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Reconfigurable Intelligent Surfaces for Energy Efficiency in Wireless Communication

Chongwen Huang, Alessio Zappone, George C. Alexandropoulos, Mérouane Debbah, Chau Yuen

arXiv:1810.06934v5cs.IT

TL;DR

The paper asks whether RIS-assisted downlink multi-user communication can improve energy efficiency despite the lower gain of passive reflectors. It develops joint transmit-power and RIS phase-shift optimization methods with a realistic power model, finding up to 300% higher EE than relay-assisted communication.

  • Problem

    RIS energy efficiency must be assessed against traditional AF relays because passive operation lowers energy consumption but also provides lower gain.

  • Method

    The paper develops a realistic RIS power model and two alternating optimization algorithms combining gradient descent or sequential fractional programming with fractional programming for transmit-power allocation.

  • Results

    300% higher EE is achieved by RIS-based communication than relay-assisted communication in the reported outdoor comparison.

  • Takeaways & Limitations

    Properly designed RIS phase shifts can yield higher EE than traditional AF relays in the evaluated outdoor multi-user MISO setting.

Abstract

from arXiv · show

The adoption of a Reconfigurable Intelligent Surface (RIS) for downlink multi-user communication from a multi-antenna base station is investigated in this paper. We develop energy-efficient designs for both the transmit power allocation and the phase shifts of the surface reflecting elements, subject to individual link budget guarantees for the mobile users. This leads to non-convex design optimization problems for which to tackle we propose two computationally affordable approaches, capitalizing on alternating maximization, gradient descent search, and sequential fractional programming. Specifically, one algorithm employs gradient descent for obtaining the RIS phase coefficients, and fractional programming for optimal transmit power allocation. Instead, the second algorithm employs sequential fractional programming for the optimization of the RIS phase shifts. In addition, a realistic power consumption model for RIS-based systems is presented, and the performance of the proposed methods is analyzed in a realistic outdoor environment. In particular, our results show that the proposed RIS-based resource allocation methods are able to provide up to $300\%$ higher energy efficiency, in comparison with the use of regular multi-antenna amplify-and-forward relaying.

I. INTRODUCTION

The paper motivates RIS-based downlink communication as an energy-efficient alternative to conventional relay systems and develops resource-allocation methods for outdoor multi-user MISO networks. It models RIS-assisted transmission, formulates joint power and phase-shift design, and evaluates the proposed algorithms.

  • Energy efficiency has become a key performance indicator for sustainable 5G and beyond wireless networks.
  • RISs can forward signals without power amplifiers by configuring reflector phase shifts for constructive combination, but their lower gain makes the EE comparison with AF relays nontrivial.
  • The study considers an outdoor downlink network where a multi-antenna base station serves single-antenna users through a discrete RIS.
  • A realistic RIS power model supports EE maximization over RIS phase shifts and downlink transmit powers under maximum-power and minimum-QoS constraints.
  • Two low-complexity, provably convergent algorithms use alternating maximization and account for the RIS phase coefficients’ unit-modulus constraint.
  • The proposed algorithms achieve higher EE than traditional relay-assisted communication in realistic outdoor numerical evaluations.

B. Total Power Consumption Model

The model defines RIS-based energy efficiency using total system power, including hardware consumption at the BS, users, and passive RIS. The resulting joint design is constrained by transmit power, QoS, phase modulus, and modeling assumptions.

  • RIS total power consumption includes BS transmit power and static hardware power at the BS, user terminals, and RIS.
  • Each RIS reflector’s consumption depends on phase-shift resolution, with 3-, 4-, 5-, and 6-bit phase shifters consuming 1.5, 4.5, 6, and 7.8mW, respectively.
  • The model does not optimize the RIS phase-quantization resolution, leaving the choice of quantization bits for future work.
  • The design jointly maximizes bit-per-Joule EE through user transmit powers and RIS phase values, defined as achievable sum rate divided by total power consumption.
  • The optimization enforces maximum transmit power, individual QoS requirements, and unit-modulus RIS coefficients that cannot amplify incoming signals.
  • The resulting optimization problem is non-convex, particularly because of the transmit-power and unit-modulus constraints.

III. ENERGY EFFICIENCY MAXIMIZATION

The proposed EE optimization separates RIS phase design and transmit-power allocation through alternating optimization. Iterative updates improve the objective while using phase-specific feasibility and unconstrained reformulations for tractable search.

