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arxiv: 2604.11069 · v1 · submitted 2026-04-13 · 💻 cs.IT · math.IT

Exact Outage Probability and Ergodic Capacity Analysis of NOMA in Rayleigh Fading Channels

Arafat Al-Dweik , Alok Kumar Shukla , Sami Muhaidat This is my paper

Pith reviewed 2026-05-10 16:35 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords NOMARayleigh fadingoutage probabilityergodic capacitysuccessive interference cancellationresidual interferencejoint PDF
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The pith

Exact NOMA analysis shows residual interference depends on power allocation, SNR and outage threshold.

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

The paper establishes that after successive interference cancellation in two-user downlink NOMA, the post-SIC noise and fading become statistically dependent, so their joint PDFs must be used instead of independent pre-SIC models. This dependence is used to derive parameter-free closed-form outage probability and single-integral or closed-form ergodic capacity expressions for the near user over Rayleigh fading. If the derivation is correct, conventional fixed residual-interference-factor models produce significant errors in outage and capacity at low-to-moderate SNRs, especially with unbalanced power allocation. The results further show that the residual interference factor is not an independent parameter but varies with power allocation, SNR and outage threshold. For two-dimensional modulations the real and imaginary noise components also become dependent after SIC.

Core claim

By noting that the noise and fading become dependent after successive interference cancellation, the exact analysis is derived by considering the joint probability density functions of the post-SIC noise and fading, which are typically considered to be independent and modeled using the same PDFs before the SIC. The derived exact PDFs are used to evaluate the impact of residual interference accurately and to obtain an exact closed-form formula for near-user outage probability together with a single-integral expression for ergodic capacity, plus a closed-form accurate expression for the latter. The formulae are parameter-free and Monte Carlo results confirm that legacy Gaussian or residual-fac

What carries the argument

Joint probability density functions of post-SIC noise and fading that capture their dependence after successive interference cancellation.

If this is right

  • Legacy Gaussian and residual-factor models can significantly misestimate outage and ergodic capacity at low-to-moderate SNRs under unbalanced power allocation.
  • The residual interference factor depends on power allocation, SNR and outage threshold and cannot be treated as an independent variable.
  • For two-dimensional modulations the real and imaginary noise components become dependent after SIC.
  • Parameter-free expressions enable more accurate performance evaluation of NOMA systems.

Where Pith is reading between the lines

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

  • System designers should recalculate the effective residual interference for each operating point rather than adopting a single bounded factor.
  • The same joint-PDF dependence may appear in multi-stage SIC or other interference-cancellation schemes and would require analogous exact analysis.
  • Hardware impairments that alter the post-SIC noise statistics would invalidate the closed-form expressions and require new joint distributions.

Load-bearing premise

The residual interference after imperfect SIC is fully captured by the joint post-SIC noise-fading distribution derived under Rayleigh fading with perfect channel knowledge at the receiver.

What would settle it

Measured outage probabilities in a real Rayleigh-fading testbed with imperfect SIC that deviate from the derived closed-form expressions while the conventional residual-factor model matches the measurements.

Figures

Figures reproduced from arXiv: 2604.11069 by Alok Kumar Shukla, Arafat Al-Dweik, Sami Muhaidat.

Figure 1
Figure 1. Figure 1: Scatter plots of the conditional and unconditional n [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The conditional and unconditional PDFs of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Scatter plots of the conditional and unconditional n [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The conditional and unconditional PDFs of [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: OP of U2 with various target rates, α1 ∈ {0.75, 0.9}. 0 20 40 10-4 10-3 10-2 10-1 100 Exact Leagacy 0 20 40 R=4 3 2 1 0.5 0.25 R=4 3 2 1 0.5 0.25 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: U2 Exact and legacy OP (7) for different target rates and power coefficients, ζ = 0. perfectly across all cases, confirming the accuracy of the analysis. Therefore, in the following figures, only the analytical results are presented to avoid overcrowding [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 9
Figure 9. Figure 9: U2 Exact and legacy OP (7) versus ζ for various values of γ¯, R = 0.5 and α1 = 0.6. Consequently, adopting the perfect SIC model in such cases makes the legacy results close to the exact ones [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: Exact and legacy EC (9) versus SNR for various values [PITH_FULL_IMAGE:figures/full_fig_p012_11.png] view at source ↗
read the original abstract

This work derives the exact outage probability (OP) and ergodic capacity (EC) for the near user (NU) in the widely adopted two-user downlink non-orthogonal multiple access (NOMA) over fading channels. By noting that the noise and fading become dependent after successive interference cancellation (SIC), the exact analysis is derived by considering the joint probability density functions (PDFs) of the post-SIC noise and fading, which are typically considered to be independent and modeled using the same PDFs before the SIC. The derived exact PDFs are used to evaluate the impact of residual interference accurately. The derived interference and noise PDFs are used to derive an exact closed-form formula for NU outage and a single-integral expression for EC. Moreover, a closed-form, accurate expression is derived for the EC. Unlike existing work, the derived formulae are parameter-free, leading to more accurate performance evaluation of such systems. Monte Carlo simulation results validate the derived analysis and demonstrate that legacy Gaussian/residual-factor models can significantly misestimate outage and EC at low-to-moderate signal-to-noise ratios (SNRs) and under unbalanced power allocation. Moreover, the obtained results show that the widely considered residual interference factor, which is bounded by [0, 1], is not sufficient to capture the actual impact of residual interference due to a SIC failure, and it cannot be treated as an independent variable because it depends on the power allocation, SNR, and outage threshold. In addition to the fading-noise dependence, for two-dimensional modulations, the real and imaginary components of the noise become dependent as well.

