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Coverage Probability of Distributed IRS Systems Under Spatially Correlated Channels

Anastasios Papazafeiropoulos, Cunhua Pan, Ahmet Elbir, Pandelis Kourtessis, Symeon Chatzinotas, John M. Senior

TL;DR

This letter suggests the use of multiple distributed intelligent reflecting surfaces (IRSs) towards a smarter control of the propagation environment and obtains the optimal coverage probability by using the deterministic equivalent (DE) analysis.

Abstract

This paper suggests the use of multiple distributed intelligent reflecting surfaces (IRSs) towards a smarter control of the propagation environment. Notably, we also take into account the inevitable correlated Rayleigh fading in IRS-assisted systems. In particular, in a single-input and single-output (SISO) system, we consider and compare two insightful scenarios, namely, a finite number of large IRSs and a large number of finite size IRSs to show which implementation method is more advantageous. In this direction, we derive the coverage probability in closed-form for both cases contingent on statistical channel state information (CSI) by using the deterministic equivalent (DE) analysis. Next, we obtain the optimal coverage probability. Among others, numerical results reveal that the addition of more surfaces outperforms the design scheme of adding more elements per surface. Moreover, in the case of uncorrelated Rayleigh fading, statistical CSI-based IRS systems do not allow the optimization of the coverage probability.

Coverage Probability of Distributed IRS Systems Under Spatially Correlated Channels

TL;DR

This letter suggests the use of multiple distributed intelligent reflecting surfaces (IRSs) towards a smarter control of the propagation environment and obtains the optimal coverage probability by using the deterministic equivalent (DE) analysis.

Abstract

This paper suggests the use of multiple distributed intelligent reflecting surfaces (IRSs) towards a smarter control of the propagation environment. Notably, we also take into account the inevitable correlated Rayleigh fading in IRS-assisted systems. In particular, in a single-input and single-output (SISO) system, we consider and compare two insightful scenarios, namely, a finite number of large IRSs and a large number of finite size IRSs to show which implementation method is more advantageous. In this direction, we derive the coverage probability in closed-form for both cases contingent on statistical channel state information (CSI) by using the deterministic equivalent (DE) analysis. Next, we obtain the optimal coverage probability. Among others, numerical results reveal that the addition of more surfaces outperforms the design scheme of adding more elements per surface. Moreover, in the case of uncorrelated Rayleigh fading, statistical CSI-based IRS systems do not allow the optimization of the coverage probability.

Paper Structure

This paper contains 10 sections, 5 theorems, 22 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. System Model
  3. Performance Analysis
  4. Main Results
  5. Finite $M$ and large $N$ analysis
  6. Large $M$ and finite $N$ analysis
  7. Reflecting beamforming optimization
  8. Numerical Results
  9. Conclusion
  10. Proof of Lemma \ref{['deriv1']}

Key Result

Lemma 1

The SNR of a SISO transmission, enabled by $M$ large IRSs with correlated Rayleigh fading is approximated by where $B_{M}\!=\!\sum^{M}_{m=1}\!\beta_{m}\!\tr\!\left( {\mathbf{R}}_{m,1}{\boldsymbol{\Phi}}_{m}{\mathbf{R}}_{m,2}{\boldsymbol{\Phi}}_{m}^{{\mathsf{H}}}\right)$ with $\beta_{m}\!=\!\beta_{m,1}$$\beta_{m,2}$.

Figures (1)

  • Figure 1: Coverage probability versus the target rate $T$ (analytical results and MC simulations) of a SISO system with correlated Rayleigh fading assisted by $(a)$$M$ IRSs each having a large number of elements ($N \to \infty$) and $(b)$ a large number of IRSs ($M\to \infty$) each having $N$ elements; $(c)$ Coverage probability versus the number of iterations for both cases.

Theorems & Definitions (14)

  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 2
  • proof
  • Proposition 2
  • ...and 4 more