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IRS Compensation of Hyper-Rayleigh Fading: How Many Elements Are Needed?

Aleksey S. Gvozdarev

TL;DR

This work tackles the problem of determining the minimum number of IRS elements needed to mitigate severe fading captured by Hyper-Rayleigh Regimes (HRRs) in IRS-assisted links under an inverse power Lomax (IPL) fading model. It develops closed-form and approximate channel statistics for a single-IRS and a total-IRS channel, uses HRR boundaries defined through an effective shape parameter $\hat{\alpha}_g$ to identify the required element counts, and reports concrete thresholds: $N_f^*=2$–$6$ to exit full-HRR and $N_n^*=4$–$14$ to reach no-HRR, depending on channel parameters. The results offer fading-centric IRS design guidance, showing that modest IRS configurations can substantially improve reliability even when sublinks remain in full-HRR, with practical implications for sub-array sizing in 6G systems. The methodology combines Fox $H$-function based exact statistics, gamma approximations, and HRR theory to deliver actionable, quantitative guidelines for IRS deployment under severe fading conditions.

Abstract

The letter introduces and studies the problem of defining the minimum number of Intelligent Reflecting Surface (IRS) elements needed to compensate for heavy fading conditions in multipath fading channels. The fading severity is quantified in terms of Hyper-Rayleigh Regimes (HRRs) (i.e., full-HRR (worst-case conditions), strong-, weak-, and no-HRR), and the channel model used (Inverse Power Lomax (IPL)) was chosen since it can account for all HRRs. The research presents the derived closed-form channel coefficient envelope statistics for the single IRS-element channel with IPL statistics in both subchannels and total IRS-assisted channel, as well as tight approximations for the channel coefficient and instantaneous signal-to-noise ratio (SNR) statistics for the latter. The derived expressions helped estimate channel parameters corresponding to the specific HRRs of the total channel and demonstrate that while both single links (i.e., ''source-IRS'' and ''IRS-destination'') are in full-HRR, the minimum number of IRS elements needed to bring the total IRS-assisted link (''source-IRS-destination'') out of full-HRR is no less than $6$ (for the whole range on the IPL scale parameter corresponding full-HRR). Furthermore, the minimum number of IRS elements required to bring the total IRS-assisted link into no-HRR is $14$ (under the same conditions).

IRS Compensation of Hyper-Rayleigh Fading: How Many Elements Are Needed?

TL;DR

This work tackles the problem of determining the minimum number of IRS elements needed to mitigate severe fading captured by Hyper-Rayleigh Regimes (HRRs) in IRS-assisted links under an inverse power Lomax (IPL) fading model. It develops closed-form and approximate channel statistics for a single-IRS and a total-IRS channel, uses HRR boundaries defined through an effective shape parameter to identify the required element counts, and reports concrete thresholds: – to exit full-HRR and – to reach no-HRR, depending on channel parameters. The results offer fading-centric IRS design guidance, showing that modest IRS configurations can substantially improve reliability even when sublinks remain in full-HRR, with practical implications for sub-array sizing in 6G systems. The methodology combines Fox -function based exact statistics, gamma approximations, and HRR theory to deliver actionable, quantitative guidelines for IRS deployment under severe fading conditions.

Abstract

The letter introduces and studies the problem of defining the minimum number of Intelligent Reflecting Surface (IRS) elements needed to compensate for heavy fading conditions in multipath fading channels. The fading severity is quantified in terms of Hyper-Rayleigh Regimes (HRRs) (i.e., full-HRR (worst-case conditions), strong-, weak-, and no-HRR), and the channel model used (Inverse Power Lomax (IPL)) was chosen since it can account for all HRRs. The research presents the derived closed-form channel coefficient envelope statistics for the single IRS-element channel with IPL statistics in both subchannels and total IRS-assisted channel, as well as tight approximations for the channel coefficient and instantaneous signal-to-noise ratio (SNR) statistics for the latter. The derived expressions helped estimate channel parameters corresponding to the specific HRRs of the total channel and demonstrate that while both single links (i.e., ''source-IRS'' and ''IRS-destination'') are in full-HRR, the minimum number of IRS elements needed to bring the total IRS-assisted link (''source-IRS-destination'') out of full-HRR is no less than (for the whole range on the IPL scale parameter corresponding full-HRR). Furthermore, the minimum number of IRS elements required to bring the total IRS-assisted link into no-HRR is (under the same conditions).
Paper Structure (16 sections, 5 theorems, 8 equations, 1 figure)

This paper contains 16 sections, 5 theorems, 8 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. IRS-assisted communication model
  4. Hyper-Rayleigh fading classification
  5. Fading channel model
  6. Problem statement
  7. Derived solution
  8. Closed-form IRS output envelope statistics
  9. Approximate IRS output SNR statistics
  10. Hyper-Rayleigh regimes of the IRS-assisted channel
  11. Derived minimum number of required IRS elements
  12. Numerical example and discussion
  13. RIS-assisted HRR mitigation
  14. Discussion
  15. Limitations of the derived results and further extensions
  16. ...and 1 more sections

Key Result

Theorem 1

The probability density function of the single IRS-element channel coefficient $Z_j$ is given by: where $\mathrm{H(\cdot)}$ is the Fox H-function (see Kil04.

Figures (1)

  • Figure 1: Exemplified IRS-assisted system performance with the results obtained from Theorem 5.

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5