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Simultaneously Transmitting And Reflecting (STAR) RIS Aided Wireless Communications

The novel concept of simultaneously transmitting and reflecting (STAR) reconfigurable intelligent surfaces (RISs) is investigated, where the incident wireless signal is divided into transmitted and reflected signals passing into both sides of the space surrounding the surface, thus facilitating a full-space manipulation of signal propagation. Based on the introduced basic signal model of `STAR', three practical operating protocols for STAR-RISs are proposed, namely energy splitting (ES), mode switching (MS), and time switching (TS). Moreover, a STAR-RIS aided downlink communication system is considered for both unicast and multicast transmission, where a multi-antenna base station (BS) sends information to two users, i.e., one on each side of the STAR-RIS. A power consumption minimization problem for the joint optimization of the active beamforming at the BS and the passive transmission and reflection beamforming at the STAR-RIS is formulated for each of the proposed operating protocols, subject to communication rate constraints of the users. For ES, the resulting highly-coupled non-convex optimization problem is solved by an iterative algorithm, which exploits the penalty method and successive convex approximation. Then, the proposed penalty-based iterative algorithm is extended to solve the mixed-integer non-convex optimization problem for MS. For TS, the optimization problem is decomposed into two subproblems, which can be consecutively solved using state-of-the-art algorithms and convex optimization techniques. Finally, our numerical results reveal that: 1) the TS and ES operating protocols are generally preferable for unicast and multicast transmission, respectively; and 2) the required power consumption for both scenarios is significantly reduced by employing the proposed STAR-RIS instead of conventional reflecting/transmiting-only RISs.


B. Motivations and Contributions
Most of the existing research contributions consider the case where the RISs are only able to reflect the incident wireless signal (referred to as conventional reflecting-only RISs). In this case, transmitter and receiver have to be located on the same side of the RIS [10][11][12][13][14][15][16][17][18][19][20][21][22][23], thus leading to a half-space SRE. However, this geographical restriction may not always be met in practice, and gravely restricts the flexibility and effectiveness of RISs, as generally users may be located on both sided of a RIS. To overcome this limitation, the novel concept of simultaneously transmitting and reflecting RISs (STAR-RISs) was proposed in [24,25]. In particular, as shown in Fig. 1, the wireless signal incident on an element of a STAR-RIS from either side of the surface is divided into two parts [26]. One part (reflected signal) is reflected to the same space as the incident signal, i.e., the reflection space, and the other part (transmitted signal) is transmitted to the opposite space as the incident signal, i.e., the transmission space. As shown in [24], by manipulating both the electric and magnetic currents of a STAR-RIS element, the transmitted and reflected signals can be reconfigured via two generally independent coefficients, namely the transmission and the reflection coefficients. Therefore, a highly flexible full-space SRE can be realized. Two prototypes, which resemble STAR-RISs, have been developed using metasurfaces [26,27]. For the prototype proposed in [26], each element was composed of a parallel resonant LC tank and small metallic loops to provide the required electric and magnetic surface reactance. Despite the above advantages, the investigation of how STAR-RISs can be integrated into wireless communication systems is still in its infancy. Based on a similar idea as STAR-RISs, an intelligent omni-surface (IOS) was proposed by the authors of [28] to achieve full-space coverage. However, unlike for STAR-RISs, for IOSs, the phase shifts for transmission (PIN) diodes installed in the IOS elements. In another early work [24], the authors focused on the investigation of hardware models and channel models for STAR-RISs. However, to the best of the authors' knowledge, efficient operating protocols for STAR-RISs and corresponding joint transmission and reflection beamforming optimization techniques for STAR-RIS aided wireless networks have not been studied, yet, which provides the main motivation for this work.
To exploit the full potential of STAR-RISs, in this paper, we propose practical protocols for the operation of STAR-RISs and investigate the joint transmission and reflection beamforming design for both unicast and multicast communication. The main contributions of this paper can be summarized as follows: • We propose three practical operating protocols for STAR-RISs, namely energy splitting • For ES, we first transform the resulting highly-coupled non-convex optimization problem into a tractable form with decoupled optimization variables. To solve the transformed nonconvex problem, we develop an efficient iterative algorithm by exploiting the penalty method and successive convex approximation (SCA). The algorithm is guaranteed to converge to a stationary point of the original optimization problem. Furthermore, we extend the proposed penalty-based iterative algorithm to solve the mixed-integer non-convex optimization problem obtained for MS. For TS, we show that the optimization problem can be decomposed into two subproblems, which can be efficiently solved using state-of-the-art algorithms and convex optimization techniques.
• Our numerical results unveil that 1) TS is generally preferable for unicast communication, while ES is superior for multicast communication; 2) the performance gain of STAR-RISs over conventional RISs increases with the number of RIS elements; 3) for ES, the performance gain of element-based amplitude control over group-based amplitude control is significant for unicast communication, while it is negligible for multicast communication.

