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Level Crossing Rate Analysis for Optimal Single-user RIS Systems

Amy S. Inwood, Peter J. Smith, Philippa A. Martin, Graeme K. Woodward

Abstract

We analyse the level crossing rate (LCR) of an uplink single-user (SU) reconfigurable intelligent surface (RIS) aided system. It is assumed that the RIS to base station (RIS-BS) channel is deployed as line-of-sight (LoS), and the user (UE)-RIS and UE-BS channels are correlated Rayleigh. For the optimal RIS reflection matrix, we derive a novel and exact analytical LCR expression for when the direct (UE-BS) channel is blocked, i.e. the RIS-only channel. Also, the existing exact expression for the direct-only channel (equivalent to classical maximal-ratio-combining (MRC)) suffers from extreme numerical precision problems when the BS has many elements. Therefore, we propose a new stable and accurate approximation to the LCR of the direct channel. The approximation is based on replacing any small similar eigenvalues of the channel correlation matrix by their average. We show that increasing the number of elements at the RIS or BS and decreasing channel correlation makes the LCR drop more rapidly for thresholds away from the mean SNR. Crucially, we find that RIS systems do not significantly amplify temporal variations in the channel. This is particularly beneficial for RIS systems considering the difficulty in acquiring channel state information (CSI).

Level Crossing Rate Analysis for Optimal Single-user RIS Systems

Abstract

We analyse the level crossing rate (LCR) of an uplink single-user (SU) reconfigurable intelligent surface (RIS) aided system. It is assumed that the RIS to base station (RIS-BS) channel is deployed as line-of-sight (LoS), and the user (UE)-RIS and UE-BS channels are correlated Rayleigh. For the optimal RIS reflection matrix, we derive a novel and exact analytical LCR expression for when the direct (UE-BS) channel is blocked, i.e. the RIS-only channel. Also, the existing exact expression for the direct-only channel (equivalent to classical maximal-ratio-combining (MRC)) suffers from extreme numerical precision problems when the BS has many elements. Therefore, we propose a new stable and accurate approximation to the LCR of the direct channel. The approximation is based on replacing any small similar eigenvalues of the channel correlation matrix by their average. We show that increasing the number of elements at the RIS or BS and decreasing channel correlation makes the LCR drop more rapidly for thresholds away from the mean SNR. Crucially, we find that RIS systems do not significantly amplify temporal variations in the channel. This is particularly beneficial for RIS systems considering the difficulty in acquiring channel state information (CSI).
Paper Structure (11 sections, 2 theorems, 36 equations, 5 figures)

This paper contains 11 sections, 2 theorems, 36 equations, 5 figures.

Table of Contents

  1. Introduction
  2. System Model
  3. Analysis
  4. LCR of the RIS Channel
  5. LCR of the Direct Channel
  6. Numerical Results
  7. Performance Validation and Variation of System Size
  8. Variation of Link Powers
  9. Impact of Correlation on LCR
  10. Special Case: Independent Channels
  11. Conclusion

Key Result

Theorem 1

The LCR of the SNR variable, $\mathrm{SNR}_\mathrm{R}(t) = cY^2(t)$, across a threshold $T$ is given by where $\omega^2$ will be given in (eq:w2), $Y(t) = \sum_{k=1}^{N}|\mathbf{h}_{\mathrm{ur},k}(t)|$, $\mathbf{h}_{\mathrm{ur}}(t) \sim \mathcal{CN}(\mathbf{0},\beta_{\mathrm{ur}}\mathbf{R}_{\mathrm{ur}})$ and the elements of $\mathbf{h}_{\mathrm{ur}}(t)$ have temporal correlation $\mathbb{E}[\ma

Figures (5)

  • Figure 1: System model showing the uplink channels at time $t$.
  • Figure 2: Plan view of the simulation system layout wu_intelligent_2019.
  • Figure 3: Simulated and analytical LCR comparison.
  • Figure 4: Full RIS system shown in Fig. \ref{['fig:LCR_Paper_System_Diagram']} with and without shadowing applied to the dominant link.
  • Figure 5: Simulated LCRs for a range of element spacings.

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • proof