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Joint Design for Simultaneously Transmitting And Reflecting (STAR) RIS Assisted NOMA Systems
Jiakuo Zuo, Yuanwei Liu, Zhiguo Ding, Lingyang Song, H. Vincent Poor
TL;DR
The paper studies joint design for STAR-RIS-NOMA systems, where full-space signal control introduces additional coupled optimization variables. It proposes a two-layer iterative algorithm that alternates beamforming and power optimization with decoding-order updates, and reports that the resulting system outperforms conventional RIS-NOMA and RIS-OMA.
Problem
Joint optimization of decoding order, power allocation, active beamforming, and STAR-RIS transmission and reflection beamforming is a challenging non-convex problem that had not been studied for STAR-RIS-NOMA.
Source: B. Motivation and Contributions · p. 2Method
A two-layer iterative algorithm alternates power allocation and beamforming for a fixed decoding order, then updates cluster decoding orders using inner-layer solutions.
Source: §VI · p. 13Results
The proposed STAR-RIS-NOMA system outperforms conventional RIS-NOMA and RIS-OMA systems, while the decoding-order scheme achieves near-optimal performance.
Source: §VI · p. 13Takeaways & Limitations
STAR-RIS-NOMA combines full-space coverage and additional transmission/reflection beamforming degrees of freedom with optimized NOMA resource allocation.
Source: §VI · p. 13The study assumes independently adjustable transmission and reflection phase shifts, whereas practical passive STAR-RISs may have coupled phase shifts.
Source: §VI · p. 13
Abstract
from arXiv · showhide
Different from traditional reflection-only reconfigurable intelligent surfaces (RISs), simultaneously transmitting and reflecting RISs (STAR-RISs) represent a novel technology, which extends the half-space coverage to full-space coverage by simultaneously transmitting and reflecting incident signals. STAR-RISs provide new degrees-of-freedom (DoF) for manipulating signal propagation. Motivated by the above, a novel STAR-RIS assisted non-orthogonal multiple access (NOMA) (STAR-RIS-NOMA) system is proposed in this paper. Our objective is to maximize the achievable sum rate by jointly optimizing the decoding order, power allocation coefficients, active beamforming, and transmission and reflection beamforming. However, the formulated problem is non-convex with intricately coupled variables. To tackle this challenge, a suboptimal two-layer iterative algorithm is proposed. Specifically, in the inner-layer iteration, for a given decoding order, the power allocation coefficients, active beamforming, transmission and reflection beamforming are optimized alternatingly. For the outer-layer iteration, the decoding order of NOMA users in each cluster is updated with the solutions obtained from the inner-layer iteration. Moreover, an efficient decoding order determination scheme is proposed based on the equivalent-combined channel gains. Simulation results are provided to demonstrate that the proposed STAR-RIS-NOMA system, aided by our proposed algorithm, outperforms conventional RIS-NOMA and RIS assisted orthogonal multiple access (RIS-OMA) systems.
I. INTRODUCTION
STAR-RISs extend RIS coverage from half-space to full-space by transmitting and reflecting signals, motivating joint STAR-RIS-NOMA optimization. The paper addresses this coupled non-convex design with a two-layer iterative approach.
Motivation
- STAR-RISs simultaneously transmit and reflect incident signals, extending conventional RIS coverage from half-space to full-space.Their transmission and reflection coefficients provide additional degrees of freedom for controlling signal propagation.Source: §I · p. 1
- Unlike reflection-only RISs, STAR-RISs remove the requirement that transmitters and receivers be located on the same side of the surface.This broadens deployment flexibility beyond the geographical constraint of conventional RISs.Source: §I · p. 1
Motivation and Contributions
- The paper investigates STAR-RIS-NOMA because joint optimization of decoding order, power allocation, active beamforming, and transmission and reflection beamforming had not been studied.The objective is to maximize achievable sum rate under coupled design variables and system constraints.Source: §I · p. 2, B. Motivation and Contributions · pp. 2–3
- The formulated optimization is non-convex because its variables are highly coupled, including the base-station and STAR-RIS beamforming designs.This coupling makes simultaneous optimization challenging.Source: B. Motivation and Contributions · pp. 2–3
- A two-layer iterative algorithm alternates power allocation and beamforming for a fixed decoding order, then updates each cluster’s decoding order.The paper also proposes an efficient decoding-order scheme and evaluates the resulting STAR-RIS-NOMA system numerically.Source: B. Motivation and Contributions · p. 3
B. Basic Signal Model
The STAR-RIS signal model splits each incident signal into transmitted and reflected components, with independently selectable phases but energy-coupled amplitudes. It combines these links with clustered downlink NOMA, where decoding order and SIC constraints affect achievable rates.