C. Organization and Notation
The rest of this paper is organized as follows: Section II introduces the basic signal model for STAR-RISs, based on which three practical protocols for operating STAR-RISs are proposed.
Section III presents the system model and the joint beamforming optimization problem formulations for STAR-RIS aided communication systems for both unicast and multicast transmission.
Efficient algorithms are developed for solving the problems formulated for each proposed operating protocol in Section IV. Section V provides numerical results to verify the effectiveness of the proposed STAR-RIS designs compared to baseline schemes. Finally, Section VI concludes the paper.

A. Basic Signal Model of STAR-RISs
As shown in Fig. 1, the wireless signal incident on a given element of the STAR-RIS is divided into a transmitted and a reflected signal. To characterize this STAR feature, let s m denote the signal incident on the mth element of the STAR-RIS, where m ∈ M {1, 2, . . . , M} and M denotes the total number of elements. The signals transmitted and reflected by the mth element can be modelled as t m = β t m e jθ t m s m and r m = β r m e jθ r m s m [24], respectively, where β t m ∈ [0, 1] , θ t m ∈ [0, 2π) and β r m ∈ [0, 1] , θ r m ∈ [0, 2π) denote the amplitude and phase shift response 1 of the mth element's transmission and reflection coefficients 2 . Note that, for each element, the phase shifts for transmission and reflection (i.e., θ t m and θ r m ) can be generally chosen independent from each other [24]. However, the amplitude adjustments for transmission and reflection (i.e., β t m and β r m ) are coupled by the law of energy conservation. This means that, for each element, the sum of the energies of the transmitted and reflected signals has to be equal to the energy of the incident signal, i.e., |t m | 2 +|r m | 2 = |s m | 2 , which leads to the following condition for the transmission and reflection amplitude coefficients of each element 3 [24]:

B. Three Practical Protocols for Operating STAR-RISs
As can be observed from (1), by properly adjusting the amplitude coefficients for transmission and reflection, a given element of a STAR-RIS can be operated in the full transmission mode (i.e., β t m = 1, β r m = 0, referred to as T mode), the full reflection mode (i.e., β t m = 0, β r m = 1, referred to as R mode), and the general simultaneous transmission and reflection mode (β t m , β r m ∈ [0, 1], referred to as T&R mode). Inspired by these observations, in this subsection, we propose three practical protocols for operating STAR-RISs in wireless communication systems, namely energy splitting (ES), mode switching (MS), and time switching (TS), as illustrated in Fig. 2.

1) Energy Splitting:
For ES, as shown in Fig. 2(a), all elements of the STAR-RIS are assumed to operate in the T&R mode, where the energy of the signal incident on each element is generally split into the energies of the transmitted and reflected signals with an energy splitting ratio of β t m : β r m . In this case, the transmission-and reflection-coefficient matrices of the STAR-RIS are given by  is enabled. However, the large number of design variables also cause a relatively high overhead for configuration information exchange between the BS and the STAR-RIS.
2) Mode Switching: For MS, as shown in Fig. 2 Accordingly, the STAR-RIS transmission-and reflection-coefficient matrices are given by and Θ MS r = diag β r 1 e jθ r 1 , β r 2 e jθ r 2 ,. . . , β r M e jθ r M , respectively, where β t m , β r m ∈ {0, 1}, β t m + β r m = 1, and θ t m , θ r m ∈ [0, 2π) , ∀m ∈ M. We note that MS STAR-RISs can be regarded as a special case of ES STAR-RISs, where the amplitude coefficients for transmission and reflection are restricted to binary values. Therefore, MS generally cannot achieve the same full-dimension transmission and reflection beamforming gain as ES, since only a subset of the elements are selected for transmission and reflection, respectively. Nevertheless, MS is still appealing in practice, since such an "on-off" type operating protocol is much easier to implement compared to the ES protocol. coefficients for TS is not coupled, and thus, easier to handle. However, the periodical switching of the elements introduces stringent requirements for time synchronization, which entails a higher hardware implementation complexity.
In Table I, we summarize the optimization variables and constraints for the ES, MS, and TS operating protocols.

