Joint Phase Shift and Beamforming Design in a Multi-User MISO STAR-RIS Assisted Downlink NOMA Network

Simultaneous transmitting and reflecting reconfigurable intelligent surfaces (STAR-RIS) has gradually been considered as a promising technology in the wireless communication networks. Besides, non-orthogonal multiple access (NOMA) is also the key technology in the sixth-generation (6G) wireless communication system. In this work, we study a multiple input single output (MISO) STAR-RIS assisted NOMA downlink network and investigate the energy efficiency (EE) maximization to achieve the tradeoff between the sum rate and the power consumption. The original formulated problem is non-convex due to the coupled beamforming vectors of the users and phase shifts of the STAR-RIS. To efficiently solve the problem, we split the original non-convex problem into the phase shift and beamforming optimization problems and then solve them alternatively. In the phase shift optimization, fractional programming (FP) is applied to transform the sum rate maximum problem to convex semidefinite relaxation (SDR) one with the rank-one constraint. After this, a novel sequential rank-one constraint relaxation (SROCR) is proposed to convert the rank-one constraint into a convex one, which can effectively overcome the inadequacy of Gaussian randomization, i.e., quality of the solutions and computational complexity. Similarly, FP is applied to solve the beamforming problem by transforming it to SDR problem. It turns out that the optimal solution of the SDR beamforming optimization problem can be guaranteed to be rank-one by the mathematical proof and experiments. The simulation results demonstrate the STAR-RIS NOMA system can achieve the superior performance in EE.

innovative device in plenty of fields such as multiple input single output (MISO) networks, multiple input multiple output (MIMO) systems to greatly improve the sum rate and energy efficiency in the wireless communication networks [1], [2]. An RIS is composed of a number of low-cost passive reflecting elements. the incident signals can be smartly adjusted by the microcontroller connected to the base station (BS) with a low power consumption [3], [4] which increases the coverage range in the future six-generation (6G) communication system [5]. Moreover, the RIS can be equipped to achieve a virtual link between the transmitter (Tx) and the Receiver (Rx) if the users are in the dead service zone or the direct links are blocked [3]. However, the RIS can only reflect the incident signal. This property requires Tx and Rx to be located at the same side of the RIS and a single RIS can only achieve the half-space coverage communciation [6], [7]. Thanks to the reflective-transmissive meta-surfaces technology [7], the simultaneous transmission and reflecting reconfigurable intelligent surface (STAR-RIS) has been developed as a promising technique to compensate the inadequacy of the RIS since it is able to extend the half-space coverage to full-space one. Specifically, the STAR-RIS splits the 3D space into two regions, namely transmission region (T region) and reflection region (R region). Different from the RIS, the STAR-RIS consists of transmitting and reflecting elements, these two types of elements are able to transmit and reflect the incident signals, respectively and build the virtual links with Rx in transmission and reflection regions. Besides, non-orthogonal multiple access (NOMA) also plays an important role in the 6G system [8]. Compared with orthogonal multiple access (OMA), where the users can only be allowed to transmit or receive the signals on their own channels, NOMA enables users to share the same time slot, frequency domain and channel code [8], [9]. This characteristic significantly improves the spectrum efficiency [1] and enables massive connectivity of the devices in the networks [10]. Inspired by the aforementioned benefits, the aim of this work is to design a multi-user STAR-RIS assisted NOMA network and analyze its energy efficiency performance.
A. Related Works maximization problem was formulated based on the RISassisted downlink system with two-user case, where the beamforming optimization problem and phase shift optimization problem were alternatively optimized. The successive convex approximation (SCA) and semidefinite relaxation (SDR) were applied as the mathematical tools to tackle the optimization problems. In [11], the authors proposed an RIS UAV-NOMA downlink network to achieve the sum rate maximization in two-user case. SCA and SDR were also applied to tackle the optimization of beamforming vectors and phase shifts. Different from [1] and [11], in [10], the RIS-assisted NOMA networks were extended to the multi-user case and the proposed scheme based on the problem relaxation and SCA and it achieved the superior performance in energy efficiency as well. Besides, [12] and [13] also studied the power consumption minimization problem in an RIS-assisted multi-cluster systems, where the concept of central and edge users are proposed. Particularly, [13] applied a novel algorithm based on second-order cone programming (SOCP) and alternating direction method of multipliers (ADMM) to optimize the beamforming of the users.