STAR-RIS operation
- Each STAR-RIS element divides an incident signal into transmitted and reflected signals, supporting transmission, reflection, or simultaneous transmission and reflection modes.The paper focuses on the energy-splitting protocol.Source: B. Basic Signal Model · pp. 3–4
- Transmission and reflection phase shifts can be selected independently, whereas their amplitude coefficients are coupled by energy conservation.For each element, the transmitted and reflected signal energies must sum to the incident signal energy.Source: B. Basic Signal Model · p. 4
System configuration
- The BS uses active beamforming to broadcast C independently superposed data streams to K clustered single-antenna users through the STAR-RIS.The direct BS-user links are blocked, and each cluster has an associated active beamforming vector and user power allocation coefficients.Source: A. System Model of The Proposed… · pp. 4–5
- A user's combined BS-RIS-user channel depends on the BS-RIS channel, the RIS-user channel, and a transmission or reflection coefficient matrix determined by its space.Users in the transmission space use Θt, while users in the reflection space use Θr.Source: A. System Model of The Proposed… · p. 5
NOMA reception
- Within each cluster, users apply successive interference cancellation, making the decoding order a key variable because RIS-modified channels and inter-cluster interference affect it.The possible decoding orders are not determined solely by conventional channel gains, and SIC decoding-rate conditions must be satisfied.Source: A. System Model of The Proposed… · p. 5
- The achievable sum rate aggregates users' rates across clusters, while SIC decoding conditions depend on active beamforming, power allocation, and transmission and reflection beamforming.Thus, the signal model links rate performance to both BS-side and STAR-RIS-side design variables.Source: A. System Model of The Proposed… · p. 5
B. Problem Formulation
The paper formulates a joint optimization problem for maximizing the achievable sum rate through decoding-order, power-allocation, active-beamforming, and transmission/reflection-beamforming design. The STAR-RIS formulation is more challenging than traditional RIS optimization because its transmission and reflection coefficients are highly coupled.
- The objective jointly optimizes decoding order, power allocation coefficients, active beamforming, and transmission and reflection beamforming to maximize the achievable sum rate.Source: B. Problem Formulation · p. 5
- The formulated problem includes minimum QoS, SIC, total transmit-power, cluster power-allocation, STAR-RIS coefficient, and decoding-order constraints.The decoding-order variables belong to the combination set of all possible decoding orders.Source: B. Problem Formulation · p. 6
- STAR-RIS optimization is more challenging than traditional RIS optimization because transmission and reflection coefficients are highly coupled.Traditional RIS optimization is a special case obtained by turning off transmission and retaining only reflection.Source: B. Problem Formulation · p. 6
IV. SOLUTION OF THE PROBLEM
The proposed solution uses an outer iteration to update decoding orders and an inner iteration to optimize the remaining variables for each order. Decoding orders are determined from equivalent-combined channel gains, enabling the SIC constraints to be removed under the optimal order.
- A two-layer iterative algorithm alternates outer-layer decoding-order updates with inner-layer optimization of power allocation, active beamforming, and transmission/reflection beamforming.The inner layer solves the joint optimization for a given decoding order.Source: §IV · p. 6
Equivalent-Combined Channel Gain based Decoding Order
- For fixed active and transmission/reflection beamforming, the optimal decoding order in each cluster is determined by the equivalent-combined channel gain.The equivalent-combined channel gain is denoted by Γc,Dc(k).Source: A. Equivalent-Combined Channel Gain… · p. 6
- The decoding order depends on active, transmission, and reflection beamforming vectors, while power allocation coefficients do not affect it.Source: A. Equivalent-Combined Channel Gain… · p. 6
- Under the optimal decoding order, the SIC condition is guaranteed, allowing the algorithm to iteratively update the order and solve a reduced problem without SIC constraints.The reduced problem is solved under the given decoding order.Source: A. Equivalent-Combined Channel Gain… · p. 6
Power Allocation Coefficients Optimization
- With beamforming vectors fixed, the power-allocation optimization decomposes into C decoupled cluster subproblems.The decomposition follows because inter-cluster interference is unrelated to the power-allocation coefficients.Source: B. Power Allocation Coefficients… · p. 6
- Lemma 2 gives a feasibility condition for the power-allocation subproblem, while Theorem 1 provides its optimal objective value and coefficients when feasible.Source: B. Power Allocation Coefficients… · p. 6
C. Active Beamforming Optimization
Active beamforming optimization is handled by reformulating the problem with slack variables and semidefinite matrices, then applying successive convex approximation. The resulting convex SDP provides a lower-bound approximation and is guaranteed to converge to a locally optimal solution.