III. SYSTEM MODEL AND PROBLEM FORMULATION
Based on the proposed operating protocols, in this section, we present the system model of a STAR-RIS aided downlink communication system and formulate joint active and passive beamforming optimization problems for both unicast and multicast transmission.

A. System Model
As shown in Fig. 3, we consider a narrow-band STAR-RIS aided downlink communication system operating over frequency-flat channels, where an N-antenna BS communicates with multiple single-antenna users with the aid of a STAR-RIS comprising M STAR elements. As previously discussed, the STAR-RIS can create full-space coverage by simultaneously transmitting and reflecting the incident signal. We refer to users that are located in the transmission space as T users, while users that are located in the reflection space are referred to as R users. As depicted in Fig. 3, in this paper, we assume that the direct communication links between the BS and the users are blocked by obstacles 4 , as this is one of the most challenging scenarios for conventional communication systems. The STAR-RIS is deployed to provide communication service for the users in the resulting signal dead zone by establishing additional transmission and reflection links. For simplicity of presentation and revealing fundamental design insights, in the remainder of the paper, we consider a two-user setup 5 , i.e., the STAR-RIS aided communication system serves one T user and one R user. Let G ∈ C M ×N , v H t ∈ C 1×M , and v H r ∈ C 1×M represent the narrow-band quasi-static fading channels from the BS to the STAR-RIS, the STAR-RIS to T user, and the STAR-RIS to R user, respectively. In order to be able to unveil the maximum performance gain enabled by STAR-RISs, the perfect CSI of all channels is assumed to be available at the BS 6 .
In this paper, we study the joint design of the active beamforming at the BS and the passive transmission and reflection beamforming at the STAR-RIS for both unicast and multicast communication. For unicast, the BS sends independent information to the T and R users. For multicast, the BS sends the same information to both users. For both scenarios, the three proposed STAR-RIS operating protocols are considered.

B. Unicast Communication and Problem Formulation
In unicast transmission, let w k and x k denote the active beamforming vector and the informationbearing symbol for user k ∈ {t, r} at the BS, respectively.

1) ES and MS:
When the ES or MS protocols are employed at the STAR-RIS, the received signal at user k ∈ {t, r} is given by where E |x k | 2 = 1 and n k ∼ CN (0, σ 2 k ) denotes the additive white Gaussian noise (AWGN) at user k. Therefore, the achievable communication rate of user k ∈ {t, r} for ES and MS is given by where k = r, if k = t; and k = t, otherwise.
2) TS: For TS, the BS consecutively sends information to the two users in the T and R periods.
Thus, the corresponding achievable communication rate of user k ∈ {t, r} can be expressed as where λ k in (4) is due to the fact that the BS sends information to user k employing w k in λ k fraction of the total communication time. The transmit power of user k is increased by 1 λ k to ensure a fair comparison with the ES and MS protocols.
For unicast transmission, we aim to minimize the total power consumption of the BS by jointly optimizing the active beamforming at the BS and the passive transmission and reflection beamforming at the STAR-RIS for a given operating protocol, while satisfying the QoS requirement of both users. Then, the optimization problem can be formulated as follows: where X ∈ {ES, MS, TS} indicates the employed STAR-RIS operating protocol, F X characterizes the corresponding feasible set for the transmission-and reflection-coefficient matrices, and R k denotes the minimum rate requirement of user k. The time allocation variables, λ k , and constraint (5d) are only valid when the TS protocol is employed, i.e., X = TS. Let For the ES and MS protocols, assuming that N ≥ 2, a sufficient condition for the feasibility of problem (5) for any finite R k is that Rank (H) = 2, where a zero-forcing solution can be easily constructed for the active beamforming [10]. Given the independent STAR-RIS-user channels, v H k , and the different transmission-and reflection-coefficient matrices, Θ ES/MS k , such a rank condition can be satisfied with a high probability. For the TS operating protocol, problem (5) is always feasible for any finite R k since there is no inter-user interference during the T and R periods, respectively. Remark 1. Compared to the power consumption minimization problem for conventional reflectingonly RISs [10], the main challenges for solving (5) can be summarized as follows. First, STAR-RISs require the optimization of two types of passive beamforming (i.e., transmission and reflection beamforming), while for conventional reflecting-only RISs, only the reflection beamforming has to be designed. Mathematically, the optimization problem for conventional reflecting-only RISs is a special case of optimization problem (5) for STAR-RISs, where the transmission function is turned off and only the R user is served. Second, for the proposed ES and MS protocols, transmission and reflection beamforming is coupled together, which further complicates resource allocation compared to conventional reflecting-only RISs. Therefore, the considered problem (5) for STAR-RISs is much more challenging to solve than that for conventional reflecting-only RISs.

