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Exact BER Performance Analysis for Downlink NOMA Systems Over Nakagami-m Fading Channels

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Pith's one-line read This paper derives exact bit-error-rate expressions for two- and three-user downlink NOMA over Nakagami-m fading with imperfect successive interference cancellation.

desk verdict The exact BER expressions for the three-user case and the Rayleigh closed-forms violate probability bounds (BER > 1/2 at zero SNR, negative at high SNR), so the central claim fails as written despite a systematic derivation. read the letter →

arxiv 1908.01357 v1 pith:7QQELFUV submitted 2019-08-04 cs.IT math.IT

Tasneem Assaf , Arafat Al-Dweik , Mohamed El Moursi , Hatem Zeineldin This is my paper · ORCID
classification cs.ITmath.IT MSC 94A1262G30
keywords NOMAbiterrorratesuccessiveinterferencecancellationNakagami-mfadingoptimalpowerallocationfairnessQPSKorderstatistics
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

This paper tries to put the bit error rate (BER) of power-domain downlink NOMA on an exact footing. It derives per-user BER expressions for two- and three-user systems that use Gray-coded QPSK over Nakagami-$m$ flat fading, and it does so under imperfect successive interference cancellation, where an earlier user's decision error leaves residual interference. The conditional error probability of each user is shown to be a short weighted sum of Gaussian Q functions, and averaging over the ordered fading statistics yields the BER formulas. The paper then uses those formulas to find optimal power coefficients for two goals: equal BER across users and minimum average BER. If the derivation is correct, system designers can predict each user's exact reliability and choose power splits for constrained-power applications such as satellite links.

What carries the argument

The argument is carried by the superimposed QPSK constellation and by the ordered Nakagami-$m$ order-statistic density. Each transmitted superposition is one of 16 points for $N=2$ and 64 points for $N=3$, and every error event reduces to the probability that a Gaussian noise component crosses one of the amplitude levels $A_{u_1u_2u_3}=u_1\sqrt{\beta_1}+u_2\sqrt{\beta_2}+u_3\sqrt{\beta_3}$, $u_i\in\{0,1,-1,2\}$, so the conditional error rate is a Q function $Q(\sqrt{\gamma})$. The averaging step uses the ordered PDF in (93), which contains powers of the lower incomplete gamma function; the infinite-series expansion (94)--(95) and binomial expansion (96) turn it into a gamma-like density, and Craig's integral representation of the Q function together with the identity $\int_0^\infty x^t e^{-bx}\,dx = t!/b^{t+1}$ produces the final average-BER expressions.

What would settle it

Run a two- or three-user Monte Carlo BER test with strongly asymmetric average channel powers, such as $\Omega_1=1$, $\Omega_2=5$, and $\Omega_3=25$, at a fixed SNR and power split, and compare with (36)--(37) or (82)--(87): a clear mismatch would show that the i.i.d.-power ordering assumption, not the conditional Q-function derivation, is the limit of the claim.

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Extended reading notes

Core claim

The central claim is that the exact BER of each user in a downlink power-domain NOMA system with QPSK and imperfect SIC is a signed sum of Gaussian Q functions whose arguments are functions of the power coefficients. For $N=2$, the second user's conditional BER is a five-term sum with coefficient vector $[2,1,-1,-1,1]$; for $N=3$, the second user needs ten terms and the third user eighteen terms. The conditional sums are then averaged over the ordered Nakagami-$m$ channel statistics, giving the expressions (36), (37), and (82)--(84), and for the Rayleigh special case $m=1$ the closed forms (38), (39), and (85)--(87). Throughout, "imperfect SIC" means the receiver explicitly accounts for the possibility that a previously decoded user's symbol is wrong, so the resulting terms reflect both correct and incorrect SIC outcomes. The paper reports that these analytical results match Monte Carlo simulation closely, and that the same expressions allow power allocation either to equalize BER among users or to minimize the average BER.

Load-bearing premise

The load-bearing assumption is that all user links experience the same Nakagami-$m$ fading law with the same average power, so a single order-statistic formula can describe the ordering of weak to strong users; if the users sit at different distances with different average received powers, the averaging formulas stop applying.

