On the Optimality of Reconfigurable Intelligent Surfaces (RISs): Passive Beamforming, Modulation, and Resource Allocation

• RIS meta-atoms manipulate incident electromagnetic waves in frequency, amplitude, and phase, then reflect them toward a desired destination without additional signal processing.Source: §I · p. 2
• RISs are electronically operated metasurfaces controlled by software through a dedicated link between the base station and RIS controller.Source: §I · p. 2
• RIS reflecting elements can be spaced between λ/8 and λ/4, whereas conventional massive antenna arrays typically require spacing greater than λ/2.Source: §I · p. 2
• The resulting compact deployment enables many reflecting elements, precise reflection-wave control, and coherent alignment with the desired channel.Source: §I · p. 2
• RISs are presented as promising for future 6G systems and compatible with techniques including SWIPT and physical-layer security.Source: §I · p. 2

• Earlier work also introduced RF-chain-free RIS transmitters supporting PSK and QAM, while RIS transmitters could create undesired interference for existing cellular users.Source: A. Related Works · p. 3
• Previous studies proposed RIS beamformers for transmit-power minimization, ergodic rate, energy efficiency, and idealized SER analysis.Source: A. Related Works · p. 3
• Many prior analyses assumed reflection-coefficient amplitudes always equal to one, although practical reflecting circuits can dissipate incident electromagnetic power.Source: A. Related Works · p. 3
• Continuous phase shifts require infinite control bits per reflecting element, exceeding the capacity of a practical RIS control link.Source: A. Related Works · p. 3
• The paper therefore calls for analysis verifying asymptotic optimality with limited RIS control capacity and practical reflection coefficients.Source: A. Related Works · pp. 3–4

• The paper analyzes achievable-rate optimality and supports the proposed schemes through simulations comparing them with their upper bounds.Source: B. Contributions · p. 4
• The proposed passive beamformer achieves asymptotic SNR optimality despite reflection-power loss and the number of RIS control bits, including with one-bit control.Source: B. Contributions · p. 4
• The modulation scheme increases RIS-system sum-rate, avoids interference with existing users, and has achievable SNR converging to that of conventional massive MIMO or MIMO relay systems.Source: B. Contributions · p. 4
• The resource allocation algorithm jointly uses the proposed beamforming and modulation schemes to maximize average sum-rate subject to minimum user-rate requirements.Source: B. Contributions · p. 4
• A practical RIS can achieve the asymptotic performance of an ideal RIS as the number of reflecting elements increases without bound.Source: B. Contributions · p. 4

• The network contains a multi-antenna BS, single-antenna UEs, and multiple RISs with N reflecting elements, dividing users into direct UEs and RIS-connected UEs.Source: §II · p. 5
• Users with CQI above a threshold receive direct BS transmission, whereas users with poor direct channels are assigned RISs through a dedicated control link.Source: §II · pp. 5–6
• Each RIS controller sets element voltages that determine reflection phases, while the model includes reflection power loss and amplitude–phase coupling from practical varactors.Source: §II · pp. 6, 8
• An RIS cannot coherently align simultaneously with all RIS-connected users, and the model neglects multiply reflected signals because their power is negligible under severe path loss and reflection loss.Source: §II · p. 6
• The analysis assumes quasi-static block fading, approximately constant channel and SNR within each resource block, Rician channels, and perfect CSI at the BS.Source: §II · pp. 6–7
• The practical design goal is to maximize achievable rate and asymptotically match ideal-RIS SNR scaling O(N^2), despite limited control capacity and nonideal reflection coefficients.Source: §II · pp. 8–9

A. Passive Beamformer Design

• Ideal RIS analyses assume lossless reflection and unlimited control-link capacity, whereas this work evaluates optimality under practical reflection and control constraints.Source: §III · p. 9
• The squared RIS SNR gain is described as yielding twice the achievable-rate performance of conventional array systems without additional radio resources.Source: §III · p. 9
• The proposed passive beamformer achieves the ideal RIS SNR order O(N^2) asymptotically with only one control bit per reflecting element.Source: §III · p. 9

B. Modulation for Unscheduled RUE

• The modulation scheme uses ambient RF signals to transmit data without additional radio resources and achieves SNR order O(N), like conventional massive MIMO or MIMO relay systems.Source: §III · p. 9

Resource Allocation

• The resource allocation algorithm jointly schedules users and controls power to maximize the sum-rate of the considered RIS-based MISO system.Source: §III · p. 9

Reflection phase selection

• The algorithm sequentially selects each discrete reflection phase by maximizing the Euclidean norm of the accumulated vector sum, avoiding exhaustive search over 2^(bN) configurations.It first selects the BS antenna with the largest channel gain, then evaluates phase candidates for each reflecting element.Source: A. Passive Beamformer Design · pp. 10–11