C. Multicast Communication and Problem Formulation 1) ES and MS:
For ES and MS, in multicast transmission, the BS employs one active beamforming vector, denoted by w c , to convey the same symbol, s c , to the T and R users.
As a result, the corresponding received signal at user k ∈ {t, r} is given by Accordingly, the achievable communication rate of user k ∈ {t, r} can be expressed as 2) TS: Since for TS the users are served in different orthogonal time slots, the BS can still employ different active beamforming vectors, which are denoted by w c,t and w c,r , to send the same symbol to the T and R users. Therefore, for multicast communication with TS, the achievable communication rate at user k ∈ {t, r} is given by We note that, for TS, the multicast communication rate in (8) is identical to the unicast communication rate in (4). This is because, regardless of whether unicast or multicast communication is considered, the proposed TS protocol can serve only one user in each time instant.
Since the information has to be delivered to both users, the system performance of multicast communication is limited by the user achieving the smaller communication rate. As a result, the effective system multicasting rate for the considered operating protocols is given by where X ∈ {ES, MS, TS}. Then, for multicast transmission, the joint beamforming optimization problem for ES, MS, and TS can be formulated as where (10b) denotes the minimum system multicast rate constraint with required communication rate R c . We note that λ k and constraint (10d) are only valid for TS. As there is no inter-user interference in multicast, for each operating protocol, problem (10) is always feasible for any finite R c .

D. Discussion
Note that the formulated joint beamforming optimization problems for multicast transmission can be regarded as special cases of those for unicast transmission. More specifically, for ES and MS, problem (10) for multicast transmission can be obtained from problem (5) for unicast transmission by setting w t = w r = w c , R t = R r = R c and removing the inter-user interference term in the denominator of the unicast communication rate expression in (3). For TS, problem (10) for multicast transmission can be directly obtained from problem (5) for unicast transmission by setting w t = w c,t , w r = w c,r , and R t = R r = R c . Motivated by this, in the following, we mainly focus on joint beamforming optimization problem (5) for unicast transmission since problem (10) for multicast transmission can be solved in a similar manner. However, problem (5) is a non-convex optimization problem since the left-hand-side (LHS) of (5b) is not concave, as Θ X k and {w k } are highly-coupled. Moreover, the feasible set, F X , is in general non-convex.
Therefore, it is difficult to obtain a globally optimal solution for such a challenging problem. In the following, we develop efficient algorithms to find a high-quality suboptimal solution for the three considered operating protocols.

IV. SOLUTION OF JOINT BEAMFORMING OPTIMIZATION PROBLEMS
In this section, we first propose a penalty-based iterative algorithm for solving the joint beamforming optimization problem for ES. This algorithm is then further extended to solve the problem for MS. Then, we show that the problem for TS can be decomposed into two subproblems, namely a transmission/reflection coefficient design problem and a resource allocation problem, which can be solved in a relatively straightforward manner.

