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Sum of Squared Extended $\eta$-$\mu$ and $\kappa$-$\mu$ RVs: A New Framework Applied to FR3 and Sub-THz Systems

T0 review · 2 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The paper claims that sums of squared i.i.d. Extended $\eta$-$\mu$ and $\kappa$-$\mu$ fading variables have exact Gamma-function PDFs and CDFs that make multi-antenna FR3 and sub-THz analysis computationally fast.

desk verdict This paper gives a useful Gamma-series tool for sums of squared generalized fading RVs, but the load-bearing residue step is asserted rather than proven, so it deserves refereeing with a request to fill that gap. read the letter →

arxiv 2502.02092 v1 pith:PR3OHEPC submitted 2025-02-04 cs.IT eess.SPmath.IT

Gustavo Rodrigues de Lima Tejerina , Italo Atzeni This is my paper · ORCID
classification cs.ITeess.SPmath.IT
keywords generalizedfadingmodelsExtendedη-µdistributionκ-µFR3sub-THzbandsumofsquaredrandomvariablesmaximumratiotransmissionGamma-functionseries
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 claims that the sum $W=\sum_{n=1}^{N} W_n$ of $N$ squared, independent and identically distributed Extended $\eta$-$\mu$ random variables has a probability density function and cumulative distribution function given exactly by the Gamma-function series (13) and (14), and that the corresponding PDF and CDF for squared i.i.d. $\kappa$-$\mu$ variables are given by (23) and (24). The coefficients $h_m$ and $k_m$ are defined recursively, so evaluating the series for a massive number of antennas means truncating a few hundred memoized recursion steps rather than evaluating Fox-H or Horn functions. If correct, this replaces the special-function expressions of [21], [22], [29] with expressions cheap enough for FR3 and sub-THz downlink analysis with hundreds to thousands of antennas. The paper also derives outage, coverage, bit-error, and symbol-error probabilities, including high-SNR asymptotics, and validates the series against Monte Carlo simulation.

What carries the argument

The load-bearing mechanism is the MGF-based contour method: rewrite the exponential and hypergeometric (or Bessel) factors in the Laplace transform as Mellin-Barnes-type contour integrals, interchange the order of integration, evaluate the resulting double contour integral by the residue theorem at the two pole families that give a double power series, then use the Cauchy product identity of [33, Eq. (0.314)] to fold the double series into a single series with recursively defined coefficients $h_m$ (12) and $k_m$ (22). Here Extended $\eta$-$\mu$ is the NLoS-oriented fading model with power-ratio $\eta$, cluster count $\mu$, and in-phase/quadrature cluster ratio $p$ (envelope PDF (4)), and $\kappa$-$\mu$ is the LoS-oriented model with LoS-to-scatter power ratio $\kappa$ and cluster count $\mu$ (envelope PDF (20)). The inverse Laplace transform of each term is a Gamma function, which is why the final PDF/CDF are Gamma-function series; the same machinery yields a second, unconditionally convergent set of expressions and the ${}_1F_1$/${}_2F_2$ truncation bounds.

What would settle it

Evaluate the single-variable MGF in (6) by direct quadrature at a non-trivial parameter point, say $\eta=0.6$, $\mu=0.5$, $p=0.5$, $\hat{w}=1$, $s=1$, and compare with the residue-based series (9) truncated at increasing $m$; if the two disagree beyond quadrature accuracy, the contour step behind (13)-(14) is not valid. A second check is to compare the CDF (14) with Monte Carlo for $N=256$ under parameters where the convergence bounds are tightest.

Watch

Extended reading notes

Core claim

The central discovery is that the moment-generating function of each squared fading variable can be written as a double contour integral (8), that this integral can be evaluated by residues at the poles $t = m-l+\mu$ and $v = -l$ to give a double series (9), and that the double series collapses, through the Cauchy product, into a single power series in $1/(s\hat{w})$ whose coefficients obey the simple recurrences (12) and (22). Multiplying $N$ copies of the MGF gives another power series of the same form, and termwise inverse Laplace transform yields the PDF and CDF in (13)-(14) and (23)-(24), expressed entirely through Gamma functions. For the Extended $\eta$-$\mu$ case the paper works from the envelope PDF of [31], and for the $\kappa$-$\mu$ case from the PDF of [34]; in both cases the alternative MGF forms (16) and (25) give unconditionally convergent versions of the performance metrics. The paper shows absolute convergence of the series by bounding the coefficients and reducing the bound to finite Kummer hypergeometric values, and it supplies truncation-error bounds as ${}_2F_2$ expressions.

