pith. sign in

arxiv: 2604.25331 · v1 · submitted 2026-04-28 · 📡 eess.SP

Performance Analysis of HAPS-RIS-Assisted MIMO Systems Under Phase-Dependent Amplitude Response Using Saddle Point Approximation

Tayfun Yilmaz , Haci Ilhan , Ali Gorcin , Halim Yanikomeroglu This is my paper

Pith reviewed 2026-05-07 15:34 UTC · model grok-4.3

classification 📡 eess.SP
keywords HAPSRISMIMOcascaded Rician fadingsaddle point approximationoutage probabilityphase-dependent amplitudediscrete phase shifts
0
0 comments X

The pith

A saddle point approximation derives closed-form outage probabilities for HAPS-RIS MIMO systems with phase-dependent impairments.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a tractable statistical model for MIMO systems assisted by reconfigurable intelligent surfaces mounted on high-altitude platforms. Because the RIS introduces coupling in the cascaded Rician channels, standard independent assumptions fail, so the authors adopt a line-of-sight aligned precoding strategy that expresses the SNR as a non-central quadratic form with structured covariance. They then apply the saddle point approximation to obtain closed-form PDF, CDF, and outage probability expressions while incorporating discrete phase shifts and phase-dependent amplitude responses. This framework enables direct performance evaluation for these systems without heavy reliance on simulations or invalid independence assumptions.

Core claim

By modeling the received SNR as a non-central quadratic form with structured covariance matrix under LoS-aligned precoding, a saddle point approximation framework is developed that yields closed-form PDF, CDF, and outage probability expressions for HAPS-RIS MIMO systems over cascaded Rician fading, while accounting for discrete phase shifts and phase-dependent amplitude responses.

What carries the argument

Saddle point approximation applied to the non-central quadratic form of the SNR with structured covariance matrix arising from LoS-aligned precoding.

If this is right

  • Closed-form expressions allow direct computation of outage probability without numerical integration.
  • The model incorporates practical RIS impairments including phase-dependent amplitude responses.
  • The approach handles dependence in cascaded channels that violates standard Wishart assumptions.
  • Validation against Monte Carlo simulations confirms the derived distributions match empirical results.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Designers could use the closed-form outage expressions to optimize RIS phase configurations or HAPS positioning for target reliability levels.
  • The SPA technique might extend to performance analysis of RIS-assisted links in terrestrial or satellite environments with similar cascaded channels.
  • The framework could support real-time link adaptation by evaluating outage under different discrete phase resolutions.

Load-bearing premise

The saddle point approximation accurately represents the distribution of the non-central quadratic form with the structured covariance matrix under the LoS-aligned precoding strategy.

What would settle it

Monte Carlo simulations producing outage probabilities that deviate substantially from the derived closed-form expressions across varying RIS element counts or Rician K-factors would falsify the accuracy of the SPA framework.

Figures

Figures reproduced from arXiv: 2604.25331 by Ali Gorcin, Haci Ilhan, Halim Yanikomeroglu, Tayfun Yilmaz.

Figure 1
Figure 1. Figure 1: Proposed HAPS-RIS assisted system model. view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of the SPA-based outage analysis with Monte Carlo simulations for random and optimized RIS phase configurations. view at source ↗
read the original abstract

This letter proposes a novel mathematical framework for the statistical characterization of reconfigurable intelligent surface (RIS)-mounted high-altitude platform station (HAPS)-assisted MIMO systems over cascaded Rician fading channels. Due to the inherent coupling introduced by the RIS, the resulting cascaded channel does not satisfy the independence assumptions required for conventional Wishart-based modeling, which motivates a tractable alternative approach. By adopting a line-of-sight (LoS)-aligned precoding strategy, the received signal-to-noise ratio (SNR) is represented as a non-central quadratic form with a structured covariance matrix. Exploiting this structure, a saddle point approximation (SPA)-based framework is developed to characterize the SNR distribution. Closed-form expressions for the probability density function (PDF), cumulative distribution function (CDF), and outage probability are derived. The proposed framework further incorporates practical RIS hardware impairments, including discrete phase shifts and phase-dependent amplitude responses. The accuracy of the proposed analysis is validated through Monte Carlo simulations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript proposes a mathematical framework for statistical characterization of HAPS-RIS-assisted MIMO systems over cascaded Rician fading channels. By using LoS-aligned precoding, the SNR is modeled as a non-central quadratic form with structured covariance; a saddlepoint approximation is then applied to derive expressions for the PDF, CDF, and outage probability while incorporating RIS impairments (discrete phase shifts and phase-dependent amplitude responses). Accuracy is checked via Monte Carlo simulations.

