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arxiv: 2604.04702 · v1 · submitted 2026-04-06 · 📡 eess.SP

Performance Analysis of STAR-RIS-Assisted NOMA Wireless Systems with Realistic Indoor Outdoor THz Channel Models

Ngoc Phuc Le , Mohamed-Slim Alouini This is my paper

Pith reviewed 2026-05-10 19:19 UTC · model grok-4.3

classification 📡 eess.SP
keywords STAR-RISNOMATerahertzoutage probabilityergodic capacitychannel modelinghardware impairments
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The pith

Closed-form outage probability and ergodic capacity expressions are derived for a STAR-RIS-assisted NOMA THz system under realistic indoor-outdoor channel models.

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

The paper sets out to obtain exact and approximate closed-form expressions for outage probability and ergodic capacity in a near-field STAR-RIS-aided downlink NOMA Terahertz system. It models indoor links with the α-μ distribution and outdoor links with Gaussian-mixture or mixture-of-gamma distributions drawn from measurement campaigns. These distributions allow the authors to first find the PDF and CDF of the relevant weighted sum of variates, then translate those into performance metrics that include high-SNR asymptotics and a separate low-SNR capacity analysis. The same framework is used to quantify the effects of hardware impairments and the choice between energy-splitting and mode-switching STAR-RIS protocols. A reader would care because the resulting formulas replace lengthy numerical integration or simulation with direct algebraic evaluation for system design at THz frequencies.

Core claim

By deriving the exact PDF and CDF of a weighted sum of α-μ variates and the corresponding approximate expressions for Gaussian-mixture variates, the authors obtain closed-form outage probability and ergodic capacity formulas (plus their high-SNR asymptotic expansions) for the proposed STAR-RIS NOMA THz system; they further supply a low-SNR capacity characterization and evaluate the impact of hardware impairments together with the energy-splitting and mode-switching STAR-RIS protocols.

What carries the argument

The PDF and CDF of the weighted sum of the fading random variables under the α-μ (indoor) and Gaussian-mixture (outdoor) models, which are then inserted into the standard integral expressions for outage and ergodic capacity.

If this is right

  • System designers can compute outage and capacity directly from algebraic formulas rather than Monte-Carlo simulation for any given set of distances, powers, and STAR-RIS coefficients.
  • The high-SNR asymptotic expressions reveal the diversity order and coding gain as functions of the channel parameters and the number of STAR-RIS elements.
  • The low-SNR capacity analysis quantifies how much rate is achievable when transmit power is severely limited, a regime relevant to THz devices.
  • Numerical comparison of energy-splitting versus mode-switching protocols shows which protocol yields lower outage for given hardware-impairment levels.

Where Pith is reading between the lines

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

  • The closed-form expressions could be embedded inside an optimizer to tune STAR-RIS phase shifts and NOMA power coefficients for target reliability in a given indoor-outdoor scenario.
  • Because the derivations separate the indoor and outdoor links, the same machinery can be reused for hybrid deployments that contain both link types without re-deriving the entire performance metric.
  • If hardware impairment parameters are measured per device rather than assumed constant, the same formulas immediately give device-specific performance predictions.

Load-bearing premise

The α-μ and Gaussian-mixture distributions accurately represent the small-scale fading measured in real indoor and outdoor THz environments.

What would settle it

Field measurements of outage probability or ergodic rate in an actual STAR-RIS NOMA THz deployment that lie outside the confidence intervals predicted by the closed-form expressions would show that the channel models or the derived sums do not match reality.

Figures

Figures reproduced from arXiv: 2604.04702 by Mohamed-Slim Alouini, Ngoc Phuc Le.

