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REVIEW 4 major objections 4 minor 35 references

Energy Efficiency Analysis of Active RIS-enhanced Wireless Network under Power-Sum Constraint

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper derives an energy-efficiency crossover: active RIS wins below roughly 1024 elements, passive RIS wins above.

desk verdict A real crossover question and the right large-N frame, but the PA-factor derivation has a load-bearing algebraic error (q4 is dimensionally wrong and reused), so N0≈2^10 is not reproducible; worth refereeing as a major-revision candidate. read the letter →

arxiv 2506.02823 v1 pith:OO7BO7Y7 submitted 2025-06-03 cs.PF

Jingdie Xin , Yan Wang , Feng Shu , Feng Zhao , Yifan Zhao , Hao Jiang This is my paper
classification cs.PF
keywords activereconfigurableintelligentsurfacepassiveRISenergyefficiencylawoflargenumberspowerallocationRayleighfadingcrossoverthresholdwirelessnetwork
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

The paper tries to settle a practical design question: when should a network operator choose an active reconfigurable intelligent surface (RIS), which amplifies signals but burns extra power, over a passive one? Using the law of large numbers to approximate the signal-to-noise ratio in Rayleigh fading, it derives a closed-form energy-efficiency (EE) expression as a function of the number $N$ of RIS elements, the power allocation factor, the total power, and two noise variances. It then sets the EE of the active system equal to that of the passive system and solves for the crossover element count. The paper's central result is that the crossover sits near $N_0\approx 2^{10}$: below about 1024 elements the active RIS is more energy-efficient, and above that the passive RIS is more energy-efficient. If correct, this gives operators a simple deployment rule based on the size of the RIS array.

What carries the argument

The argument is carried by a law-of-large-numbers asymptotic approximation: sums over the $N$ RIS reflection elements are replaced by their expectations, turning the SNR and total power consumption into deterministic closed forms, Eq. (25) for active RIS and Eq. (74) for passive RIS. With these forms, equality of the two efficiencies becomes a single nonlinear equation in $\alpha=1/N$, and the paper applies Newton's method, bisection, and simulated annealing to locate its root, which is the crossover $N_0$.

What would settle it

Run an exact Monte Carlo simulation that keeps the direct-path amplitude $|h|$ as a random variable (no mean replacement) for $N$ near $10^3$, with the paper's parameter set, and check whether the active-versus-passive EE crossover still lies at $N_0\approx 2^{10}$; a shift of more than a few hundred elements would falsify the threshold as stated.

Watch

Extended reading notes

Core claim

The central discovery is a quantitative threshold for the active-versus-passive RIS energy-efficiency trade-off. Under a power-sum constraint, where the base station and the RIS share a fixed total power, the paper shows that active RIS energy efficiency first rises and then falls as the number of elements $N$ grows, because the amplifier adds noise whose impact scales with $N$, while passive RIS efficiency rises monotonically. Solving $\mathrm{EE}_{\mathrm{active}}(N)=\mathrm{EE}_{\mathrm{passive}}(N)$ with $N$ as the variable yields a unique crossover $N_0\approx 2^{10}$; for $N<N_0$ the active RIS achieves higher EE, and for $N>N_0$ the passive RIS does. The comparison is made on the same total power budget, and the asymptotic EE expressions agree with exact Monte Carlo simulation for medium and large $N$.

Load-bearing premise

The derivation replaces the single direct-path channel amplitude $|h|$ by its mean value, but $|h|$ is one random draw and does not average out as $N$ grows, so the crossover location could shift if this averaging is inaccurate for medium-scale $N$.

Editorial extensions

If this is right

  • Below roughly 1024 RIS elements, deploying an active RIS yields higher energy efficiency than a passive one on the same total power budget.
  • Above the crossover, passive RIS becomes the greener choice, so RIS size alone can drive the active/passive decision.
  • The optimal power allocation factor $\beta$ has a closed form derived from a first-order Taylor expansion and Ferrari's method, letting operators set BS-versus-RIS power splitting without search.
  • Increasing total power $P_t$ beyond its optimum reduces EE, since capacity grows logarithmically while power grows linearly.
  • As the noise variances at the RIS and the user shrink, EE saturates at a constant value rather than growing without bound.