  • III. ENERGY EFFICIENCY MAXIMIZATION: The formulation uses Zero-Forcing transmission under perfect channel knowledge, while activating more reflectors increases energy consumption.
  • III. ENERGY EFFICIENCY MAXIMIZATION: Alternating optimization separately and iteratively solves for RIS phases with fixed powers and powers with fixed RIS phases.
  • III. ENERGY EFFICIENCY MAXIMIZATION: Each alternating iteration improves EE and converges in objective value because the objective is upper-bounded on the feasible set.
  • Optimization with respect to the RIS Elements Values Φ: For fixed transmit powers, RIS phase design becomes a feasibility problem with unit-modulus constraints and a constant objective.
  • Optimization with respect to the RIS Elements Values Φ: The phase problem is challenging because its objective is non-differentiable and its feasible set is non-convex.
  • Gradient Descent Approach:: Representing RIS coefficients as φ_n=e^jθ_n converts the phase design into an unconstrained problem suitable for gradient search toward a stationary point.

1) Gradient Descent Approach:

The gradient-descent approach reformulates RIS phase optimization as an unconstrained problem and iteratively searches for phase updates that decrease its objective. It uses analytical gradients, line-search step sizes, and a modified conjugate-gradient direction rule.

  • 1) Gradient Descent Approach:: RIS phase optimization is reformulated as an unconstrained minimization that gradient search can monotonically decrease to a stationary point.The formulation removes the explicit phase constraints before applying gradient-based optimization.
  • 1) Gradient Descent Approach:: The objective gradient is derived by expanding the quadratic form y^HAy with respect to the RIS phase variables.The derivation exploits the Hermitian property of A and an index map for the matrix elements.
  • 1) Gradient Descent Approach:: Each phase iterate moves along a descent direction with a positive step size selected by minimizing the resulting one-dimensional objective.The step size can be obtained through line search, while a quadratic approximation offers a lower-complexity alternative.
  • 1) Gradient Descent Approach:: The phase-dependent quadratic expansion is represented using constant, linear, and quadratic terms to simplify the step-size calculation.The approximation is characterized by terms z0, z1, and z2, with z1 ≥ 0 and z2 > 0.
  • 1) Gradient Descent Approach:: The method uses Polak-Ribiere-Polyak conjugate-gradient updates to construct successive descent directions.The direction update is modified when the approximate step-size rule does not guarantee descent.

2) Sequential Fractional Programming:

The sequential fractional programming approach handles the RIS unit-modulus constraints by constructing successive approximate problems with suitable upper bounds. Its phase updates align the free components with a transformed quadratic-form vector, while convergence yields first-order optimality properties under stated conditions.

  • 2) Sequential Fractional Programming:: Sequential fractional programming tackles the non-convex RIS phase problem through a sequence of approximate subproblems.The method is also described as majorization-minimization or inner approximation.
  • 2) Sequential Fractional Programming:: Under three surrogate-function conditions, the optimal objective sequence decreases monotonically and converges with first-order optimality satisfied upon convergence.The conditions require an upper bound, equality at the current point, and matching gradients there.
  • 2) Sequential Fractional Programming:: For the RIS phase objective y^HAy, a positive-semidefinite quadratic bound supplies the surrogate used by the sequential fractional programming method.The bound is constructed with M = λmax I_N^2, where λmax is the maximum eigenvalue of A.
  • 2) Sequential Fractional Programming:: The vectorized phase matrix contains unit-modulus diagonal entries and zero off-diagonal entries, preserving the RIS structure during optimization.Only the diagonal components corresponding to reflector phases are free variables.
  • 2) Sequential Fractional Programming:: The constrained surrogate is solved by aligning the phases of nonzero vector components with the corresponding entries of (λmaxI_N^2 − A)y(t).All other vector components remain zero under the structural constraints.
  • 2) Sequential Fractional Programming:: The phase-update solution follows directly because only unit-modulus component phases remain free after vectorization.The resulting alignment maximizes the surrogate objective subject to the RIS constraints.

B. Optimization with respect to the Power Allocation P

With RIS phases fixed, transmit-power optimization becomes a constrained single-ratio maximization that can be solved globally using Dinkelbach’s algorithm. The power and phase solutions are then alternated until convergence, although global optimality is not guaranteed.