Significance. If the SPA expressions are accurate and the modeling choices hold, the work would provide a useful analytical tool for outage and distribution analysis in RIS-assisted aerial MIMO links where standard Wishart assumptions fail due to channel coupling. The explicit treatment of hardware impairments adds deployment relevance.

major comments (1)
  1. [SPA framework and abstract] Abstract and SPA derivation: the claim that 'closed-form expressions for the PDF, CDF, and outage probability are derived' via saddlepoint approximation is not supported by the standard form of the method. The cumulant generating function K(s) of the non-central quadratic form leads to the saddlepoint equation K'(s*) = x, which must be solved numerically for each evaluation point x; the resulting PDF ≈ exp(K(s*)−s*x)/sqrt(2π K''(s*)) and its integrated CDF therefore remain implicit. The LoS-aligned structure simplifies Σ but does not yield an analytic inverse for s*, so the expressions are semi-analytic rather than closed-form. This directly affects the central statistical-characterization claim.
minor comments (2)
  1. [Simulation results] Figure captions and legends should explicitly distinguish Monte Carlo curves from the SPA analytical curves and state the number of realizations used.
  2. [System model] Notation for the structured covariance matrix and the quadratic-form parameters should be introduced once in a dedicated subsection and used consistently thereafter.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and valuable feedback on our manuscript. The comment on the characterization of the SPA-derived expressions is well-taken, and we address it directly below. We are committed to revising the manuscript to ensure precise terminology.

read point-by-point responses

  1. Referee: Abstract and SPA derivation: the claim that 'closed-form expressions for the PDF, CDF, and outage probability are derived' via saddlepoint approximation is not supported by the standard form of the method. The cumulant generating function K(s) of the non-central quadratic form leads to the saddlepoint equation K'(s*) = x, which must be solved numerically for each evaluation point x; the resulting PDF ≈ exp(K(s*)−s*x)/sqrt(2π K''(s*)) and its integrated CDF therefore remain implicit. The LoS-aligned structure simplifies Σ but does not yield an analytic inverse for s*, so the expressions are semi-analytic rather than closed-form. This directly affects the central statistical-characterization claim.

    Authors: We agree with the referee that the saddlepoint approximation requires numerically solving the saddlepoint equation K'(s*) = x for each evaluation point x, and that the resulting PDF and CDF expressions are therefore semi-analytic rather than purely closed-form. While the LoS-aligned precoding yields a structured covariance that simplifies the cumulant generating function K(s), it does not admit an analytic solution for the saddlepoint s*. In the wireless communications literature, SPA-based results are occasionally described as closed-form when they avoid infinite series or integrals, but this usage can be imprecise. To correct this, we will revise the abstract to read: 'Expressions for the PDF, CDF, and outage probability are derived using the saddlepoint approximation.' We will add a clarifying sentence in Section III explaining that the saddlepoint equation is solved numerically for each argument, and update the introduction and conclusion accordingly. These changes will be incorporated in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

Standard SPA application to modeled quadratic form; no reduction to fitted inputs or self-referential definitions

full rationale

The paper models the cascaded Rician channel under LoS-aligned precoding as a non-central quadratic form with structured covariance, then applies the saddlepoint approximation to obtain PDF/CDF/outage expressions. This follows the standard cumulant-generating-function route for quadratic forms without any step that defines the target distribution in terms of itself or renames a fit as a prediction. No load-bearing self-citations or uniqueness theorems imported from the authors' prior work are required for the core derivation; the framework is self-contained once the channel model and SPA are accepted. Minor self-citation of standard SPA literature does not affect the score.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard domain assumptions from wireless fading literature plus the choice of saddle point approximation; no new physical entities are postulated.

axioms (2)
  • domain assumption Cascaded RIS channel does not satisfy independence assumptions required for conventional Wishart-based modeling
    Explicitly stated as the motivation for adopting the SPA approach.
  • domain assumption LoS-aligned precoding produces a non-central quadratic form with structured covariance matrix
    Used to represent the received SNR before applying the saddle point approximation.

pith-pipeline@v0.9.0 · 5486 in / 1239 out tokens · 66304 ms · 2026-05-07T15:34:37.930342+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

  1. [1]

    HAPS for 6G networks: Potential use cases, open challenges, and possible solutions,

    O. Abbasi, A. Yadav, H. Yanikomeroglu, N.-D. Dao, G. Senarath, and P. Zhu, “HAPS for 6G networks: Potential use cases, open challenges, and possible solutions,”IEEE Wireless Communications, vol. 31, no. 3, pp. 324–331, 2024

    work page 2024

  2. [2]

    RIS meets aerodynamic HAPS: A multi- objective optimization approach,

    A. Azizi and A. Farhang, “RIS meets aerodynamic HAPS: A multi- objective optimization approach,”IEEE Wireless Communications Let- ters, vol. 12, no. 11, pp. 1851–1855, 2023

    work page 2023

  3. [3]

