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Fair Beam Allocations through Reconfigurable Intelligent Surfaces

Rujing Xiong, Ke Yin, Tiebin Mi, Jialong Lu, Kai Wan, Robert Caiming Qiu

TL;DR

This paper proposes the Moreau-Yosida approximation (MA) algorithm, which pursues the optimal beamforming designs based on arbitrary utility functions in the user's received power, and leverages the passive characteristics of RIS, modeling the reflecting signals and articulating a comprehensive max-min optimization framework.

Abstract

A fair beam allocation framework through reconfigurable intelligent surfaces (RISs) is proposed, incorporating the Max-min criterion. This framework focuses on designing explicit beamforming functionalities through optimization. Firstly, realistic models, grounded in geometrical optics, are introduced to characterize the input/output behaviors of RISs, effectively bridging the gap between the requirements on explicit beamforming operations and their practical implementations. Then, a highly efficient algorithm is developed for Max-min optimizations involving quadratic forms. Leveraging the Moreau-Yosida approximation, we successfully reformulate the original problem and propose iterations to attain the optimal solution. A comprehensive analysis of the algorithm's convergence is provided. Importantly, this approach exhibits excellent extensibility, making it readily applicable to address a broader class of Max-min optimization problems. Finally, numerical and prototype experiments are conducted to validate the effectiveness of the framework. With the proposed beam allocation framework and algorithm, we clarify that several crucial redistribution functionalities of RISs, such as explicit beam-splitting, fair beam allocation, and wide-beam generation, can be effectively implemented. These explicit beamforming functionalities have not been thoroughly examined previously.

Fair Beam Allocations through Reconfigurable Intelligent Surfaces

TL;DR

This paper proposes the Moreau-Yosida approximation (MA) algorithm, which pursues the optimal beamforming designs based on arbitrary utility functions in the user's received power, and leverages the passive characteristics of RIS, modeling the reflecting signals and articulating a comprehensive max-min optimization framework.

Abstract

A fair beam allocation framework through reconfigurable intelligent surfaces (RISs) is proposed, incorporating the Max-min criterion. This framework focuses on designing explicit beamforming functionalities through optimization. Firstly, realistic models, grounded in geometrical optics, are introduced to characterize the input/output behaviors of RISs, effectively bridging the gap between the requirements on explicit beamforming operations and their practical implementations. Then, a highly efficient algorithm is developed for Max-min optimizations involving quadratic forms. Leveraging the Moreau-Yosida approximation, we successfully reformulate the original problem and propose iterations to attain the optimal solution. A comprehensive analysis of the algorithm's convergence is provided. Importantly, this approach exhibits excellent extensibility, making it readily applicable to address a broader class of Max-min optimization problems. Finally, numerical and prototype experiments are conducted to validate the effectiveness of the framework. With the proposed beam allocation framework and algorithm, we clarify that several crucial redistribution functionalities of RISs, such as explicit beam-splitting, fair beam allocation, and wide-beam generation, can be effectively implemented. These explicit beamforming functionalities have not been thoroughly examined previously.
Paper Structure (18 sections, 2 theorems, 32 equations, 19 figures, 1 table, 1 algorithm)

This paper contains 18 sections, 2 theorems, 32 equations, 19 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. Outline
  4. Notations
  5. Physical modeling and problem formulation
  6. The Response of RISs to External Illuminations
  7. Max-min Fair Beam Allocations Problem
  8. Moreau-Yosida Approximation-based Method
  9. Flexible beam allocation functionalities
  10. Beam-Spliting
  11. Fair Beam Allocation
  12. Wide-beam Generation
  13. Power Level Control
  14. Performance Evaluation and Comparison
  15. Power Boosting
  16. ...and 3 more sections

Key Result

Theorem 1

Suppose $\{f_k(\mathbf{w})\}_{k=1}^K$ are differentiable functions bounded from below. Then for every $\epsilon>0$, if $\mu>\frac{1}{2\epsilon}$, then there exists some non-negative non-zero vector $\hat{\mathbf{p}}=[\hat{p}_1,\dots,\hat{p}_K]^T$ such that the minimizer $\hat{\boldsymbol{\omega}}$ o where $\hat{\mathbf{w}}=(e^{j\hat{\boldsymbol{\omega}}})^H$.

Figures (19)

  • Figure 1: RIS-aided multi-user wireless communication.
  • Figure 2: The RIS is illuminated by a single incident EM wave from $(r^\text{i}, \theta^\text{i}, \phi^\text{i})$.
  • Figure 3: Incident EM waves uniformly scatter in all directions over the hemisphere of reflection.
  • Figure 4: Geometry used for field calculations of a line source along the y-axis.
  • Figure 5: Phase discrepancy due to the interelement path length difference for linear RISs. The RIS is illuminated by uniform plane waves lying in the $yoz$ plane. The incident EM wave (in blue) is originating from $\theta^\text{i}$. The scattered electric field is observed at $(r^\text{s}, \theta^\text{s})$ (in red). The incident and scattered path length differences between the origin and the unit at $(n-1)d$ are $d_{\mathbf{p}_n}^\text{i} = - (n-1)d \sin \theta^\text{i}$ and $d_{\mathbf{p}_n}^\text{s} = - (n-1)d \sin \theta^\text{s}$, respectively.
  • ...and 14 more figures

Theorems & Definitions (6)

  • Remark 1
  • Theorem 1
  • proof
  • Remark 2
  • Theorem 2
  • proof