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Robust IRS-Element Activation for Energy Efficiency Optimization in IRS-Assisted Communication Systems With Imperfect CSI

Christos N. Efrem, Ioannis Krikidis

TL;DR

This work addresses maximizing the worst-case energy efficiency in IRS-assisted communications under bounded CSI errors by optimally activating IRS elements and configuring phase shifts. It derives closed-form worst-case SNR expressions for both continuous and discrete phase shifts and formulates robust optimization problems with a minimum SNR constraint. It then proposes a dynamic-programming-based global optimization for continuous shifts with $O(L\log L)$ complexity and a convex-relaxation-based method for discrete shifts with $O(L^{3.5})$ complexity, each accompanied by performance guarantees. Numerical results demonstrate orders-of-magnitude speedups over exhaustive search and clear EE improvements over the all-elements-on baseline, highlighting practical scalability for large IRSs.

Abstract

In this paper, we study an intelligent reflecting surface (IRS)-aided communication system with single-antenna transmitter and receiver, under imperfect channel state information (CSI). More specifically, we deal with the robust selection of binary (on/off) states of the IRS elements in order to maximize the worst-case energy efficiency (EE), given a bounded CSI uncertainty, while satisfying a minimum signal-to-noise ratio (SNR). In addition, we consider not only continuous but also discrete IRS phase shifts. First, we derive closed-form expressions of the worst-case SNRs, and then formulate the robust (discrete) optimization problems for each case. In the case of continuous phase shifts, we design a dynamic programming (DP) algorithm that is theoretically guaranteed to achieve the global maximum with polynomial complexity $O(L\,{\log L})$, where $L$ is the number of IRS elements. In the case of discrete phase shifts, we develop a convex-relaxation-based method (CRBM) to obtain a feasible (sub-optimal) solution in polynomial time $O(L^{3.5})$, with a posteriori performance guarantee. Furthermore, numerical simulations provide useful insights and confirm the theoretical results. In particular, the proposed algorithms are several orders of magnitude faster than the exhaustive search when $L$ is large, thus being highly scalable and suitable for practical applications. Moreover, both algorithms outperform a baseline scheme, namely, the activation of all IRS elements.

Robust IRS-Element Activation for Energy Efficiency Optimization in IRS-Assisted Communication Systems With Imperfect CSI

TL;DR

This work addresses maximizing the worst-case energy efficiency in IRS-assisted communications under bounded CSI errors by optimally activating IRS elements and configuring phase shifts. It derives closed-form worst-case SNR expressions for both continuous and discrete phase shifts and formulates robust optimization problems with a minimum SNR constraint. It then proposes a dynamic-programming-based global optimization for continuous shifts with complexity and a convex-relaxation-based method for discrete shifts with complexity, each accompanied by performance guarantees. Numerical results demonstrate orders-of-magnitude speedups over exhaustive search and clear EE improvements over the all-elements-on baseline, highlighting practical scalability for large IRSs.

Abstract

In this paper, we study an intelligent reflecting surface (IRS)-aided communication system with single-antenna transmitter and receiver, under imperfect channel state information (CSI). More specifically, we deal with the robust selection of binary (on/off) states of the IRS elements in order to maximize the worst-case energy efficiency (EE), given a bounded CSI uncertainty, while satisfying a minimum signal-to-noise ratio (SNR). In addition, we consider not only continuous but also discrete IRS phase shifts. First, we derive closed-form expressions of the worst-case SNRs, and then formulate the robust (discrete) optimization problems for each case. In the case of continuous phase shifts, we design a dynamic programming (DP) algorithm that is theoretically guaranteed to achieve the global maximum with polynomial complexity , where is the number of IRS elements. In the case of discrete phase shifts, we develop a convex-relaxation-based method (CRBM) to obtain a feasible (sub-optimal) solution in polynomial time , with a posteriori performance guarantee. Furthermore, numerical simulations provide useful insights and confirm the theoretical results. In particular, the proposed algorithms are several orders of magnitude faster than the exhaustive search when is large, thus being highly scalable and suitable for practical applications. Moreover, both algorithms outperform a baseline scheme, namely, the activation of all IRS elements.
Paper Structure (11 sections, 3 theorems, 71 equations, 5 figures, 2 algorithms)

This paper contains 11 sections, 3 theorems, 71 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Motivation and State-of-the-Art
  3. Key Contributions
  4. Outline
  5. Mathematical Notation
  6. System Model
  7. Signal Transmission Model
  8. Imperfect CSI Model
  9. IRS Phase-Shift Design
  10. Continuous Phase Shifts
  11. Discrete Phase Shifts

Key Result

Theorem 1

Given that ${\boldsymbol{\phi}}^{\star} \in [0,2\pi)^L$, the closest point rule in equation:discrete_phase-shifts for selecting discrete phase shifts is equivalent to where $\operatorname{round}(x) = \lfloor x + 1/2 \rfloor$. In addition, the closed-form expression equation:discrete_phase-shifts_closed-form requires $O(L \log K) = O(L b)$ arithmetic operations, thus achieving an exponential speed

Theorems & Definitions (6)

  • Remark 1
  • Theorem 1
  • Remark 2
  • Proposition 1
  • Proposition 2
  • Remark 3