Reading between the lines

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

  • The crossover $N_0\approx 2^{10}$ is reported for one parameter set (noise variances at $-70$ dBm, specific path-loss exponents); under a different noise budget or power level the threshold would likely move, so the useful output is the crossing mechanism rather than the exact number.
  • The same equal-EE equation could be solved for the total power $P_t$ or noise variance instead of $N$, giving complementary deployment thresholds.
  • Because the law-of-large-numbers step averages the direct-path amplitude, a finite-$N$ exact analysis that avoids that averaging would be a sharper test of the crossover; the paper's own asymptotic-versus-exact agreement for medium and large $N$ suggests the error is small there but is not shown for the crossover region.
  • The framework could extend to discrete phase shifters, imperfect channel knowledge, or multi-antenna nodes, where the amplification-noise penalty of active RIS would interact with those effects.
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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

4 major / 4 minor

Summary. The paper analyzes the energy efficiency (EE) of an active RIS-assisted wireless network under a power-sum constraint. Using the law of large numbers, it derives an asymptotic EE expression as a function of the power allocation factor β, the number of RIS elements N, total power Pt, and noise variances σr² and σu². It then studies the effect of each parameter, provides a closed-form expression for the optimal β via Taylor expansion and Ferrari's method, and defines a crossover equation between the EE of active and passive RIS systems. Three root-finding algorithms (Newton, bisection, simulated annealing) are proposed, and the paper claims that active RIS outperforms passive RIS for N below about 2^10 and is worse above that threshold.

Significance. If the crossover result is correct, it would provide a simple and practically useful deployment rule for choosing between active and passive RIS. The paper has genuine strengths: the derivation is not circular, no parameter is fitted to force the threshold, exact and asymptotic EE expressions are compared in simulation, and the proposed numerical methods are standard and clearly described. However, the central claim is built on approximations whose validity conditions are violated at the paper's own parameters, and several coefficient definitions in the derivation are internally inconsistent. These issues prevent the reported N0≈2^10 from being reproduced from the manuscript as written, so the significance of the claimed threshold cannot yet be assessed reliably.

major comments (4)
  1. [Eqs. (27)–(28)] The definition q4 = -A5 N Pt in Eq. (28) is internally inconsistent. In the total-power denominator q4β + q9 of Eq. (27), the β coefficient must be -A5 N Pt, as follows from Eq. (23). But inside the square root q4β² + q5β + q6, the β² coefficient extracted from Eq. (25) is -A5 N Pt². A single symbol q4 cannot have both values. Because the subsequent coefficients c1, c2, l1–l5 in Eqs. (31)–(36) use q4 through q3q4/(2√q6), the closed-form optimal β is wrong. This inconsistency must be corrected and all derived coefficients re-derived before the crossover computation can be trusted.
  2. [Eqs. (29)–(30)] The Taylor expansions are not valid at the paper's operating point. With Pt = 26 dBm, σr² = σu² = -70 dBm, and the stated path-loss exponents, the corrected small parameter satisfies Δx ≈ -39.3β² + 38.3β, which is about 4 at the reported optimum β ≈ 0.85; the argument of the logarithm is γ ≈ 10^4, so the first-order expansion log2(1+γ) ≈ γ/ln2 is grossly invalid. The closed-form β obtained from Ferrari's method is therefore not justified, and any 'Taylor approximate' curve in Fig. 3 is not a controlled approximation. Since the crossover equation (56) depends on β, the claimed threshold N0 ≈ 2^10 is not established by the manuscript. In addition, Eq. (29) as printed drops the factor √q6, so the equality '√q6(1+…) = 1+…' is dimensionally wrong; the correct form is √q6 + (q4β² + q5β)/(2√q6).
  3. [Eq. (50)] The coefficient m2 in Eq. (50) omits the factor βPt that multiplies the square-root term in Eq. (25). In Eq. (49), the cross-product term is βPt A4√(N X)/Dden, with X = A5βPt²(1−β) − βPtσr² + Ptσr², but m2 as printed is A4√X/Dden. This gives m2 the dimension of 1/W while m1 and m3 are dimensionless, and it underweights the cross term by a factor βPt in the EE comparison (56), so the computed root N0 shifts even if all other equations are corrected.
  4. [Section VI / Fig. 8] The parameters needed to reproduce the claimed N0 ≈ 2^10 are not given. Section VI specifies geometry, path-loss exponents, and μ, but not the bandwidth B, the hardware power constants Pc,n, P0,RIS, P^p_c,n, and P^p_0,RIS, nor the value of β used when evaluating EEa(N) in Eq. (56). Since the active-RIS total power in Eq. (23) and the crossover equation depend on these quantities, a reader cannot verify Fig. 8 or the central threshold claim from the manuscript as written.
minor comments (4)
  1. [Eq. (22)] The single direct-path amplitude |h| is replaced by its expectation, although there is only one realization of this Rayleigh random variable and the law of large numbers does not apply to it. For the stated geometry and N ≈ 1024 this term is very small relative to the RIS term, so this is not the main defect, but the approximation should be justified or bounded, especially for smaller N.
  2. [Eq. (12)] The total power model subtracts the received power Pin from the reflected power Pout and then adds μβPt + Pc. For some parameter values this expression can become negative, so the physical interpretation of the power model should be clarified.
  3. [Fig. 2] The legend contains the typo 'Asympototic'; it should read 'Asymptotic'.
  4. [References] The simulated-annealing method is attributed to reference [24], which is a paper on power allocation and beamforming; a standard numerical-analysis textbook reference would be more appropriate for this classical method.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the active/passive EE crossover N0 is solved from independently derived formulas, not fitted or defined into existence.