Load-bearing premise

Everything rests on the claim that the double contour integral in the MGF derivation collapses exactly into residues at two chosen pole families, a step the paper asserts by reference rather than demonstrating; if that step is wrong, the PDF and CDF series are not proven exact.

Editorial extensions

If this is right

  • The PDF and CDF series (13)-(14) and (23)-(24) are exact for every valid set of fading parameters and for any number $N$ of antennas, with no integer-parameter restriction.
  • The derived outage, coverage, BEP, and M-PSK/M-QAM SEP expressions inherit the Gamma-function form, and the second representations (16) and (25) converge for any parameter choices.
  • Absolute-convergence proofs and truncation-error bounds justify stopping the series at a few hundred terms, keeping numerical error below a prescribed threshold.
  • In the FR3 downlink example, 256 antennas give near-perfect coverage out to roughly 300-450 m at 15 GHz, and in the sub-THz example 1024 antennas give coverage out to roughly 700-1200 m at 140 GHz.
  • The high-SNR asymptotics identify the diversity order as $N\mu$ (antennas times multipath clusters), and an eightfold increase in antennas approximately doubles the usable carrier frequency.

Reading between the lines

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

  • Beyond the two models treated here, the same power-series-plus-recurrence template should apply to other generalized fading densities with a contour representation, such as $\alpha$-$\eta$-$\kappa$-$\mu$ or fluctuating two-ray models; this extension is not claimed in the paper.
  • The truncation bounds involve Kummer and ${}_2F_2$ functions that stay finite, so the series is likely practical well past the 1024-antenna examples; a timing and memory benchmark at $N=2048$ or $4096$ would test that scaling claim directly.
  • Because the $m=0$ terms dominate at high SNR, a designer could use the asymptotic diversity order $N\mu$ to budget antennas against multipath clusters before running the full series, a shortcut the paper leaves implicit.
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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

2 major / 4 minor

Summary. The manuscript derives infinite-series representations for the PDF and CDF of the sum of N i.i.d. squared Extended η-µ and κ-µ random variables, with coefficients defined recursively. It uses these series to obtain outage, BEP, and SEP expressions for M-PSK/M-QAM in closed form, together with asymptotic high-SNR formulas. The framework is applied to an FR3 downlink modeled by Extended η-µ and a sub-THz downlink modeled by κ-µ, and validated by Monte Carlo simulations and by comparison with existing special-function representations [21], [29].

Significance. If the claimed exactness holds, the paper provides a substantial practical advance: it replaces Fox-H/Horn-function representations with sums of Gamma functions that can be evaluated for massive antenna counts, with stated convergence and truncation bounds and with runtime gains of orders of magnitude. The paper also offers useful system-level insights (e.g., antenna count vs. carrier-frequency scaling). The main limitation is that the central residue-theorem step is asserted, not proved; because this step feeds the exactness of all subsequent expressions, the paper's central claim is currently unproven though supported numerically.

major comments (2)
  1. [Section II-A, Eq. (8) to Eq. (9)] The transition from the double contour integral in (8) to the residue series in (9) is the load-bearing step for both the Extended η-µ and κ-µ results, but it is not demonstrated. The text states that the contours are 'defined to guarantee convergence and circumvent duplicate poles' and that both integrals are solved by the residue theorem at the poles t → m−l+µ and v → −l, citing [27]; however, the integrand in (8) includes the factor Γ(µp/(1+p)−v)/Γ(µ−v), which is absent in [27]. The pole families of this integrand (t = −n, v = −n, t+v = µ+n, v = µp/(1+p)+n, and the zeros at v = µ+n) are not enumerated, the decay of the integrand at infinity is not shown, and it is not established that the residues at all other enclosed poles vanish or cancel. Since (9) is the sole source of exactness for (13)–(14) and, by the same argument, for (21)–(24), I request a complete contour-deformation proof, including the choice of contours L∗t and L∗v, a list of enclosed poles and their residues, and estimates for the remaining contour integrals, or an alternative derivation of (9) via analytic continuation of the Mellin-Barnes integral.
  2. [Section III-C, Eqs. (60) and (62)] The incomplete Beta function terms in (60) and (62) are written with negative first arguments, e.g., B(−ξ/(gq ˆw); Nµ+m, 1/2−Nµ−m) and B(−˜K/(gq ˆw); Nµ+m, 1/2−Nµ−m). In the standard definition, the incomplete Beta function B(x; a, b) is defined for 0 ≤ x ≤ 1 (with a,b>0), so these expressions are not defined without specifying an analytic continuation. If the authors intend a generalized incomplete Beta function or an analytic continuation, this must be stated explicitly and justified. Additionally, the SEP formulas (53), (60), (55), and (62) are presented without derivation; I ask for a sketch of the integration steps from (51) and (58) and for a direct numerical check against the integrals in (51) and (58) over a grid of parameters, because the simulation markers in Figs. 7 and 13 alone do not isolate all intermediate algebraic steps.
minor comments (4)
  1. [Section II-D] The computational complexity statement appears internally inconsistent: the recursion-tree count is given as 2ϵ−1 while the memoized count is ϵ(ϵ+1)−1, which exceeds the recursion-tree count for ϵ>1. Please clarify what quantity is being counted (e.g., number of h_m evaluations, number of terms in the inner sums, or number of floating-point operations) and correct the formulas.
  2. [Section II-C] The derivation of the bounds (33) and (35) relies on several steps summarized as 'after some mathematical manipulations'; please expand these steps or provide a supplementary derivation, since the bound on |hm| in (30)–(32) is essential to the convergence argument.
  3. [Section III-A] The asymptotic outage expressions (38) and (39) are stated as just the m=0 term of (14) and (24); please add the condition w/ˆw→0 under which this is the dominant term, or a short justification that the higher-order terms are negligible.
  4. [Table I] The phrase 'no restrictions on the convergence' for the second column should be reconciled with the analytic-continuation caveats for the Gauss hypergeometric function in (47)–(48) and (53)/(55); please state the exact convergence conditions for each expression.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the series coefficients are obtained by contour integration of the defining MGFs, with no fitted quantities and no load-bearing self-citations.

