Rate Splitting Multiple Access for Sum-Rate Maximization in IRS Aided Uplink Communications

In this paper, an intelligent reflecting surface (IRS) aided uplink (UL) rate-splitting multiple access (RSMA) system is investigated for dead-zone users where the direct link between the users and the base station (BS) is unavailable and the UL transmission is carried out only through IRS. In the considered RSMA system, a message of each user is split into several sub-messages and each part contributes to the rate of that user and depending upon split proportions BS decodes them using appropriate decoding order. The problem of sum-rate maximization is formulated to jointly design the optimal power allocation at each UL user, passive beamforming at the IRS under optimal decoding order of sub-messages. Due to non-convexity and discrete non-linear programming of the formulated problem, the original problem is intractable and hence, we decouple the problem into different sub-problems in which the problems of power allocation and passive beamforming are alternatively solved under using successive convex approximation and Riemaniann conjugate gradient algorithms, respectively. Moreover, the decoding order strategy is analytically derived which confirm that the optimal decoding order strategy depend upon decreasing order of channel gain of users and increasing order of split proportions of sub-messages. Later, the unified solution based on block-coordinate descent (BCD) algorithm is proposed. Simulation results validate that the proposed decoding order scheme attains performance closer to the optimal solution with low computational complexity. Moreover, the proposed IRS aided RMSA system outperforms the system with non-orthogonal multiple access (NOMA) and orthogonal multiple access (OMA) schemes in terms of achievable sum-rate throughput.


I. INTRODUCTION
T HE ever rising demand of high data rates and upsurge in growth of mobile devices in recent years have led to the inevitable search for very high spectrum and power efficient technologies [1], [2], [3], [4], [5]. As a possible candidate technique to radio access of the future wireless networks, rate-splitting multiple access (RSMA) has recently been recognized as a promising solution due to its superior spectral efficiency and user fairness [5], [6], [7], [8], [9], [10], [11]. Particularly, the RSMA flexibly splits the rate of each user into various messages with different proportion by controlling the interference and treating interference as noise through optimal decoding of messages [10], [11], [12]. The configurable split of messages according to priority enables effective power control and better spectral efficiency for both uplink (UL) and downlink (DL) transmission as compared to existing multiple access schemes such as non-orthogonal multiple access (NOMA) and space-division multiple access (SDMA) [8], [11], [12], [13]. However, the optimal performance of RSMA for UL transmission depends upon appropriate selection of the decoding order for successive interference cancellation at the receiver [13].
The intelligent reflecting surface (IRS) has gained growing attention in recent years for various wireless communication applications owing to its appealing advantages such as improved coverage capabilities, enhanced spectral and energy efficiency and ease in reconfiguration and deployment [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26]. The array of multiple low cost elements at IRS reflects the signal by altering the phase of the received signal. The IRS shapes the impinging signal by controlling the phase shift of each reflecting element instead of employing a power amplifier and effectively tune the channel vector of the users. The phase shift induces a passive beamforming at the IRS by reconfiguring the propagation environment [20], [21], [22], [23], [24], [25]. Particularly, the IRS aided network are beneficial for dead zone users where the users fail to establish direct communication link with the base-station (BS) [26], [27]. Many wireless communication scenarios occur in harsh environments or unfavourable conditions where multiple users lie in dead zones. Such scenarios are commonly encountered in dense urban settings as well as in transportation networks, indoor environments, cellular data-offloading in rural areas [28], [29].
Moreover, IRS has a great potential to enhance the combined channel strengths, coverage capability and further improve spectral efficiency, while RSMA can achieve higher spectrum-efficiency gain than conventional space-division multiple access (SDMA), non-orthogonal multiple access (NOMA) and orthogonal (OMA) schemes. The spectral efficiency of the RSMA and NOMA schemes strongly depend upon the interference management and decoding order of the messages. Importantly, the phase-shift control at IRS has a great potential to enhance the combined channel strengths, coverage capability by coherently improving the received signal power or effective interference mitigation. In other words, IRS reconfigure the phase shifts of RIS elements and customize the channel gain for enhanced signal strength of desired user and mitigate undesired interferences especially for multi-user scenario [26], [29]. So, the interplay between IRS and RSMA secure a smart reconfigurable radio environment [30] which can provide better interference management control and deliver improved spectral efficiency. There exist many research works on IRS-aided NOMA system, however, the IRS-aided RSMA system is still in its infancy. To this end, the key motivation of this work lies behind the investigation of IRS-aided RSMA system to enhance the sum-rate throughput of dead zone users where direct link between BS-users is severely blocked due to unfavourable conditions. Note that the maximum achievable sum-rate throughput is referred as spectral efficiency (SE) throughout the paper.

