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Energy-Efficient Wireless Communications with Distributed Reconfigurable Intelligent Surfaces

Zhaohui Yang, Mingzhe Chen, Walid Saad, Wei Xu, Mohammad Shikh-Bahaei, H. Vincent Poor, Shuguang Cui

arXiv:2005.00269v1eess.SPcs.IT

TL;DR

The paper studies energy-efficient resource allocation in wireless networks with distributed RISs, jointly controlling RIS activation and reflection design with transmit beamforming under minimum-rate constraints. It develops iterative algorithms for single-user and multi-user cases, combining successive convex approximation with dual or low-complexity RIS on-off optimization. The proposed scheme achieves up to 33% and 68% energy-efficiency gains over conventional RIS and AF relay schemes, respectively.

  • Problem

    The paper addresses energy-efficiency optimization for wireless networks with distributed RISs while satisfying users’ minimum-rate requirements and controlling RIS activation, reflection coefficients, and transmit beamforming.

  • Method

    The paper proposes iterative single-user and multi-user algorithms using successive convex approximation, closed-form power optimization, dual RIS control, and low-complexity search for multi-RIS activation.

  • Results

    Up to 33% and 68% energy-efficiency gains are reported over the conventional RIS scheme and AF relay scheme, respectively.

  • Takeaways & Limitations

    Distributed RIS energy efficiency can be improved by jointly optimizing RIS on-off status, reflection phases, and transmit-side resource allocation.

Abstract

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This paper investigates the problem of resource allocation for a wireless communication network with distributed reconfigurable intelligent surfaces (RISs). In this network, multiple RISs are spatially distributed to serve wireless users and the energy efficiency of the network is maximized by dynamically controlling the on-off status of each RIS as well as optimizing the reflection coefficients matrix of the RISs. This problem is posed as a joint optimization problem of transmit beamforming and RIS control, whose goal is to maximize the energy efficiency under minimum rate constraints of the users. To solve this problem, two iterative algorithms are proposed for the single-user case and multi-user case. For the single-user case, the phase optimization problem is solved by using a successive convex approximation method, which admits a closed-form solution at each step. Moreover, the optimal RIS on-off status is obtained by using the dual method. For the multi-user case, a low-complexity greedy searching method is proposed to solve the RIS on-off optimization problem. Simulation results show that the proposed scheme achieves up to 33\% and 68\% gains in terms of the energy efficiency in both single-user and multi-user cases compared to the conventional RIS scheme and amplify-and-forward relay scheme, respectively.

I. INTRODUCTION

The paper addresses energy-efficient resource allocation for wireless networks with distributed RISs, which can improve coverage and energy efficiency but introduce network-optimization challenges. It jointly optimizes RIS operation, phase shifts, and transmit beamforming under user-rate and power constraints, with separate low-complexity algorithms for single-user and multi-user settings.

  • Motivation: Existing energy-efficiency optimization considered jointly controlling base-station transmit power and RIS phase shifts, but only a single RIS was modeled.The paper extends this setting to multiple distributed RISs that can cooperatively enhance network coverage.
  • Motivation: Distributed RISs can provide robust transmission and multiple received-signal paths, while potentially consuming less energy than amplify-and-forward relays.Their spatial separation supports robust data transmission, and multiple paths can increase received signal strength.
  • Problem formulation: The paper formulates energy-efficiency maximization with dynamically controlled RIS on-off statuses, RIS phase shifts, and transmit beamforming under minimum-rate, transmit-power, and unit-modulus constraints.The formulation targets distributed RIS wireless networks and jointly controls the main resource-allocation variables.
  • Proposed algorithms: For a single user, the iterative low-complexity algorithm uses successive convex approximation for phase optimization, closed-form power optimization, and the dual method for RIS on-off control.The resulting solution is described as suboptimal for the joint problem, while the RIS on-off subproblem is solved optimally by the dual method.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The system is a downlink MISO network with multiple spatially distributed RISs assisting communication between a multi-antenna BS and single-antenna users. RIS activation and phase shifts affect both received performance and total energy consumption.

