signal when R (II) K,1 ≥R s,1 , which prevents error propagation in SIC processing x K,1 → x 0 → x K,2 . When R (II) K,1 <R s,1 , U K keeps silence and only the GBU U 0 transmits. Thus, the GBU U 0 does not encounter outage when the RSMA-SGF scheme operates in Case II. As such, the outage events occur to the transmissions of x 0 in Cases I and III are the same as in OMA, and the GBU U 0 achieves the same outage performance as in OMA when the RSMA-SGF scheme is applied.

III. OUTAGE PERFORMANCE ANALYSIS
In the RSMA-SGF transmission system, since the GBU's outage performance is guaranteed to be the same as in OMA, we mainly focus on the outage performance of the admitted GFU. In this section, we first introduce the non-outage zone to clarify the advantage of the RSMA-SGF scheme. Then, we derive the analytical expression for the outage probability in closed-form and investigate the asymptotic outage performance in the high SNR region.
For the uplink MACs in which the two-user U 0 and U K are admitted simultaneously, the non-outage zone is defined bȳ If a target rate pair {R 0 ,R s } lies in the non-outage zone, both U 0 and U K can attain the achievable rates R 0 ≥R 0 and R K ≥R K such that neither U 0 nor U K encounters the outage event. Otherwise, both U 0 and U K are in outage. R K } is called the outage zone when both U 0 and U K are admitted simultaneously. In Fig. 2, an example of the non-outage zone achieved by the RSMA-SGF scheme is illustrated. For the comparison purpose, the non-outage zone achieved by the NOMA-SGF scheme is also illustrated in which the SIC decoding orders x 0 → x K and x K → x 0 are respectively utilized [6]. In this example, we assume 0 < τ < P s |h 1 | 2 . From Fig. 2, we can see that the non-outage zones achieved by the NOMA-SGF scheme using the SIC decoding orders x 0 → x K and x K → x 0 are identical to those achieved by the RSMA-SGF scheme in Cases I and III, respectively. Thus, in Cases I and III, the RSMA-SGF scheme can achieve the same outage performance as that of the NOMA-SGF scheme. Nevertheless, the NOMA-SGF scheme cannot prevent U 0 and U K from being in outage when the target rate pair {R 0 ,R K } lies in the triangle ABC, which is beyond the nonoutage zone of the NOMA-SGF scheme. Fortunately, the points on the line AB in Fig. 2 can be achieved by ratesplitting. Specifically, an additional non-outage zone, the triangle ABC, is achieved when the RSMA-SGF scheme operates in Case II. Thus, the non-outage zone of the RSMA-SGF scheme is extended compared to that of the NOMA-SGF scheme, which shows that the RSMA-SGF scheme can support more target rate pairs than that of the NOMA-SGF scheme.
With respect to the operations of the RSMA-SGF scheme in Cases I, II, and III, the outage probability of the admitted GFU can be written as where P out respectively denote the probability of that U K encounters outage in Cases I, II, and III, which can be expressed as and P (III) with P (II,0) and for k = 1, 2, . . . , K − 1, The following theorem provides an exact expression for the admitted GFU's outage probability achieved by the RSMA-SGF scheme.
Theorem 1: Assume that K ≥ 2, the outage probability of the admitted GFU is given by where K n denotes the binomial coefficient, ε s 2R s − 1, out , as derived in Appendix A, all the outage events reflect the fact that the upper bound on the channel gain |h K | 2 should be greater than the lower bound on a specific GFU's channel gain, which is always and naturally guaranteed by |h 0 | 2 < η 0 (1 + ε s ) without imposing additional constraints on ε 0 and ε s , so that the derived expression for the outage probability is applicable to all the feasible ε 0 and ε s . On the contrary, for the NOMA-SGF scheme, the analytical expression for the outage probability in [6] is applicable only for ε 0 ε s < 1. As an example, the outage and non-outage zones achieved by the RSMA-SGF and the NOMA-SGF schemes are illustrated in Fig. 3. It can be seen that the outage zone constrained by ε 0 ε s < 1 is a small portion of the whole outage zone for the NOMA-SGF scheme and a similar phenomenon happens to the RSMA-SGF scheme as well. Therefore, the analytical results provided in Theorem 1 are more general due to its applicability to all the feasible values of ε 0 and ε s . In contrast to the RSMA-SGF scheme, the NOMA-SGF scheme results in a worse outage performance due to the extended outage zone. When the considered system consists of a single GFU and GBU, the two users are paired directly. Then, the GFU's outage probability achieved by the RSMA-SGF scheme can be written as: Corollary 1: Assume that K = 1, the admitted GFU's outage probability is given by and its approximation in the high SNR region is given by Proof: A proof can be found in [15]. Corollary 2: Assuming that K ≥ 2, the outage probability experienced by the admitted GFU can be approximated in the high SNR region as follows:  (34) in the high SNR region. Thus, a diversity gain of K is achieved by the RSMA-SGF scheme. Remark 6: Corollaries 2 and 3 demonstrate that the RSMA-SGF scheme ensures an achievable diversity gain proportional to the number of the GFUs without resulting in an outage floor.