A. Related Works
Several works have investigated the system performance with respect to spectral efficiency, energy efficiency, outage probability, network coverage capability in IRS-assisted wireless systems. The joint problem of active and passive beamforming design at the base station (BS) and IRS, respectively was investigated in [25], [31], and [32] for IRS aided multiuser downlink (DL) system in order to maximize the sum-rate with minimum transmit power. The authors in [33] utilize an IRS to enhance the coverage of multiuser milli-meter wave communications and maximize the sum-rate under joint optimization of active and passive beamforming. In [34] and [35], the problem of sum-rate maximization for IRS-assisted multipleinput multiple-output (MIMO) and multiple-input singleoutput (MISO) downlink (DL) communication was studied. Further, an energy-efficient resource allocation scheme in an IRS-assisted DL multiuser communications system was investigated in [23] under the constraints of transmit power allocation, phase shifts at IRS and individual link budget. The authors in [36], [37], and [38] proposed algorithms to improve the spectrum and energy-efficiency fairness for a multiuser IRS aided network. In [39] and [40], the problem of maximizing energy efficiency under joint power control and passive beamforming at IRS was investigated for user-user and user-BS communication, respectively. The non-trivial tradeoff between the energy efficiency (EE) and the spectral efficiency (SE) was studied in [21] and [41] for IRS aided multiuser MIMO uplink (UL) and DL communications. In [42] and [42], the authors concluded that the high SINR and high number of IRS elements improve outage probability, ergodic capacity and rate throughput.
Many of the above mentioned works perform IRS-assisted sum-rate maximization of multi-user linear precoding transmission, i.e., SDMA scheme which always relies on multiple transmitting/receiving antenna. There exist other works which are based on NOMA scheme. The joint power allocation and phase shifts optimization problems for sum-rate maximization and power minimization were studied for the DL IRS-NOMA multi-user systems in [15], [17], [22], [44], [45], and [46], respectively, which validated the superior performance of NOMA over orthogonal multiple access (OMA) scheme. The authors in [29] studied an IRS-assisted uplink NOMA system with an objective of maximizing the sum-rate of all users under individual power and passive beamforming constraints. In [16] and [47], an alternate problem for an IRS aided DL wireless power transfer and UL wireless information transmission was studied which considered the sum-rate maximization problem for jointly optimizing the transmit time, power, and phase shift. Moreover, in [19], [26], and [48], the problem of maximizing the system energy efficiency was investigated by jointly optimizing the transmit beamforming at the BS and the reflecting beamforming at the IRS. The authors in [14], [18], and [49] studied the outage performance of IRS-aided NOMA systems which validate that the IRS reflected communication links provide better system performance than the direct communication links in high fading environments.
Overall, as compared to conventional OMA scheme, the NOMA-IRS aided network provides better spectral efficiency and coverage capability especially when direct link between communicating nodes is absent [26], [29]. Nevertheless, the recent studies reveal that the system performance can be further improved using RSMA which splits the rate of user into various sub-messages [8], [9], [10], [11], [13], [37], [50], [51], [52], [53]. NOMA and SDMA are subsets of RSMA [11], [12]. Particularly, RSMA effectively utilizes resources and allocates them to the various messages due to which it gains better performance than NOMA [13]. The authors in [52] considered the maximizing the minimum transmission rate among multiple single-antenna users for IRS aided multiuser MISO RSMA downlink system via optimizing the transmit beamformers at the transmitter and the phase sifters at the IRS. In [53], the problem of energy efficiency maximization was investigated by controlling the active and passive beamforming at the BS and at the IRS, respectively. The closed-form expressions for the outage probability of cell-edge users and near users were derived in [51] for IRS assisted DL communication which target to gain optimal resource allocation. It was shown that RSMA performs better than NOMA under various system parameters such as the number of IRS reflecting elements and the node density. However, the message split of each UL users considerably increases the complexity of successive interference cancellation (SIC) design at the receiver [53]. Moreover, the optimal selection of decoding order is difficult and need to be resolved carefully [13].

B. Motivation and Contributions
Most of the existing works utilizing IRS aided communication considered either NOMA scheme or SDMA scheme for sum-rate maximization. Very few works in the literature focus on the consideration of RSMA in IRS aided network [51], [52], [53]. To the best of the author's knowledge, the IRS aided UL RSMA system has not been investigated yet from neither a theoretical nor a resource allocation perspective. Besides, the optimal selection of the decoding order of sub-messages in the considered IRS aided UL RSMA system becomes critical as the overall channel gain of user-BS is subject to vary under passive beamforming design. The appropriate selection of decoding order plays an important role in optimal performance of RSMA for UL transmission in terms of interference management and sum-rate metric [13]. Motivated by this background, we investigate RSMA for an IRS-aided UL system which target to achieve improved spectral efficiency under optimal resource allocation and near-optimal decoding order. The major contributions of this work are as follows: 1) We consider the problem of sum-rate maximization under the constraints of total transmit power, passive beamforming at IRS, decoding order and minimum quality-of-service (QoS) per sub-messages for an IRS aided UL RSMA system where the direct link between the users and the BS is absent due to harsh non-line of sight propagation condition. For the formulated sum-rate maximization problem, a sufficient and necessary feasibility condition is discussed w.r.t. transmit power and QoS constraints. 2) In the considered system, a message of each user is split into several sub-messages and each part contributes to the rate of that user and depending upon sub-messages priorities (split proportions) the BS decodes them using appropriate decoding order. We analytically derive a strategy of the decoding order which state that the optimal decoding order for the considered UL RSMA depend is set based on descending order of channel gain of the users and ascending order of predetermined proportional indices of sub-messages. The proposed decoding order achieve performance closer to the exhaustive search based decoding order [13] with low computational complexity. 3) Due to the strong coupling of the variables, the formulated sum-rate maximization problem is non-convex and thus, it is hard to solve directly. To make the problem tractable, we decouple the original problem into different sub-problems. The power allocation sub-problem and passive beamforming design sub-problem are solved separately with fixed decoding order using successive convex approximation (SCA) method and Riemannian conjugate gradient method, respectively. Later, we propose a unified solution based on block-coordinate descent (BCD) algorithm. 4) The computational complexity and numerical simulations are provided to demonstrate the effectiveness of the proposed solution. As a benchmark, we compare the performance of the proposed RSMA system with the conventional NOMA and OMA schemes subject to convergence, node density, noise power, and the number of IRS elements etc. Overall, the proposed power allocation, passive beamforming and unified solution algorithm converge within few iterations and give near optimal solution with lower computational complexity. Moreover, the proposed solution outperforms the conventional NOMA and OMA schemes in terms of sumrate metric.

