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Reconfigurable Holographic Surface for Simultaneous Wireless Information and Power Transfer

Yuan Guo, Wen Chen, Ziwei Liu, Chaoying Huang, Zhendong Li, Ying Wang

arXiv:2609.19030v1eess.SP

TL;DR

Limited spectrum and energy resources, together with propagation loss for separated SWIPT users, motivate an RHS-enabled SWIPT design. The paper jointly optimizes BS and RHS beamforming; its RHS scheme achieves the highest sum-rate among four schemes, while its low-complexity algorithm substantially reduces runtime with comparable performance.

  • Problem

    Limited spectrum and energy resources constrain next-generation wireless networks, while separated SWIPT can require high BS transmit power to meet EH requirements under severe path loss.

    Source: §I · p. 1

  • Method

    The approach jointly optimizes BS digital and RHS holographic beamforming for ID-user sum-rate subject to individual EH requirements, using WMMSE and MM with an ADMM-based low-complexity algorithm.

    Source: A. Motivations and Contributions · p. 2, §V · p. 12

  • Results

    RHS opt. consistently achieves the highest sum-rate among four schemes, while SOCP and Low. com. converge within 20 iterations and achieve nearly identical performance.

    Source: §IV · pp. 11–12

  • Takeaways & Limitations

    Alg. 1 takes roughly one to two orders of magnitude less time than CVX while maintaining comparable performance.

    Source: §IV · pp. 10–11

Abstract

from arXiv · show

In this paper, we propose to use a novel reconfigurable holographic surface (RHS) to improve the performance of simultaneous wireless information and power transfer (SWIPT) by exploiting the additional spatial degrees of freedom (DoFs) enabled by the holographic interference principle. Specifically, we study an RHS-empowered SWIPT system in which the digital beamformer at the base station (BS) and the holographic beamformer at the RHS are jointly optimized to maximize the weighted sum-rate of information-decoding (ID) users while guaranteeing a minimum harvested energy requirement for each energy-harvesting (EH) user. Due to the non-convexity of the optimization problem, the weighted sum-rate maximization problem is challenging to solve. We first adopt the weighted minimum mean squared error (WMMSE) method to transform the objective into a more tractable form. We then develop an iterative optimization framework, where the BS digital beamforming and the RHS holographic beamforming subproblems are solved via the majorization-minimization (MM) method. Since solving each subproblem can incur prohibitive complexity as the variable dimension increases, we further propose low-complexity solutions for the two subproblems based on the alternating direction method of multipliers (ADMM) methodology. Numerical results validate the convergence behavior of the proposed algorithms and demonstrate that the RHS-aided BS achieves significant performance gains over a conventional fully-digital BS benchmark. Moreover, the proposed low-complexity algorithms substantially reduce computational complexity while maintaining nearly the same performance.

I. INTRODUCTION

SWIPT shares time-frequency resources for information and power, but separated systems can require high BS transmit power when EH users face severe path loss. The paper motivates jointly designed RHS-assisted beamforming to address limited spatial DoFs while meeting EH requirements, and develops iterative and low-complexity optimization methods.

I. INTRODUCTION

  • RIS element counts within a fixed aperture are limited by half-wavelength spacing, constraining spatial DoFs, while severe path loss to distant EH users can require high BS transmit power.RHS units can be spaced sub-wavelength, and an RHS can realize a large-aperture array with fewer active RF components.Source: §I · p. 1

A. Motivations and Contributions

  • Joint digital and holographic beamforming for RHS-assisted SWIPT with coexisting ID and EH users had not been systematically studied.The design maximizes ID-user sum-rate while meeting each EH user's harvested-energy requirement under real-valued RHS amplitude constraints.Source: A. Motivations and Contributions · p. 2
  • WMMSE recasts the sum-rate objective, MM reformulates the resulting subproblems as SOCPs, and ADMM-based solutions reduce the complexity of designing both beamformers.The large number of per-element real-valued amplitude constraints complicates beamforming design.Source: A. Motivations and Contributions · p. 2
  • Numerical results report improved system performance over a conventional phased-array architecture and substantially lower computational complexity than SOCP-based benchmarks for the proposed low-complexity solutions.The paper identifies these as advantages of RHS-assisted transmission and its low-complexity algorithms.Source: A. Motivations and Contributions · p. 2