Figure 1
Figure 1. Figure 1: A STAR-RIS-aided downlink NOMA THz system model. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: PDF of |HI | under different values of M. 0 10 20 30 40 50 x 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 CDF function F|H I |(x) M = 2, 3, 4, 5 Solid curve: Analytical expression, i.e., Eq. (22) Circle: Empirical data M = 2: AI = [1, 0.7] M = 3: AI = [1, 0.7, 2.5] M = 4: AI = [1, 0.7, 2.5, 1.4] M = 5: AI = [1, 0.7, 2.5, 1.4, 0.8] [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: CDF of |HI| under different values of M. and F|HI |(x) =  α1µ µ1 1 Γ(µ1) M X∞ i=0 δix iα1+Mα1µ1 Γ(iα1 + Mα1µ1 + 1), (25) where δi = X i mM=0 BmM,M i−XmM mM−1=0 BmM−1,M−1 · · · i− PMXj=3 mj m2=0 Bm2,2Bi− PM j=2 mj ,1 , (26) and Bn,m = (−1)nβ α1(n+µ1) 1 Γ(α1(n+µ1)) n!µ µ1 1 (AI,mΩ1) α1(n+µ1) . Proof : See Appendix A. As shown in Appendix A, δi can be calculated in a recursive manner. Then, (24) and (25) ar… view at source ↗
Figure 4
Figure 4. Figure 4: PDF and CDF of |HO|. where γ is the received SNR, and γth denotes the SNR threshold. With the help of the statistical distributions of the e2e derived in Section III, we obtain the closed-form expressions of the OPs of the users. Theorem 5: The OP of the indoor user is given by Pout,I (γth,I ) =     α1µ µ1 1 Γ(µ1) M P∞ i=0 δi[ΨI (γth,I )](iα1+Mα1µ1)/2 Γ(iα1+Mα1µ1+1) , if γth,I < ρI κ2 and γth,O < … view at source ↗
Figure 5
Figure 5. Figure 5: Outage probabilities versus transmit power. [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Ergodic capacities versus transmit power. [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: It can be observed that the capacities are improved [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 8
Figure 8. Figure 8: Ergodic capacities at low SNR regime. 0.05 0.1 0.15 HWI level 2 10-6 10-5 10-4 10-3 10-2 10-1 100 Outage probability Pout Analysis: Indoor User Analysis: Outdoor User Simulation 0.05 0.1 0.15 HWI level 2 0.5 1 1.5 2 2.5 3 3.5 4 Ergodic capacity (bits/s/Hz) Analysis: Indoor User Analysis: Outdoor User Simulation [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 11
Figure 11. Figure 11: Ergodic capacities in MS mode. rate between the STAR-RIS-NOMA system and the STAR-RIS OMA system, respectively. In both systems, the target rates of RI = RO = 0.5 (bit per channel use) is used as in [36]. Note that in the STAR-RIS OMA system, the SNR threshold is calculated as γ OMA th,χ = 22Rχ − 1. Also, the sum-rate is defined as the summation of the rates of the indoor user and the outdoor user. It can… view at source ↗
Figure 12
Figure 12. Figure 12: Outage probabilities: NOMA versus OMA. 20 25 30 35 40 45 Transmit power P (dBm) 0 1 2 3 4 5 6 7 8 9 10 11 Sum-rate (bits/s/Hz) STAR-RIS NOMA (solid curve) STAR-RIS-OMA (dashed curve) Simulation With HWI ( 2 = 0.08) No HWI ( 2 = 0) [PITH_FULL_IMAGE:figures/full_fig_p011_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Sum-rate: NOMA versus OMA. multiple antennas at transceivers and/or mixed near-field/far￾field users on each side of the STAR-RIS panel. Optimization of RIS phase-shifts for improved performance is also worth studying. Furthermore, reducing the passive beamforming complexity and channel estimation overhead, especially when the number of STAR-RIS elements is large, is critical. To this end, it is of intere… view at source ↗
read the original abstract

In this paper, a simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS)-aided downlink non-orthogonal multiple access (NOMA) Terahertz (THz) wireless system is proposed for indoor and outdoor transmissions. We consider a near-field communication scenario where an access-point (AP) is deployed near a STAR-RIS panel. For links from the STAR-RIS to users, $\alpha-\mu$ distribution is adopted for the indoor small-scale fading channels, whereas the outdoor channels are based on Gaussian mixture or mixture of gamma, which follows the recent practical measurement reports. To facilitate performance analysis, we derive exact expressions of a probability density function (PDF) and a cumulative distribution function (CDF) of a weighted sum of $\alpha-\mu$ variates. Approximate PDF and CDF expressions of a weighted sum of Gaussian mixture variates are derived as well. Based on these results, closed-form expressions of the outage probability and the ergodic capacity, together with their asymptotic formulas at high signal-to-noise ratio (SNR), are obtained. Moreover, we analyze the capacity of the THz system at the low SNR regime. Impacts of hardware impairments and STAR-RIS protocols (i.e., energy splitting and mode-switching) on the system performance are evaluated. All developed analytical results are validated and demonstrated via numerical 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

2 major / 2 minor

Summary. The paper proposes a STAR-RIS-aided downlink NOMA THz wireless system for indoor and outdoor transmissions in a near-field scenario. It uses α-μ distribution for indoor small-scale fading and Gaussian mixture or mixture of gamma for outdoor channels based on measurement reports. Exact PDF and CDF for weighted sum of α-μ variates and approximate for Gaussian mixture are derived. These enable closed-form outage probability and ergodic capacity expressions, high-SNR asymptotics, low-SNR capacity analysis, and evaluation of hardware impairments and STAR-RIS protocols (energy splitting, mode-switching), all validated by simulations.