C. Notations
Notations: Throughout the paper, the scalar, vectors, matrices and sets are represented by regular, bold lowercase, bold uppercase and scripts, respectively. |S| represents cardinality of the set S. ||l|| 2 indicates the 2 norm of the vector l. p (a) denotes the value of parameter p at a th iteration whereas {p i } indicates the accumulation of all variables p i , ∀i. n ∼ CN (μ, σ 2 ) denotes that n is circularly symmetric Gaussian random variable with μ mean and variance σ 2 . is Hadamard product or element-wise product of the two vectors. The complex conjugate of a complex vector a is denoted as a * .

II. SYSTEM MODEL AND PROBLEM FORMULATION
We consider an IRS aided UL RSMA system where singleantenna K users aim to communicate with a single-antenna BS. Assuming 1 that there does not exist direct link between the users and the BS due to unfavourable propagation conditions [26], [29] in an urban setting [29]. Therefore, the UL transmission is carried out via IRS with N reflection elements which is mounted on a skyscraper at particular height from the ground plane as shown in Fig. 1.

A. Signal Model
In UL RSMA system, each user splits its own message into J parts (sub-messages) and transmits them to BS simultaneously via an IRS at same time and frequency slot [13]. Then, the BS uses a SIC technique to decode the sub-messages of all users using the optimal decoding order. Denoting J = {1, . . . , J} as the set of sub-messages for each user. Let x k denotes the transmitted signal from the k th user such that where s kj is the j th sub-message of the k th user with E |s kj | 2 = 1 and p kj is the transmit power for sub-message s kj . The total transmission signal power p k of each user is limited upto P max i.e., Now, the total signal received y at the BS is given as where g UR k ∈ C N ×1 is the channel gain between the k th user and IRS, g RB ∈ C N ×1 is the channel gain between IRS and the BS and Φ = diag e jφ1 , . . . , e jφN T ∈ C N ×N is a diagonal matrix that accounting for passive beamforming at the IRS and φ n represents the phase shift at the n th element of the IRS. Furthermore, n is additive white-Gaussian noise which is modeled as complex Gaussian random variable with zero mean and uniform variance of σ 2 , i.e., n ∼ N 0, σ 2 . The IRS position along with the BS is fixed and pre-defined which ensures LOS path between IRS-users and IRS-BS [36].
To account for small-scale fading, we assume the Rician channel fading for all the channels with K f factor. Similar to [17], [26], [27], [29], [47], [54], [55], it is assumed that the channel state information (CSI) of all channels involved is perfectly known 2 at the BS and the IRS using quasi-static flat-fading channel model [32]. Now, the BS utilizes SIC to decode all the sub-messages of the users from the received signal y. There exist total M = K × J sub-messages from K users. Assuming that decoding order of sub-messages at the BS is denoted as the set π = {s kj : k ∈ K, j ∈ J } in which first element is decoded first, second element is decoded second and so on. The permutation π belong to the set Π which is set of all the possible decoding order of sub-messages. Let π kj ∈ M {1, . . . , M} denotes the decoding order of the sub-message s kj . Particularly, for the sub-message s kj , the BS successfully decodes and eliminates all the sub-messages which have low decoding order than s kj and treats remaining sub-messages as the interference (other than s kj ). So, the signal to noise-ratio (SINR) in RSMA system for 2 In this paper, we primarily focus on the study of power allocation and beamforming design on the considered IRS aided RSMA system and to characterize the maximum achievable performance using perfect knowledge of CSI. The results in this paper serve as theoretical performance upper bounds for the considered system which can provide a benchmark for the system design under imperfect CSI. The system model involving imperfect CSI or no CSI can be considered as a future scope of this work. the sub-message s kj can be given as ∀k ∈ K, ∀j ∈ J , where Q kj is the set of all the sub-messages which have greater decoding order than s kj i.e., Using (4), the achievable rate r kj for the s kj sub-message is given as ∀k ∈ K, ∀j ∈ J . Utilizing (5), the total rate for the k th UL user is given by r k = j∈J r kj .