A. Proposed Solution for ES and MS
To start with, we first transform problem (5) for ES and MS into a more tractable form. To facilitate the design, we define the transmission-and reflection-coefficient vectors for ES and MS as Then, optimization problem (5) for ES and MS can be reformulated as follows: min where Y ∈ {ES, MS} specifies the operating protocol and γ k = 2 R k − 1, ∀k ∈ {t, r}. We note that constraint (11h) is only present for MS, i.e., Y = MS. Problem (11) is still non-convex due to the highly-coupled non-convex constraint (11b) and the non-convex rank-one constraint (11d) and (11e). Moreover, for MS, the optimization problem becomes a mixed-integer non-convex problem due to binary constraint (11h). In the following, we first focus on optimization problem (11) for ES, when constraint (11h) is absent.

1) Penalty-based Algorithm for ES:
Before handling the highly-coupled non-convex constraint (11b), we first have the following lemma.
Lemma 1. For any two Hermitian matrices A ∈ H M and B ∈ H M having the same size, we have the following two equalities: Proof. We first prove equation (12a), whose right-hand-side (RHS) can be rewritten as where (a) is due to the fact that A and B are Hermitian matrices. Equation (12b) can be proved in a similar manner, and we omit the details for brevity.
Based on Lemma 1, the first and second non-convex terms in (11b) can be transformed into the following difference of convex (DC) functions: (13b) Furthermore, the non-convex rank-one constraint (11d) can be equivalently written as the following equality constraint [35]: where Next, we employ the penalty method [36] to solve problem (11). By exploiting (13a), (13b), and (14), we obtain the following optimization problem: where equality constraint (14) is relaxed to a penalty term added to the objective function, and η > 0 is the penalty factor which penalizes the objective function if Q ES k is not rank-one.
It can be verified that, when η → +∞, the solution Q ES k of problem (15) always satisfies equality constraint (14), i.e., problems (11) and (15) are equivalent [37]. However, if the initial value of the penalty factor η is chosen too large, the objective function of (15) is dominated by the penalty term, and the impact of the desired power minimization on the solution becomes negligible. To avoid this, we first initialize η with a small value to find a good starting point, and then, in the course of several iterations, we gradually increase η to a sufficiently large value to eventually obtain feasible rank-one matrices. Note that, for any given penalty factor η > 0, problem (15) is still non-convex due to the non-convexity of the objective function and the non-convex constraints (15b) and (11e). In the following, we employ SCA [38] to obtain a suboptimal solution of (11) in an iterative manner.
Note that the penalty term, Υ ES kk , and Π ES kk are in the form of DC functions. For a given point Q (n) k in the nth iteration of the SCA method, using first-order Taylor expansion, a convex upper bound for the penalty term can be obtained as follows: where u Q ES(n) k denotes the eigenvector corresponding to the largest eigenvalue of Q Similarly, for given points Q in the nth iteration of the SCA method, convex upper bounds of Υ ES kk and Π ES kk are respectively given by As a result, for given points Q , by replacing the non-convex terms with their convex upper bounds, problem (15) is transformed into the following optimization problem:

Proof. See Appendix A.
The relaxed problem (18), which is a standard convex semidefinite program (SDP), can be efficiently solved via standard convex problem solvers such as CVX [39]. Then, we propose a penalty-based iterative algorithm for solving problem (11), which comprises two loops. In the outer loop, the penalty factor is gradually increased from one iteration to the next as follows: where ω > 1. The algorithm terminates when the penalty term satisfies the following criterion: where ε 1 denotes a predefined maximum violation of equality constraint (14). Therefore, (14) will be eventually satisfied with accuracy ε 1 as η increases. In the inner loop, Q ES k , W k are jointly optimized by iteratively solving the relaxed version of problem (18) for the given penalty factor.
The objective function value of the relaxed version of (18) is non-increasing in each iteration of the inner loop, and the optimal objective function value of the relaxed version of (18) is bounded from below. Therefore, as η approaches infinity, the developed penalty-based iterative algorithm is guaranteed to converge to a stationary point of the original problem (11) [38]. The details of the developed algorithm are summarized in Algorithm 1. with the obtained optimal solutions, and n = n + 1.

