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Joint Beamforming for Continuous Transmissive RIS-Enabled Multiuser Communications

Beining Han, Shumeng Zhang, Deyou Zhang, Qingchao Li, Jun Liu, Chuang Shi

arXiv:2610.07618v1cs.ITmath.IT

TL;DR

CT-RIS channel modeling and joint beamforming remain largely unexplored, particularly because the passive surface requires modeling propagation on both sides of its continuous aperture. The paper develops a joint optimization framework for the continuous phase function and BS precoders, and reports that CT-RIS outperforms the considered benchmarks.

  • Problem

    CT-RIS channel modeling and joint beamforming remain largely unexplored, and their formulation must account for propagation on both sides of the passive continuous aperture.

    Source: §I · p. 1

  • Method

    An alternating optimization framework jointly designs finite-dimensional BS precoders and the continuous CT-RIS phase function, using closed-form precoder updates and calculus-of-variations and minorization-maximization phase updates.

    Source: §I · p. 1

  • Results

    The proposed CT-RIS outperformed the subaperture CT-RIS and element-based T-RIS, with the reported gains associated with continuous phase control and larger effective receiving area.

    Source: §V · p. 6, §IV · p. 6

  • Takeaways & Limitations

    The subaperture benchmark separates the reported advantage over element-based T-RIS into contributions from continuous phase control and larger effective receiving area.

    Source: §V · p. 6

Abstract

from arXiv · show

This paper studies continuous transmissive reconfigurable intelligent surface (CT-RIS)-enabled multiuser downlink communications, where passive beamforming is characterized by a phase function defined over the continuous aperture. We employ a distance-dependent spherical-wave model for the base station (BS)-to-CT-RIS link and introduce a finite-path model based on field-response vectors for the CT-RIS-to-user links. The resulting weighted sum rate maximization problem jointly optimizes the finite-dimensional BS precoders and the infinite-dimensional CT-RIS phase function, giving rise to a mixed finite- and infinite-dimensional non-convex optimization problem. To address this problem, we develop an alternating optimization algorithm that combines a closed-form BS precoder update with a functional phase update derived using the calculus of variations and minorization-maximization. Numerical results demonstrate that the proposed CT-RIS consistently outperforms the considered benchmarks, with the performance gains attributed to spatially continuous phase control and a larger effective receiving area.

I. INTRODUCTION

The paper motivates continuous-aperture modeling for transmissive RISs and develops a CT-RIS downlink model that combines both propagation links across the aperture. It jointly optimizes BS precoding and the continuous phase response.

I. INTRODUCTION

  • CT-RIS channel modeling and joint beamforming remain largely unexplored, and modeling must characterize propagation on both sides of the aperture and coherently integrate their contributions.Source: §I · p. 1
  • The proposed framework combines a spherical-wave BS-to-CT-RIS model with finite-path field-response vectors for CT-RIS-to-user links to form equivalent user channels.Source: §I · p. 1
  • The resulting WSR problem jointly optimizes finite-dimensional BS precoders and an infinite-dimensional phase function, and the paper addresses it with alternating optimization.Source: §I · p. 1

II. SYSTEM MODEL AND PROBLEM FORMULATION

  • The system has an N-antenna BS serving K single-antenna users through a CT-RIS, with direct BS-to-user links assumed blocked or negligible.Source: §II · p. 2

A. Transmit–Receive Signal Model

  • The CT-RIS phase response is a continuous-aperture function, and its ideal lossless phase-only transmission is coherently integrated to obtain each user's received signal.Source: A. Transmit–Receive Signal Model · p. 2

B. Channel Modeling

The BS-to-CT-RIS channel model retains element-to-aperture distance variation to represent wavefront curvature.

B. Channel Modeling

  • The BS-to-CT-RIS link uses exact distances from each BS antenna element to each aperture point under scalar free-space line-of-sight propagation.This retains point-dependent amplitude and phase variations without a plane-wave approximation.Source: 1) BS-to-CT-RIS Link · p. 2

1) BS-to-CT-RIS Link:

The channel formulation combines a distance-dependent BS-to-aperture link with finite-path CT-RIS-to-user links, then jointly optimizes precoding and the aperture phase function for WSR.

1) BS-to-CT-RIS Link:

  • The BS-to-CT-RIS model retains point-dependent amplitude and phase variations across the aperture rather than invoking a plane-wave approximation.Source: 1) BS-to-CT-RIS Link · p. 2

2) FRV-Based CT-RIS-to-User Links:

  • CT-RIS-to-user links use a finite-path field-response model with a far-field plane-wave approximation over the aperture and path-length offsets measured relative to its center.The model can accommodate both LoS and scattered paths.Source: 2) FRV-Based CT-RIS-to-User Links · pp. 2–3

C. Problem Formulation

  • The WSR formulation jointly optimizes BS precoding and the continuous phase function under a power limit, with user SINRs coupled across both variable blocks and a pointwise unit-modulus constraint.Source: C. Problem Formulation · p. 3

III. AO-COV-MM ALGORITHM FOR JOINT PRECODING AND FUNCTIONAL PHASE OPTIMIZATION

The AO-CoV-MM algorithm makes the WSR problem blockwise tractable through auxiliary variables and alternates updates for the auxiliary variables, BS precoders, and CT-RIS phase function.