B. Problem Formulation
The prime motivation of this work is to maximize the sum-rate throughput for an IRS aided RSMA system subject to QoS constraints through joint optimization of power allocation, passive beamforming at IRS and decoding order. In this paper, we primarily focus on the improvement of spectral efficiency of dead zone uplink users using RSMA involving efficient resource allocation and comprised the utilization of user-pairing schemes. 3 Indeed, the user-pairing schemes in power-domain multiple access schemes such as NOMA and RSMA improve sum-rate throughput of the system as well as individual user rates, especially when an optimal pairing scheme is proposed [44], [56]. However, the optimal selection of user paring is computationally complex as the total number of pairing sets grows swiftly with K [56]. Although, there exist low-complex user-pairing schemes based on asymmetric rates or asymmetric deployment conditions [44] which can be solved effectively using polynomial time complexity, these user-pairing does not guarantee to provide the optimal solution in poor signal quality [43].
Using (4) and (5), the joint optimization problem can be formulated as (6), shown at the bottom of the next page, where p = [. . . , p kj , . . . ] , π = [. . . , π kj , . . . ] and (C1), (C2) and (C3) are the power allocation, passive beamforming and decoding order constraints, respectively. h kj denotes predetermined non-negative index which set the split proportion for the j th sub-message such that r kj = h kj r k , ∀j ∈ J and j h kj = 1, ∀k ∈ K. (7) and r min k denotes the minimum rate requirement for the k th user. The constraint (C4) corresponds to minimum QoS constraint for each sub-message. The constraint (C5) corresponds to the prioritization of the different sub-messages, i.e., splitting the overall rate of the each user among all the sub-messages based on split proportions. Hence, the rate of the sub-messages belonging to the same user will vary depending upon the proportional indices. 4 In other words, there can exist some sub-messages which can have a higher/lower rate (or SINR) than other sub-messages of the same users. The sub-message with higher proportion will have a higher rate than other sub-messages belonging to the same user. Since the proportion indices for the sub-messages for each user are pre-determined, we can prioritize the sub-messages based on their known information of the proportional index. For example, a submessage with higher rate-proportion index can be assigned with higher priority and vice versa for lower rate-proportion index. Using the rate-proportion index, the constraint (C4) can be equivalently written as Note that the constraint (C2) is a non-convex unit-modulus constraint, constraint (C3) is a discrete variable constraint, while (C4) is a non-convex minimum rate (or SINR) requirement constraint. In result, the formulated problem (P0) is non-convex mixed integer optimization which cannot be solved in a straightforward manner. In general, there is no standard method to solve such non-convex optimization problems. Therefore, in the next section, the formulated problem (P0) is decoupled into different sub-problems which address the problems of optimal power allocation, passive beamforming and decoding order alternatively.

C. Feasibility of Problem (P0)
Before solving the problem (P0), we discuss the necessary and sufficient conditions for its feasibility w.r.t. power and 4 The proportional indices considered in this work focus on the rate-splitting of the user among its sub-messages in various proportions. Clearly, these predefined proportional indices implicitly/explicitly affect the selection of near-optimal decoding order of all the sub-messages which is validated shortly.
QoS constraints. Depending on the channel conditions, the minimum rate requirement constraint in (C4) may not be satisfied for all users under the maximum transmit power P max . Therefore, the formulated problem (P0) may be infeasible in nature i.e., there may not exist an optimal solution which satisfy all the QoS constraints. Here, we identify and provide the necessary and sufficient condition on the feasibility of the problem (P0).
For guaranteeing the minimum transmission rate requirement for each user, the signal strength gap among the sub-messages should be considered as follows: Lemma 1: In order to satisfy the minimum transmission rate for each user, the signal strength gap among the submessages should be considered such that the minimum power allocation for sub-message s kj under given QoS should be lower bounded as where γ min kj = 2 (h kj r k ) − 1 and π kj ∈ M is the decoding order of the s kj sub-message.
Proof: Refer Appendix A for proof. Using Lemma 1, the total UL power for k th user is lower bounded as The minimum transmit power condition provided in the Lemma 1 provides a necessary condition for the feasibility of the problem (P0). Nevertheless, the necessary and sufficient condition for any power optimization problem with any QoS constraint is guaranteed when the maximum transmit power is greater than the minimum power transmit power which marginally satisfy the QoS constraint [57], [58], [59] which leads to the discussion of the following argument. Lemma 2: The necessary and sufficient condition for the feasibility of the problem (P0) requires that the maximum UL : Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
power should be lower bounded as Proof: See Appendix B.

III. PROPOSED SOLUTION
The objective function in (6) is non-convex due to strong coupling of variables. Consequently, the non-convex problem (6) is difficult to transform into a convex one since the channel gain is dependent on the phase matrix Φ. Moreover, the discrete variable π make it mixed-integer non-convex problem which is challenging to solve. Hence, we decouple the original problem into different sub-problems in which the problem of power allocation for users and passive beamforming at the IRS are separately addressed for given decoding order. Later, a unified solution is presented using BCD method which jointly solves these sub-problems in an iterative manner. These sub-problems and their solutions are discussed in detail as follows:

