An Energy Efficient Design of Hybrid NOMA Based on Hybrid SIC with Power Adaptation

Chenyu Zhang; Minghui Min; Ning Wang; Shiyin Li; Yanshi Sun; Yuanwei Liu

REVIEW 1 major objections 4 minor 28 references

An Energy Efficient Design of Hybrid NOMA Based on Hybrid SIC with Power Adaptation

T0 review · 1 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Power-adaptive hybrid NOMA beats OMA in rate and energy at high SNR

desk verdict Sign error in Lemma 1 undermines the exact derivation, but the core idea is sound and the conclusion may survive after a corrected proof. read the letter →

arxiv 2507.09458 v2 pith:V4ALV7BJ submitted 2025-07-13 eess.SP

Ning Wang , Chenyu Zhang , Yanshi Sun , Minghui Min , Yuanwei Liu , Shiyin Li This is my paper · ORCID

classification eess.SP

keywords hybridNOMApoweradaptationsuccessiveinterferencecancellationenergyefficiencyuplinkoutageprobabilityasymptoticanalysisorthogonalmultipleaccess

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Hybrid NOMA splits a user's transmission between a dedicated OMA slot and a shared NOMA slot, so the central question is whether the NOMA slot pays for itself. This paper proposes adding a power-adaptation coefficient, called HSIC-PA, to the hybrid scheme and analyzes the probability $\hat P_n$ that the total rate over both slots falls short of what the user could get from pure OMA while using less energy. The paper proves that $\hat P_n \to 0$ as the signal-to-noise ratios of the legacy and opportunistic users grow with a fixed ratio, for every power-reduction coefficient $0<\beta<1/2$ and every target rate $R_m$. That means the error floors seen in fixed-SIC and hybrid-SIC-without-power-adaptation H-NOMA disappear. If the result is right, the scheme gives a higher data rate than OMA with probability approaching one while consuming less energy in the high-SNR limit.

What carries the argument

The load-bearing mechanism is the power-adaptation coefficient $\gamma$ used in Type II, Case 2 of the NOMA slot. When the opportunistic user's un-adapted received power exceeds the interference threshold $\tau_m = \max\{0, \rho_m |h_m|^2/(2^{R_m}-1)-1\}$, the receiver chooses $\gamma$ so that $\gamma \beta \rho_n |h_n|^2 = \tau_m$. This places the signal exactly at the maximum interference the legacy user can tolerate, allowing the opportunistic user to be decoded at the second SIC stage at rate $\log(1+\tau_m)$ rather than suffering the first-stage interference-limited rate $\log\bigl(1 + \beta\rho_n|h_n|^2/(\rho_m|h_m|^2+1)\bigr)$. The equality converts a bad channel draw into a deterministic rate that depends only on the legacy user's target rate, which is what removes the error floor of the earlier HSIC-NPA scheme.

What would settle it

Simulate the uplink with a fixed channel-estimation error variance $\sigma_e^2>0$ and sweep the SNR; if the measured $\hat P_n$ stops decaying and flattens above zero instead of falling like $1/\rho^n$, the exact-equality assumption is doing the work and the high-SNR dominance claim fails under imperfect CSI.

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Extended reading notes

Core claim

The central claim, stated as Theorems 1 and 2 of the paper, is that for the proposed HSIC-PA aided H-NOMA uplink, the probability $\hat P_n$ that the achievable rate over the NOMA and OMA slots combined is no larger than the pure-OMA rate tends to $0$ as $\rho_n,\rho_m \to \infty$ with $\rho_n/\rho_m = \eta$ fixed, for any $0<\beta<1/2$, any target rate $R_m$, and any power ratio. Equivalently, at high SNR the scheme outperforms OMA in rate while using less energy with probability tending to one. The paper also derives exact closed-form expressions for $\hat P_n$ in both pairing orders $m<n$ and $m>n$, and shows asymptotically that $\hat P_n$ decays like $1/\rho^n$, so the opportunistic user's channel-gain order $n$ dominates the decay. The restrictive conditions required by the earlier HSIC-NPA scheme disappear because the power-adaptation branch can force the opportunistic user's received power down to the legacy user's interference threshold.

Load-bearing premise

The result rests on perfect instantaneous channel knowledge at both transmitter and receiver and on the power adaptation coefficient $\gamma$ being set so that $\gamma \beta \rho_n |h_n|^2 = \tau_m$ exactly; if channel estimation is imperfect or $\gamma$ is quantized, the claimed rate $\log(1+\tau_m)$ and the asymptotic $\hat P_n \to 0$ are not established.