7:
until the fractional decrease of the objective function value is below a predefined threshold ǫ 1 > 0 or the maximum number of inner iterations n max is reached. if the interior point method is employed [40], where K = 2 is the number of users. Then, the overall computational complexity of Algorithm 1 is O (I out I inn (KN 3.5 + 2M 3.5 )), where I inn and I out denote the number of inner and outer iterations required for convergence, respectively.
As can be observed, the computational complexity of Algorithm 1 is polynomial in M, which facilitates its practical implementation even if the number of STAR-RIS elements is large.
2) Extended Penalty-based Algorithm for MS: Compared to ES, for MS, the formulated optimization problem in (11) involves the additional non-convex binary constraints (11h). Therefore, we only need to focus on how to tackle this new obstacle since the other non-convex terms can be handled in a similar manner as previously described. We first transform the binary constraint (11h) equivalently into the following equality constraint: Recall that because of constraint (11g) the amplitude coefficients for transmission and reflection are between 0 and 1. Thus, we always have β k m − β k m 2 ≥ 0, where equality holds if and only if β k m is 0 or 1, i.e., a binary variable. As a result, the equality constraint in (20) is satisfied only for binary variables. In the following, we extend the proposed penalty-based algorithm for ES to solve the optimization problem for MS. In particular, non-convex constraint (11b), which is highly-coupled in Q MS k , W k , and the non-convex rank-one constraint (11d) are handled in a similar manner as previously described. Moreover, by further adding equality constraint (20) as another penalty term into the objective function, we obtain the following optimization problem for MS: min where Υ MS kk , Π MS kk , ∀k ∈ {t, r} are obtained by replacing Q ES k in (13a) and (13b) by Q MS k , and χ > 0 is a new penalty factor which penalizes the objective function if β k m belongs to (0, 1). Similarly, it can be verified that, when η, χ → +∞, the solution obtained from problem (21) satisfies equality constraints (14) and (20). As problem (21) has a similar structure as problem (15), we can still employ SCA to solve this new non-convex optimization problem. For given points β (n) k in the nth iteration of the SCA method, by using first-order Taylor expansion, an upper bound for the new penalty term can be obtained as follows: By replacing the non-convex terms with their upper bounds for given points Q problem (21) can be reformulated as follows: min where the remaining non-convexity is caused by rank-one constraint (11e). We can employ again SDR and remove the non-convex rank-one constraint. The tightness of the relaxation can be proved similar to Theorem 1. As a result, the relaxed version of problem (23) is convex and can be efficiently solved via standard convex problem solvers such as CVX [39]. Similar to Algorithm 1 for ES, for MS, a two-loop penalty-based iterative algorithm for solving problem (11) can be developed. This algorithm is summarized in Algorithm 2. Since for MS there are two penalty terms, the termination criterion in the outer loop is given by Algorithm 2 Proposed penalty-based iterative algorithm for solving problem (11)  Set iteration index n = 0 for inner loop. 4: repeat: inner loop 5: For given Q , solve the relaxed version of problem (23). 6: with the obtained optimal solutions, and n = n+1. 7: until the fractional decrease of the objective function value is below a predefined threshold ǫ 2 > 0 or the maximum number of inner iterations n max is reached. 8: Update η = ωη, χ = ̟χ. 10: until the constraint violation is below a predefined threshold ε 2 > 0.
Remark 2. Although, in this paper, we focus on systems with one T user and one R user, the proposed Algorithms 1 and 2 can be extended to ES/MS STAR-RIS aided communication systems with multiple T and R users. Suppose that there are J > 2 users. The corresponding optimization problem can be obtained from optimization problem (11) by replacing the inter-user interference in the first term of (11b) (i.e., γ k Tr Q Y k H k W k H H k ) with the sum of the interuser interference caused by all J − 1 interfering users and minimizing the power consumption caused by the J active beamforming vectors. In particular, the J − 1 new non-convex inter-user interference terms can be tackled in a similar manner as previously discussed (i.e., (13a) and (17a)). Therefore, the optimization problem for ES and MS for multiple T and R users can still be solved with the proposed Algorithms 1 and 2.
(25d) As, in each time instant, only one user receives data from the BS via the STAR-RIS, for any given transmission/reflection-coefficient vector, the optimal active beamforming vector at the BS is the maximum-ratio transmission (MRT) beamformer [10], i.e., w * k = √ p k , where p k denotes the allocated transmit power for user k. By substituting {w * k } into (25), we obtain the following problem: min (26c) Since the two users are alternatingly served, the optimal transmission/reflection-coefficient vectors for problem (26) maximize the effective channel gain of each user, which yields the following subproblem: The above transmission/reflection-coefficient design problem can be efficiently solved using either the suboptimal low-complexity SDR based algorithm proposed in [10] or the globally optimal branch-and-bound based algorithm proposed in [11]. The details are omit here for brevity.
With the obtained desired solutions for the transmission/reflection-coefficient vectors, q TS * k , problem (26) is reduced to the following subproblem: min The above resource allocation problem is a convex optimization problem, since the LHS of (28b) is a jointly concave function with respect to p k and λ k , and (25d) is an affine constraint.
Therefore, problem (28) can be efficiently solved using standard convex problem solvers such as CVX [39]. Based on the above two subproblems, the procedure for solving problem (25) for TS is summarized in Algorithm 3. The complexity of Algorithm 3 is analyzed as follows.
Suppose that the suboptimal SDR based algorithm [10] is employed for solving problem (27) Algorithm 3 Procedure for solving problem (25) for TS 1: Obtain the transmission/reflection-coefficient vectors, denoted by q TS * k , ∀k ∈ {t, r} , by solving the corresponding effective channel gain maximization problem (27). 2: For given q TS * k , ∀k ∈ {t, r} , solve the convex resource allocation problem (28) to obtain the optimal solutions p * k , λ k * , ∀k ∈ {t, r} .