2) Studies Related to STAR-RIS NOMA Systems: The potentials of STAR-RIS assisted NOMA systems have also gradually been studied in more and more works. In [6], the coverage characterization of two-user STAR-RIS networks was discussed and compared between the NOMA and OMA cases. The key idea of this work was to apply Karush-Kuhn-Tucker (KKT) conditions to transform the non-convex decoding order constraint to a linear one. Besides, the authors in [7] proposed a joint power allocation and beamforming design by applying various mathematical tools, i.e., SCA, SDR and alternating optimization (AO). Morever, the authors in [8] also proposed the STAR-RIS aided NOMA and OMA networks. In this work, SDR, convex upper bound approximation, and geometry programming were used to optimize the decoding orders, beamforming coefficient vectors and power allocations to achieve the sum rate maximization of the network.

B. Motivations and Contributions
Note that the sum rate or weighted sum date (WSR) has been widely studied in STAR-RIS NOMA systems mentioned above [7], [8], [14]. However, solely pursuing the increase of the sum rate in a network is not an economical and efficient solution in the future 6G network [15], and hence we aim to exploit a better strategy which can achieve the maximum sum rate with the limited amount of the transmitting power. Inspired by the above points of view, one of the motivations of this paper is to achieve the energy efficiency maximization in a multi-cluster MISO STAR-RIS assisted downlink system, where each cluster has one near user and one far user and NOMA is applied on each cluster to reduce the interference. Besides, the number of transmitting clusters is the same as the reflecting counterparts. Without the direct links between the BS and the users, when the STAR-RIS receives the incident signals, the elements can transmit and reflect them to the users in the clusters. By tuning the electromagnetic properties of the elements in STAR-RIS, the transmitted and reflected signals can be modified. From the mathematics perspective, the RIS or STAR-RIS phase shift and beamforming optimization problems are usually related to the method of semidefinite relaxation (SDR). The rank-one constraints can be introduced after the SDR, which makes the problem NP-hard. In practice, this type of problems is challenging to solve. To tackle this type of problems, most existing works usually first solve the convex SDR problem by dropping rank-one constraints and then check whether the optimal solution was a rank-one matrix. If not, Gaussian randomization was applied to find the suboptimal solution of the original problem [16], [17], [18]. Besides, in some cases such as [18], [19], [20], [21], the solutions to the SDR problems by dropping rank constraints could happen to be rank-one mathematically. However, in this system, plenty of the constriants in the proposed optimization problems indicate that it might be difficult to generate the feasible points by Gaussian randomization. To obtain the desired suboptimal solution, more random points have to be sampled, which dramatically increases the complexity of the algorithm. Therefore, in this work, one novel mathematical tool and mathematical proof are adopted to replace the Gaussian randomization to tackle the rank-one constraints in the phase shift and beamforming optimization problems, respectively. The contributions can be summarized below: r We propose a multi-cluster MISO STAR-RIS assisted downlink NOMA network to find the maximum energy efficiency by optimizing phase shifts of the STAR-RIS and beamforming vectors of the users. However, these two types of variables are coupled with each other and it is difficult to formulate the closed form of each variable. To address this dilemma, we decouple the original problem to two suboptimal problems and solve them alternatively.
r To solve the phase shift optimization problem, fractional programming [22], [23] is first exploited to address the non-convexity issue of objective function and constraints. Then, we transform the problem to an SDR one with rank-one constraints. Instead of applying the Gaussian randomization, we propose another method called a sequential rank one constraint relaxation (SROCR) to address the rank-one issue and obtain a suboptimal solution [24].
r The challenge to solve the beamforming optimization problem is due to the fact that the objective function is non-convex fractional form. The key idea is to first apply the novel fractional programming combined with SDR and convert the QoS constraints to a concave/convex form. After this, instead of applying Dinkelbach's method to transform the objective function which can bring extra computational complexity, we apply fractional programming directly to transform the objective function to a concave form and it will greatly increase the efficiency of the algorithm. In this paper, different from the phase shift optimization, the optimal solution of the SDR beamforming problem dropping rank constraints can be proven to be rank-one, which avoids the extra steps to tackle the rank issue.