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 / 3 minor

Summary. The manuscript derives exact closed-form outage probability (OP) and ergodic capacity (EC) expressions for the near user in two-user downlink NOMA over Rayleigh fading. It models the dependence between post-SIC residual interference, noise, and fading via joint PDFs (instead of assuming independence as in legacy models), yielding parameter-free formulas. Monte Carlo simulations validate the analysis and demonstrate that conventional residual-interference-factor models misestimate performance at low-to-moderate SNRs and under unbalanced power allocation; the paper further shows that the effective residual factor depends on power coefficients, SNR, and outage threshold. A side observation notes dependence between real and imaginary noise components for 2D modulations.

Significance. If the joint-PDF derivations are correct, the work supplies a more accurate, parameter-free analytical framework for NOMA performance that improves upon widely used Gaussian/residual-factor approximations, particularly at low-to-moderate SNR. This has direct value for system design and link-budget calculations in practical wireless networks.

major comments (2)
  1. [§III] §III (joint PDF derivation): the central claim that the final OP and EC expressions are exactly parameter-free rests on the specific post-SIC residual model under perfect CSI; the manuscript should explicitly state the perfect-CSI assumption and note that any channel-estimation error would alter the joint distribution, thereby qualifying the exactness claim.
  2. [§IV] §IV (EC derivation): the single-integral EC expression is useful, but the additional claim of a closed-form accurate expression requires a clear statement of whether the closed form is exact or involves further approximation; if the latter, the parameter-free advantage over legacy models needs re-examination.
minor comments (3)
  1. [Abstract / §V] The abstract and §V mention dependence of real/imaginary noise components for 2D modulations; this observation should be briefly justified or referenced in the main text to avoid appearing as an afterthought.
  2. [§II] Notation for power-allocation coefficients (a1, a2) and residual terms should be introduced once and used consistently; a short table of symbols would improve readability.
  3. [§V] Monte Carlo validation figures would benefit from explicit statement of the number of trials and any confidence intervals to allow readers to assess statistical agreement with the closed-form curves.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and positive recommendation. We address each major comment below.

read point-by-point responses

  1. Referee: [§III] §III (joint PDF derivation): the central claim that the final OP and EC expressions are exactly parameter-free rests on the specific post-SIC residual model under perfect CSI; the manuscript should explicitly state the perfect-CSI assumption and note that any channel-estimation error would alter the joint distribution, thereby qualifying the exactness claim.

    Authors: We agree that the perfect-CSI assumption underlying the post-SIC residual model should be stated explicitly. In the revised manuscript we will add a sentence in Section III clarifying that the analysis assumes perfect channel state information at the receivers and that channel-estimation errors would change the joint distribution of post-SIC noise and fading, thereby qualifying the exactness claim for the parameter-free expressions. revision: yes

  2. Referee: [§IV] §IV (EC derivation): the single-integral EC expression is useful, but the additional claim of a closed-form accurate expression requires a clear statement of whether the closed form is exact or involves further approximation; if the latter, the parameter-free advantage over legacy models needs re-examination.

    Authors: The single-integral EC expression is exact, obtained directly from the joint PDF without further approximation, and is parameter-free. The additional closed-form expression is an accurate approximation derived via series expansion. In the revision we will explicitly distinguish these two results, confirm that the parameter-free property applies to the exact single-integral form, and re-examine the comparison with legacy models to ensure the advantage is stated appropriately for each expression. revision: partial

Circularity Check

0 steps flagged

No circularity: joint post-SIC PDF derivation is independent and self-contained.

full rationale

The paper derives the joint PDF of post-SIC noise and fading from the received signal model y = h(√(a1P)s1 + √(a2P)s2) + n under Rayleigh fading and imperfect SIC, then uses that PDF to obtain closed-form OP and integral EC expressions. No equations reduce to fitted parameters renamed as predictions, no self-citations are invoked as load-bearing uniqueness theorems, and no ansatz or known result is smuggled in via prior work by the same authors. The conclusion that the conventional residual factor β cannot be treated as independent follows directly from the derived dependence on power allocation, SNR, and threshold within the model; it does not presuppose the final result. The analysis is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard Rayleigh fading statistics and the definition of successive interference cancellation; no free parameters are introduced in the final expressions, and no new entities are postulated.

axioms (2)
  • domain assumption The wireless channel follows Rayleigh fading with the usual complex Gaussian distribution.
    Invoked when stating the fading model for the two-user downlink NOMA scenario.
  • domain assumption Successive interference cancellation is performed with the far-user signal treated as known interference that is subtracted.
    Central to the post-SIC noise model.

pith-pipeline@v0.9.0 · 5598 in / 1497 out tokens · 22923 ms · 2026-05-10T16:35:21.234907+00:00 · methodology

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