  • B. Optimization with respect to the Power Allocation P: For fixed RIS phases, the transmit-power subproblem has a concave numerator, affine denominator, and affine QoS and power constraints.This structure makes the problem a single-ratio maximization problem.
  • B. Optimization with respect to the Power Allocation P: Dinkelbach’s algorithm globally solves the fixed-phase power-allocation problem with limited complexity.The feasible set is defined by the individual QoS and power constraints.
  • B. Optimization with respect to the Power Allocation P: The complete algorithms alternately update RIS phases and transmit powers until convergence, increasing energy efficiency at each iteration.The power-update method is embedded with the phase-update method in the combined algorithms.
  • B. Optimization with respect to the Power Allocation P: Global optimality is not guaranteed because the joint problem is non-convex and the phase-optimization methods may not find globally optimal phase matrices.Objective-value convergence follows from monotonic improvement and boundedness, not from joint convexity.

C. Sum Rate Maximization

Sum-rate maximization is obtained as a special case of the energy-efficiency formulation by setting ξ = 0. This makes the denominator constant and reduces power optimization to a convex nonfractional problem.

  • C. Sum Rate Maximization: Sum-rate maximization is the ξ = 0 special case of the considered energy-efficiency maximization problem.The sum rate is the numerator of the energy-efficiency objective.
  • C. Sum Rate Maximization: With ξ = 0, the transmit-power subproblem becomes nonfractional and convex, so Algorithm 1 requires only one iteration.The phase-optimization framework remains specialized from the energy-efficiency algorithm.

D. Computational Complexity

The algorithms’ complexity is governed by alternating-maximization iterations and the costs of optimizing RIS phases and transmit powers. Numerical results indicate convergence in a few iterations, while Dinkelbach’s method has super-linear convergence.

  • Complexity drivers: Both algorithms’ complexity depends on the number of alternating-maximization iterations and the complexity of their subproblems.The alternating-maximization iteration count is denoted Ialt.
  • Gradient-based EE Maximization Algorithm: Algorithm 2 has complexity O(Ialt(IgdN^2 + IDK^p)), with phase optimization scaling through gradient updates and transmit-power optimization through Dinkelbach iterations.Here, 1 ≤ p ≤ 4, reflecting the polynomial complexity of the convex transmit-power subproblems.
  • Convergence: The numerical results confirm convergence in a few iterations, while Dinkelbach’s algorithm is known to have a super-linear convergence rate.Closed-form expressions for Ialt, Igd, and ID as functions of system parameters are considered prohibitive.

IV. NUMERICAL RESULTS

The numerical evaluation models RIS-based multi-user MISO communication under realistic propagation, hardware, and QoS assumptions. It uses an alternating sequential-programming algorithm and reports feasibility across QoS settings, relaxing constraints only in very few infeasible cases.

  • Algorithm: The numerical study uses the sequential programming EE maximization algorithm, alternating optimization of RIS phase shifts and transmit power.The algorithm initializes equal power allocation and iteratively updates Φ and P until convergence or infeasibility.
  • Simulation setup: The simulations model a multi-antenna base station, an N-element RIS, and randomly placed single-antenna users in a 100m×100m outdoor environment.Results average over 10^3 independent user-position and channel realizations generated according to the 3GPP propagation environment.
  • QoS assumptions: Equal user rate constraints are set relative to the genie-case rate under mutually orthogonal channels and uniform power allocation.The QoS constraints therefore vary with Pmax so the feasibility rate remains approximately constant with Pmax.
  • QoS assumptions: The study reports feasibility rates for different QoS fractions at Pmax = 20 dBW.Table II summarizes the resulting feasibility rates.
  • Feasibility: In very few unfeasible scenarios, the authors relax the QoS constraint and use the corresponding solution, with negligible impact on the reported results.This treatment is justified by the high feasibility rate observed in the simulations.

A. Benchmark: Amplify-and-Forward Relay

The benchmark replaces the RIS with a conventional N-antenna amplify-and-forward relay and compares energy and spectral efficiency under matched channel realizations. The relay requires numerical exhaustive search for its matrix optimization and incurs noise amplification and RF power consumption absent from the RIS case.