full rationale

The derivation chain is self-contained. Section III derives the asymptotic SNR and EE from the system model (Eqs. (1)-(14)) by applying the law of large numbers and the Rayleigh moment formulas (Eqs. (19)-(24)), so Eq. (25) is a closed-form consequence of the stated model rather than an imported prediction. The PA factor beta in Section IV-A is produced by optimizing the paper's own Taylor-approximated EE through Ferrari's method, not by calibrating a parameter to the threshold N0. The crossover in Section V is obtained by equating two separately derived EE functions, EEa(N) in Eq. (49) and EEp(N) in Eqs. (51)-(54), and solving Eq. (56); no parameter in m1-m9 is fitted to make the root equal 2^10. The only notable self-citation is Ref. [26] for the uniform active-RIS amplification factor in Eq. (13), but that assumption is explicitly stated and comes from a peer-reviewed prior work; the threshold is not forced by that citation. Reproducibility weaknesses (e.g., the Taylor expansion in Eq. (29), omitted hardware-power constants, and the unspecified beta used in Fig. 8) are correctness concerns, not circularity.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

No new physical entities are introduced; active RIS, Rayleigh channels, and power models are taken from prior work. The analysis leans on standard probability approximations and on circuit-power constants that are not fully specified, which limits the generality of the claimed crossover.

free parameters (4)
  • per-element RIS circuit power P_c,n = not specified
    Appears in the EE denominators (Eqs. 23, 25, 49) and affects the crossover N0, but no value is given in Section VI.
  • static power constants P_0 and P_0,RIS (combined as A7) = not specified
    A7 = P0 + P0,RIS appears in Eq. (26) and in all EE curves, but is never assigned numerical values.
  • bandwidth B = not specified
    B multiplies every EE expression (Eq. 14) and therefore scales all plotted EE values.
  • power allocation factor beta in the crossover simulation = not specified for Figs. 7-8
    The threshold N0 is computed from EE_a(N), which depends on beta, but the paper does not state which beta (optimal or fixed) is used in the crossover plots.
assumptions (7)
  • standard math Sums of N independent channel products converge to their expectation as N grows (weak law of large numbers).
    Used in Section III, Eqs. (15)-(18), to convert sums into E[|f||g|] and similar terms.
  • domain assumption All channels are Rayleigh fading with parameters alpha_f, alpha_g, alpha_h and are mutually independent.
    Invoked in Section III, Eqs. (19)-(20), to evaluate the expectations.
  • domain assumption Continuous phase shifters at the RIS perfectly align all phases, and the direct-path phase is set to zero.
    Used in Eq. (4) and (5); any phase quantization error is neglected.
  • domain assumption All active RIS elements share the same amplification factor lambda given by Eq. (15).
    Following [26], the amplitude is assumed identical across elements, which allows pulling lambda out of sums.
  • ad hoc to paper The direct-path amplitude |h| in the SNR numerator can be replaced by its mean E|h| even though it is a single random variable.
    Eq. (22) uses E|h| and E|h|^2 terms for a quantity that is not averaged over N; no error bound is provided.
  • ad hoc to paper The Taylor expansions of the square root and the logarithm are valid because Delta x and gamma are small.
    Eqs. (29)-(30) assume Delta x to 0 and gamma(beta) to 0, but the simulation parameters can make Delta x of order 100; no validity condition is stated.
  • domain assumption Total power consumption of the active RIS system is P_out - P_in + mu beta P_t + P_c, so received power reduces consumption.
    Eq. (12); this sign convention is physically unusual and is not justified, though it is small in the plotted regime.