IV. SIMULATION RESULTS
In this section, we present simulation results to verify the accuracy of the analytical results and the superior outage performance of the RSMA-SGF scheme. For the purpose of comparisons, the admitted GFU's outage probabilities achieved by the NOMA-SGF scheme [6] and the NOMA-SGF scheme with power control (NOMA-SGF-PC) [13] are also presented. In the simulation, the channel coefficients are randomly generated according to i.i.d. circular symmetric complex Gaussian random distribution with zero mean and unit variance and the transmission data rate is measured in bits per channel use (BPCU).
The outage performance achieved by the RSMA-SGF scheme is compared with those of the NOMA-SGF and  Fig. 4, where we setR 0 = 2.5 BPCU,R s = 1.5 BPCU, and P s = P 0 15 , i.e., the transmit SNR of U 0 is 11.76 dB higher than that of the GFUs. From Fig. 4, we can see that the RSMA-SGF scheme achieves the smallest outage probabilities among the three schemes for both K = 1 and K = 5. The results in Fig. 4 also verify the accuracy of the derived analytical expressions. As transmit SNR increases, the outage probabilities achieved by both the RSMA-SGF and NOMA-SGF-PC schemes decrease monotonically, whereas an outage probability floor occurs for the NOMA-SGF scheme in the high SNR regime. The reason for the outage probability floor is that the achievable rate attained by the NOMA-SGF scheme satisfies R (II) NOMA = R (III) K → log 2 (1 + P s /P 0 ) as P s → ∞ and P 0 → ∞ under the condition of 0 < τ < P s |h 1 | 2 in Case II, which always results in an outage event if log 2 (1 + P s /P 0 ) <R s occurs. Although the NOMA-SGF-PC scheme avoids the outage probability floor, the achieved outage probability is still higher than that of the RSMA-SGF scheme. At the outage probability level of 10 −3 , the RSMA-SGF scheme achieves a 3 dB transmit SNR gain over that of the NOMA-SGF-PC scheme. Therefore, the results in Fig. 4 verify that the RSMA-SGF scheme achieves the superior outage performance compared to the NOMA-SGF and NOMA-SGF-PC schemes.

NOMA-SGF-PC schemes in
In Fig. 5, we investigate the outage performance achieved by the SGF schemes for various transmit powers. In particular, we set a fixed transmit power P 0 = 15 dB and vary P s from 0 dB to 45 dB, which reflects that the GFUs can have greater and lower transmit SNRs than that of U 0 . The results in Fig. 5 also verify the accuracy of the derived analytical expressions. From Fig. 5, we can see that the RSMA-SGF scheme achieves the smallest outage probabilities in the whole SNR region. At the outage probability level of 10 −3 , the RSMA-SGF scheme achieves a 7 dB transmit SNR gain over that of the NOMA-SGF-PC scheme. As transmit SNR increases, the outage probability achieved by the RSMA-SGF scheme decreases monotonically for both  K = 1 and K = 5. Also, the NOMA-SGF scheme achieves the worst outage probabilities in the whole SNR region. Especially in the high SNR region, the outage probability floor occurs for the NOMA-SGF scheme due to a limited asymptotic achievable rate R (II) NOMA as resulted by (20). The results in Fig. 5 verify that the superior outage performance of the RSMA-SGF scheme is irrespective of whether the GFUs are stronger users or weaker users compared to U 0 .
In Fig. 6, we examine the accuracy of the derived analytical expressions for the outage probability. In Fig. 6, we setR 0 = 2 BPCU,R s = 1.5 BPCU, and P s = P 0 15 . In Fig. 6, for the curves corresponding to the expressions provided in Corollaries 2 and 3, we denote them by "approximation (33)" and "approximation (34)", respectively. The curves in Fig. 6 verify that the accuracy of derived analytical expression in Theorem 1. For the approximated expressions, we can see that the curves of the "approximation (33)" match well with the exact results in the high SNR region, whereas the curves of the "approximation (34)" match well with the exact results in the high SNR region only for small K values. For large K values (K = 4 in this example), a gap exists between the curves of the "approximation (34)" and exact results. The  reason for this phenomenon is that we ignore the terms being proportional to 1 P K+1 s and 1 P K+2 s in the expression in (34). The impact of the target rate on the outage probability is investigated in Fig. 7. For the simulation results in Fig. 7, we set P 0 = 10 dB, P s = 15 dB, andR 0 =R s . It is seen in Fig. 7 that for given transmit SNR values, the RSMA-SGF scheme achieves the smallest outage probabilities in the considered whole target rate region. As the target rate increases, the outage probability values achieved by all the SGF schemes increase and approach 1.
The impact of the number of the GFUs on the outage probability is investigated in Fig. 8. For the simulation corresponding to Fig. 8, we set {R 0 = 1.5 BPCU,R s = 2 BPCU} and consider two cases of transmit SNR settings {P 0 = 20 dB, P s = 10 dB} and {P 0 = 20 dB, P s = 10 dB}. As K increases, the outage probabilities achieved by the three SGF schemes decreases. Furthermore, Fig. 8 verifies that the multiuser diversity is achieved by the RSMA-SGF scheme. Consequently, among the three SGF schemes, the RSMA-SGF scheme achieves the smallest outage probability for different numbers of the GFUs.