A. Power Allocation
For given phase matrix Φ and π, the sum-rate maximization problem (6) can be recast as (12), shown at the bottom of the page, which can be alternatively written as where Q + kj is the set of all the sub-messages which have greater than equal to decoding order than s kj i.e., Q + kj = {(l, m) : π lm ≥ π kj }.
Utilizing the fairness proportion constraint (C5) and the slack variables {r k }, the problem (P1) can be simplified as where r = [r 1 , . . . , r K ] T and h kj is the fairness proportion for the j th sub-message. However, the problem in (14) is still non-convex due to the constraint (C6). Particularly, the left side of (C6) can be written as difference of convex function (or approximate convex functions) as where However, the constraint (C6) is still non-convex due to nonconvexity of ξ kj . Therefore, we adopt SCA method in order to convert the non-convexity of (C6) into a convex one. The SCA method iteratively approximates ξ kj into its equivalent convex forms using given power allocation from the previous iteration. Let p (a−1) kj be the power allocation obtained at the (a − 1) th iteration then ξ kj can be approximated at the a th iteration as (16), shown at the bottom of the next page. Now, the constraint (C6) can be approximately written as Ultimately, the problem in (14) can be written into approximate convex form as The objective function in (18) is affine and the constraints (C1) and ( C4) are linear. Since the left hand side (LHS) term of constraint ( C6) is the logarithmic of affine functions which is concave, hence the constraint ( C6) is convex. Therefore, the problem (18) is convex in terms of variables ≤ a or a ≤ A max do 4: Solve (18) and calculate the sum rate R (a) = k r (a) j 5: a=a+1 6: end while 7: Output:R = R (a) ,p = p (a) and thus the solution can be obtained using standard convex optimization methods [60] or toolbox [61]. With given initial power allocation values, {p (0) kj } the problem (18) is iteratively solved for maximum A max iterations or until convergence for sum-rate R = k r k . Let a denotes the convergence factor.
The proposed SCA method for achieving the optimal power allocation for given phase matrix Φ o and decoding order π o is summarized in Algorithm 1.

B. Passive Beamforming
Under given power and decoding order satisfying constraint ( C4), the problem for passive beamforming i.e., optimal phase shift determination at the IRS can be formulated from (6) as (19), shown at the bottom of the page, where θ = [θ 1 . . . θ N ] T = diag (Φ). However, the problem (19) is challenging to solve due to non-convexity of the objective function and unit-modulus constraint (C2). To the best of the authors' knowledge, the global optimal solution of such non-convex optimization problems with unit modulus constraints is generally intractable [54]. In order to efficiently solve (19), we develop a Riemannian conjugate gradient (RCG) method based on manifold optimization, which directly solves (19) to provide a local optimal solution. Importantly, the constraint (C2) can be viewed as smooth Riemannian manifold [31] M such that Now, the set of all the tangent vectors which are tangent to the manifold M at the point θ gives tangent space T θ M which is defined as Particularly, a manifold is a topological space that locally resembles Euclidean space near each point and hence, the optimization over a Riemannian manifold is analogous to that in the Euclidean space [54]. To this end, we propose an iterative conjugate gradient algorithm for the passive beamforming based on Riemannian manifold optimization to locally solve (19). In order to facilitate the conjugate gradient algorithm, the Euclidean gradient is firstly determined from the objective function to determine Riemannian gradient. Denoting the objective function in (19) as whose vectorized Euclidean gradient can be expressed as (23), shown at the bottom of the next page.
As per [62], the Riemannian gradient of any function f c (θ) at point θ is calculated based on the orthogonal projection of the Euclidean gradient Δf c (θ) onto tangent space T θ M such that (24) where P θ (z) is the orthogonal projection of z on the tangent space. The expression in (24) is straightforward to obtain as referred to in all the relevant literature of complex manifold optimization [27], [55].
The conjugate gradient method finds the local optimal point on the manifold iteratively. Noting that the retraction helps to restrict the search on the manifold M with given search direction as where α is a search step size and d = [d 1 , . . . , d N ] T is the search direction. The step size α can be chosen by a line-search algorithm using the Armijo backtracking line search as given in Definition 4.2.2 of [62]. The update rule of the search direction of the conjugate gradient for the (b + 1) th iteration can be obtained in similar to the Euclidean space as Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
where g (b+1) is the Riemannian gradient and λ (b+1) is the Polak-Ribiere parameter which is calculated as The update points for the phase shifts are given by using retraction operations. The passive beamforming based on the RCG is summarized in Algorithm 2 which converges to the critical point [ Choose Armijo backtracking line search step size α (b)

C. Decoding Order Selection
The optimal selection of decoding order effectively cancels out interference from other sub-messages and improves the overall rate-throughput and hence, it is required to be considered. Using (6), the problem of the optimal selection of decoding order can be given as The problem in (30) is difficult to solve due to the non-convex objective function and strong coupling of discrete variables. The optimal decoding order setting in problem (32) can be viewed as mixed integer non-linear programming which is nearly intractable. There exists no standard methods to solve such problems. Most of the decoding orders of sub-messages from the users are carried out based on their channel gain [14], [29], [63]. The authors in [13] proposed an exhaustive search algorithm based method to ensure the optimal decoding order for UL-RSMA. Undoubtedly, the exhaustive search algorithm is optimal as it executes the proposed power allocation and beamforming design in IRS-RSMA system for all possible decoding order schemes and selects that decoding order which ensures best achievable rate-throughput. However, the overall computational complexity of exhaustive search rises exponentially with the number of sub-message. Conclusively, the exhaustive search is although optimal, but it is in-efficient in terms of computational complexity. Therefore, we proposed a decoding order of M sub-messages at the BS based on the channel gain and the split proportions of the sub-message of each users and its optimality is validated in the simulation section.
Lemma 3: The decoding order of sub-messages is set according to the descending order of ν kj is the effective channel gain between k th user and the BS.
Proof: See Appendix C for the proof of Lemma 3 Utilizing Lemma 2, it can be concluded that descending order of channel gain with ascending order of rate proportion can provide near-optimal decoding order which is validated in the numerical simulation section shortly.