Editorial extensions

If this is right

  • At $m=1$ (Rayleigh fading), the BER formulas become closed forms, so per-user error rates and power-optimization costs can be evaluated without numerical integration.
  • The fairness-optimal power allocation gives almost all the power to the weakest user at high signal-to-noise ratio (over 98% for $N=2$ at 30 dB), so equalizing BER is expensive and requires accurate knowledge of operating SNR.
  • The exact results differ from the union-bound approximation by up to about 3 dB in the low-SNR regime, so designers using the bound can underestimate the power needed to hit a target BER.
  • Imperfect SIC has a large effect on the farthest user in the three-user case; assuming perfect SIC overstates reliability most where SIC errors are common.

Reading between the lines

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

  • The same conditioning recipe suggests a general structural pattern: for any $N$-user system the per-user conditional BER should be a signed sum of Q functions over constellation amplitudes, with coefficients fixed by the combinatorics of SIC error events; the two- and three-user formulas are then instances of one larger family.
  • Because the average-BER formulas use an ordered density built from identical link distributions, a natural extension is to non-identically distributed Nakagami links, where the users have different average powers; that would require an order-statistic density for non-i.i.d. channels and would likely change the power-allocation tables.
  • The near-monopoly of power assigned to the weakest user under fairness suggests that in practice a NOMA system might combine the exact BER expressions with adaptive modulation to avoid extreme power splits.
  • The closed-form Rayleigh results could be inverted to give, for a target BER, the required power split as a function of SNR; such an explicit design rule is not written out in the paper but follows directly from (38), (39), (85)--(87).
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper analyzes the bit error rate (BER) of downlink power-domain NOMA with imperfect successive interference cancellation over Nakagami-m flat fading channels, for two-user and three-user scenarios. Exact conditional BER expressions are derived by averaging over the QPSK symbol combinations and the SIC error events, and the average BER is obtained by averaging over ordered Nakagami-m statistics. For the special case m=1 (Rayleigh fading), closed-form average BER expressions are claimed for all users. The derived BER expressions are then used to formulate two optimal power allocation problems, one minimizing the average BER and one achieving equal BER among users. Numerical results and Monte Carlo simulations are presented for various m and power allocations, with claims of perfect agreement between analysis and simulation.

Significance. If correct, the paper would provide the first exact BER analysis of downlink NOMA with imperfect SIC over Nakagami-m fading, including closed forms for Rayleigh fading, and would enable BER-aware optimal power allocation. The conditional-error-probability derivation is a substantial combinatorial exercise, and the general-m averaging framework using ordered Nakagami-m statistics is a useful contribution. However, the claimed Rayleigh closed forms contain serious algebraic errors that make them invalid as probabilities, so the central contribution is not currently established. The paper also silently assumes i.i.d. channel statistics, which limits the scope of the claims.