V. CONCLUSION
In this paper, we have conducted the outage performance analysis for the RSMA-SGF transmission system. By applying rate-splitting under the CR principle, the RSMA-SGF scheme effectively utilizes the transmit power to attain the maximum achievable rate for the admitted GFU. Without introducing intolerable interference to the GBU, the RSMA-SGF scheme has significantly extended the non-outage zone compared to the NOMA-SGF scheme. We have derived exact and approximated expressions for the outage probability of the admitted GFU and revealed that the full multiuser diversity gain can be achieved by the RSMA-SGF scheme. Simulation results have verified the superior outage performance achieved by the RSMA-SGF scheme.

APPENDIX A PROOF OF THEOREM 1
To derive the admitted GFU's outage probability, we evaluate the probability terms P (I) out , P (II) out , and P (III) out , respectively. In Case II, P (II) is derived by evaluating the different P (II,k) out taking into account the corresponding order statistics.

A. EVALUATION OF P (II,0) OUT
To evaluate P (II,0) out , let us introduce S 0 as Then, P (II,0) out in (26) can be rewritten as where ε s = 2R s − 1 and η 0 = ε 0 P 0 . In (A.2), the expectation on S 0 is conducted over Since the upper bound on |h K | 2 should be larger than the lower bound on |h 1 | 2 , the expectation in (A.2) should consider the hidden constraint |h 0 | 2 < η 0 (1 + ε s ) as well, so that P (II,0) out is rewritten as The joint probability density function (PDF) of the order statistics |h 1 | 2 and |h K | 2 is given by [44] where x < y and ϕ 0 = K! (K−2)! . Then, S 0 can be evaluated as follows:

B. EVALUATION OF P (II,K )
out , which can be rewritten as: where S k is defined by In (A.10), the expectation is taken over η 0 < |h 0 | 2 < η 0 (1 + ε s ) considering the relationship between the upper and lower bounds on the channel gains. The joint PDF of three order statistics, h k , h k+1 , and h K , is given by [44] Using (A.12), S k can be expressed in terms of |h 0 | 2 as follows: After some algebraic manipulations, S k can be further evaluated as follows: (A.14) By substituting (A.14) into (A.10), P (II,k) out can be evaluated as:

C. EVALUATION OF P (II,K−1) OUT
The probability term P (II,K−1) out can be expressed as: By extracting the hidden constraint on the upper and lower bounds on |h K | 2 from (A.17), i.e., can be rewritten as follows: where S K−1 denotes probability inside the expectation in (A.17). There are two order statistics h K−1 and h K involving in S K−1 with the joint PDF [44] where x ≤ y. Using (A.19), S K−1 can be evaluated as follows: In Case I, the determination of P (I) out involves two independent random variables |h 0 | 2 and |h K | 2 . Recalling the expression in (23), P (I) out can be rewritten as follows: By comparing and ε s P s , it can be seen that out can be evaluated as follows: out is a function of two independent random variables |h 0 | 2 and |h K | 2 . In Case III, the achievable rate is R (III) K = log 2 1 + P s |h K | 2 P 0 |h 0 | 2 +1 and τ = 0. Then, P (III) out can be evaluated as follows: By combing the obtained expressions for P out , and P (III) out , we arrive at (28).

APPENDIX B PROOF OF COROLLARY 1
Considering that P (I) out , P (II) out , and P (III) out depend on k, the high SNR approximations for P (I) out , P (II) out , and P (III) out will be derived separately in the following subsections.

A. HIGH SNR APPROXIMATION FOR P (II,0) OUT
Based on the derived closed-form expression in (A.9), P (II,0) out can be rewritten as follows: By applying the approximation e −x ≈ 1 − x for x → 0 and using the definitions of μ 1 and μ 2 , as P 0 = P s → ∞, P (II,0) out can be approximated as follows: where step (e) is obtained by applying K n K n (−1) n a n = (1 − a) K . By applying the binomial expansion, P can be further expressed as follows: where step (f) follows high SNR approximations. Applying the binomial expansions to (B.6), the high SNR approximation for P (II,k) out can be obtained as follows: By applying the approximations e −x ≈ 1 − x for x → 0, as P 0 = P s → ∞, P (II,K−1) out can be approximated as follows: . (B.10) Following similar algebraic manipulations as for deriving the high SNR approximation of P , the high SNR approximation for P out is obtained as (33).