Significance. This work contributes analytical closed-form expressions for performance metrics in a complex THz system incorporating STAR-RIS and NOMA with realistic channel models. If accurate, it offers significant value for researchers and engineers designing future THz networks by providing tractable expressions that can be used for optimization and quick performance prediction, reducing reliance on simulations. The consideration of practical aspects like hardware impairments and different RIS protocols strengthens its applicability.

major comments (2)
  1. Section III (Channel Models): The selection of α-μ and Gaussian mixture/mixture-of-gamma models is justified by citing external measurement reports, but the manuscript does not verify or discuss applicability to the specific near-field STAR-RIS geometry, panel size, or NOMA user distances. Since the PDF/CDF derivations for the weighted sum (Section IV) and all subsequent outage/capacity expressions rest directly on these distributions, this is load-bearing for the central claim of analyzing a 'realistic' THz system.
  2. Section IV (Performance Analysis): The exact closed-form PDF/CDF for the weighted sum of α-μ variates is stated without expanded derivation steps or intermediate results (e.g., the integral transformations or special-function representations used). This makes it difficult to confirm the correctness of the outage probability (Eq. (XX)) and ergodic capacity expressions that follow. For the approximate Gaussian-mixture case, error bounds or convergence analysis for the approximation should be added, as tail accuracy directly affects outage results at practical target levels.
minor comments (2)
  1. Abstract and Section II: Minor inconsistency in outdoor channel description ('Gaussian mixture or mixture of gamma'); ensure uniform terminology and parameter definitions for the weighting coefficients tied to NOMA power allocation.
  2. Section V (Numerical Results): Simulation figures would benefit from explicit parameter tables or captions listing all SNR, power allocation, and impairment values used, to facilitate exact reproduction of the validation curves.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough and constructive review of our manuscript. We address each major comment below with clarifications and indicate the revisions we will make.

read point-by-point responses

  1. Referee: Section III (Channel Models): The selection of α-μ and Gaussian mixture/mixture-of-gamma models is justified by citing external measurement reports, but the manuscript does not verify or discuss applicability to the specific near-field STAR-RIS geometry, panel size, or NOMA user distances. Since the PDF/CDF derivations for the weighted sum (Section IV) and all subsequent outage/capacity expressions rest directly on these distributions, this is load-bearing for the central claim of analyzing a 'realistic' THz system.

    Authors: The α-μ and Gaussian mixture models are adopted directly from recent THz measurement campaigns reported in the cited references, which cover indoor and outdoor propagation at similar frequencies. The near-field geometry is incorporated via the explicit distance-dependent path-loss terms and the specific AP-RIS-user distances used in the NOMA setup. However, the manuscript does not contain an explicit discussion of how well the measurement conditions match the STAR-RIS panel size and near-field regime considered here. We will add a short paragraph in Section III that justifies the model choice by referencing additional near-field THz literature and noting that the statistical parameters (α, μ, mixture weights) are taken from measurements performed under comparable link distances and environments. This addresses the applicability concern without requiring new measurements. revision: partial

  2. Referee: Section IV (Performance Analysis): The exact closed-form PDF/CDF for the weighted sum of α-μ variates is stated without expanded derivation steps or intermediate results (e.g., the integral transformations or special-function representations used). This makes it difficult to confirm the correctness of the outage probability (Eq. (XX)) and ergodic capacity expressions that follow. For the approximate Gaussian-mixture case, error bounds or convergence analysis for the approximation should be added, as tail accuracy directly affects outage results at practical target levels.

    Authors: We agree that additional derivation detail will improve verifiability. The exact PDF/CDF of the weighted sum of α-μ random variables is obtained via the moment-generating-function method, followed by an inverse Laplace transform expressed with the Meijer-G function; we will move the full step-by-step derivation (including all integral transformations and the final Meijer-G representation) to an expanded Appendix. For the Gaussian-mixture approximation, we will insert a new subsection that provides a convergence argument based on the number of mixture components and includes numerical results comparing the approximate tail probabilities against the exact mixture at the outage thresholds used in the paper. These additions will directly address the concern about tail accuracy. revision: yes

Circularity Check

0 steps flagged

No circularity: derivations use externally cited channel models and perform independent mathematical steps

full rationale

The paper selects α-μ (indoor) and Gaussian-mixture/mixture-of-gamma (outdoor) distributions explicitly because they 'follow the recent practical measurement reports,' then derives PDF/CDF expressions for weighted sums of these variates via standard integral manipulations and approximations. Closed-form outage probability, ergodic capacity, high-SNR asymptotics, and low-SNR analysis follow directly from those PDF/CDF results. No parameter is fitted to the paper's own performance metrics and then relabeled as a prediction; no uniqueness theorem or ansatz is imported via self-citation; the central claims rest on independent external data sources and algebraic derivations that do not reduce to the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on the accuracy of externally measured channel distributions and standard assumptions of statistical independence among user links; no new entities are postulated.

axioms (3)
  • domain assumption Indoor small-scale fading follows the α-μ distribution
    Adopted directly from recent practical THz measurement reports as stated in the abstract.
  • domain assumption Outdoor channels follow Gaussian mixture or mixture of gamma distributions
    Taken from practical measurement reports cited in the abstract.
  • standard math User channels are statistically independent
    Standard assumption required to obtain the joint PDF/CDF of the weighted sum.

pith-pipeline@v0.9.0 · 5543 in / 1522 out tokens · 58383 ms · 2026-05-10T19:19:10.036206+00:00 · methodology

discussion (0)

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Reference graph

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