r By comprehensively considering the quality of the solutions and the complexity of the algorithm, the proposed scheme can achieve better energy efficiency performance than the benchmark schemes, such as SDR by Gaussian randomization. Besides, this work also demonstrates the great value of the combination between the NOMA technology and STAR-RIS. For instance, the simulation results show that the energy efficiency performance of the STAR-RIS NOMA system can outperform that of the other systems, such as RIS NOMA, STAR-RIS OMA, RIS OMA. Furthermore, the proposed scheme also adapts to other types of the optimization problems, which involve non-convex quadratic fractional objective functions and constraints or have the rank-one constraints. Meanwhile, the proposed method can also be extended in the networks with more than two users in a cluster by a given pre-defined decoding order.

C. Notations
In this paper, the bold capital letters and bold lowercase letter refer the the metrics and vectors, respectively. The transpose, hermitian, and trace of a matrix A are A T , A H and Tr(A), respectively. A 0 means that A is a positive semidefinite matrix. Besides, · 2 represents the 2-norm of the metrix or vector. a ∼ CN (0, σ 2 ) denotes a follows the Gaussian distribution with zero mean and σ 2 variance. a † represents the conjugate of the complex number a. diag(a) represents the diagonal matrix of the vector a. j represent the imaginary unit of the complex numbers. Except for the mathematical notations, there are also some particular variables and notations presenting in Table I for the convenience of the readers.

II. SYSTEM MODEL
We consider a multi-cluster MISO STAR-RIS system. BS is equipped with M transmit antennas. Firstly, the base station (BS), STAR-RIS and users are located in the Cartesian coordinate system. the STAR-RIS elements of this system are N . There are totally 4 K users and 2 K clusters in this system, where 2 K clusters are evenly distributed in transmission and reflection regions. Each cluster contains one near user and one far user. l ∈ {t, r} refers to the transmitting or reflection region, i ∈ { , ς} refers to the near user or far user. As is shown in Fig. 1, each region has totally K clusters and they are notated from 1 to K. U k,l,i refers to the i user in the kth cluster of the region l, k ∈ 1, . . . , K. The phase shift vector consists of the phase shift angle vector θ l = β l e jθ 1,l , . . . , e jθ N,l , which adjusts the phase shift of the incident signals and √ β l , who controls the power allocation of the l elements. Note that β t and β r are coupled with each other and satisfy β t + β r = 1. Because of the existence of the block, there is no direct link between the BS and users. G ∈ C N ×M is the channel gain from the BS to the STAR-RIS. h H k,l,i ∈ C 1×N refers to the channel gain from the STAR-RIS to the i user of the kth cluster in the region l. For example, h k,t, refers to the channel gain from the STAR-RIS to the near user in the kth cluster in the transmission region. Therefore the total channel gain for the i user of the kth cluster in region l, h k,l,i can be expressed as is the beamforming vector of the i user of the kth cluster in the l region. n k ∼ CN (0, σ 2 ) is the additive white Gaussian noise (AWGN). For traditional NOMA scheme such as [10], successive interference cancellation (SIC) is implemented with a given decoding order of all K users {U 1 , . . . , U K }. Specifically, given two of these K users, denoted by U k and U j , k < j, the U k 's signal is decoded before U j 's one. By applying the SIC, all the interference of the users whose indexes are less than k can be removed when decoding U k 's signal. However, if using such a scheme, it requires that each user's signal has to be decoded at itself and all the users who have a higher decoding priority than it. Thus, the decoding complexity of this system can be very high with a larger number of the users. To decrease the complexity during the decoding process, we group the users into clusters and apply NOMA only within each cluster. Specifically, the decoding order of this network is decided according to the user's decoding capability. In this paper, we assume that the near user can have a higher decoding capability than the far user within a cluster. Therefore, by applying NOMA in each cluster, the far user's signal is decoded before the near user. For the near user's signal, SIC can be employed to cancel the intra-cluster interference from the far user in its cluster. In contrast, for the far user, it cannot decode the near user's