Asymptotic optimality

• Algorithm 1 achieves instantaneous SNR O (N2) for all complex channels and any number of RIS control bits b ≥1, with asymptotic complexity O (N) as N increases.Source: A. Passive Beamformer Design · p. 11

Average SNR analysis

• The lower bound of average SNR increases as O (N^2), while gains from the remaining M −1 BS antennas become negligible relative to the selected antenna as N increases.With limited control bits, the average SNR converges to that of the maximum-gain transmit antenna selection.Source: A. Passive Beamformer Design · pp. 12–13

Practical implementation

• Because MRT in a single-user MISO system requires multiple RF chains and higher cost and hardware complexity, transmit antenna selection can achieve full multi-antenna diversity at low cost and complexity, with performance converging to MRT as N increases.Source: A. Passive Beamformer Design · p. 13

Asymptotic SNR

• The average SNR scales as O(N^2) for spatially correlated Rician fading channels.The i.i.d. Rayleigh case is treated as a special case of the broader analysis.Source: A. Passive Beamformer Design · p. 14

Complexity

• Algorithm 1 has O(N) complexity, substantially lower than the O(N^2) complexity of [31] for large N.The lower complexity is identified as a significant advantage over [31].Source: A. Passive Beamformer Design · p. 14

Scheduling diversity

• The instantaneous-SNR variance grows asymptotically as O(N^3), producing scheduling diversity for large N.This motivates a resource-allocation algorithm intended to achieve maximum scheduling diversity in Section IV-C.Source: A. Passive Beamformer Design · p. 14

Error performance

• The SER associated with Algorithm 1 decays exponentially as a function of N.For M = 1, the derived average SNR agrees with the result in.Source: A. Passive Beamformer Design · p. 14

Modulation Design

• The modulation scheme controls each unscheduled RIS phase to convey data to its corresponding unscheduled RUE while preserving scheduled-user transmission.Source: B. Modulation for Unscheduled RUE · pp. 14–15

Interference-Free Operation

• Keeping each unscheduled RIS phase unchanged from DMRS through data transmission makes the effective CSI constant and yields zero interference to scheduled UEs.Source: B. Modulation for Unscheduled RUE · p. 15

Resource and Rate Benefits

• Each unscheduled RIS transmits one symbol during channel coherence time, reducing control-link burden while providing additional data rates without extra radio resources.Source: B. Modulation for Unscheduled RUE · p. 15

Modulation Design

• The proposed modulated reflection matrix uses Mo-PSK symbols, whose phase can be estimated from CSI, precoding, and broadcast modulation information.Source: B. Modulation for Unscheduled RUE · p. 16

Asymptotic Performance

• Theorem 1 gives unscheduled RUEs average SNR order O(N), while scheduled RUEs retain O(N^2) asymptotic SNR.Source: B. Modulation for Unscheduled RUE · p. 17
• The average SER decreases with N and approaches zero as N tends to infinity, supporting reliable communication at unscheduled RUEs.Source: B. Modulation for Unscheduled RUE · p. 17

C. Resource Allocation Algorithm

• The proposed algorithm jointly optimizes user scheduling and transmission power to maximize average sum-rate under minimum-rate and power constraints.The formulation accounts for time-varying channel states, quantized CSI, per-UE rate requirements, and average and instantaneous BS power limits.Source: C. Resource Allocation Algorithm · pp. 18–19
• The algorithm combines the proposed passive beamforming and uRUE modulation schemes to calculate instantaneous rates while modeling interference from neighboring RISs.Interference from RISs to scheduled DUEs is omitted under the OFDMA assumption, whereas undesired reflections from other RISs affect uRUEs.Source: C. Resource Allocation Algorithm · p. 18
• The optimization is mixed integer and generally hard, so the method relaxes it and uses a dual formulation whose duality gap vanishes as N →∞.The relaxed objective and constraints become continuous, enabling asymptotically exact dual optimization.Source: C. Resource Allocation Algorithm · pp. 19–20
• For fixed dual multipliers, Proposition 2 obtains optimal scheduling and power decisions, while the algorithm updates the multipliers using stochastic subgradients.The per-slot decisions are computed from the dual function, and the multipliers are iteratively updated with step size Δt = 1/t.Source: C. Resource Allocation Algorithm · pp. 20–22
• Algorithm 1 achieves O (N2) in instantaneous SNR regardless of RIS control-bit count, while its design must be updated at each time slot, resulting in high computational complexity.Source: C. Resource Allocation Algorithm · p. 22
• With step size Δt = 1/t, the stochastic subgradient iterations converge, and the proposed solution asymptotically reaches the maximum average sum-rate under the constraints.This conclusion follows from convergence of the subgradient procedure together with the vanishing duality gap as N increases.Source: C. Resource Allocation Algorithm · p. 22