full rationale

The derivation chain is self-contained with respect to the claimed sum-statistics results. The PDF/CDF series in (13)-(14) and (23)-(24) are obtained by: (i) taking the defining envelope PDFs from [31, Eq. (14)] and [34, Eq. (1)]; (ii) applying a Laplace transform to obtain the single-RV MGF; (iii) evaluating the resulting double contour integral by residues following the method of [27]; (iv) raising the MGF to the N-th power for i.i.d. sums and applying [33, Eq. (0.314)] to organize the Cauchy product; and (v) inverting the Laplace transform termwise. No parameter in the final series is fitted to the target PDF/CDF or to any performance metric; the coefficients h_m and k_m are recursively defined solely in terms of the model parameters (eta, mu, p, kappa, N). The asymptotic formulas, e.g., (38), (39), (43), and (44), are leading-order truncations of the same exact series, not independent predictions. The self-citations [26] and [31] are to the authors' prior definition and study of the Extended eta-mu model; they supply the input model, not a surrogate for the sum-statistics derivation. The reference to [27] is to an external published residue method; even if that residue step were incomplete, it would be a mathematical gap (the evaluation from (8) to (9) is asserted rather than proven), not a circular reduction, because the assertion does not assume the conclusion it is used to prove. Figures 1-2 compare the series against simulation and against the existing solutions in [21] and [29], providing external numerical checks rather than circular support. No step reduces by construction to its own output, so the circularity score is 0.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central mathematical representation does not fit any parameters to data, but the application sections rely on hand-picked fading parameters for FR3 and sub-THz scenarios, and the core derivation leans on an unshown residue-evaluation step. No new physical entities are introduced.

free parameters (2)
  • FR3 Extended eta-mu scenario parameters = eta = 1.5, mu = 0.5, p = 0.75 (also variants in Fig. 6)
    Chosen by hand to represent severe fading; not estimated from FR3 measurements. The system-level conclusions in Section IV-A depend on these values.
  • Sub-THz kappa-mu scenario parameters = kappa = 0.5, mu = 0.5 (also variants in Fig. 12)
    Chosen by hand to represent severe fading; not estimated from sub-THz measurements. The system-level conclusions in Section IV-B depend on these values.
assumptions (4)
  • domain assumption The Extended eta-mu envelope PDF in (4), taken from [31], is the correct statistical description of FR3 fading.
    No FR3 measurement data is used; the parameters are chosen by hand in Section IV-A.
  • domain assumption The kappa-mu envelope PDF in (20), taken from [34], is the correct statistical description of sub-THz fading.
    No sub-THz measurement data is used; the parameters are chosen by hand in Section IV-B.
  • domain assumption The fading coefficients g_n are independent and identically distributed, with common scale w_hat.
    This is required for the MGF product step (11) and (21) in Section II-A and II-B.
  • ad hoc to paper The double contour integral in (8) can be evaluated by residues at the stated poles, with all other singularities contributing zero.
    This is the unproved residue-theorem step in Section II-A that underlies the exact series representations (9)-(14).