Editorial extensions

If this is right

At high SNR, any user paired under the scheme can be served at a higher rate than pure OMA with less energy, with probability tending to one, for every $0<\beta<1/2$ and every target rate $R_m$.
The error floors that appear in fixed-SIC and HSIC-NPA H-NOMA under specific target-rate and power-ratio conditions are eliminated by the power-adaptation branch.
The failure probability $\hat P_n$ decays exponentially at rate $n$, the channel-gain order of the opportunistic user, so system performance depends more on the opportunistic user than on its legacy partner.
Because the dominance holds for every power ratio $\eta$ and every $R_m$, user pairing and power allocation become simpler at high SNR: no restrictive condition needs to be enforced.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Inference: A practical implementation would need to quantize $\gamma$ and tolerate imperfect CSI; the analysis implies the loss appears as an error floor rather than just a slower decay, because the exact equality is what unlocks the clean second-stage rate.
Inference: The same scale-down-to-threshold trick should carry over to downlink H-NOMA or relay-aided NOMA wherever the receiver can compute $\tau_m$ and feed it back; the proof structure needs only the exact equality to be reachable.
Inference: The scheme only reduces the opportunistic user's power; allowing $\gamma>1$ when the legacy channel is weak could extend the gain to lower SNR, but that would require re-deriving the interference-threshold analysis.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The paper combines hybrid SIC with power adaptation in hybrid NOMA and claims that the probability of H-NOMA underperforming OMA while using less energy goes to zero at high SNR with no restrictive conditions. That is a real extension of the prior HSIC-NPA result, and if true it removes a known error floor. The scheme itself is a natural fusion of [25] and [28]; the new asymptotic claim is specific and testable.

The good news: the problem is well posed, the decomposition of the failure probability is systematic, and the simulations appear to match the analysis in the plotted regimes. The energy accounting is clear, and the comparison with the NPA baseline is honest.

The soft spot is load-bearing. In Lemma 1, the inequality (1+τ_m)(1+βρ_n y) ≤ 1+ρ_n y is rearranged to y ≥ Θ without checking the sign of the denominator. When τ_m > (1−β)/β — which is the typical case at high SNR because τ_m grows with |h_m|^2 — the inequality has no positive solution, yet the expression in (17) counts the region Φ < y < Ω as failure. That adds a positive-probability region that is not actually a failure event. The true failure probability is smaller than the derived expression. So the exact expression for P_hat_n in Theorems 1 and 2 is not the probability defined in (13). The asymptotic claim that P_hat_n → 0 may survive because the spurious term decays, but the proof as written does not establish it. The authors need to correct Lemma 1 and redo the subsequent integrals, or provide a different argument.

Minor points: the paper calls Gauss-Chebyshev quadrature expressions closed-form; that's a terminology stretch. Several components (P1, P2,2) are imported from [25] without re-derivation, so the independent check is harder. No code or data is released, but the simulation/analysis agreement in the figures suggests the numerics are reproducible in principle. The perfect-CSI assumption for the power adaptation equality is acknowledged and is standard for this line of work.

Who is this for? Researchers in NOMA and H-NOMA energy efficiency will find the scheme interesting. It deserves a serious referee, but not in its current form. I'd recommend sending to review with a request for major revision: fix the sign error, provide full derivations (or at least the missing proofs), and either release code or show enough intermediate steps to allow independent verification.

Referee Report

1 major / 4 minor

Summary. The manuscript proposes an uplink hybrid NOMA (H-NOMA) scheme that combines hybrid successive interference cancellation (HSIC) with power adaptation (PA), and claims that the probability \hat P_n of the total achievable rate over the NOMA and OMA slots being no larger than the pure-OMA rate tends to zero as the signal-to-noise ratios grow, for any \beta<1/2, any target rate, and any power ratio. The theoretical part derives closed-form-looking expressions for \hat P_n in the two user-ordering cases m<n and m>n, and asymptotic approximations showing \hat P_n \to 0. The paper builds its derivation on Lemma 1, which decomposes the Type II probability into two terms using thresholds \Phi, \Omega, \Theta, and \Psi. Numerical simulations are presented as verifying the analysis.