- The active beamforming problem introduces slack variables Ac,k and Bc,k to reformulate the rate-related constraints.Source: C. Active Beamforming Optimization · p. 7
- The reformulation expresses channel-gain terms using Wc and Hc,k through the trace relation |hc,kwc|2 = Tr(WcHc,k).Source: C. Active Beamforming Optimization · p. 7
- The active beamforming formulation is non-convex because its rate and rank-one constraints are non-convex.The rate constraint involves joint convex logarithmic terms in the slack variables.Source: C. Active Beamforming Optimization · p. 7
Successive Convex Approximation
- Successive convex approximation replaces the rate expression with a linear lower bound and iteratively solves the resulting approximation.The linearized rate function is a lower bound over Ac,k and Bc,k.Source: C. Active Beamforming Optimization · p. 7
- Dropping the rank-one constraint does not change the optimal rank because the relaxed problem always yields rank(Wc) = 1.Source: C. Active Beamforming Optimization · p. 7
Successive Convex Approximation
- The relaxed problem is a standard convex SDP, and the SCA-based algorithm is guaranteed to converge to a locally optimal solution.Source: C. Active Beamforming Optimization · p. 7
D. Transmission and Reflection Beamforming Optimization
Transmission and reflection beamforming are optimized with fixed active beamforming and power allocation by relaxing the rank-one constraint and progressively tightening it. The resulting convex SDP is solved iteratively through sequential constraint relaxation.
- With active beamforming vectors fixed, the transmission and reflection beamforming problem is formulated for the vectors ut and ur associated with transmission-side and reflection-side users.Source: D. Transmission and Reflection… · p. 8
- The rank-one constraint on Up is relaxed using a parameter ε(τ2) that controls the ratio between its maximum eigenvalue and trace.Source: D. Transmission and Reflection… · p. 8
- ε(τ2) = 0 drops the rank-one constraint, whereas ε(τ2) = 1 is equivalent to enforcing it.Source: D. Transmission and Reflection… · p. 8
Sequential Constraint Relaxation
- Increasing ε(τ2) sequentially from 0 to 1 gradually approaches a rank-one solution.Source: D. Transmission and Reflection… · p. 8
- The relaxed transmission/reflection optimization becomes a standard convex SDP solved efficiently by numerical solvers such as CVX.Source: D. Transmission and Reflection… · p. 8
- Algorithm 2 repeatedly solves the relaxed SDP, updates the relaxation parameter, and stops when the parameter and objective value converge.Source: D. Transmission and Reflection… · p. 8
E. Proposed Algorithm, Convergence and Complexity
Algorithm 3 uses a two-layer iterative procedure: inner iterations alternate optimization of beamforming and power allocation, while outer iterations update NOMA decoding orders.
Proposed Algorithm
- Algorithm 3 solves the original optimization problem through inner and outer iterative layers.The inner layer addresses joint resource and beamforming optimization, while the outer layer updates decoding orders.Source: E. Proposed Algorithm, Convergence and… · pp. 8–9
- The algorithm calculates equivalent-combined channel gains and updates each cluster’s decoding order accordingly.The decoding-order update is based on Lemma 1.Source: E. Proposed Algorithm, Convergence and… · p. 9
- For fixed decoding orders, power allocation, active beamforming, and transmission and reflection beamforming are updated successively.These updates use Theorem 1, Algorithm 1, and Algorithm 2, respectively.Source: E. Proposed Algorithm, Convergence and… · p. 9
- The inner iteration continues until the objective value converges, after which the resulting optimization variables are recorded.The recorded solutions include power allocation factors, active beamforming vectors, and transmission/reflection beamforming vectors.Source: E. Proposed Algorithm, Convergence and… · p. 9
1) Convergence analysis:
The proposed algorithm converges because alternating updates make the inner objective non-decreasing and bounded, while the outer achievable sum rate is also non-decreasing.
Convergence analysis
- For a fixed decoding order, the power allocation and beamforming variables are solved alternatingly in the inner layer.The variables include power allocation coefficients, active beamforming vectors, and transmission and reflection beamforming vectors.Source: 1) Convergence analysis · p. 9
- The inner-layer objective value is monotonically non-decreasing after each iteration and is upper bounded by the achievable sum rate.Therefore, the inner-layer iteration is guaranteed to converge.Source: 1) Convergence analysis · p. 9
- The achievable sum rate is monotonically non-decreasing across outer-layer iterations.This follows from the updates in the specified algorithm steps.Source: 1) Convergence analysis · p. 9
- Because both inner- and outer-layer iterations converge, Algorithm 3 converges.The convergence conclusion follows from the separate monotonicity and boundedness arguments.Source: 1) Convergence analysis · p. 9
Complexity analysis
- The algorithm’s complexity mainly depends on Algorithms 1 and 2, which solve active and transmission/reflection beamforming subproblems.The analysis therefore focuses on the complexity of those two component algorithms.Source: 2) Complexity analysis · p. 9
2) Complexity analysis:
The complexity analysis decomposes Algorithm 3 into its active-beamforming and transmission/reflection-beamforming components, while simulations evaluate a three-cluster STAR-RIS-NOMA setting under specified channel assumptions.