A. Simulation Setup
where d BR and d RU,k denote the distances between the BS and the STAR-RIS and between the STAR-RIS and user k, α BR and α RU denote the corresponding path loss exponents, K BR and  Table II. Without loss of generality, we assume that the two users for unicast transmission have the same QoS requirements, i.e., R t = R r = R 0 log 2 (1 + γ 0 ) and γ 0 is the minimum required signal-to-interference-plus-noise ratio (SINR). The communication requirement for multicast transmission is set to R c log 2 (1 + γ c ), where γ c is the minimum required SINR. The power consumption results shown (i.e., Figs. 6-8) were obtained by averaging over 100 channel realizations. Specifically, we first randomly generated 100 user distributions, and then obtained the corresponding channel realizations. User noise power −90 dBm η, χ Initialized penalty factors for Algorithms 1 and 2 10 −4 ω, ̟ Scaling factors for Algorithms 1 and 2 10 ǫ 1 , ǫ 2 Convergence tolerance for SCA 10 −2 ε 1 , ε 2 Accuracy for equality constraints 10 −7 n max Maximum number of inner iterations of Algorithms 1 and 2 30

B. Baseline Schemes
To verify the effectiveness of the proposed STAR-RIS concept and the corresponding operating protocols, we compare with the following two baseline schemes. • Baseline scheme 2 (also referred to as uniform energy splitting (UES)): In this case, we assume that all elements of the ES STAR-RIS employ the same amplitude coefficients for transmission and reflection, respectively, i.e., β k m = β k , β r m = β r ∀m ∈ M, where 0 ≤ β k , β r ≤ 1 and β t + β r = 1. UES can be regarded as a special case of the ES STAR-RIS employing a group/surface-wise amplitude design. The resulting optimization problem can be solved by applying Algorithm 1 with the above linear equality constraints.

C. Convergence of Algorithms 1 and 2
In Fig. 5, we investigate the convergence behavior and the violation of the equality constraints, i.e., (19) and (24)

D. Required Power Consumption Versus Number of RIS Elements
In Fig. 6, we investigate the required power consumption versus the number of RIS elements, M, for both unicast and multicast transmission. Besides the three proposed operating protocols, the results obtained for the two baseline schemes are provided for comparison. We set N = 2, As can be seen from Fig. 6 Regarding the performance comparison between STAR-RISs and baseline scheme 1, as can be observed from Fig. 6(a), independent of the adopted operating protocol, STAR-RISs always outperform conventional RISs for unicast communication. The reasons behind this can be explained as follows. Since conventional RISs employ fixed numbers of transmission and reflection elements, they cannot exploit the same DoFs as available to the STAR-RISs to enhance the desired signal strength and mitigate inter-user interference. Therefore, conventional RISs always suffer from the worst performance for unicast communication. However, in Fig. 6(b), for multicast communication, we observe that conventional RISs can achieve a higher performance than TS   For baseline scheme 2, a special case of ES, it is interesting to observe that there is a noticeable performance gap between ES and UES for unicast communication in Fig. 6(a). However, the performance gap is negligible for multicast communication in Fig. 6    UES is a promising operating protocol for multicast communication, where it achieves similar performance as ES. However, compared with ES, the group-wise amplitude control of UES reduces the overhead caused by exchanging configuration information between the BS and the STAR-RIS, which is a major bottleneck for the practical implementation of RISs.