D. Unified Solution
Here, we present a joint solution for the optimal power allocation to users and passive beamforming at the IRS using BCD. Particularly, the sub-problems of power allocation and phase shift determination are solved alternatively in iterative manner with decoding order as considered in Lemma 3. With initial power allocation, we execute phase matrix determination at IRS using Algorithm 2. Later, the obtained phase matrix is utilized for the power allocation in Algorithm 1. In next iteration, the obtained power value in the previous iteration is utilize for phase matrix calculations and this process continues until a convergence ( c ) is achieved. The output of each optimization process in current iteration serves an input to the next iteration. Algorithm 3 summarizes the provided unified solution. Since this algorithm is based on the BCD method Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
≤ a or c ≤ C max do 4: Given power allocation p (c) , execute Algorithm 2 and obtain Φ (c+1)

6:
Update c = c + 1 7: end while 8: Output:R f = R (c) and the sub-problems for updating each block of variables are convex, the algorithm convergence can be guaranteed.

IV. COMPUTATIONAL COMPLEXITY
This section describes the computational complexity of the proposed algorithms and solution. The SCA based power allocation problem in (18) has 2M variables and 4M + K constraints where M = KJ. With consideration of worst case scenario, let the SCA algorithm converges at maximum iterations A max . So, the worst case computational complexity of SCA based power allocation algorithm can be given as O A max (KJ + K) 2 (4KJ + K) . Further, the RCG algorithm for passive beamforming design entails worst-case complexity of O (2N ) 1.5 [32]. Let, the unified solution converges in maximum C max iterations then the overall complexity of the proposed algorithm can be given as Hence, the overall complexity mostly depends on the number of UNs, the sub-messages and slight on number of IRS elements too. The optimal decoding order is selected among only four orders and hence, the selection of optimal decoding does not significantly alter the computational complexity. However, the exhaustive search method for decoding order discussed in [17] involves (KJ)! search operations which exponentially increase the computational complexity. Further, the overall worst case complexity of IRS aided UL-NOMA system in [29] can be given as O C max A max 8K 3 + N 3.5 . The RSMA has slightly increased complexity than NOMA due to J parts splitting of rate. However, our proposed solution has comparatively low-computation complexity w.r.t to number of reflecting elements as compared to [14], [29]. The another important parameter affecting complexity of proposed solution is the convergence rate of the SCA algorithm and the BCD algorithm. The fast convergence rate of the proposed algorithm is validated shortly in the next section.