III. AO-COV-MM ALGORITHM FOR JOINT PRECODING AND FUNCTIONAL PHASE OPTIMIZATION

  • AO-CoV-MM iteratively updates auxiliary variables, BS precoders, and the CT-RIS phase function.Source: §III · p. 3

A. Equivalent Reformulation of the WSR Problem

B. Block Update Rules

The block updates alternate among auxiliary variables, BS precoders, and the CT-RIS phase function. The auxiliary-variable and precoder subproblems are solved optimally for fixed values of the remaining variables.

B. Block Update Rules

  • The auxiliary variables, BS precoders, and CT-RIS phase function are updated sequentially, with an MM step for the phase function guaranteeing a nondecreasing transformed objective.The auxiliary-variable and BS-precoding subproblems are optimally solved for fixed values of the other variables.Source: B. Block Update Rules · p. 3

1) Auxiliary Variable Update:

  • For fixed precoders and phase function, the auxiliary variables are obtained using the Lagrangian-dual and complex-quadratic transforms.Source: 1) Auxiliary Variable Update · p. 3

2) BS Precoder Update:

  • The BS-precoder subproblem is a concave quadratic maximization over a convex feasible set, and its optimal dual variable can be found by one-dimensional bisection when required.If P(0) ≤ PT, the optimal dual variable is zero; otherwise, bisection finds the solution to P(ν⋆) = PT.Source: 2) BS Precoder Update · p. 4

3) CT-RIS Phase-Function Update:

The CT-RIS phase-function update uses calculus of variations to characterize stationarity and minorization-maximization to construct a pointwise update. The MM-based update guarantees a nondecreasing transformed objective.

3) CT-RIS Phase-Function Update:

  • The phase function is represented as z(s) = e^-jθ(s), and calculus of variations characterizes its functional stationarity condition.The variation considers θ(s) mapped to θ(s) + εU(s), with U(s) an arbitrary real-valued function.Source: 3) CT-RIS Phase-Function Update · p. 4
  • Because Rz depends on z itself, the stationarity condition is implicit and does not directly provide a monotonic update.Source: 3) CT-RIS Phase-Function Update · p. 4
  • The MM construction forms a tight minorizer and maximizes it subject to |z(s)| = 1 to obtain a pointwise phase update.The minorizer is tight at z = zt; the update uses Φt(s) = a(s) − Rzt(s).Source: 3) CT-RIS Phase-Function Update · p. 4
  • The resulting phase-function update guarantees a nondecreasing transformed objective within the overall AO-CoV-MM procedure.Source: 3) CT-RIS Phase-Function Update · p. 4

1) Convergence:

The update order yields a nondecreasing sequence of transformed objective values, which is bounded and converges to a finite value. This convergence does not guarantee global optimality.

1) Convergence:

  • The auxiliary-variable, BS-precoder, and MM phase-function updates make the transformed objective nondecreasing across iterations.The phase step's tight minorization ensures it does not decrease Fθ or F0.Source: 1) Convergence · pp. 4–5
  • The transformed-objective sequence is upper bounded and converges to a finite value because the WSR is finite under the stated constraints.The passage identifies bounded transmit power, unit-modulus CT-RIS transmission, and positive noise variance as conditions ensuring finite WSR.Source: 1) Convergence · p. 5
  • Convergence of the objective-value sequence does not establish global optimality.Source: 1) Convergence · p. 5

2) Computational Complexity:

The numerical results compare the proposed CT-RIS with subaperture and element-based T-RIS benchmarks across transmit power, aperture size, and user count. The section also reports computational costs for the algorithm’s main updates.

2) Computational Complexity:

  • Using M × M Gauss–Legendre quadrature nodes, channel reconstruction costs O(M^2KN), auxiliary-variable updates O(K^2N), and constructing Q costs O(KN^2).The BS-precoder update costs O(KN^2 + TbN^3), where Tb is the number of bisection iterations.Source: 2) Computational Complexity · p. 5

IV. NUMERICAL RESULTS

  • Across transmit powers, the proposed CT-RIS achieves the highest WSR, followed by the subaperture CT-RIS and element-based T-RIS.The reported gaps reflect both effective receiving area and spatially continuous phase control.Source: Figure 2 · p. 5, §IV · p. 6
  • As aperture area increases, WSR rises monotonically for all schemes, while the proposed CT-RIS remains highest across the considered sizes.The performance gaps remain evident across aperture sizes.Source: Figure 3 · p. 6, §IV · p. 6
  • As the number of users increases under a fixed total transmit-power budget, WSR decreases for all schemes, but the proposed CT-RIS retains a clear advantage over both benchmarks.The decrease reflects increased difficulty of multiuser interference management.Source: Figure 4 · p. 6, §IV · p. 6

V. CONCLUSION

The paper develops joint beamforming for CT-RIS-enabled multiuser downlink communications using an AO-CoV-MM algorithm. Its benchmarks attribute the proposed CT-RIS’s WSR advantage to spatially continuous phase control and larger effective receiving area.

V. CONCLUSION

  • The proposed CT-RIS outperforms the subaperture CT-RIS, while the subaperture CT-RIS outperforms the element-based T-RIS.The paper attributes these gains, respectively, to spatially continuous phase control and larger effective receiving area.Source: §V · p. 6
  • The framework jointly optimizes BS precoders and the spatially continuous CT-RIS phase function to maximize WSR.It addresses the resulting mixed finite- and infinite-dimensional non-convex problem with an AO-CoV-MM algorithm.Source: §V · p. 6
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