Beamforming evaluation

• Algorithm 1 with transmit-antenna selection approaches MRT performance as N increases, although the ratio decreases with the number of BS antennas M.The simulations verify that antenna selection asymptotically achieves MRT performance.Source: Figure 6 · p. 23, §IV · p. 24

Modulation evaluation

• The proposed modulation’s simulated SER closely follows the asymptotic result and decreases linearly with N, converging to zero as N grows.When N doubles, the SNR difference remains 3 dB; at target SER 2·10^-2, the required SNRs range from 5 dB at N = 16 to −10 dB at N = 512.Source: Figure 7 · p. 24, §IV · pp. 24–25

Resource-allocation evaluation

• Algorithm 2 satisfies transmission-power and minimum-rate constraints regardless of N and converges within about 1.0 s, indicating that its optimal sum-rate is reached within a few seconds.The convergence experiment uses 1000 TTIs of 1 ms each.Source: Figure 9 · p. 25, §IV · p. 25
• The additional sum-rate grows approximately linearly with the number of RUEs and increasingly affects average sum-rate as N and K grow, supporting massive connectivity.The simulations compare average data rates with and without minimum-rate requirements.Source: §IV · pp. 25–26
• Without minimum-rate requirements, Algorithm 2 favors more RUEs and achieves higher average sum-rate but produces unfair individual rates; enforcing the requirements reverses this tradeoff.With minimum-rate constraints, all individual rates satisfy the requirements while average sum-rate decreases.Source: §IV · p. 25, Figure 10 · p. 26, Figure 11 · p. 26

Scope and limitations

• The analysis addresses practical RISs with reflection-coefficient constraints and limited RIS control-link capacity.The stated future scope includes multi-user MIMO OFDM, finite reflecting elements, and multi-hop reflections.Source: §V · pp. 26–27

Passive beamforming

• The passive beamformer achieves asymptotic SNR optimality with discrete reflection phases, practical reflection-power loss, and even one-bit RIS control.Its performance matches the ideal-RIS benchmark as the number of reflecting elements grows.Source: §V · p. 26

RIS modulation

• The proposed RIS modulation increases achievable sum-rate over a conventional network without RIS and approaches the SNR of massive MIMO or MIMO relay systems.Its approximated symbol error ratio closely agrees with simulations.Source: §V · p. 26

Resource allocation

• The resource allocation algorithm jointly supports scheduling and power control, asymptotically attaining optimal performance while satisfying each user equipment's minimum rate.It also controls the tradeoff between individual fairness and maximum system performance.Source: §V · pp. 26–27

Simulation results

• The proposed schemes converge to their asymptotic upper bounds as the number of RIS reflecting elements tends to infinity.The resource algorithm's performance increases with the number of user equipments because of additional rate from RIS-assisted users.Source: §V · pp. 26–27

APPENDIX A

• The resulting expression converges to the claimed asymptotic form, completing the scaling-law proof.The proof uses moments of non-central chi-squared variables.Source: Appendix A · pp. 27–28
• The appendix shows that dominant positive-element sums scale as O(N), while other terms become negligible in the asymptotic limit.The RIS is modeled as a two-dimensional uniform planar array under the stated scaling analysis.Source: Appendix A · p. 28

APPENDIX B

• The RIS-assisted user receives diversity gain proportional to N_s, yielding N_s-fold desired-signal power when the same symbols are transmitted during the data period.This connects repeated RIS transmission to the derived SNR expression.Source: Appendix B · p. 28

2 Var

• The average SNR at the RIS-assisted user scales as O(N), including in the large-antenna analysis.The derivation considers chi-squared and beta-prime distributions and central-limit approximations.Source: §2 · pp. 29–30
• The appendix derives the average SER of the proposed M0-PSK signal from the moment-generating-function analysis.A special case considers independent identically distributed Rayleigh fading.Source: §2 · pp. 29–30
• The distributional analysis shows that the relevant large-antenna random variables converge to Gaussian or chi-squared forms used to obtain the asymptotic moments.The instantaneous SNR is described as monotonic in the number of base-station antennas.Source: §2 · pp. 29–30

APPENDIX C PROOF OF PROPOSITION 2

• The scheduling variable is binary across the full power range, so optimal scheduling and power can be found jointly.The integer solution is selected after determining the optimal power values.Source: Appendix C · p. 30
• The rate functions are strictly concave in transmission power, enabling a unique power maximizer under the power constraint.The maximizer is obtained by setting the first derivative to zero.Source: Appendix C · p. 30
• The resulting power and scheduling choices are globally optimal under the stated power constraint.The proof concludes that the selected transmission power is optimal.Source: Appendix C · p. 30