V. NUMERICAL SIMULATION AND DISCUSSION
In this section, the performance of the proposed IRS aided UL RSMA system is examined through extensive computer simulations. The simulation results are averaged by 1000 Monte-Carlo simulations. For simulations, K = 4 UL users are uniformly deployed on ground plane of 100 × 100 m 2 . The position of BS is fixed on ground plane at [−50m, −50m, 0] T , while the IRS is pre-deployed on the building at height 30 m which is assumed to be located at origin [0, 0, 30m] T . The number of elements at the IRS are set as N = 10. The distance dependent path loss model is given by where d is distance between communicating devices, L o = −10dB is the path loss at reference distance d o = 1 m and γ = 2.2 is the path loss exponent.
To account for small-scale fading, we consider all channels as Rician distributed due to fact that it contains both LOS and NLOS components. Thus, the channels are expressed as where K f = 10 is Rician factor, G L = 1 is LOS component and G N is NLOS component that follows Rayleigh distribution with parameter L = 1. The elements of G are multiplied with square root of distance-dependent path-loss model. The noise power σ 2 is set as −100 dB. The signal from each user is split into two sub-messages, i.e., J = 2. The convergence factors a , b and c are set as 10 −4 and the maximum number of iterations for SCA and BCD algorithms are set to 10, i.e., a = b = c = 10 −4 and A max = C max = 10. The UL power budget is set as p max = 1 W. The sub-message priority indices for rate-splitting is set as h kj = 0.5, ∀k ∈ K, ∀j ∈ J . The phase shift is initially selected as random value and the initial transmit power is set as p kj = p max /J, ∀k ∈ K, ∀j ∈ J . The minimum QoS for UL users are set as r min k = 0.5bps/Hz, ∀k ∈ K. All the UL users, the IRS and BS are assumed to operate under same time/frequency band with bandwidth 1Hz. We refer our proposed unified solution in the Algorithm 3 as IRS-RSMA. For performance benchmark and evaluation, we solve the equivalent sum-rate maximization problem for OMA and NOMA schemes for IRS assisted networks which are labelled as IRS-OMA and IRS-NOMA, respectively.
Firstly, we examine the convergence behaviour of the proposed SCA based power allocation (SCA-PA-IRS-RSMA) and RCG based phase-shift design (RCG-PSD-IRS-RSMA) IRS for IRS aided RSMA system and IRS aided NOMA system with respect to maximum number of iterations A max and B max as illustrated in Fig. 2a and 2b, respectively. In order to observe convergence of SCA based Algorithm 1 and RCG based Algorithm 2, we fixed the alternate parameter i.e., phase shift and power allocation, respectively and observe the achievable rate throughput (objective function f c in (22) for Algorithm 2) with varying maximum number of iterations. It can be observed that Algorithm 1 converges within 5-6 iterations and Algorithm 2 converges within 15-20 iterations which validate the quick convergence of the proposed power allocation and phase-shif design i.e., passive beamforming design. Moreover, the performance of the proposed SCA based power allocation and RCG based passive beamforming design for IRS-NOMA and IRS-RSMA schemes are compared with respect to equivalent exhaustive search algorithms with varying step sizes as shown in Fig. 2a and Fig. 2b. The selection of step size for search operation plays an important role as small step sizes ensure higher optimality at the expense of increased computational search to get an optimal solution under a given decoding order which confirms that the computational complexity of exhaustive search methods rapidly increases with a slight increase in the number of sub-messages and IRS elements. Overall, the proposed solutions for SCA based power allocation and RCG based beamforming are although sub-optimal, however they provide effective solutions in terms of complexity especially for large numbers of users (submessages) and IRS elements N > 50. And, the large number of IRS elements is especially advantageous for dead-zone users, which is shortly validated in the subsequent discussion.
We also examined the convergence behaviour of the proposed unified solution with respect to maximum number of BCD iterations, C max as shown in Fig. 2. Particularly, we compare two different BCD solutions such that a) power allocation solved first and later beamforming design denoted as BCD1-JNT-IRS-RSMA and BCD1-JNT-IRS-NOMA b) solving passive beamforming first and later solving the power allocation denoted as BCD2-JNT-IRS-RSMA and BCD2-JNT-IRS-NOMA for IRS-RSMA and IRS-NOMA systems, respectively. Interestingly, both the unified solution converge to the same value. Hence, the selection of optimization order to alternatively solve the passive beamforming and the power allocation is not too significant in terms of achievable sum-rate . Although, the performance of the proposed BCD solution for joint solution is sub-optimal as shown in Fig. 2, the proposed search operation gains the advantage of lower computational complexity when compared to the exhaustive search as the proposed solution attain convergence within 3-4 iterations. Overall, the proposed IRS aided RSMA solution achieves better throughput than the equivalent NOMA and OMA solution.
Next, we examine the performance of the proposed IRS-RSMA solution w.r.t. varying maximum transmit power p max as shown in Fig. 3. It is quite obvious that increasing in maximum transmission power for each user increases the rate-throughput. However, the RSMA system achieves better performance as compared to equivalent NOMA and OMA schemes due to the efficient utilization of power resource. Fig. 4 illustrates the performance behaviour of the proposed IRS-RSMA solution w.r.t. to number of sub-messages. The IRS-RSMA can be considered as special case of NOMA where the power is split to multiple sub-messages for each user. The power splitting ensures maximum spectral efficiency by effectively allocating the transmit power to various submessages which achieves maximal rate region. Increasing in sub-messages for each users increases the effective increase in sub-messages as shown in Fig. 4. However, the increase  in sub-messages increases the challenges for SIC at the receiver and resource management for multiple sub-message transmission, while the complexity in SIC design for RSMA is greater than NOMA. We also compared the performance of the IRS-NOMA and IRS-RSMA system by setting ascending or descending order of channel gain based decoding order as shown in Fig. 3 and 4. It can be observed that the descending order of channel gain attains performance closer to the exhaustive search method discussed in [13] when compared to ascending order. Specifically, the decoding order of users based on ascending order provides the near-optimal performance when the user channels are sufficiently aligned and there exit a large disparity in their channel strengths. While, the ascending order based decoding for users allocate high rate (power) to strongest channel and allocate nearly equal rate to the other users and thus, it achieves better overall system performance especially when there does not exist disparity among the channel gains [64]. Generally, in IRS aided systems, the channels are strong and there does not exist disparity among channel gain of users due to multiple IRS reflectors [53]. Further, we heuristically examine the optimality of the proposed decoding order through