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Pith. "Pith review of Energy Efficiency Analysis of Active RIS-enhanced Wireless Network under Power-Sum Constraint." pith.science (2026). https://pith.science/paper/OO7BO7Y7

@misc{pith2026250602823,
  author       = {Pith},
  title        = {Pith review of: Energy Efficiency Analysis of Active RIS-enhanced Wireless Network under Power-Sum Constraint},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OO7BO7Y7}},
  note         = {Machine review of arXiv:2506.02823}
}
read the original abstract

Recently, as a green wireless technology, active reconfigurable intelligent surface (RIS) attracts numerous research activities due to its amplifying ability to combat the double-fading effect compared to passive one. How about its energy efficiency (EE) over passive one? Below, the EE of active RIS-aided wireless network in Rayleigh fading channels is analyzed. Using the law of large numbers, EE is derived as a function of five factors: power allocation factor, the number (N) of RIS elements, the total power, the noise variances at RIS and at user. To evaluate each factor's impact, the simple EE function for the concerning factor is given with others fixed. To assess the impact of N on EE, we establish an equation with the EE of active RIS equaling that of passive one, and three methods, bisection, Newton's method, and simulated annealing, are designed to find the roots of this equation. Simulation results show that as N tends to medium-scale or large-scale, the asymptotic performance formula is consistent with the exact EE expression well. As N varies from small-scale to large-scale, the active RIS intersects passive one at some point. When N< N_0, active RIS performs better than passive one in terms of EE. Otherwise, there is a converse conclusion.

Figures

Figures reproduced from arXiv: 2506.02823 by the authors.

Figure 1
Figure 1. The model of wireless communication system assisted by active RIS. [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. EE versus the number N of RIS element. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 100 200 300 400 500 600 700 EE (bit/J) Asymptotic P t = 26 dBm Taylor approximate P t = 26 dBm Asymptotic P t = 23 dBm Taylor approximate P t = 23 dBm 0.8485 0.85 618 620 0.85 0.8504 246.5 247 247.5 [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. EE versus the PA factor β. characteristics, and the first-order Taylor approximation can fit the EE curve well. Particularly, for different total power Pt configurations, the system has a unique optimal PA factor β that maximizes EE, and the candidate solutions at this point are determined in (39) [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: EE versus the noise σ 2 r at active RIS. 10 20 30 40 50 60 70 80 90 100 P t (dBm) 0 0.5 1 1.5 2 2.5 3 3.5 EE (bit/J) 106 N = 256 N = 512 N = 1024 N = 2048 N = 4096 [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 5
Figure 5. Figure 5: EE versus the total power Pt [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: depicts the curves of EE versus σ 2 u . It can be seen that the trend in [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]
Figure 7
Figure 7. Figure 7: The absolute value of the difference in EE between active and passive RIS-assisted systems [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]
Figure 8
Figure 8. Figure 8: The difference of the EE both active and passive RIS-assisted systems [PITH_FULL_IMAGE:figures/full_fig_p021_8.png]

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