A. RHS Holographic Interference Principle

  • An RHS uses guided reference waves that leak through sub-wavelength radiating elements to form a desired wavefront, with real-valued amplitude control.The feeds are connected to RF chains, and the reference waves propagate along the waveguide and sequentially excite the radiating elements.Source: A. RHS Holographic Interference… · p. 3
  • For multiple desired beams, the holographic principle extends through weighted superposition of single-beam holographic patterns.This multi-beam extension is also known as HDMA.Source: A. RHS Holographic Interference… · p. 3

B. System Model

The system uses one RHS-equipped BS to serve single-antenna ID and EH users. Its beamforming model uses real-valued RHS amplitude coefficients and assumes perfectly estimated CSI, with mutual coupling incorporated into the effective channel.

B. System Model

  • One RHS-equipped BS simultaneously serves K single-antenna ID users and G single-antenna EH users.The user groups are indexed by K and G, respectively.Source: B. System Model · p. 3
  • The RHS holographic beamformer is a diagonal matrix of element amplitudes, each constrained between 0 and 1, while Q models phase shifts from feeds to radiating units.The digital beamformers are associated with the ID and EH users; transmitted information and energy signals are modeled as independent, unit-variance circularly symmetric complex Gaussian signals.Source: B. System Model · p. 4
  • The design assumes perfectly estimated CSI as a performance benchmark, and treats the geometry- and impedance-determined mutual-coupling matrix as known and absorbed into the effective channel.Robust beamforming under imperfect CSI is identified as future work.Source: B. System Model · p. 4

C. Performance Metrics

The section defines the ID-rate and linear EH metrics, then formulates joint beamforming to maximize ID-user weighted sum-rate subject to EH and power constraints. It transforms the non-convex problem and develops MM- and ADMM-based updates for the BS digital beamformer.

C. Performance Metrics; 2) Energy harvesting metric:

D. Problem Formulation

  • The problem jointly optimizes BS digital and RHS holographic beamforming to maximize weighted ID-user sum-rate while meeting each EH threshold and the BS transmit-power constraint.The objective and constraints are non-convex.Source: D. Problem Formulation · pp. 4–5

A. Problem Reformulation

  • WMMSE reformulates the fractional sum-rate using auxiliary variables β_k and ω_k, and BCA addresses the coupled optimization variables.The transformation remains valid because the RHS-related matrices preserve the received-signal input-output structure required by WMMSE.Source: A. Problem Reformulation · p. 5

B. Optimizing auxiliary variables

  • For fixed beamformers, the auxiliary variables have analytically obtainable optimal updates.The paper uses the WMMSE-derived updates in equations (14)–(15).Source: B. Optimizing auxiliary variables · p. 5

C. Updating The Digital Beamformer

  • The digital-beamformer subproblem's non-convex constraints are linearized with MM, yielding a tight lower bound and an SOCP solvable by standard numerical solvers.The linearization uses a feasible solution from the previous iteration.Source: C. Updating The Digital Beamformer · p. 6
  • Because the SOCP variable dimension scales with L(K + G), solving it with CVX can be prohibitively complex.The paper introduces auxiliary variables to decouple constraints before proposing its ADMM-based update.Source: C. Updating The Digital Beamformer · p. 6
  • The ADMM method provides an iterative closed-form update for the digital beamformer, with independent subproblems enabling parallel updates of the G variables.The method applies block coordinate descent within ADMM.Source: C. Updating The Digital Beamformer · pp. 6–7

D. Updating The Holographic Beamformer

The holographic-beamformer subproblem is first convexified using MM. The paper then applies ADMM with auxiliary variables and sequential block updates to obtain a lower-complexity solution.

