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Multi-RIS-aided Wireless Systems: Statistical Characterization and Performance Analysis

Tri Nhu Do, Georges Kaddoum, Thanh Luan Nguyen, Daniel Benevides da Costa, Zygmunt J. Haas

arXiv:2104.01912v2cs.IT

TL;DR

Distributed multi-RIS systems require statistical models that accommodate differing RIS sizes and independent but non-identically distributed channels. The paper proposes ERA and ORA and applies a method-of-moments framework to derive approximate channel distributions and OP/EC expressions. ERA outperforms ORA in OP and EC, while RIS element allocation and locations significantly affect performance.

  • Problem

    Accurate fading characterization remains open for distributed multi-RIS systems because prior analyses often use simplified, i.i.d., or deterministic channel models.

  • Method

    The paper proposes ERA and ORA and uses a method-of-moments framework to approximate end-to-end channel magnitudes with Gamma or Log-Normal models.

  • Results

    ERA outperforms ORA in outage probability and ergodic capacity, while Gamma and Log-Normal approximations provide better OP approximations than Gaussian models.

  • Takeaways & Limitations

    Reflecting-element allocation and RIS locations significantly influence both schemes, and ORA can achieve higher energy efficiency in specific target spectral-efficiency ranges.

Abstract

from arXiv · show

In this paper, we study the statistical characterization and modeling of distributed multi-reconfigurable intelligent surface (RIS)-aided wireless systems. Specifically, we consider a practical system model where the RISs with different geometric sizes are distributively deployed, and wireless channels associated to different RISs are assumed to be independent but not identically distributed (i.n.i.d.). We propose two purpose-oriented multi-RIS-aided schemes, namely, the exhaustive RIS-aided (ERA) and opportunistic RIS-aided (ORA) schemes. A mathematical framework, which relies on the method of moments, is proposed to statistically characterize the end-to-end (e2e) channels of these schemes. It is shown that either a Gamma distribution or a Log-Normal distribution can be used to approximate the distribution of the magnitude of the e2e channel coefficients in both schemes. With these findings, we evaluate the performance of the two schemes in terms of outage probability (OP) and ergodic capacity (EC), where tight approximate closed-form expressions for the OP and EC are derived. Representative results show that the ERA scheme outperforms the ORA scheme in terms of OP and EC. In addition, under i.n.i.d. fading channels, the reflecting element settings and location settings of RISs have a significant impact on the system performance of both the ERA or ORA schemes.

I. INTRODUCTION

The paper addresses statistical modeling of distributed multi-RIS systems with independently but non-identically distributed channels by proposing ERA and ORA schemes and a method-of-moments framework. It derives approximate channel distributions and performance expressions, showing ERA generally performs better in OP and EC while infrastructure settings materially affect both schemes.

  • I. INTRODUCTION: Existing multi-RIS analyses often assume i.i.d., deterministic, or simplified channels, leaving accurate fading characterization for distributed systems unresolved.The paper highlights the need to model independent but non-identically distributed channels across RISs while retaining small-scale fading.
  • I. INTRODUCTION: The paper proposes ERA, where all RISs participate, and ORA, where only one scheduled RIS assists transmission.ERA can improve performance but requires greater complexity and fronthaul/backhaul load, whereas ORA can use other RISs for other users or purposes.
  • I. INTRODUCTION: A method-of-moments framework statistically characterizes the magnitude of the end-to-end channel and estimates candidate distribution parameters by matching candidate and population moments.The framework also verifies approximation accuracy by comparing the simulated true distribution with the matched distribution.
  • I. INTRODUCTION: The framework finds Gamma or Log-Normal approximations for ERA and a Log-Normal approximation for ORA under the considered fading environment.These distributional findings support subsequent OP and EC analysis.
  • I. INTRODUCTION: ERA outperforms ORA in OP and EC, although ORA achieves higher EE in a specific target-SE range and ERA is more robust to infrastructure changes.The number and allocation of reflecting elements and RIS locations significantly affect performance; centralized and distributed deployments have setting-dependent advantages.

B. The Opportunistic RIS-aided Scheme

The ORA scheme selects one RIS to assist direct transmission, reducing receiver processing and resource usage while requiring the same number of CSI estimations as ERA. Its channel characterization uses a flexible moment-matching framework for complex, non-identically distributed components.

  • Scheme operation: ORA schedules only the most appropriate RIS to assist the direct transmission, reducing receiver-side complexity and energy usage.The receiver processes substantially fewer reflecting signals than in ERA.
  • RIS selection: The selected RIS is chosen by maximizing the end-to-end received SNR after ideal phase-shift configuration.Each reflecting element is configured to align the reflected signal with the direct-link phase.
  • Resource usage: ORA requires the same number of CSI estimations as ERA but processes (L_n + 1) signals instead of (N × L_n + 1).This difference in participating passive elements creates a significant energy-efficiency gap between the schemes.
  • Statistical characterization: The e2e channel magnitude for ORA is represented by R = h_0 + M_V, whose exact distribution is difficult to derive because of its complicated structure.The proposed three-step framework addresses this by selecting candidate distributions, estimating parameters through moments, and verifying accuracy.
  • Statistical characterization: The framework applies to non-identically distributed channel components and can iteratively approximate distributions of individual SINR components.Repeated distribution fitting can reduce accuracy, although additional reflecting elements may compensate for this inaccuracy.