performance comparison of the proposed decoding order w.r.t exhaustive search and other decoding orders. Particularly, we analyze the sum-rate throughput of the RSMA system with only two sub-messages J = 2 and 4 different decoding order π 1 , π 2 , π 3 and π 4 as discussed in Table I and compare them with exhaustive search (ES) method as shown in Fig. 5. It can be found that the selection of optimal decoding order among these 4 decoding order depends on the sub-message proportional indices. The π 2 and π 4 decoding provide better performance when h 1 < h 2 This indicates that the decoding order based on descending order of channel gain of the users and ascending order of sub-message proportions, i.e., the proposed decoding order in Lemma 2 attain performance spectral efficiency closer to the optimal solution (exhaustive search) with low computational complexity and attain improved performance when compared to other decoding orders. Importantly, the equal split proportion of rate into sub-messages gains better performance when compared to different split proportions as π 2 and π 4 gain equal performance when h 1 = h 2 . Hence, it can be concluded that the near-optimal design of proportional indices {h kj } can achieve efficient rate-splitting among the sub-messages for the user which can enhance the overall performance of the UL RSMA system. We in the proposed work consider that the proportional indices are already set using a heuristic approach. 5 Fig . 6 shows the performance behavior of proposed IRS system w.r.t. to varying number of IRS elements, i.e., phase shifters. As the number of IRS elements increases, the effective  channel gain between users and the BS is improved and hence the achievable sum-rate increases. In other words, the extra phase shifters can reflect more power of the signal received from the users to BS which leads to a power gain. Particularly, it provides higher flexibility in resource allocation and stronger beamforming gain which improve sumrate throughput. Moreover, we also compare the results with random phase shift which are denoted as RND-PSD-IRS-RSMA and RND-PSD-IRS-NOMA. The near optimal phase shift determination (passive beamforming) provides better rate throughput than schemes with random phase shift. The passive beamforming at the IRS nearly improves the sum-rate for both NOMA and RSMA by 20%. Now, we examine the sum-rate throughput of the proposed solution with varying noise power as shown in Fig. 7. Intuitively, the system performance of the proposed solution degrades with an increase in noise power. Although RSMA gains better performance than NOMA and OMA over the entire range of noise power, the performance of RSMA degrades much faster than NOMA with an increase in noise power. It is due to the fact that the increase in noise power reduces the SINR for both the sub-messages. This further increases the channel noise power over inter-node interference in RSMA and NOMA schemes. In a high noise environment, the performance of RSMA is approximately closer to the NOMA as the channel noise power dominates the performance as illustrated in Fig. 7. In the previous scenario, the bandwidth of the system is set as 1 Hz. Next, we analyze the effect of varying bandwidth on the system performance as validated in Fig. 8, where σ 2 = −180 + 10 log 10 B (dB). It is shown that  the increase in bandwidth, B results in increases in overall rate throughput at lower bandwidth. However, beyond particular bandwidth say 5-6 Mhz, the spectral efficiency improvement is minor as the slope of sum-rate throughput start decreasing due to increase in the noise power [13]. Nevertheless, the overall performance of the uplink RSMA is still better than NOMA and OMA schemes for all the considered scenarios. Fig. 9 shows the system performance analysis of the proposed IRS-RSMA scheme and equivalent NOMA and OMA scheme under varying numbers of users. The RSMA effectively utilizes the spectrum and allocates resources optimally and hence the sum-rate throughput (overall rate for the system) is better than NOMA and OMA schemes. However, the average rate throughput for NOMA and RSMA per user decreases with an increase in the rate-users. Particularly, the increase in number of users results in high inter-node interference due to which the rate per user decreases. With a high number of users K > 8, the inter-node interference provide high noise level in signal equivalent to or higher than channel noise power due to which the rate per user decay or saturate as shown in Fig. 9. However, in practical scenarios, the SIC for decoding the sub-messages gets tighter as the number of users or submessages increases and this increases the hardware design complexity at the receiver, especially under a low transmission power budget. The user-paring schemes in RSMA system is particularly beneficial for high-performance than NOMA and OMA scheme as validated in [13].
Finally, we examine the performance of the considered IRS system w.r.t to varying path loss exponent (PLE) as shown in Fig. 10. Assuming that there exist a weak direct path between users and the base-station with fix PLE as γ d = 4.5 where user can send signal via direct path or indirect path, i.e, IRS assisted path with high uplink transmission power. Although this consideration is not valid for this system model having only dead-zone users, the direct link is considered as the performance benchmark for indirect links. Next, we examine the rate-throughput for both NOMA and RSMA schemes with the varying values of PLE γ for user-IRS and IRS-BS links. It is quite obvious that the increase in γ will result in decrease in rate-throughput for both IRS aided NOMA and IRS-RSMA schemes as the high values of PLE attenuates the signal heavily by reducing the overall channel gain [13]. Particularly, the effect become worse for an small IRS model (N = 10) where the IRS-assisted performance is inferior to direct link when γ > 3 as shown in Fig. 10. This effect can be minimized upto some-extent by utilizing large IRS (N = 100), however, the performance of large IRS is suppressed under large scale fading especially when γ > 3.6. Hence, the IRS is beneficial for the dead-zone users when large scale IRS with N > 100 are deployed with preferably small scale-fading.

VI. CONCLUSION
In this paper, the problem of sum-rate maximization for an IRS aided UL RSMA system was investigated subject to the constraints of transmit power for user and passive beamforming at the IRS. Due to the non-convexity of the joint optimization problem, it was decoupled into different sub-problems which addressed power allocation and passive beamforming alternatively using SCA and RCG algorithms, respectively under fixed decoding order. Particularly, the considered RSMA splits the rate for each user by dividing message of each user into multiple parts with different proportions. An near-optimal decoding order was analytically derived which confirms that the decoding order based on descending order of channel conditions and ascending order of split proportions of the sub-messages of the user gain better sum-rate throughput as compared to other decoding order. Simulation results validated that the proposed solution has fast convergence rate and the IRS-aided RSMA system achieved better spectral efficiency as compared to the NOMA and OMA systems for considered all the network scenarios.
There exist infinite possible combinations in the set P r ,