D. Updating The Holographic Beamformer

  • MM convexifies the holographic-beamformer constraints, producing a convex problem that can be solved with CVX.The convexification linearizes the left-hand term of constraint (42b).Source: D. Updating The Holographic Beamformer · p. 7
  • To reduce solution complexity, ADMM introduces auxiliary variables φ to decouple the constraints and forms an augmented Lagrangian.The resulting variables are updated by block coordinate descent in the order ψ, {φ_g}, φ_0, and τ.Source: D. Updating The Holographic Beamformer · pp. 7–8
  • With other variables fixed, ψ is updated by solving an unconstrained convex problem.The update has an optimal solution obtained directly from the resulting quadratic problem.Source: D. Updating The Holographic Beamformer · p. 8
  • The φ_g variables decouple and can be updated independently, while φ_0 splits into N element-wise convex quadratic programs with bound constraints.The φ_0 subproblems have directly obtainable optimal values.Source: D. Updating The Holographic Beamformer · p. 8

E. Convergence, Feasible Initialization, and Complexity Analysis

The algorithm's objective sequence is non-decreasing because the MM surrogate constraints are tight at the current iterate and each beamforming subproblem is solved exactly.

E. Convergence, Feasible Initialization, and Complexity Analysis

  • The objective sequence is non-decreasing: tight MM surrogate constraints preserve the previous iterate's feasibility, so exact beamforming-subproblem solutions cannot reduce the objective.The sequence is upper-bounded because the feasible set is compact.Source: 1) Convergence analysis · p. 8

1) Convergence analysis:

The convergence analysis states that the objective sequence converges and that every limit point of the beamforming iterates is a stationary point of (P1).

1) Convergence analysis:

  • Every limit point of the beamforming iterates is a stationary point of (P1).The objective sequence is upper-bounded and convergent because the feasible set is compact.Source: 1) Convergence analysis · p. 9

2) Feasible initialization:

Because Alg. 3 requires a feasible initial point, the authors construct one through an auxiliary feasibility problem. Feasibility is certified by the nonnegative optimal value of a reformulated problem.

3) Feasible initialization:

  • Alg. 3 requires a feasible initial point because its MM-based surrogate constraints are built around the current iterate.The initial-point procedure addresses this requirement.Source: 2) Feasible initialization · p. 9
  • The authors formulate (Pinit,1) to find beamformers and an RHS configuration satisfying the EH-user thresholds.The displayed constraints require Qg to meet Qt for every EH user.Source: 2) Feasible initialization · p. 9
  • Since (Pinit,1) lacks an objective, the authors replace it with (Pinit,2), whose optimal value tests whether a feasible initialization exists.A nonnegative optimum yields a solution that satisfies (Pinit,1) and can initialize Alg. 3.Source: 2) Feasible initialization · p. 9

3) Complexity analysis:

The complexity analysis gives separate costs for the SOCP-based and low-complexity ADMM-based methods. The SOCP expression includes the number of outer iterations.

3) Complexity analysis:

  • The SOCP-based method has overall complexity O(Iout((L(K + G))^3.5 + N^3.5)), with Iout denoting the number of outer iterations.The expression combines the costs of solving (P3) and (P10).Source: 3) Complexity analysis · p. 9
  • For the low-complexity ADMM method, solving (P3) has approximate complexity O(L^3(K + G)^3).Source: 3) Complexity analysis · p. 9
  • For the low-complexity ADMM method, solving (P10) has dominant complexity O(N^3).Source: 3) Complexity analysis · p. 9

IV. NUMERICAL RESULTS

Simulations compare convergence and runtime of the proposed algorithms, then evaluate RHS beamforming against alternative designs. Results show comparable performance between low-complexity and solver-based methods, and higher sum-rates for optimized RHS designs than for the conventional phased-array benchmark.