Complexity analysis
- Algorithm 3’s total complexity depends on the outer- and inner-layer iteration numbers together with the complexities of Algorithms 1 and 2.The outer- and inner-layer iteration numbers are identified as 3 in the supplied passage.Source: 2) Complexity analysis · p. 9
Simulation setting
- The simulation uses three clusters with three users per cluster, placing one cluster in reflection space and two in transmission space.The BS and STAR-RIS are positioned at (0, 0, 20) meters and (0, 30, 20) meters, respectively.Source: §V · p. 9, Figure 3 · p. 10
- All channels use Rician fading, with path loss modeled through a reference loss, link distance, and path loss exponent.The BS-RIS and RIS-user channels are modeled using the stated path-loss and Rician-fading assumptions.Source: §V · p. 9
- The simulations assume equal user QoS requirements with Rmin_c,k = 0.1 bits/s/Hz.The supplied settings also specify ℓBR = ℓRU = 2.2, εBR = εRU = −30 dB, κBR = κRU = 3 dB, and σ2 = −90 dBm.Source: §V · p. 10
A. Convergence of The Proposed Algorithms
The proposed algorithms converge through nested iterations, with convergence behavior depending on the number of STAR-RIS elements, antennas, and transmit power. The outer-layer decoding-order update converges quickly, while the proposed system improves achievable sum rate over traditional RIS-NOMA.
- Algorithm 1 converges within 12 iterations under different numbers of BS antennas.Source: A. Convergence of The Proposed… · p. 10
- Algorithm 2 requires more iterations as M increases because more transmission and reflection coefficients must be optimized.Source: A. Convergence of The Proposed… · p. 10
- The inner-layer iteration requires more iterations as M and Pmax increase, reflecting larger coefficient sets and feasible active-beamforming regions.Source: A. Convergence of The Proposed… · p. 10
- The outer-layer iteration converges within a small number of iterations despite its worst-case dependence on all possible decoding orders.Source: A. Convergence of The Proposed… · p. 10
- The proposed STAR-RIS-NOMA system consistently outperforms traditional RIS-NOMA as the total transmit power budget increases.Source: E. Impact of The Total Transmit Power… · p. 12
APPENDIX A: PROOF OF PROPOSITION 1
The appendix derives conditions for NOMA decoding order and minimum power allocation, then characterizes the optimal allocation under users’ QoS requirements. It also states the resulting objective expression for the allocation problem.
- Equivalent-combined channel gains of users in a cluster must satisfy an ordering condition under the optimal decoding order.Source: Appendix A · p. 13
- The derived inequality establishes the successive-interference-cancellation condition for the users.Source: Appendix A · p. 13, Appendix B · p. 13
- Minimum power allocation coefficients are chosen so that all users meet their minimum QoS requirements.Source: Appendix B · p. 13
- Power is first allocated to weaker equivalent-combined-channel users at their minimum requirements, while remaining power goes to the higher-decoding-order user.Source: Appendix C · pp. 13–14
- The optimal objective value is expressed using the derived minimum-rate terms and optimal power allocation factors.Source: Appendix C · p. 14
APPENDIX D: PROOF OF THEOREM 2
The proof removes the rank-one constraint from the beamforming problem, solves the resulting jointly convex formulation through its dual problem, and uses KKT conditions to establish the beamforming solution’s structure.
- The beamforming problem without the rank-one constraint is jointly convex and has zero duality gap with its dual formulation.Source: Appendix D · p. 14
- The Lagrangian introduces multipliers for QoS, power, and positive-semidefinite constraints on the beamforming matrix.Source: Appendix D · p. 14
- KKT conditions characterize the optimal beamforming matrix and its associated optimal Lagrange multipliers.Source: Appendix D · p. 14
- At least one optimal multiplier is strictly positive because the corresponding power constraint is active.Source: Appendix D · p. 14
- The proof uses the rank of the dual matrix together with QoS feasibility to exclude a zero beamforming solution.Source: Appendix D · p. 14