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Scalable Beamforming Design for Multi-RIS-Aided MU-MIMO Systems with Imperfect CSIT

Mintaek Oh, Jinseok Choi

Abstract

This paper presents a scalable beamforming design for maximizing the spectral efficiency (SE) of multi-reconfigurable intelligent surface (RIS)-aided communications through joint optimization of the precoder and RIS phase shifts in multi-user multiple-input multiple-output (MU-MIMO) systems under imperfect channel state information at the transmitter (CSIT). To address key challenges of the joint optimization problem, we first decompose it into two subproblems by deriving a proper lower bound. We then leverage a generalized power iteration (GPI) approach to identify a superior local optimal precoding solution. We further extend this approach to the RIS design using regularization; we set a RIS regularization function to efficiently handle the unit-modulus constraints, and also find the superior local optimal solution for RIS phase shifts under the GPI-based optimization framework. Subsequently, we propose an alternating optimization method. Our proposed algorithm offers scalable multi-RIS beamforming in terms of computational complexity that scales linearly with the number of RISs, while achieving superior performance. We further reduce the complexity with respect to the number of RIS elements by using diagonal approximation of the channel error covariance and avoiding direct matrix inversion. Simulations validate the proposed algorithm in terms of both the sum SE performance and the scalability.

Scalable Beamforming Design for Multi-RIS-Aided MU-MIMO Systems with Imperfect CSIT

Abstract

This paper presents a scalable beamforming design for maximizing the spectral efficiency (SE) of multi-reconfigurable intelligent surface (RIS)-aided communications through joint optimization of the precoder and RIS phase shifts in multi-user multiple-input multiple-output (MU-MIMO) systems under imperfect channel state information at the transmitter (CSIT). To address key challenges of the joint optimization problem, we first decompose it into two subproblems by deriving a proper lower bound. We then leverage a generalized power iteration (GPI) approach to identify a superior local optimal precoding solution. We further extend this approach to the RIS design using regularization; we set a RIS regularization function to efficiently handle the unit-modulus constraints, and also find the superior local optimal solution for RIS phase shifts under the GPI-based optimization framework. Subsequently, we propose an alternating optimization method. Our proposed algorithm offers scalable multi-RIS beamforming in terms of computational complexity that scales linearly with the number of RISs, while achieving superior performance. We further reduce the complexity with respect to the number of RIS elements by using diagonal approximation of the channel error covariance and avoiding direct matrix inversion. Simulations validate the proposed algorithm in terms of both the sum SE performance and the scalability.
Paper Structure (19 sections, 3 theorems, 60 equations, 11 figures, 2 tables, 3 algorithms)

This paper contains 19 sections, 3 theorems, 60 equations, 11 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Contributions
  4. System Model And Problem Formulation
  5. Signal Model
  6. Channel Acquisition Model
  7. Performance Metrics and Problem Formulation
  8. Proposed Beamforming Design
  9. Optimizing Precoder ${{\bf F}}$
  10. Optimizing RIS Phase Shifts ${\boldsymbol{\Phi}}$
  11. Joint Optimization Algorithm
  12. Complexity Analysis
  13. Reduced-Complexity Approach
  14. Numerical Results
  15. Simulation Environments
  16. ...and 4 more sections

Key Result

Proposition 1

Using the LMMSE estimator, we consider all channel estimation errors as uncorrelated noise with the transmitted signal. Accordingly, the lower bound of the instantaneous SE is derived as where ${\boldsymbol{\Xi}}_k = \sum_{\ell=1}^{L}\left({\boldsymbol {\phi}}^{\sf T}_{\ell} \otimes {\bf I}_N\right) {\bf R}_{k,\ell}^{\sf e}\left({\boldsymbol {\phi}}^{\sf T}_{\ell} \otimes {\bf I}_N\right)^{\sf H}

Figures (11)

  • Figure 1: An illustration of the multi-RIS-aided downlink MU-MIMO system architecture.
  • Figure 2: An illustration of the comparison between the true minimum function and the LogSumExp approximation.
  • Figure 3: An illustration of the computationally efficient matrix inversion of $\bar{{\bf D}}(\bar{{\bf w}})$ in Algorithm 2 via block-wise decomposition.
  • Figure 4: A schematic of the multiuser network with two RISs
  • Figure 5: The sum SE versus the maximum transmit power $P$ dBm for $N = 16$ BS antennas, $K = 4$ users, and $M = 64$ RIS phase shifts.
  • ...and 6 more figures

Theorems & Definitions (7)

  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 1: Complexity Comparison and Scalability