Simulation setup

  • The simulation setup uses K = 2 information-decoding users, G = 2 energy-harvesting users, L = 6 RF chains, and a 5 × 5 RHS.Receiver noise at EH users is neglected in the linear harvested-energy calculation.Source: §IV · p. 9

Digital-beamforming algorithm

  • For digital-beamforming Alg. 1, ρ in [0.6, 1.4] reaches 10^-8 precision within 40 iterations, with runtime roughly one to two orders below CVX.The objective reaches 10^-4 precision within 40 iterations.Source: §IV · p. 10, Figure 5 · p. 10, Figure 6 · p. 10

Holographic-beamforming algorithm

  • For holographic-beamforming Alg. 2, the objective difference from CVX falls below 10^-4 within 50 iterations, indicating comparable performance.With a suitable η, the algorithm reaches a stationary point within 50 ADMM iterations at 10^-7 tolerance.Source: §IV · p. 10, Figure 7 · p. 10, Figure 8 · p. 11
  • Alg. 2 typically uses one to two orders of magnitude less runtime per iteration than CVX while maintaining comparable performance on (P10).Runtime for both methods increases with the number of RHS units.Source: §IV · pp. 10–11, Figure 9 · p. 11

Overall convergence

  • The SOCP and low-complexity algorithms converge within 20 iterations and achieve nearly identical performance across the tested settings.Sum-rate increases as the number of RHS elements and/or RF chains increases.Source: §IV · p. 11, Figure 10 · p. 11

Sum-rate versus transmit power

  • Across 6–20 dBm, RHS opt. achieves the highest sum-rate, while RHS opt. and RHS dis. 1-bit substantially outperform Tra. BS.RHS dis. 1-bit performs below RHS opt. but above RHS fix.Source: §IV · pp. 11–12, Figure 11 · p. 11

Sum-rate versus RF chains and antenna spacing

  • Increasing RF chains and reducing antenna spacing both benefit sum-rate; RHS opt. outperforms RHS fix. across the considered RF-chain range.The paper attributes the shorter-spacing advantage to higher spatial resolution.Source: §IV · p. 12, Figure 12 · p. 12

V. CONCLUSIONS

The conclusion presents joint BS and RHS beamforming for SWIPT, using WMMSE and MM to address the non-convex design and ADMM to reduce runtime. Simulations report gains over a conventional BS and lower runtime without performance degradation relative to the solver-based approach.

Conclusion

  • The design jointly optimizes BS digital and RHS holographic beamforming to maximize ID-user sum-rate while meeting each EH user's energy requirement.The RHS imposes real-domain amplitude constraints.Source: §V · p. 12
  • WMMSE and MM transform the non-convex problem, while an ADMM algorithm provides analytical variable updates to reduce computational complexity.The solver-based approach becomes prohibitively complex as RF-chain or RHS-unit counts increase.Source: §V · p. 12
  • Simulations show RHS-enabled BS gains over a conventional BS and lower runtime for the low-complexity method without performance degradation versus the solver-based solution.Source: §V · p. 12

APPENDIX

The appendix establishes numerical properties of an ADMM matrix update and outlines a Lagrangian solution for (P8).

A. Invertibility and Numerical Stability of the Matrix Inversions

  • The ADMM penalty parameter ρ adds diagonal loading, making the matrix Hermitian positive definite and nonsingular while improving numerical stability.Its minimum eigenvalue is bounded below by ρ/(2(G + 1)).Source: A. Invertibility and Numerical… · p. 12

B. The Solution of (P8)

  • For (P8), the solution is derived from the Lagrangian by differentiating with respect to f0, applying the power constraint, and considering cases based on whether µ is positive.µ is the Lagrangian multiplier, and the constraint expression is described as monotonically decreasing in µ.Source: B. The Solution of (P8) · pp. 12–13
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