major comments (3)
  1. [Sec. III.C, Eq. (39); Sec. IV.D, Eqs. (85)-(87)] The Rayleigh closed-form expressions (39), (85), (86), and (87) are not valid BER probabilities. For (39), if all γ_{2,c}=Γ and Γ→0, each bracket tends to 1, so the expression tends to (1/2)∑_{c=1}^5 v_c = 1; but the conditional BER in (35) at zero SNR is (1/2)∑ v_c Q(0) = 1/2. Similarly, (85) tends to 1 as Γ→0, whereas the conditional expression (44) gives 1/2; (86) also tends to 1 as Γ→0; and (87) tends to (1/4)·7·(-1/2) = -7/8 as Γ→∞ because ∑ v_c = 7 and the bracket tends to -1/2. A BER must lie in [0,1] and, for Gray-coded QPSK at zero SNR, must equal 1/2. Thus these closed forms are internally inconsistent with the conditional expressions from which they are supposedly derived, and the simulation matches claimed in Sec. VI for (39),(85)-(87) cannot be correct as stated.
  2. [Appendix I, Eq. (93); Sec. VI] The ordered channel PDF in (93) is derived under the assumption that the N channel gains are i.i.d. Nakagami-m random variables, which is why it factors as f(α)[F(α)]^{n-1}[1-F(α)]^{N-n}. The system model in Sec. II does not state this i.i.d. assumption, and in a downlink NOMA setting users generally have different average powers (different Ω_n). For non-identically distributed Nakagami-m gains, the order-statistics PDF does not have this simple product form, so the average BER expressions (36)-(37), (82)-(84), and (101) do not apply to the general downlink NOMA scenario. The paper should either explicitly restrict all claims and simulations to the i.i.d. case or extend the analysis to non-identical fading parameters.
  3. [Sec. VI, Figs. 6-12] The simulation validation is not reproducible from the information given: no number of Monte Carlo trials, no description of how the ordered channels are generated when users have different average SNRs, and no code or pseudocode for the SIC detector are provided. Moreover, the claimed 'perfect match' between the analytical curves (39),(85)-(87) and simulation cannot be true given the invalid probability limits identified above. The figures and the optimization tables in Sec. V must be regenerated with corrected closed forms, and the simulation methodology should be described in enough detail to allow independent verification.
minor comments (5)
  1. [Fig. 7 caption] The caption says 'N = 2' but the text and the figure refer to the three-user scenario; this should be 'N = 3'.
  2. [Eq. (11)] The subscript in γ_{1,2} is written as α_{2n}; it should be α_1 (the channel gain of the first user).
  3. [Appendix I, Eq. (92)] The function Φ(a,z) is called the lower incomplete gamma function in (92) but is described as the upper incomplete Gamma function in the surrounding text; the terminology should be made consistent.
  4. [Appendix I, Eqs. (94)-(95)] The infinite series representation of [Φ(m,·)]^μ uses coefficients S_i that depend on μ, but this dependence is not indicated in (99), where S_i appears for each k without a subscript denoting the exponent n+k-1; the notation should be clarified.
  5. [Sec. VI] The phrase 'perfectly match' should be replaced with a quantitative statement of agreement (e.g., maximum deviation or confidence bounds), since no simulation parameters or error bars are given.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: BER expressions are derived by averaging conditional Q-function decision probabilities over the ordered Nakagami-m density, with no target result used as input.

full rationale

The derivation chain is self-contained. The conditional BERs in Secs. III and IV are built from exact nearest-neighbor decision regions of the superimposed QPSK constellation (e.g., Eq. (12): PU1 = 1/2[Q(sqrt(gamma_1,1)) + Q(sqrt(gamma_1,2))], and Eq. (35) for U2), and the average BER in Appendix I (Eq. (101)) is obtained by integrating these Q-functions against the ordered Nakagami-m density (Eqs. (90)-(93)). The Rayleigh closed forms (38), (39), (85)-(87) are algebraic specializations m=1 of those averages, not fitted quantities. The optimal power allocation in Sec. V uses the derived BER as the objective function in a constrained optimization; this is an application of the derived result, not a hidden input. Monte Carlo simulations in Sec. VI are external numerical validation. The self-citations in the paper ([2] and [21]) are background and comparison material: the union bound of [21] is benchmarked against the exact expression in Fig. 8, not used to justify it. The reader's noted i.i.d. channel-order limitation and the algebraic validity of the printed Rayleigh formulas are correctness/scope concerns, not circularity: no equation is defined in terms of the quantity it claims to predict, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard communications assumptions (AWGN, perfect phase compensation, i.i.d. ordered Nakagami-m fading) and on the convergence of the infinite series representation of the lower incomplete gamma function. No free parameters are fitted; the power allocation coefficients are optimization outputs, not fitting parameters.

assumptions (5)
  • domain assumption User channel gains are i.i.d. Nakagami-m before ordering.
    The ordered PDF in (93) combines a single f(alpha) and F(alpha); real deployments with unequal average SNRs would break this.
  • domain assumption Perfect channel phase compensation at receivers.
    The model removes phase before detection (Sec. II), ignoring phase estimation errors.
  • standard math AWGN is circularly symmetric with independent equal-variance real and imaginary parts.
    Used to reduce 2D error events to 1D Gaussian Q functions.
  • standard math Infinite series for [Phi(m,x)]^mu in (94) converges and can be integrated termwise.
    The exactness of the general-m BER expressions depends on this unstated convergence and interchange condition.
  • domain assumption b12 errors do not affect U2 detection and b12/b22 errors do not affect U3 detection.
    Stated in Sec. III.B as 'it is assumed that Pb12 = 1' and analogous for U3; valid for QPSK because I/Q decisions are separable, but it is a stated simplification.