signal. So the intra-cluster interference from the near user when decoding the far user's signal cannot be removed. This scheme enables each near user and far user's signals to be decoded only once and twice, respectively and greatly decreases the decoding compleixty of the network. γ k,l, is the SINR for the kth near user in region l. γ k,l, →ς is the SINR for signal of the kth far user decoded at the near user. γ k,l,ς→ς is the SINR for signal of the far user of the kth cluster decoded by itself. After SIC, the SINR for each user with different decoding situations could be expressed as followed: where where w is the collection of all the beamforming vectors, i, i ∈ { , ς} and l ∈ {t, r}, l = l. In this network, the SIC can be carried out by satisfying the conditions that the near user's signal of the kth cluster can be decoded by itself and the far user's signal can be decoded by itself and the near user in the same cluster successfully. Therefore, the achievable SINR of the far user in the kth cluster at l region should be γ k,l,ς = min(γ k,l,ς→ς , γ k,l, →ς ) and the achievable data rate for the near and far users can be expressed as followed: Define the power amplifier coefficient and power consumption for internal circuit are η and P c , respectively. The energy efficiency optimization problem for the entire users can be formulated as: where θ is denoted as a collection of the phase shift vectors, respectively and f (w, θ) is and (5b)-(5d) are to satisfy the quality of survice (QoS) requirements, which is to ensure that the data rate of each user should be greater than a certain value. γ k,l, ,min , and γ k,l,ς,min are constant values. (5e) is to guarantee the total transmitting power consumption of the beamforming P is limited by the maximum power budget P max .

III. PROPOSED OPTIMIZATION METHOD
In this section, we aim to design a scheme to effectively transform the non-convex optimization problem (P5) to convex and propose the corresponding algorithms. From the formulation of the original problem, it can be observed that there are totally two types of variables to be optimized, one is the phase shifts of the STAR-RIS elements and another is the beamforming of the users. However, it is difficult to solve this problem directly because two types of variables are coupled with each other. To address the coupling issue, we propose the alternating optimization scheme to split the original problem to phase shift and beamforming optimization problems, respectively and alternatively solve them. Besides, the original non-convex objective function and constraints can also greatly increase the difficulty to tackle this problem. Therefore, it is necessary to transform the original problem to a solvable convex optimization problem by applying mathematical techniques.

A. Phase Shift Optimization
In the phase shift optimization, we first convert the original problem to a semidefinite relaxation (SDR) one with rank-one constraints. After the transformation of the SDR of the original problem, we apply the fractional programming method to transform the non-convex objective function and QoS constraints to convex by introducing the auxiliary variables. To solve the non-convex rank-one constraints, sequential rank-one constraint relaxation (SROCR) is proposed to partially relax the rank-one constraints.
From the proposed optimization problem, it can be observed that the phase shift optimization is actually a sum rate maximum problem related to the the phase shifts of the elements because the denominator part of the objective function is only decided by the beamforming vectors. Thus, the constraint (5e) can be temporarily neglected. In the first step, given the fixed beamforming vectors w, we can define a collection of constant vectors a j,l, i k,l,i , where a j,l, i k,l,i = diag h H k,l,i Gw j,l, i and the phase shift vector v l = β l [v 1,l , . . . , v N,l ] H = e jθ 1,l , . . . , e jθ N,l H , where k, j = 1, . . . , K, l, l ∈ {t, r} and i, i ∈ { , ς}. Therefore the channel gain of the kth i user in region l decoded by j th i user in region l can be reformulated as v H l a j,l, i k,l,i . Therefore, the SINR for each user with different decoding situations could be expressed as followed: where The sum rate function is not concave because the fractional terms are inside the logarithms. To solve this problem, we introduce a collection of auxiliary slack variables {z k,l,i } to make the SINR expression out of the logarithms. Therefore, (5b), (5c), and (5d) can be transformed as followed in this network: and z k,l, ≥ γ k,l, ,min , (11a) Now the original