  • A. Transmission Model: The considered network contains one BS, K users, and L RISs, with M BS antennas and one antenna per user.Each RIS l has N_l reflecting elements and can assist BS–user communication.
  • A. Transmission Model: The BS transmits unit-power user symbols using beamforming vectors w_k.The transmitted signal is formed at the BS from the user symbols and beamforming vectors.
  • RIS L: Each RIS applies element-wise phase shifts through a diagonal reflection matrix Θ_l.The phase of each reflecting element lies in [0, 2π].
  • RIS L: The received signal and SINR depend on the BS–user, BS–RIS, and RIS–user channel responses and additive Gaussian noise.The sum-rate is computed from the users’ SINRs over channel bandwidth B.
  • B. Power Consumption Model: Total system power includes BS transmit power, BS and user circuit power, and the power consumed by active RIS reflecting elements.RIS consumption is weighted by each RIS’s activation status and number of reflecting elements.

C. Problem Formulation

The paper formulates energy-efficiency maximization as a joint optimization of RIS phases, transmit beamforming, and RIS activation under user-rate and BS-power constraints. The resulting problem is a difficult MINLP, motivating separate single-user and multi-user algorithms.

  • C. Problem Formulation: The objective jointly optimizes RIS reflection coefficients, the beamforming vector, and the RIS on-off vector.The goal is to maximize energy efficiency under minimum user-rate requirements and a total power constraint.
  • C. Problem Formulation: The optimization variables include the phase vector θ, beamforming vector w, and binary RIS activation vector x.R_k denotes the minimum data-rate requirement for user k, while P_max limits BS transmit power.
  • C. Problem Formulation: Each RIS phase coefficient satisfies a unit-modulus constraint, while the BS transmit power satisfies the total power constraint.The unit-modulus condition follows from |e^{jθ_ln}|=1.
  • C. Problem Formulation: The resulting formulation is a mixed-integer nonlinear program even for the single-user case.Its global optimum is generally difficult to obtain.
  • C. Problem Formulation: Two iterative algorithms are proposed to obtain suboptimal solutions for the single-user and multi-user cases.The algorithms address the respective optimization problems separately.
  • C. Problem Formulation: For one user, the problem is solved by alternately optimizing phase and power for fixed RIS activation, then updating the RIS on-off vector.Maximum-ratio transmission is optimal at the BS because there is no multi-user interference.

A. Joint Phase and Power Optimization

For fixed RIS activation, the single-user method alternates phase and power optimization. It uses SCA for phase design and provides convergence to a locally optimal solution of the original nonconvex phase problem.

  • 1) First stage:: With a fixed RIS on-off vector, the first stage jointly optimizes the phase vector and transmit power before updating RIS activation.The phase and power subproblems are handled in separate stages.
  • 1) First stage:: The phase vector is chosen to maximize the effective channel gain, aligning the signals passing through the RISs.The resulting phase choice is independent of the channel amplitude.
  • 1) First stage:: The phase variable is represented through a vector containing the RIS phase entries, with dimensions determined by the total RIS elements.The formulation also includes the stacked optimized phase and RIS-dependent channel terms.
  • 1) First stage:: Earlier SDR and successive-refinement approaches are described as computationally demanding for phase optimization.SDR requires high complexity for a rank-one solution, while successive refinement may require many iterations.
  • 1) First stage:: SCA approximates the nonconvex phase objective with a first-order Taylor expression and solves the resulting convex problem iteratively.The algorithm updates the phase-related variable until the objective converges.
  • 1) First stage:: The SCA phase algorithm has monotonically non-decreasing objective values and converges to a KKT point of the original nonconvex problem.The paper characterizes this as convergence to a locally optimal solution.

2) Second stage:

The second stage optimizes single-user transmit power after phase optimization. Its objective is unimodal, enabling a closed-form optimum characterized by three candidate power values.

  • 2) Second stage:: The Dinkelbach method could solve the energy-efficiency problem through repeated convex subproblems, but the paper derives a closed-form solution instead.The closed-form derivation avoids the additional computational complexity associated with those repeated subproblems.
  • 2) Second stage:: The objective first increases and then decreases with transmit power, so it has a unique stationary optimum within the relevant range.This follows from the monotonic behavior of the auxiliary function f_1(p_1).
  • 2) Second stage:: The optimal power is selected among the minimum transmit power, the zero-derivative power, and the maximum-power boundary.The resulting power-control expression is given in closed form.