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Pith. "Pith review of Exact BER Performance Analysis for Downlink NOMA Systems Over Nakagami-m Fading Channels." pith.science (2026). https://pith.science/paper/7QQELFUV

@misc{pith2026190801357,
  author       = {Pith},
  title        = {Pith review of: Exact BER Performance Analysis for Downlink NOMA Systems Over Nakagami-m Fading Channels},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7QQELFUV}},
  note         = {Machine review of arXiv:1908.01357}
}
read the original abstract

In this paper, the performance of a promising technology for the next generation wireless communications, non-orthogonal multiple access (NOMA), is investigated. In particular, the bit error rate (BER) performance of downlink NOMA systems over Nakagami-m flat fading channels, is presented. Under various conditions and scenarios, the exact BER of downlink NOMA systems considering successive interference cancellation (SIC) is derived. The transmitted signals are randomly generated from quadrature phase shift keying (QPSK) and two NOMA systems are considered; two users' and three users' systems. The obtained BER expressions are then used to evaluate the optimal power allocation for two different objectives, achieving fairness and minimizing average BER. The two objectives can be used in a variety of applications such as satellite applications with constrained transmitted power. Numerical results and Monte Carlo simulations perfectly match with the derived BER analytical results and provide valuable insight into the advantages of optimal power allocation which show the full potential of downlink NOMA systems.

Figures

Figures reproduced from arXiv: 1908.01357 by the authors.

Figure 1
Figure 1. The constellation diagram of the transmitted symbol [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Equivelant constellation of xsic|sˆ1=s1 and xsic|sˆ16=s1 of the second user, N = 2. by a0, a1, a2 and a3. Substituting the results of the 16 cases in (7) gives P11 = 1 2 h Q √γ1,1  + Q √γ1,2 i. It is also straightforward to show that P12 = P11. Therefore, the conditional BER of the first user is given: PU1 = 1 2 [Pb11 + P b12 ] = 1 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. b. To detect its own symbol s2, the second user should follow the SIC process described in (13). It should be noted that the error probability of the second user is not affected by the detection result of the second bit of the first user b12. The BER of U2 depends on s1, sˆ1, s2 and s3. Therefore, the average BER for b2i is the average of all possible combi￾nations, Pb2i = X g,l,k,v Pb2i | s (g) 1 ,s (k) 2 ,s (v) 3 … view at source ↗
Figures from the paper (9 more)
Figure 3
Figure 3. Figure 3: The transmitted superimposed signal constellation for [PITH_FULL_IMAGE:figures/full_fig_p007_3.png]
Figure 4
Figure 4. Figure 4: Equivelant constellation of (a) xsic|ˆb11=b11 and (b) xsic|ˆb116=b11 of the second user, N = 3. The error probability of b21 can be evaluated as Pb21 |A324 P  ˆb (0) 11 | b (0) 11 ,s (0) 2 ,s (0) 3  = P(α2A´1´1´1 ≤ n2 ≤ α2A0´1´1 ) = Q √γ3,5  − Q √γ3,4  . (54) where…
Figure 5
Figure 5. Figure 5: Equivelant constellation of the third user, [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: BER for the first and second users, N = 2, m = 0.5, 1, 2 and 3, and Ω = 1. assigned for the users with the channel gains α1 < α2 < · · · < αN , respectively. The problem in (88b) is a constrained non￾linear optimization problem which is solved using the Interior￾Point …
Figure 7
Figure 7. Figure 7: BER for the first, second, and third users in the [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: Exact BER and union bound for the first and second users, [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: Perfect and imperfect BER for the first, second, and third users in the N=3, [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
Figure 11
Figure 11. Figure 11: BER for the first, and second users at various [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]
Figure 12
Figure 12. Figure 12: BER for the first, second, and third users at various [PITH_FULL_IMAGE:figures/full_fig_p015_12.png]

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    J. Craig, “A new, simple and exact result for calculating the probability of error for two-dimensional signal constellations,” in Proc. Military Commun. Conf. (MILCOM91) , Nov. 1991. Tasneem Assaf (S’14) received the B.Sc. degree in com- munications engineering from Khalifa Un...

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    He joined Vestas Wind Systems, Arhus, Denmark, in the Technology R&D with the Wind Power Plant Group

    He joined McGill University as a Postdoctoral Fellow with the Power Electronics Group. He joined Vestas Wind Systems, Arhus, Denmark, in the Technology R&D with the Wind Power Plant Group. He was with TRANSCO, UAE, as a Senior Study and Planning Engineer. He is currently a Pro...

    1999

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.