Movable-Element STARS-Assisted NOMA Communications

I. INTRODUCTION

• Fixed-position RIS/STARS elements limit use of spatial channel variation, while movable elements can adjust their positions to improve channel conditions; NOMA users share resource blocks using power-domain signal separation.STARS can transmit and reflect signals for full-space coverage, and NOMA decoding order depends on user channel conditions.Source: §I · p. 1

B. Motivations and Contributions

• Prior work studied RIS/STARS-assisted NOMA and movable-element RIS/STARS systems, but integrating ME-STARS with NOMA remains under-explored.The paper identifies joint beamforming and position design, as well as applying SIC in the reconfigurable propagation environment, as challenges.Source: 1) Studies on RIS/STARS-assisted NOMA… · pp. 1–2, 2) Studies on PAA-assisted… · p. 2, B. Motivations and Contributions · p. 2
• The proposed system places movable elements on a STARS and jointly designs BS beamforming, ME-STARS beamforming, and element positions to maximize sum achievable rate subject to NOMA SIC decoding.The system model assumes energy-splitting operation, and the authors note that limited ME speed suits slowly varying channels such as IoT scenarios.Source: B. Motivations and Contributions · p. 2, A. System Model · p. 3
• An alternating-optimization algorithm divides the coupled problem into three subproblems, using SCA for BS beamforming and penalty-based methods with gradient descent for ME-STARS beamforming and positions.The position constraints are transformed into an unconstrained formulation using penalties and a Sigmoid function before gradient descent.Source: B. Motivations and Contributions · pp. 2–3
• Simulations report considerable sum-rate improvement over STARS with fixed-position elements, with a significant ME-STARS advantage even for restricted moving regions.The passage states these comparisons without giving numerical values.Source: B. Motivations and Contributions · p. 3

A. System Model

• The downlink model has a BS with M antennas serving K single-antenna users through an N-element ME-STARS, whose users lie on transmission or reflection sides.The element positions are constrained to a feasible moving region.Source: A. System Model · p. 3

B. Signal Model

• Direct BS-user links are assumed blocked, while the BS-ME-STARS and ME-STARS-user channels are modeled as line-of-sight links.The model allows multiple paths on both channel segments.Source: B. Signal Model · p. 3
• The model assumes perfect CSI at the BS, while the impact of imperfect CSI is identified as future work.The paper notes that perfect CSI acquisition is challenging in practice.Source: B. Signal Model · p. 4
• Users farther from the ME-STARS receive lower decoding orders, using distance as a proxy because user channel-gain differences mainly arise from the STARS-user channels.The paper avoids exhaustive decoding-order search, whose number of candidates grows as O(K!).Source: B. Signal Model · p. 4
• A user's achievable rate is limited by the rate at which that user and users with higher decoding orders can decode its message, supporting successful SIC.The model also imposes a rate-fairness constraint for users with lower decoding orders.Source: B. Signal Model · pp. 4–5

C. Problem Formulation

• The objective is to maximize the sum achievable rate for K users through joint design of BS beamforming, ME-STARS beamforming, and ME positions.The formulation includes user-rate and SIC decoding constraints.Source: C. Problem Formulation · p. 5
• The constraints cover successful SIC decoding, total transmit power, rate fairness, element phase and energy conservation, feasible ME locations, and minimum spacing between adjacent elements.The BS power limit is denoted PB, and Dmin is the minimum distance.Source: C. Problem Formulation · p. 5
• The optimization is highly coupled and non-convex, with the jointly optimized ME positions adding complexity alongside BS and STARS beamforming.The paper introduces a more tractable formulation before developing an alternating-optimization algorithm.Source: C. Problem Formulation · p. 5

III. PROPOSED SOLUTION

• The alternating-optimization framework solves the three subproblems iteratively until the sum achievable rate stabilizes.The subproblems are BS beamforming, ME-STARS beamforming, and ME positions optimization.Source: §III · p. 5

A. BS Beamforming Optimization

• The BS beamforming problem uses a first-order Taylor approximation to handle its nonconvex constraint.The approximation is applied to the left-hand term of constraint (16b).Source: A. BS Beamforming Optimization · pp. 5–6
• The optimal W_k has rank at most one even when the rank-one constraint is omitted, so problem (18) can be solved as a convex problem.The paper identifies standard solvers such as CVX for this formulation.Source: A. BS Beamforming Optimization · p. 6

B. ME-STARS Beamforming Optimization

• The rank-one constraint is expressed as Tr(U_s) − ||U_s||_2 = 0 and incorporated into the objective as a penalty term.The penalty weight κ is increased iteratively to limit constraint violation.Source: B. ME-STARS Beamforming Optimization · p. 6
• A first-order Taylor expansion approximates the nonconvex largest-eigenvalue term in the penalty formulation.The approximation provides a linear upper bound for ||U_s||_2.Source: B. ME-STARS Beamforming Optimization · p. 6
• A relatively small initial penalty weight is increased in an outer loop while the beamforming matrices are updated in an inner loop.The outer loop continues until the rank-one equality violation is within the prescribed tolerance.Source: B. ME-STARS Beamforming Optimization · p. 7
• The transformed problem is convex and can be solved with standard convex solvers such as CVX.The procedure is presented as a penalty-based SCA algorithm.Source: B. ME-STARS Beamforming Optimization · p. 7