B. Statistical Channel Characterization of the ERA Scheme Based on Gamma Distribution

For ERA, the e2e channel magnitude is approximated by a Gamma distribution whose parameters are obtained from moments, enabling corresponding power-gain and outage-probability characterizations.

  • Gamma approximation: The true ERA channel magnitude Z is accurately approximated by a Gamma distribution.The approximation uses parameters α_Z and β_Z.
  • Parameter estimation: The Gamma parameters α_Z and β_Z are estimated by matching moments of the candidate distribution to moments of the true channel magnitude.The resulting approximate PDF and CDF of Z follow from the Gamma model.
  • Power-gain characterization: The squared ERA channel magnitude Z^2 can be represented using a Generalized Gamma distribution derived from the Gamma approximation of Z.Its distribution functions use parameters related to α_Z and β_Z.
  • Outage probability: ERA outage probability is defined as the probability that instantaneous mutual information falls below the target spectral-efficiency threshold R_th.The Gamma approximation yields an approximate closed-form outage-probability expression.

2) Ergodic Capacity:

The paper derives approximate closed-form ergodic-capacity expressions for ERA using Gamma and Log-Normal channel approximations and analyzes the associated transformed channel distributions.

  • Gamma-based analysis: The ERA ergodic capacity is obtained by averaging instantaneous capacity over channel realizations and invoking the Gamma-based channel characterization.This produces an approximate closed-form expression for EC.
  • Log-Normal characterization: The ERA channel magnitude can alternatively be approximated by a Log-Normal distribution with parameters estimated by matching its first two moments.The Log-Normal model is characterized by ν_Z and ζ_Z.
  • Log-Normal characterization: The method of moments determines the Log-Normal parameters from the first two moments of the true channel magnitude.The required moments are derived from a general k-th-moment expression for Z.
  • Capacity approximation: A separate Log-Normal approximation is applied to Z^2 to support tractable ergodic-capacity analysis.This avoids a non-closed-form integral that would result from directly transforming the Log-Normal model of Z.

IV. PERFORMANCE ANALYSIS OF THE ORA SCHEME

The ORA performance analysis characterizes the selected channel magnitude through distributions of the maximum RIS contribution and the direct-link component, yielding approximate closed-form PDF and CDF expressions.

  • Channel characterization: The ORA e2e SNR is expressed using the direct-link magnitude and the maximum contribution among the RISs.The selected RIS changes the statistical characterization through a maximum operation.
  • Channel characterization: Approximate closed-form expressions are derived for the CDF and PDF of the ORA channel magnitude R.These expressions combine the distributions of h_0 and M_V.
  • Maximum-RIS contribution: The distribution of the maximum RIS contribution is derived from the independent but non-identically distributed variables V_k.The derivation uses the CDF of M_V and the pairwise disjoint events associated with the maximizing RIS.
  • Approximation procedure: An integral arising in the ORA distribution derivation is approximated using the M-staircase approximation.Differentiation then yields the PDF of R and the PDF of M_V.

1) Outage Probability:

The ORA scheme is analyzed using approximate Gamma and Log-Normal models for its end-to-end channel, enabling approximate outage-probability and ergodic-capacity expressions. The Gamma-based capacity integral lacks a closed form, motivating the Log-Normal characterization.

  • Outage Probability: An approximate closed-form expression for ORA outage probability is obtained using Theorem 3.
  • Ergodic Capacity: The ORA ergodic capacity is defined as E[log2(1 + ¯ρR2)] and analyzed through an approximate closed-form expression.
  • Ergodic Capacity: The Gamma-based ORA ergodic-capacity integral has no closed-form solution, motivating a Log-Normal approximation.
  • Statistical Channel Characterization: Theorem 4 approximates the true ORA channel magnitude distribution R by LogNormal(νR, ζR), with parameters estimated from its moments.

1) Outage Probability:

The paper validates its distributional and performance analysis through simulations of channel distributions, outage probability, and ergodic capacity. The approximate results agree well with simulations, while Gamma and Log-Normal models outperform Gaussian modeling for low-outage accuracy.

  • Results and Discussions: Theoretical and simulation results for ERA and ORA outage probability and ergodic capacity are well corroborated in Fig. 3.
  • Results and Discussions: Gamma and Log-Normal distributions provide better approximations than the Gaussian distribution for the evaluated channel distributions.
  • Results and Discussions: For OP below 10^-2, Gaussian modeling shows a tangible gap from true values because of inaccurate left-tail approximation.
  • Results and Discussions: EC analytical and simulation results remain well corroborated across Gamma, Log-Normal, and Gaussian approximations, including high transmission power.