    Reconfigurable intelligent surface-assisted haps relaying communication networks for multiusers under AF protocol: A performance analysis,

    K. O. Odeyemi, P. A. Owolawi, and O. O. Olakanmi, “Reconfigurable intelligent surface-assisted haps relaying communication networks for multiusers under AF protocol: A performance analysis,”IEEE Access, vol. 10, pp. 14 857–14 869, 2022

    work page 2022

  4. [4]

    En- hancing physical layer security in RIS-aided HAPS for non-terrestrial networks,

    H. Memarian, S. Mohammad Razavizadeh, and A. Kuhestani, “En- hancing physical layer security in RIS-aided HAPS for non-terrestrial networks,”IEEE Access, vol. 13, pp. 103 010–103 018, 2025

    work page 2025

  5. [5]

    Outage probability analysis of RIS-assisted UA V communications with direct link and ISI,

    A. Al-Rimawi, A. Al-Dweik, R. Hamila, and A. Gouissem, “Outage probability analysis of RIS-assisted UA V communications with direct link and ISI,”IEEE Transactions on Aerospace and Electronic Systems, pp. 1–8, 2025

    work page 2025

  6. [6]

    En- hanced hybrid HAP-assisted NOMA-RIS systems for next-generation wireless networks,

    K. A. Alnajjar, A. Abdelaziz Salem, S. Ansari, and M. El-Tarhuni, “En- hanced hybrid HAP-assisted NOMA-RIS systems for next-generation wireless networks,”IEEE Open Journal of the Communications Society, vol. 6, pp. 10 694–10 705, 2025

    work page 2025

  7. [7]

    Ergodic capacity analysis of reconfigurable intelligent surface assisted MIMO systems over Rayleigh-Rician channels,

    M. Abbasi Mosleh, F. Heliot, and R. Tafazolli, “Ergodic capacity analysis of reconfigurable intelligent surface assisted MIMO systems over Rayleigh-Rician channels,”IEEE Communications Letters, vol. 27, no. 1, pp. 75–79, 2023

    work page 2023

  8. [8]

    Characterization of capacity and outage of RIS-aided downlink systems under Rician fading,

    K. K. Kota, P. D. Mankar, and H. S. Dhillon, “Characterization of capacity and outage of RIS-aided downlink systems under Rician fading,”IEEE Wireless Communications Letters, vol. 14, no. 3, pp. 631– 635, 2025

    work page 2025

  9. [9]

    Ergodic capacity analysis of RIS-aided MIMO systems under amplitude-phase coupling,

    M. A. Mosleh, F. H´eliot, G. Gradoni, and R. Tafazolli, “Ergodic capacity analysis of RIS-aided MIMO systems under amplitude-phase coupling,”IEEE Transactions on V ehicular Technology, pp. 1–16, 2026

    work page 2026

  10. [10]

    Beamforming for massive MIMO aerial communications: A robust and scalable DRL approach,

    H. Khoshkbari, G. Kaddoum, O. Abbasi, B. Selim, and H. Yanikomeroglu, “Beamforming for massive MIMO aerial communications: A robust and scalable DRL approach,”IEEE Transactions on Communications, vol. 74, pp. 261–275, 2026

    work page 2026

  11. [11]

    Intelligent reflecting surface: Practical phase shift model and beamforming optimization,

    S. Abeywickrama, R. Zhang, Q. Wu, and C. Yuen, “Intelligent reflecting surface: Practical phase shift model and beamforming optimization,” IEEE Transactions on Communications, vol. 68, no. 9, pp. 5849–5863, 2020

    work page 2020

  12. [12]

    Precoding matrix indicator in the 5G NR protocol: A tutorial on 3GPP beamforming codebooks,

    B. Ning, H. Yin, S. Liu, H. Deng, S. Yang, Y . Zhang, W. Mei, D. Gesbert, J. Park, R. W. Heath, and E. Bj¨ornson, “Precoding matrix indicator in the 5G NR protocol: A tutorial on 3GPP beamforming codebooks,”IEEE Communications Surveys and Tutorials, pp. 1–1, 2026

    work page 2026

  13. [13]

    Ergodic capacity analysis of reconfigurable intelligent surface assisted MIMO systems with a practical phase shift and amplitude response,

    M. A. Mosleh, F. H ´eliot, and R. Tafazolli, “Ergodic capacity analysis of reconfigurable intelligent surface assisted MIMO systems with a practical phase shift and amplitude response,”IEEE Transactions on V ehicular Technology, vol. 73, no. 8, pp. 11 441–11 457, 2024

    work page 2024