B. RIS On-Off Optimization

The RIS on-off problem is a nonconvex fractional mixed-integer program. A dual-based reformulation and Dinkelbach updates yield an optimal RIS activation solution for the single-user case.

  • Problem formulation: The RIS on-off optimization is difficult because it has a fractional objective and a nonconvex constraint involving binary activation variables.The problem is described as a nonconvex MINLP.
  • Fractional optimization: Introducing parameter λ transforms the fractional objective into a parametric objective whose root H(λ)=0 is found using Dinkelbach iterations.For fixed λ, the transformed problem is optimized before updating λ.
  • Convex reformulation: Binary products are represented with auxiliary variables z_lm and linear constraints, then the relaxed problem becomes convex.The reformulation uses z_lm=x_lx_m and relaxes x_l from {0,1} to [0,1].
  • Dual solution: The dual method obtains an integer solution despite relaxation, guaranteeing optimality and feasibility for the original problem.The resulting activation variables remain either 0 or 1.
  • RIS activation rule: RIS l is activated when its coefficient C_l is positive, meaning its rate benefit exceeds the additional power consumption.When C_l>0, keeping RIS l active improves energy efficiency.
  • Iterative algorithm: The overall single-user algorithm alternates phase optimization, power control, and RIS on-off optimization until the objective converges.Algorithm 3 uses Algorithm 1 for phases and Algorithm 2 for RIS activation.

C. Complexity Analysis

The proposed single-user solution has quadratic growth with the number of RISs and lower complexity than the referenced SDR-based algorithm.

  • Phase optimization: The phase optimization costs O(T1QM), where T1 is the number of iterations of Algorithm 1.The factor QM is incurred at each phase-optimization iteration.
  • RIS on-off optimization: The RIS on-off optimization has complexity O(T2T3L^2), combining primal-dual iterations with Dinkelbach parameter updates.T2 counts primal-dual iterations and T3 counts λ-update iterations.
  • Overall complexity: The total complexity is O(T0T1QM + T0T2T3L^2), where T0 is the number of outer iterations.The RIS-dependent term grows quadratically with the total number of RISs.

IV. ENERGY EFFICIENCY OPTIMIZATION WITH MULTIPLE USERS

For multiple users, the paper proposes a low-complexity iterative method that alternately optimizes RIS phases, beamforming, and RIS activation while enforcing minimum user rates.

  • Overall approach: The multi-user energy-efficiency problem is addressed with an iterative algorithm that alternates phase, beamforming, and RIS on-off optimization.The approach is explicitly characterized as low complexity.
  • Phase optimization: With beamforming and RIS activation fixed, phase optimization is handled by slack variables, penalty terms, first-order approximations, and SCA.The penalty formulation enforces unit-modulus phase coefficients at the optimum.
  • Rate constraints: Slack vectors ensure selected constraints hold with equality at the optimum, while separate constraints guarantee each user’s minimum rate requirement.These properties are stated for the reformulated phase-related problem.
  • Phase optimization: The phase subproblem uses convex approximations solved iteratively, with user-rate constraints retained in the formulation.The approximated convex problem is solved at each SCA iteration.

B. Beamforming Optimization

The beamforming subproblem is approximated into convex fractional programs and solved iteratively, while multi-user RIS activation uses a convergent greedy deactivation procedure.

  • Beamforming formulation: Given phase and RIS activation vectors, the beamforming problem is reformulated using slack variables and convex approximations.The reformulation addresses nonconvex constraints in the original beamforming problem.
  • Beamforming solution: The resulting concave-over-convex problem is solved optimally with Dinkelbach iterations inside the SCA procedure.The beamforming optimization therefore combines SCA with Dinkelbach updates.
  • RIS activation challenge: The multi-user RIS activation problem is a nonlinear integer optimization problem that is generally difficult to solve globally.The difficulty arises from the integer RIS on-off vector.
  • Greedy procedure: The greedy algorithm starts with all RISs active and repeatedly tests turning off one active RIS, retaining only feasible candidates.Infeasible candidates receive objective value zero.
  • Greedy procedure: At each iteration, the algorithm removes the RIS whose deactivation gives the highest energy efficiency when that value exceeds the current objective.The active set and reference objective are updated after each accepted deactivation.
  • Convergence: The greedy method terminates when no single RIS deactivation improves energy efficiency and outputs the remaining active set.The algorithm’s objective value increases monotonically and is bounded above, so it converges.