Significance. If the asymptotic claim were rigorously established, it would be a meaningful advance: it would show that the proposed HSIC-PA H-NOMA scheme achieves a higher rate than OMA with probability approaching one at high SNR while consuming less energy, without the restrictive conditions that cause error floors in earlier H-NOMA designs (e.g., [25]). The paper provides a substantial set of analytical expressions, piecewise tables, and simulation curves, and the system model is clearly described. However, the central derivation rests on a lemma that contains an algebraic sign error, so the claimed exact expression for \hat P_n is not the probability defined in Eq. (13), and the asymptotic proof is not supported as written. The qualitative conclusion may still be correct, but the supplied analysis does not establish it.

major comments (1)

[Lemma 1, Eq. (17), and Appendix A] The rearrangement of the rate inequality in the Type II, Case 2 branch is incorrect for \tau_m > (1-\beta)/\beta. Starting from (1+\tau_m)(1+\beta\rho_n y) \le 1+\rho_n y, where y=|h_n|^2, we obtain \tau_m \le \rho_n y (1-\beta-\beta\tau_m). If \tau_m < (1-\beta)/\beta, this gives y \ge \tau_m/[\rho_n(1-\beta-\beta\tau_m)] = \Theta. If \tau_m > (1-\beta)/\beta, the right-hand side is negative, so the inequality has no solution for y>0; the correct condition is an empty set, not y \ge \Theta with a negative \Theta. Equation (17) keeps the region |h_n|^2 > \Theta, |h_n|^2 < \Omega, |h_n|^2 > \Phi, |h_m|^2 > \alpha_m, which for \tau_m > (1-\beta)/\beta includes a positive-probability set that does not satisfy the failure condition. Since at high SNR \tau_m = \rho_m |h_m|^2/\varepsilon_m - 1 tends to infinity for typical nonzero |h_m|^2, the erroneous regime is the dominant one rather than a corner case. Consequently, the exact expression for \hat P_n in Theorems 1 and 2 is not the probability defined in Eq. (13), and the asymptotic claim \hat P_n \to 0 is not established by the supplied derivation. This is a load-bearing error that requires a full re-derivation of Lemma 1 and all subsequent results that depend on it.

minor comments (4)

[Eq. (22) and Eq. (49)] The definitions of c_p and \hat c_p contain a typo: the binomial coefficient uses l where it should use p. For example, Eq. (22) writes c_p = \binom{n-m-1}{l}(-1)^{n-m-1-p}, but the summation index is p, so it should be \binom{n-m-1}{p}(-1)^{n-m-1-p}.
[Abstract and Section III] The paper repeatedly calls the derived expressions "closed-form", but several key terms (S2, S3, S5, V5, V7, etc.) rely on Gauss-Chebyshev quadrature with the parameter n_c, which is a numerical integration approximation, not a closed form. The language should be adjusted to "semi-analytical" or "numerically evaluated" where quadrature is used.
[Conclusion] The conclusion contains a sentence fragment: "particularly in achieving \hat P_n \to 0 in the high SNR regime under all conditions. guaranteed performance superiority." The second phrase should be integrated into a complete sentence.
[General] The paper relies heavily on results from [25] for P1 and P2,2 without re-deriving them; the manuscript would be easier to verify if those expressions were summarized or briefly re-derived in an appendix, or if the exact statements from [25] were stated explicitly.

Circularity Check

0 steps flagged · score 2.0 of 10

No material circularity: the HSIC-PA probability is derived from the model equations; the only self-citation is the baseline [25] terms, which are independent prior results rather than the target claim.

full rationale

The derivation chain is largely self-contained algebra on the stated model. The probability \hat P_n in Eq. (13) is not defined as the desired answer; it is computed from the event {T \hat R_n^NOMA + T R_n^OMA ≤ T R_n} by first splitting on βρ_n|h_n|^2 ≤ τ_m (Eq. 14) and then evaluating P_I and P_II with the explicit rates in Eqs. (5)-(10). The asymptotic PT→0 claims follow from Taylor expansions of the Rayleigh order-statistic densities in Appendices C and E, not from assuming the conclusion. No parameters are fitted to data: γ is chosen per channel realization to satisfy γβρ_n|h_n|^2 = τ_m, so the fitted-input-called-prediction pattern does not apply. The one load-bearing import is P1 and P2,2 from the authors' prior [25]; these are published baseline HSIC-NPA terms, the assumptions of [25] do not include the HSIC-PA result, and the present conclusion is not equivalent to them. That is legitimate prior-work support, not circularity under the hard rules. The skeptical point that Appendix A's reduction of Eq. (83) to Eq. (17) omits the sign condition on 1−β|h_m|^2/α_m, and the Section V acknowledgment of perfect-CSI assumptions, are correctness and scope limitations rather than circular-dependency issues. The score of 2 reflects only the mild self-citation burden from [25]; there is no circularity in the central derivation.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted to the data; beta, gamma, R_m, and eta are system or design inputs. The analysis assumes i.i.d. Rayleigh fading, perfect CSI, ideal SIC, and standard order-statistics and quadrature tools. The quadrature order n_c used in the analytical curves is not stated, which is an implementation detail rather than a fitted physical parameter.