A. The Difference in Lower Tail Between Gamma and LogNormal Distributions

The Gamma and Log-Normal approximations differ mainly in their lower-tail behavior, while their accuracy becomes comparable for sufficiently large shape parameters and reflecting-element counts. The section also relates these approximations and channel decompositions to performance trends in distributed RIS systems.

  • A. The Difference in Lower Tail Between Gamma and LogNormal Distributions: The lower-tail discrepancy between Gamma and Log-Normal distributions is very small, making their OP estimates nearly identical.The difference arises specifically from the distributions’ left tails.
  • A. The Difference in Lower Tail Between Gamma and LogNormal Distributions: When α ≥30, the lower tails of the Gamma and Log-Normal distributions are similar.For the considered system, the ERA shape parameter αZ lies between 77 and 120.
  • A. The Difference in Lower Tail Between Gamma and LogNormal Distributions: Under i.n.i.d. fading, reflecting-element settings significantly affect OP and EC, with ORA more sensitive to setting changes than ERA.Changing from L4 to L2 requires 1.8 dBm additional transmit power for ERA and 6.3 dBm for ORA to reach OP = 10^-4.
  • A. The Difference in Lower Tail Between Gamma and LogNormal Distributions: For element setting L3, ORA has higher energy efficiency when Rth ≤ 12 b/s/Hz, while ERA is higher when Rth ≥ 12 b/s/Hz.The crossing points shift to 12.5 and 15.5 b/s/Hz for settings L2 and L4, respectively.
  • A. The Difference in Lower Tail Between Gamma and LogNormal Distributions: For small reflecting-element counts, Gamma provides lower KL divergence and KS distance than Log-Normal, whereas their approximation accuracies become comparable above 100 elements.The KS test uses p = 0.05, s = 10^3 samples, and critical value Ds,p = 0.043.

VI. CONCLUSIONS

The paper concludes that moment-based modeling characterizes distributed multi-RIS end-to-end fading and supports performance analysis of ERA and ORA. ERA generally provides better OP and EC, while ORA can offer higher energy efficiency under some target spectral efficiencies.

  • VI. CONCLUSIONS: The framework models ERA magnitudes with Gamma or Log-Normal distributions and ORA magnitudes with a Log-Normal model plus approximate CDF and PDF expressions.These models support subsequent OP and EC analysis.
  • VI. CONCLUSIONS: ERA outperforms ORA in terms of OP and EC, but ORA achieves better energy efficiency for some target SE values.The conclusion also reports significant effects from element allocation and RIS positions for a fixed total number of elements.
  • VI. CONCLUSIONS: A centralized large-RIS system can yield better EC than ERA near the source or destination, while ERA outperforms it otherwise.This comparison depends on the centralized RIS location.

APPENDIX A PROOF OF THEOREM 1

Appendix A derives moments for the end-to-end channel components by starting from exact individual-element distributions and then using Gamma approximations to obtain tractable channel statistics. The proof combines independence, multinomial and binomial expansions, and moment matching.

  • APPENDIX A PROOF OF THEOREM 1: The individual reflected component Unl is derived from two independent Nakagami-m channels and follows a generalized-K distribution.Its shaping parameters are mgn and mhn.
  • APPENDIX A PROOF OF THEOREM 1: The exact PDF of Unl is used to derive its kth moment, which is then fitted to a Gamma distribution because the exact subsequent PDF becomes intractable.The approximation enables tractable CDF and PDF derivations for later channel components.

APPENDIX B DERIVATION OF (25)

Appendix B derives the ERA ergodic-capacity expression by rewriting the expectation and applying identities for powers, incomplete Gamma functions, Meijer-G functions, and a general integral.

  • APPENDIX B DERIVATION OF (25): The ERA ergodic capacity is first rewritten as an expectation involving the channel magnitude.The derivation then transforms this expression using an identity for (1+x)^-ξ.
  • APPENDIX B DERIVATION OF (25): Incomplete-Gamma and Meijer-G identities convert the intermediate expression into a closed-form representation.A general-integral identity completes the final transformation.

APPENDIX C PROOF OF LEMMA 2

The proof of Lemma 2 derives an approximate closed-form expression for μMV(k) through algebraic rearrangements, special-function identities, and integral representations.

  • Proof of Lemma 2: The proof uses gamma-function identities and integral formulas to evaluate terms appearing in the derivation.It invokes the gamma integral and a re-expression of γ(·, ·).
  • Proof of Lemma 2: The derivation applies mathematical manipulations and rearrangements to express intermediate quantities including μMV(k).The proof repeatedly rewrites the relevant expressions before obtaining the target result.
  • Proof of Lemma 2: The derivation introduces the Pochhammer symbol and notes that i is independent from t in equation (102).The Pochhammer symbol is defined as ⟨x⟩n ≜ Γ(x + n)/Γ(x).
  • Proof of Lemma 2: Using the series and integral representations of the Lauricella function Type-A, the proof obtains the approximate closed-form expression for μMV(k) in (57).This completes the proof of Lemma 2.
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