E. Required Power Consumption Versus Number of BS Antennas
In Fig. 7, we study the required power consumption versus the number of BS antennas, N, for the proposed operating protocols for STAR-RISs and the baselines for both unicast and multicast communication. We set M = 10, γ 0 = 0 dB for unicast communication, and M = 10, γ c = 10 dB for multicast communication. Fig. 7 shows that the required power consumption for all schemes decreases as N increases thanks to a higher active beamforming gain. For unicast communication in Fig. 7(a), the proposed STAR-RISs outperform conventional RISs since more DoFs for transmission and reflection design can be exploited. As we consider a small M, for multicast communication in Fig. 7(b), the loss in DoFs for employing conventional RISs is not significant. Thus, in this case, conventional RISs outperform the TS STAR-RIS because of the more efficient exploitation of the time resources. As can be observed in Fig. 7(a), there is a considerable performance gap between UES and ES for unicast communication, while UES and ES achieve a similar performance for multicast communication in Fig. 7(b). This is consistent with the results in Fig. 6.

F. Required Power Consumption Versus QoS Requirements
In Fig. 8, we study the required power consumption versus the minimum required SINR, γ 0 and γ c , for both unicast and multicast transmission. We set N = 2 and M = 10 for all considered schemes. For unicast communication in Fig. 8(a), we observe that TS achieves the   best performance for small γ 0 (i.e., when γ 0 < 12 dB) but is outperformed by the other schemes for large γ 0 . This is because the interference-free communication for TS is achieved at the expense of an inefficient use of the communication time, which becomes a major performance bottleneck for large γ 0 . In contrast, since for ES, MS, and the two baseline schemes, both users are served during the entire communication time, they achieve a higher performance than TS for large γ 0 despite the inter-user interference. Similarly, for multicast communication, we can observe from Fig. 8 The results obtained in this paper confirm the effectiveness of employing STAR-RISs for improving the performance of wireless networks, which motivates related future research on STAR-RISs, such as channel estimation and deployment design. More specifically, although the channel acquisition methods proposed for conventional RISs can be applied for STAR-RIS with TS, considerable pilot overhead is required to consecutively estimate the transmission and reflection channels. To tackle this issue, channel acquisition methods based on the ES protocol have to be developed to simultaneously estimate both channels with less pilot overhead. Moreover, the full-space coverage created by STAR-RISs also imposes new challenges for deployment.
Different from conventional reflecting-only RISs, whose optimal deployment strategy requires only to assign either the users or the BSs to its local reflection region [42,43], the deployment locations and orientations of STAR-RISs have to be further optimized to balance the numbers users on both sides of the STAR-RIS, i.e., the number of T users and the number of R users.
APPENDIX A: PROOF OF THEOREM 1 The relaxed version of problem (18) without rank-one constraint (11e) is jointly concave with respect to W k , Q ES k , β k and satisfies Slater's constraint qualification [41]. Therefore, strong duality holds and the Lagrangian function is given by where τ is the collection of all terms which do not depend on {W k }, and µ k and Y k are the Lagrange multipliers associated with constraints (18b) and (11f), respectively. Based on the Karush-Kuhn-Tucker (KKT) conditions with respect to W k , the structure of the optimal W * k can be characterized as follows: where µ * k and Y * k denote the optimal Lagrange multipliers and ∇ W * k L is the gradient of L with respect to W * k . Then, ∇ W * k L = 0 can be further expressed as where Ω * k is given by By exploiting the results in [17, Appendix A], it can be proved that Rank (Y * k ) = N − 1. Furthermore, (31b) implies that Rank (Y * k ) + Rank (W * k ) ≤ N. Therefore, Rank (W * k ) ≤ 1 has to hold. Thus, due to the QoS constraint in (18b), Rank (W * k ) = 1 for the optimal solution. This completes the proof.