C. MEs Positions Optimization

• A sigmoid mapping transforms real-valued auxiliary variables into the confined ME movement region.The transformed positions are used to express the minimum-distance constraint equivalently.Source: C. MEs Positions Optimization · p. 7
• Penalty terms convert the constrained position problem into an unconstrained objective, which is smoothed with log-sum-exp before gradient descent.The log-sum-exp function approximates the nondifferentiable maximum; the approximation is more precise at lower smoothing factor ζ.Source: C. MEs Positions Optimization · pp. 7–8
• Gradient descent updates the ME positions along the gradient direction using an iteration-dependent step size.A backtracking line search adjusts the step size until the Armijo condition is satisfied.Source: C. MEs Positions Optimization · pp. 8–9
• The nested iterative procedure updates the penalty and smoothing parameters until the minimum-distance, resource-allocation, and minimum-rate constraints are satisfied.The inner loop stops when objective increments are small enough or its iteration limit is reached.Source: C. MEs Positions Optimization · pp. 8–9

D. Convergence and Complexity Analysis of the Proposed AO Algorithm

• Each overall iteration optimizes the BS beamforming vectors, ME-STARS beamforming coefficients, and ME positions by solving their respective subproblems.The corresponding problems are (18), (23), and (29).Source: D. Convergence and Complexity Analysis… · p. 9
• After the first iteration, each subproblem uses the previous iteration’s solutions as its input local points.The first iteration is the stated exception.Source: D. Convergence and Complexity Analysis… · p. 9
• Sequential refinement of the coupled variables is described as improving overall system performance through cumulative improvements.Source: D. Convergence and Complexity Analysis… · p. 9

• The argument uses locally tight first-order Taylor approximations, a lower bound from reformulated problem (23), and GDA updates that increase function values.Source: 1) Convergence Analysis · p. 9
• The objective value of problem (11) is non-decreasing after alternatively solving problems (16), (19), and (26).Source: 1) Convergence Analysis · p. 9
• Because the objective is upper bounded, this monotonicity ensures Algorithm 4 converges to a stationary point of problem (11).Source: 1) Convergence Analysis · p. 9

• Algorithm 4’s complexity is mainly determined by Algorithms 1, 2, and 3; Algorithms 1 and 2 use an interior-point method for the standard convex problem.Source: 2) Complexity Analysis · p. 9
• Algorithm 3’s complexity depends mainly on gradient calculation and includes a backtracking line search whose cost depends on its iteration count.Source: 2) Complexity Analysis · p. 9
• The computational complexities of Algorithms 1, 2, and 3 are polynomial in M, N, and K, and Algorithm 4 is described as practically feasible.Source: 2) Complexity Analysis · p. 9

IV. NUMERICAL RESULTS

• The AO algorithm’s sum rate stabilizes within 5 outer iterations, while convergence takes more iterations as the number of MEs increases.The convergence curves use a single stochastic channel realization.Source: §IV · p. 10, Figure 2 · p. 10
• ME-STARS outperforms RPE-STARS and FPE-STARS by around 8.81% and 11.25% on average, respectively, as BS maximum transmit power varies.The comparison is reported for sum achievable rate versus BS maximum transmit power.Source: §IV · p. 10, Figure 3 · p. 11
• With normalized region size A/λ of 3 and 4, ME-STARS sum rate is 8.65% and 10.60% higher than FPE-STARS, respectively.Sum rate improves as the moving region expands, but the gain gradually converges; maximal sum rate can be reached within a finite region size.Source: §IV · p. 11, Figure 4 · p. 11
• As the number of ME elements N grows, all schemes’ sum rates increase, while ME-STARS remains superior and its advantage over FPE-STARS shrinks.The text attributes eventual sum-rate saturation to the constrained moving region limiting how many MEs can occupy locally optimal positions.Source: §IV · p. 11, Figure 5 · p. 11
• As channel path count L increases, every scheme’s sum rate increases, and ME-STARS achieves the highest performance for all reported L values.The paper attributes the trend to higher multiuser-channel rank and more local maxima that provide MEs-based schemes with favorable positions.Source: §IV · pp. 11–12, Figure 6 · p. 11

V. CONCLUSION

• The objective is to maximize sum achievable rate by jointly optimizing BS beamforming, passive STARS beamforming, and movable-element positions.The formulation includes the NOMA SIC decoding constraint.Source: §V · p. 12
• Alternating optimization decomposes the coupled non-convex problem into three subproblems, using SCA for beamforming and penalty terms with gradient descent for element positions.The positions are adjusted within a confined region.Source: §V · p. 12
• Simulations show ME-STARS-assisted NOMA outperforms fixed-position STARS, with the sum-rate gain attributed to exploiting extra spatial degrees-of-freedom.Robust algorithm design against imperfect CSI, especially in dynamic environments, is identified as future research.Source: §V · p. 12

APPENDIX A DETAILED DERIVATION OF ∂f

• The derivatives of r(q_n) and R_j→k with respect to q_n depend on calculating the derivative of υ_j,k(q_n).The passage identifies this derivative as a vector of partial derivatives.Source: Appendix A · p. 12
• The appendix gives the partial derivative of υ_j,k(q_n) with respect to y_n and combines intermediate expressions to obtain further derivatives.It also states that multiplication factors are associated with x_n and y_n.Source: Appendix A · p. 12
• The resulting equations are used to obtain the gradient vector of f at q_n.The appendix refers to equations (A.5) and (A.6) in this derivation.Source: Appendix A · p. 12