EE problem can be converted to following form max v,β,z K k=1 l∈{t,r} i∈{ ,ς} Note that (P12) is still not a convex optimization problem. On the one hand, although the fractional forms have been removed out of the logarithms and the objective function become convex, the non-convex fractional forms still exist in the constraints. On the other hand, for each STAR-RIS elements, the amplitude of each phase shift has to be β l but this type of equality constraint is not affine, which cannot be solved by some optimization toolboxes. Therefore, more mathematical techniques are needed to further transform it to convex. Inspired by [1], by introducing a new variable matrix V l = v H l v l , which is a positive semidefinite and rank-one matrix. After this, it can be found that v l a j,l, i k,l,i 2 is equivalent to The SINR for each user can be rewritten as: and (9a) and (9b) can be reformulated as: By transforming the quadric expressions to affine, (12c) can be equivalent that each diagonal element of V l is one and it should be the rank-one matrix. Meanwhile, to tackle the non-convex QoS constraints, the fractional programming method can be utilized to transform them to convex ones. Inspired by the Corollary 3 of [22], by introducing several real auxiliary variables ξ k,l , ζ k,l and δ k,l , (13a), (13b), (13c) can be equivalent to where ξ, ζ, δ are the collection of ξ k,l , ζ k,l , δ k,l and The optimal solutions for ξ k,l , ζ k,l and δ k,l can be found by finding partial derivatives ∂g k,l, ξ k,l , ∂g k,l,ς→ς ζ k,l and ∂g k,l, →ς δ k,l as followed: By substituting (17) into (15), we can obtain the same expressions with (13). Therefore, it can be observed that the problem can be alternatively solved between {V l , β l } and ξ, ζ, δ. However, since SDR is applied to transform the quadratic expressions to the affine trace form, the rank constraints are introduced to the problem formulation: V l (n, n) = 1, l ∈ {t, r}, n = 1, · · · N, (18c) Now, all the objective function and constraints in (P18) are convex except (18e). According to [18], Gaussian randomization is usually a very effective technique to address the rank-one constraints [25], which is to firstly solve the convex SDR problem and obtain an optimal solution by dropping the rank-one constraints. Then, by generating several approximated feasible vectors according to [3], we can obtain a suboptimal solution of the original problem. The advantage of this method is that it can be easily implemented in practice and a high quality suboptimal solution can be obtained in some problems. For example, the authors in [18] pointed that the Gaussian randomization can guarantee π 4 −approximation of the optimal value of the standard semidefinite programming (SDP) problem dropping rank-one constraints if the number of the approximated samples is large enough. However, the inadequacy of this method is also obvious. If the solution of the SDR problem is a high-rank matrix, it will be difficult to generate an approximated solution to satisfy all the other constraints of the problem [24]. To overcome such a delimma, the author in [24] also proposed a novel algorithm called sequential rank-one constraint relaxation (SROCR). It is designed to find a locally optimal solution by partially relaxing the rank-one constraints. Firstly, we combine V t and V r to one variable matrix V ∈ C 2N ×2 N , which is where Due to the fact that V is a rank-one matrix, (18e) can be equivalent to the following form which one e max is the largest eigenvalue of the matrix V. Further, we can write that From the characteristic of the rank-one matrix, the optimal solution ϕ * can be obtained by finding the unit eigenvector corresponding to the maximum eigenvalue of V. Inspired by this idea, we can find the optimal V * by alternatively optimizing {{V, β}, ϕ}. Now, we can relax the rank-one constraint (18e) by introducing ω ∈ [0, 1] in the following: where ϕ * can be obtained in last iteration. (P18) has been transformed to a convex problem: In summary, by alternatively optimizing {V, β} and ξ, ζ, δ, ϕ, we can obtain a rank-one suboptimal solution V * and v * l can be obtained by the eigenvalue decomposition.

B. Beamforming Optimization
In the beamforming optimization, we also first convert the objective function and QoS constraints to convex by applying fractional programming and SDR and introducing the relevant variables. Different from the phase shift optimization, it can be proven that the optimal solution of the SDR problem is rank one. Therefore, no more steps are needed to be implemented to tackle rank-one constraints.