D. Complexity Analysis

The multi-user algorithm decomposes the optimization into phase, beamforming, and RIS on-off subproblems, with total complexity determined by their iterative solutions. Its stated complexity is lower than the SDR-based algorithm in [29].

  • Complexity decomposition: The multi-user optimization alternates phase optimization, beamforming optimization, and RIS on-off status optimization.These components are iterated until the objective value converges.
  • Phase optimization: The phase optimization uses SCA, with complexity O(K3.5 log2(1/ǫ1)).Here, ǫ1 is the SCA accuracy for the phase problem.
  • Beamforming optimization: The beamforming optimization uses SCA with complexity O(TK3.5 log2(1/ǫ2)).T is the number of Dinkelbach iterations and ǫ2 is the SCA accuracy for beamforming.
  • RIS on-off optimization: The RIS on-off optimization has complexity O(L2QM).The objective calculation costs O(QM), and O(L2) iterations are used.
  • Overall complexity: The total complexity is O(S3K3.5 log2(1/ǫ1)+S3TK3.5 log2(1/ǫ2) + S3L2QM).S3 denotes the number of iterations for the overall algorithm.
  • Comparison: The proposed Algorithm 5 has lower complexity than the SDR-based algorithm in [29].

V. SIMULATION RESULTS AND ANALYSIS

Simulations evaluate distributed RISs under varied power, rate, deployment, antenna, and user settings. The proposed DRIS scheme generally improves energy efficiency over centralized RIS and AF relay baselines, while distributed deployment benefits become especially clear as the number of RISs increases.

  • Simulation setup: The simulations use K users in a 300 m × 300 m area, with the BS at the center and L RISs placed around it.Unless otherwise specified, Pmax = 50 dBm, M = 8, L = 8, N = 4, and R = 1 Mbps.
  • Simulation setup: The proposed DRIS scheme is compared with a centralized RIS scheme, CRIS, and an amplify-and-forward relay scheme, AFR.CRIS uses one central RIS, while AFR uses the distributed deployment.
  • Convergence: Eight iterations are sufficient for the proposed algorithm to converge, demonstrating fast convergence.The convergence study uses different initial solutions.
  • Maximum transmit power: 27% and 68% energy efficiency gains are achieved by DRIS over CRIS and AFR, respectively, at high BS maximum transmit power.Energy efficiency first increases and then stabilizes as maximum transmit power rises.
  • Sum-rate: 26% higher sum-rate is achieved by DRIS compared with CRIS as BS maximum transmit power varies.AFR achieves the best sum-rate because it actively retransmits the received signal.
  • Minimum rate demand: 33% and 67% energy efficiency gains are achieved by DRIS over CRIS and AFR, respectively, as minimum rate demand varies.Energy efficiency decreases rapidly at high rate demands because the BS must transmit with high power.
  • RIS deployment: 4.3 Mbits/Joule is achieved with N = 4 and L = 12, compared with 4.0 Mbits/Joule for N = 12 and L = 4.This supports higher energy efficiency when L > N under equal total reflecting elements.

VI. CONCLUSION

The paper jointly optimizes RIS phase shifts, BS beamforming, and RIS activation to maximize energy efficiency under system constraints. Two low-complexity iterative algorithms are proposed, and numerical results show improved energy efficiency over conventional schemes, especially in demanding settings.

  • Problem and objective: The resource-allocation problem jointly optimizes RIS phase shifts, BS transmit beamforming, and RIS on-off status.The objective is system energy efficiency under minimum rate, maximum transmit power, and unit-modulus constraints.
  • Proposed algorithms: Two low-complexity iterative algorithms provide suboptimal solutions for the single-user and multi-user cases.
  • Single-user case: The single-user phase optimization uses SCA and obtains a closed-form solution at each step.
  • Results: Numerical results show that the proposed scheme outperforms conventional schemes in energy efficiency, especially at small maximum transmit power and large rate demands.
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