assumptions (4)

domain assumption User channels |h_i|^2 are i.i.d. exponential (Rayleigh fading) and are perfectly ordered; the joint order-statistic PDF has the product form used in Appendices B and D.
Invoked in Eq. (86) and Eq. (96). If the fading is not Rayleigh, the exact CDF and the high-SNR Taylor expansions f(x) = e^{-x} approximately 1 and F(x) = 1 - e^{-x} approximately x do not hold.
domain assumption Perfect CSI and slow time-varying channels allow the BS to know |h_m|^2 and |h_n|^2 and to set \gamma so that \gamma\beta\rho_n|h_n|^2 = \tau_m exactly.
Used to define Type II Case 2 and Eq. (7). Without exact CSI the rate log(1+\tau_m) is unattainable and the asymptotic result is not established. The paper states this as a limitation in Section V.
domain assumption Ideal SIC with no error propagation; a user decoded at the second SIC stage achieves log(1 plus its received SNR).
Assumed in Eqs. (5)-(9). Imperfect SIC would introduce error floors, directly undermining the claim that \hat P_n tends to 0.
standard math Standard order-statistics, multinomial, binomial, Taylor expansion, and Gauss-Chebyshev quadrature tools are applied correctly.
Used throughout Appendices A-E. The typo in the definition of c_p in Eq. (22) means these applications are not fully checked.

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Pith. "Pith review of An Energy Efficient Design of Hybrid NOMA Based on Hybrid SIC with Power Adaptation." pith.science (2026). https://pith.science/paper/V4ALV7BJ

@misc{pith2026250709458,
  author       = {Pith},
  title        = {Pith review of: An Energy Efficient Design of Hybrid NOMA Based on Hybrid SIC with Power Adaptation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V4ALV7BJ}},
  note         = {Machine review of arXiv:2507.09458}
}

read the original abstract

Recently, hybrid non-orthogonal multiple access (H-NOMA) technology, which effectively utilizes both NOMA and orthogonal multiple access (OMA) technologies through flexible resource allocation in a single transmission, has demonstrated immense potential for enhancing the performance of wireless communication systems. To further release the potential of HNOMA, this paper proposes a novel design of H-NOMA which jointly incorporates hybrid successive interference cancellation (HSIC) and power adaptation (PA) in the NOMA transmission phase. To reveal the potential of the proposed HSIC-PA aided H-NOMA scheme, closed-form expression for the probability of the event that H-NOMA can achieve a higher data rate than pure OMA by consuming less energy is rigorously derived. Furthermore, the asymptotic analysis demonstrates that the probability of the proposed H-NOMA scheme approaches 1 in the high signal-to-noise ratio (SNR) regime without any constraints on either users' target rates or transmit power ratios. This represents a significant improvement over conventional H-NOMA schemes, which require specific restrictive conditions to achieve probability 1 at high SNRs as shown in existing work. The above observation indicates that with less energy consumption, the proposed HSIC-PA aided H-NOMA can achieve a higher data rate than pure OMA with probability 1 at high SNRs, and hence a higher energy efficiency. Finally, numerical results are provided to verify the accuracy of the analysis and also demonstrate the superior performance of the proposed H-NOMA scheme.

Figures

Figures reproduced from arXiv: 2507.09458 by the authors.

**Figure 2.** PT versus SNR for case m > n with different m. Rm = 0.35 BPCU, M = 5, η = ρn/ρm = 1, n = 1 and β = 1 4 the lowest among all choices of n at high SNRs. Accordingly, the curve corresponding to n = 5 has the largest slope. 0 5 10 15 20 10-4 10-3 10-2 10-1 100 (a) η = ρn/ρm = 7, n = 5, β = 1 3 0 5 10 15 20 25 30 35 40 10-4 10-3 10-2 10-1 100 (b) η = ρn/ρm = 7, m = 1, β = 1 3 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 1.** PT versus SNR for case m < n with different n. Rm = 0.2 bits per channel use (BPCU), M = 5, η = ρn/ρm = 1, m = 1 and β = 1 4 [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗

**Figure 5.** Comparisons of HSI-NPA and HSIC-PA aided hybrid [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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