Since the phase shifts of the STAR-RIS elements has been solved which can be considered fixed and beamforming vectors of the users have become the targeted optimization variables, the total channel gain for each i user of the kth cluster in the region l can be expressed by (1). Similar to the phase shift optimization, we still use SDR to transform the quadratic beamforming expressions to affine. let H k,l,i = h H k,l,i h k,l,i and W k,l,i = w k,l,i w H k,l,i , H k,l,i and W k,l,i is are rank-one matrice. Therefore, the SINR for each user can be rewritten as γ k,l, →ς = Tr(H k,l, W k,l,ς ) σ 2 +Tr(H k,l, W k,l, )+Ω k,l, (W)+Ω k,l, (W) , where where W is the collection of the beamforming matrices {W k,l,i } i, i ∈ { , ς} and l ∈ {t, r}, l = l. Different from the phase shift optimization, the objective function is fractional form which the variables exist both in the nominator and denominator. Firstly, the nominator part is still a sum rate expression. Therefore, the same method in the phase shift optimization problem can also be used to remove the fractional terms out of the logarithms by introducing several slack auxiliary variables {α k,l, , α k,l,ς } such that and Therefore, the beamforming optimization problem can be reformulated as: where P = K k=1 l∈{t,r} i∈{ ,ς} From the problem formulation, it can be observed that (28a) is a concave/convex objective function which can be solved by the Dinkelbach's method to obtain the optimal solution. However, the complexity of the algorithm can be greatly increased since the massive iterations will be inevitably introduced by the Dinkelbach's method. Therefore, in this work, the proposed optimization method for the beamforming aims to obtain the optimal solution with a relatively low complexity compared with the Dinkelbach's method. The first step is to apply the fractional programming method to transform the objective function to a concave form by introducing y such that max W,α 2y K k=1 l∈{t,r} i∈{ ,ς} where y is updated by each iteration in the algorithm. Next, to solve the QoS constraint (26a), (26b) and (26c), we can still apply fractional programming method to be transformed to the following constraints: max τ 2τ k,l Tr(H k,l,ς W k,l,ς )−τ 2 k,l σ 2 +Tr(H k,l,ς W k,l, ) max λ 2λ k,l Tr(H k,l, W k,l,ς )−λ 2 k,l σ 2 +Tr(H k,l, W k,l, ) where I k,l,i (W) = Ω k,l,i (W) +Ω k,l,i (W).
Therefore, the beamforming optimization problem can be further written as:  It can be observed that the (28e) is still non-convex rank-one constriants. Lemma 1: If solving (P33) dropping (28e), it can be guaranteed that any optimal beamforming solution W k,l,i is rank-one.
Proof: Please see the Appendix A. Therefore, the optimal EE can be obtained by alternatively  optimizing W k,l,i and y, μ, τ , λ.

C. Algorithm Design
Before solving the phase shift and beamforming optimization problems alternatively, the feasible initial points for phase shift and beamforming vectors w (0) and Θ (0) are supposed to be searched to guarantee the convergence of the algorithms we design to realize the EE maximazation. Firstly, we are able to generate a series of ramdom phase shift angles whose range is from 0 to 2π and β l which satisfies β t + β r = 1 of the elements. For finding a feasible beamforming initial solution, we can apply the fractional programming for the complex case [22]. Back to the original problem formulation (P5), consider there is a feasible beamforming solution w, (5b), (5c), (5d) can be equivalent to the following form: f k,l, →ς λ = 2 λ † k,l h H k,l, w k,l,ς Algorithm 2: Alternating Optimization Algorithm.
1: Initialize: Given the initial w (0) and . Define EE (0) = 0 2: repeat 3: Phase Shift Optimization with SROCR: Given the fixed beamforming vectors w (l) and EE (0) 1 4: repeat 5: update ξ, ζ, δ and solve the Optimization Problem (P23). 6: if solvable: Obtain the Optimal V (n+1) and let else Let V (n+1) = V (n) and To discuss the complexity of Algorithm 2, we can first analyze two sub-algorithms, respectively and then combine them together. To simplify the analysis, we assume that the phase shift optimization problem is solvable in each iteration. Therefore, it can take a complexity of O(KN 3 ) when solving (P23) with the same method applied in Algorithm 1. Assuming two thresholds in this sub-algorithm are assumed to be 1 and 2 for EE and ω, respectively, the total complexity of phase shift sub-algorithm is O(log 1 1 log 1 2 KN 3 ) in the worst case. Similarly, it can take a complexity of O(log 1 3 KM 3 ) for beamforming sub-algorithm by assuming the convergence of EE in the beamforming optimization sub-algorithm is 3 in the worst case. After adding these two complexity together, we can get the total complexity of Algorithm 2 is O(log 1 4 log 1 1 log 1 2 KN 3 + log 1 4 log 1 3 KM 3 ) by assuming the convergence threshold of the outer iterations is 4 .