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Cite this review

Pith. "Pith review of Sum of Squared Extended $\eta$-$\mu$ and $\kappa$-$\mu$ RVs: A New Framework Applied to FR3 and Sub-THz Systems." pith.science (2026). https://pith.science/paper/PR3OHEPC

@misc{pith2026250202092,
  author       = {Pith},
  title        = {Pith review of: Sum of Squared Extended $\eta$-$\mu$ and $\kappa$-$\mu$ RVs: A New Framework Applied to FR3 and Sub-THz Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PR3OHEPC}},
  note         = {Machine review of arXiv:2502.02092}
}
abstract

The analysis of systems operating in future frequency ranges calls for a proper statistical channel characterization through generalized fading models. In this paper, we adopt the Extended $\eta$-$\mu$ and $\kappa$-$\mu$ models to characterize the propagation in FR3 and the sub-THz band, respectively. For these models, we develop a new exact representation of the sum of squared independent and identically distributed random variables, which can be used to express the power of the received signal in multi-antenna systems. Unlike existing ones, the proposed analytical framework is remarkably tractable and computationally efficient, and thus can be conveniently employed to analyze systems with massive antenna arrays. For both the Extended $\eta$-$\mu$ and $\kappa$-$\mu$ distributions, we derive novel expressions for the probability density function and cumulative distribution function, we analyze their convergence and truncation error, and we discuss the computational complexity and implementation aspects. Moreover, we derive expressions for the outage and coverage probability, bit error probability for coherent binary modulations, and symbol error probability for M-ary phase-shift keying and quadrature amplitude modulation. Lastly, we provide an extensive performance evaluation of FR3 and sub-THz systems focusing on a downlink scenario where a single-antenna user is served by a base station employing maximum ratio transmission.

Figures

Figures reproduced from arXiv: 2502.02092 by the authors.

Figure 1
Figure 1. PDF of the sum of squared i.i.d. Extended η-µ RVs, with N = 16, wˆ = 1, and the following combinations of parameters: (i) η → 0, µ = 0.5, p = 1 (Nakagami-m); (ii) η = 0.6, µ = 0.5, p = 0.5; (iii) η = 0.25, µ = 1.25, p = 1.1, and (iv) η = 1.5, µ = 2, p = 0.5. Kummer confluent hypergeometric function 1F1 in (36) is finite for any choice of parameters in our framework. Moreover, we obtain the upper bound on the truncat… view at source ↗
Figure 3
Figure 3. Extended η-µ model: Downlink coverage probability versus distance, with η = 1.5, µ = 0.5, p = 0.75, fc = 15 GHz, Pt = 30 dBm, N ∈ {16, 32, 64, 128, 256}, α = 0, and γth ∈ {0, 5} dB. 20 25 30 35 40 10−6 10−5 10−4 10−3 10−2 10−1 1 N = 16 = 32 = 64 = 128 = 256 Pt [dBm] BEP α = 0 α = 0.3 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Extended η-µ model: Downlink BEP for coherent BPSK versus transmit power, with η = 1.5, µ = 0.5, p = 0.75, fc = 15 GHz, d = 250 m, N ∈ {16, 32, 64, 128, 256}, and α ∈ {0, 0.3}. and eventually the expression fails to converge to the correct value [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figures from the paper (4 more)
Figure 7
Figure 7. Figure 7: Extended η-µ model: Downlink SEP for M-PSK and M-QAM versus transmit power, with η = 1.5, µ = 0.5, p = 0.75, fc = 15 GHz, d = 250 m, Pt = 30 dBm, N = 256, α = 0, MPSK ∈ {4, 8, 16}, and MQAM ∈ {8, 16, 32, 64}. 20 25 30 35 40 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 1 Pt …
Figure 10
Figure 10. Figure 10: κ-µ model: Downlink BEP for coherent BPSK versus transmit power, with κ = 0.5, µ = 0.5, fc = 140 GHz, d = 300 m, N ∈ {64, 128, 256, 512, 1024}, and α ∈ {0, 0.3}. analytical results are computed with 100 terms, remarkably taking less than 5 s to generate the whole plot…
Figure 13
Figure 13. Figure 13: plots the SEP for M-PSK and M-QAM versus the transmit power, with N = 512, MPSK ∈ {4, 8, 16}, and MQAM ∈ {8, 16, 32, 64}. In this setting, the SEP is lower than that of the FR3 system in Section IV-A (cf [PITH_FULL_IMAGE:figures/full_fig_p012_13.png]
Figure 14
Figure 14. Figure 14: κ-µ model: Downlink BEP for coherent BPSK and SEP for 4-PSK and 8-QAM versus transmit power, with κ = 0.5, µ = 0.5, fc = 140 GHz, d = 50 m, N = 32, and α = 0. C. Comparison Between FR3 and Sub-THz Systems Based on the results in Sections IV-A and IV-B, we highlight th…

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.