  • Benchmark setup: The benchmark uses a conventional N-antenna AF relay in place of the RIS and reuses the same user positions and channel realizations for a fair comparison.The relay is represented by an N × N complex diagonal AF matrix V.
  • Reported comparisons: Figures 3 and 4 compare average spectral efficiency and average energy efficiency of RIS and AF relay systems versus Pmax.The comparisons use Rmin = 0 bps/Hz with configurations M = 32, K = 16, N = 16 and M = 16, K = 8, N = 8.
  • Benchmark setup: Unlike the RIS phase matrix, the relay matrix V has diagonal elements without unit-modulus constraints and is subject to a maximum relay power constraint.This gives the AF relay an additional power-control dimension relative to RIS phase shifts.
  • Power and noise: AF relaying amplifies thermal noise because V is not unitary and consumes RF power to amplify the incoming signal.These effects distinguish the relay power-consumption model from the RIS design case.
  • Optimization problem: The AF benchmark formulates joint energy-efficiency maximization over transmit power P and relay matrix V.It assumes equal end-to-end transmission duration for relay and RIS systems, so no pre-log factor is used.
  • Optimization method: For fixed V, transmit-power optimization follows the RIS approach, whereas optimizing V for fixed P is handled by numerical exhaustive search.The V subproblem is more challenging because of its relay-power constraint.

B. RIS vs AF relay Performance Comparison

The RIS-based designs achieve substantially higher energy efficiency than AF relaying, despite lower spectral efficiency at low transmit powers. The two proposed RIS algorithms perform similarly, with the SFP-based method slightly better.

  • Spectral efficiency: The AF relay outperforms the RIS in spectral efficiency at low Pmax because it has dedicated transmit circuitry and no unit-modulus constraint.At Pmax = 30dBm, RIS loses about 40bps/Hz for M = 32, K = 16, N = 16 and 20bps/Hz for M = 16, K = 8, N = 8.
  • Spectral efficiency: As Pmax increases, the spectral-efficiency gap between RIS and AF relay becomes smaller because relay transmit power becomes less relevant to SE.
  • Algorithm comparison: Both proposed RIS algorithms perform similarly, with the SFP-based algorithm achieving slightly better performance.This similarity holds for both SE and EE results.
  • Energy efficiency: 300%: RIS energy efficiency is larger than AF-relay energy efficiency when Pmax ≥32dBm.The paper attributes this advantage to the RIS-based system's much lower energy consumption.
  • Energy efficiency: For Pmax ≥32dBm, energy efficiency saturates because excess BS transmit power is not used when it would reduce EE.

C. Impact of the QoS Constraints

QoS requirements shape RIS performance across transmit-power regimes: stricter minimum rates can improve SE but reduce EE faster at larger power budgets. The experiments also show distinct SE- and EE-optimal designs, near-optimal SE from the SFP approach, and an EE-optimal RIS size governed by rate gains and RIS power costs.

  • C. Impact of the QoS Constraints: At low Pmax, QoS-constrained designs are often infeasible because the base station lacks sufficient transmit power to meet users’ rate requests.In these cases, the designed solutions coincide at very low SE values.
  • C. Impact of the QoS Constraints: Increasing Rmin increases achievable SE and outperforms the saturating unconstrained Rmin = 0bps/Hz case.The SE curve has a higher slope for larger Rmin.
  • C. Impact of the QoS Constraints: At larger Pmax, stricter QoS constraints cause EE to decrease faster because excess base-station power is used to meet common user-rate requirements.The EE behavior follows the same QoS trend as SE but reflects the additional power expenditure.
  • D. Comparison between the SE and EE Maximizing Designs: For Pmax ≤ 15dBm, EE- and SE-maximizing designs perform similarly, whereas above 15dBm, SE maximization uses all available power but EE maximization stops increasing power beyond a threshold.The comparison includes unconstrained and QoS-constrained EE maximization, SE maximization, and full-power allocation.
  • E. Impact of the number of RIS Elements: An EE-optimal number of RIS elements exists because larger RIS structures trade rate benefits against their energy-consumption cost.The operating point also depends on the numbers of users and base-station antennas and on each RIS element’s power consumption.
  • E. Impact of the number of RIS Elements: The SFP-based approach achieves near-optimal SE, while increasing the number of RIS elements increases achievable SE under the reported settings.The SE comparison uses SNR = 20dB, M = 64, K = N, and Rmin = 2bps/Hz against numerical global optimization.
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