IV. SIMULATION RESULTS
In this section, we present the simulation results to demonstrate the superior performance of the proposed algorithms. Firstly, we set the coordinates of the transmitter in the BS and centre of the STAR-RIS are (0, 0, 20) and (0, 50, 10), respectively. Besides, all the users are deployed in the T region and R region. The radiuses of these two half-circle regions are both 20 m and the radius of each cluster is 5 m. Without loss of generality, we consider that the total bandwidth of the network is 100 MHz [13] and the noise spectrum density is −170 dBm/Hz [7]. Besides, the Racian fading is applied for all the channels involved in this system. For example, the Racian fading channel channel F can be expressed as where κ denotes the Racian factor and it is set to 2. The F LoS and F nLoS denote the line of sight (LoS) and non line of sight (nLoS) components, respectively. The path loss exponent for all the links is set to 2.5 [1]. The minimum SINR for each user is 10 dBm. The internal circuit power P c and power amplifier coefficient η are set to 20 dBm and 0.8, respectively [10].

A. The Convergence of the Algorithms
In Fig. 2, we demonstrate the convergence the proposed Algorithm 2 by the numerical analysis. From the figure, we can find that the phase shift sub-algorithm can converge within 30 iterations for all the five cases. Except for the convergence of the algorithm, it can also be observed that the Algorithm 2 sometimes does not converge to the maximum value during the iterations. For example, in the case of M = 8, N = 50, the optimal value reaches 36.1547 bps/Hz at the eighth iteration. However, it drops to 35.1003 bps/Hz at the next iteration. It is because that the rank of the optimal V at the eighth iteration is not one and it does not satisfy one of the convergence conditions of the this sub-algorithm.  Figs. 3 and 4 demonstrate the convergence of the Algorithm 3 both in energy efficiency and power consumption. From the figure, it can be found that the beamforming sub-algorithm is able to converge within 12 iterations. Besides, interestingly, during each iteration, the ranks of all the beamforming variables can be verified to be always one after the optimization, which demonstrates the correctness of the Lemma 1 from the perspective of experiments. In Fig. 4, the energy efficiency is always increasing as the number of iterations grows while the transmitting power consumption initially underpins a sharp decrease. However, in some cases such as M = 12, N = 40, after reaching the minimum, which is 10.327 dBm at the seventh iteration, it slowly rises to 10.3827 dBm and eventually converges at the 10.4253 dBm. Therefore, it cannot be guaranteed that the proposed beamforming algorithm can achieve the energy efficiency maximization and power consumption minimization simultaneously in this network as when the energy efficiency reaches to the highest point and converges, the value of the total  transmitting power might be higher than the minimum value during the iterations. Fig. 5 demonstrates the convergence of the proposed alternating algorithm. It can be found that there is a sharp increase when the outer iteration is from zero to one. After several iterations and slow increasing of the EE values, the energy efficiency of these five cases finally converges within 6 iterations. Fig. 6 shows the relationship between the energy efficiency and the number of elements. Firstly, four different schemes are proposed as benchmarks, namely SDR by Gaussian randomization [18], RIS NOMA, STAR-RIS random phase shift, STAR-RIS OMA. For RIS systems, one transmitting-only RIS and reflecting-only RIS are deployed adjacent to the original position of STAR-RIS to realize the full-space communication. Note that since two RISs are independent, β t and β r are not coupled with each other. Therefore, β t and β r can be randomly generated with guaranteeing β t + β r = 1. Meanwhile, to guarantee the number of elements for RIS systems is same as that of STAR-RIS, each RIS is equipped N 2 elements [7]. In the figure, it can be observed the STAR-RIS NOMA scheme by either the SROCR or Gaussian randomization can outperform other schemes including RIS NOMA, random phase shift and STAR-RIS OMA in energy efficiency with the same number of elements. Especially, the STAR-RIS NOMA can outperforms the RIS NOMA system. It is because the STAR-RIS NOMA system can have a higher degree-of-freedom (DoF). This DoF reflects on the amplitude coefficients, which means that STAR-RIS can optimize β l and θ l jointly to adjust the transmitted and reflected signals and mitigate inter-cluster and intra-cluster interferences. This property allows a better performance of the STAR-RIS NOMA than RIS NOMA system in energy efficiency [7]. Besides, it can be also observed that the energy efficiency can increase as the number of elements increases. It is because a higher number of elements is able to produce higher transmitting and reflecting gains [7]. From the perspective of the the mathematical tools, the performance of the proposed scheme can always be better than that of the SDR by Gaussian randomization. Besides, From the perspective of computational complexity, assuming the convergence threshold of the proposed algorithm is equal to that of the Gaussian randomization, different from the SROCR scheme, Gaussian randomization requires additional compleixty of O(LN 3 ) to generate L points since Gaussian distribution involves eigenvalue decomposition (EVD) of the SDR optimal solution of V. The complexity of EVD is O(b 3 ) for a matrix B ∈ C b×b [13]. Therefore, it is not worthwhile spending such an extra complexity solving the phase shift optimization problem by using Gaussian randomization in this network. we can also find that the STAR-RIS NOMA system performs better than the RIS NOMA case. Meanwhile, similar to the relationship between STAR-RIS NOMA and RIS NOMA, it can be also found that STAR-RIS OMA outperforms RIS OMA. It is because with the fixed summation of β t and β r , the STAR-RIS can have a better performance in EE by flexibly tuning β and phase shifts of the elements jointly while only phase shifts can be tuned in the RIS network. Fig. 8 shows the relationship between the energy efficiency and the maximum power budget. From the figure, we can find that the energy efficiency increases as the maximum power budget increases. However, as P max further increases, there is no obvious increase in energy efficiency for all the four cases. Make STAR-RIS NOMA as an example, the energy efficiency rises from 218.6660 bps/Hz/Joule to 264.0331 bps/Hz/Joule when P max grows from 0 dBm to 15 dBm. However, when P max is 20 dBm, the energy efficiency for STAR-RIS NOMA is 265.1941 bps/Hz/Joule, which is almost the same as the value of P max =15 dBm. It is because when the P max is relatively small, the domain of the beamforming is limited. Therefore, the transmitting power of the beamforming is likely to be equal to P max to satisfy the QoS constraints and realize the energy efficiency maximization. As the P max increases, the total channel gains of the users can increase and energy efficiency is likely to improve because the domain of the beamforming expands. When P max is large enough, the transmitting power does not have to be equal to P max to meet QoS constraints and maximize the energy efficiency. Therefore, the energy efficiency can be kept the same in total no matter how the P max increases.

V. CONCLUSION
In this paper, we investigated a novel scheme in a multi-user MISO STAR-RIS assisted downlink NOMA network, which is to achieve the energy efficiency maximization by alternatively optimizing the phase shifts and beamforming vectors. Firstly, to efficiently solve the non-convex original problem, we first splited it into the phase shift and beamforming optimization problems. In these two problems, both SDR and fractional programming were introduced to convert the quadratic non-convex QoS constraints to convex. the difference was in the phase shift optimization, the rank constraints could not be neglected and they were solved by SROCR. However, in the beamforming optimization, the rank of the optimal SDR beamforming solutions could be proven to be equal to one by both theoretical analysis and experimental results. The simulation results demonstrated the STAR-RIS NOMA network can outperform the RIS NOMA, STAR-RIS OMA, and RIS OMA systems. Besides, the simulation also demonstrated that the proposed SROCR scheme can have a better performance than the Gaussian randomization method by comprehensively considering the difference between the optimal energy efficiency they can reach and the complexity of the algorithm.