{"id":"f093a032-d8dd-4bf2-b7d2-29de37ee177c","arxiv_id":"2502.06980","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For a CAPA under isotropic scattering, the received SNR is approximately a weighted sum of DOF=2L/lambda independent exponential random variables, yielding closed-form SNR and capacity expressions.","lead":"This paper derives closed-form formulas for the signal-to-noise ratio distribution and average capacity of a continuous-aperture array (CAPA) fading channel under isotropic scattering. It shows that the SNR behaves like a weighted sum of about 2L/lambda independent exponential random variables, and tests the formulas with simulations.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Matched-filter SNR is mis-derived in Eq. (17): the SNR is (P/σ²)∫|g|², not (P/σ²)|∫g|², so Eq. (18) and the subsequent PDF/capacity formulas do not follow as stated.","rationale":"The paper's contribution is a closed-form SNR distribution and capacity for CAPA under isotropic scattering. The Landau eigenvalue machinery and the sum-of-exponentials representation are mathematically sound, and the isotropic Gaussian model is a reasonable idealization. However, the derivation of the quantity whose distribution is computed contains a concrete algebraic error: the matched filter defined by the paper yields an output energy of ∫|h|², not |∫h|², and the paper's Eq. (17) also asserts a false distributional equality between these two distinct functionals. This is not a matter of model realism but internal consistency: the same system model and filter definition contradict Eq. (17). The reader's verdict of REJECT is therefore appropriate; my concern is more specific than the declared weakest assumption (the Gaussian isotropic premise), which is why I mark agreement as partial. The result is likely repairable by inserting the P/σ² scaling and re-expressing the PDF, so the rejection is on the submitted derivation, not on the underlying approach.","tokens_in":6816,"tokens_out":11704,"duration_ms":107618,"concrete_test":"Re-derive Eq. (17) from Eqs. (1)-(3): substitute j(t)=h*/√∫|h|² and compute |∫ h j|². The result is ∫|h|², not |∫h|². Then evaluate both sides of Eq. (17) on Monte Carlo realizations of g with Rg(z,z')=sinc(k0(z-z')) for L=32λ; if the empirical means of |∫g|² and ∫|g|² differ by the predicted factor ~32, the distributional equality is false and Eqs. (18)-(22) should be re-derived with the corrected SNR.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Eq. (17) is the load-bearing step. From Eq. (3), with j(t)=h*(r,t)/√(∫_A |h|² dt), the matched-filter output is |∫_A h j dt|² = ∫_A |h|² dt. The paper instead writes γ = (P/σ²)|∫_A h dt|² and then asserts |∫ g|² d= ∫ |g|². These are not equivalent: for the sinc-correlated process with Rg(z,z')=sinc(k0(z-z')), E|∫g|² = ∫∫ Rg ≈ λ/2 for L≫λ, while E∫|g|² = L. Thus Eq. (17) would imply a saturating SNR, contradicting the linear-in-L growth that the Landau truncation is meant to capture. Even if Eq. (18) is read as the normalized sum, the printed equality omits the P/σ² factor, and Eq. (19) is the PDF of the unscaled sum, not of the SNR of Eqs. (1)-(3). Consequently the central closed forms are not derived from the stated system model; a repair (γ d= (P/σ²)∑σ_l|Φ_l|²) is likely straightforward, but it is not what the paper proves.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript analyzes the statistics of the received SNR of a continuous-aperture array (CAPA) under isotropic scattering. Using a planar-wave EM channel model from the holographic-MIMO literature, it derives the spatial autocorrelation of the channel response as a sinc kernel, invokes Landau's eigenvalue theorem to show that the kernel's eigenvalues have a step-like behavior with DOF = 2L/λ, and then truncates the Karhunen–Loève expansion at DOF terms. On this basis it claims that the matched-filter SNR is statistically equivalent to a finite weighted sum of independent exponential random variables, and it gives closed-form expressions for the SNR PDF and for the average capacity, together with a high-SNR expansion. Numerical results plot the eigenvalues and compare the analytical capacity with simulations and with conventional MIMO.","tokens_in":7106,"tokens_out":4653,"duration_ms":42569,"significance":"The topic is timely: CAPA and holographic MIMO channel statistics are of active interest, and a simple closed-form description of fading statistics for electrically large continuous apertures would be valuable. The paper has notable strengths: it builds on an established electromagnetic channel model, uses no fitted parameters, and its numerical eigenvalue plots support the step-like spectrum predicted by Landau's theorem. However, the central matched-filter SNR derivation contains a mathematical error that, as submitted, invalidates the claimed PDF and capacity formulas. The error appears to be readily repairable, but the manuscript as it stands does not prove its main result.","major_comments":[{"comment":"The matched-filter SNR calculation is wrong. With j(t) = h*(r,t)/√(∫_A |h(r,t)|² dt), the matched-filter output satisfies |∫_A h(r,t) j(t) dt|² = ∫_A |h(r,t)|² dt, so the SNR is (P/σ²)∫_A |h(r,t)|² dt, not (P/σ²)|∫_A h(r,t) dt|² as written immediately after Eq. (3). This is not a minor notational slip: it changes the SNR by the ratio |∫ h|²/∫|h|², which is not equal to unity for the random channel considered here.","section":"§2.1, Eq. (3)"},{"comment":"The asserted equivalence |∫_A h(r,t) dt|² d= ∫ |g(z)|² dz in Eq. (17) is false for the sinc-correlated Gaussian process derived in §3.2. For L ≫ λ, E|∫g|² = ∫∫ sinc(k₀(z−z′)) dz dz′ ≈ λ/2, whereas E∫|g|² = L; the former saturates while the latter grows linearly in L. The Karhunen–Loève expansion in Eq. (16) applies to ∫|g|², not to |∫g|², so the truncation at DOF terms in Eq. (18) is not established for the printed SNR formula. With the correct matched-filter SNR, the result can be repaired as γ d= (P/σ²)∑_{ℓ=1}^{DOF} σ_ℓ|Φ_ℓ|², but Eq. (18) as written omits the P/σ² factor and Eq. (19) is the PDF of the unscaled sum, not of the SNR in Eq. (3). Consequently Eqs. (19)–(24) do not follow from the stated system model without additional scaling and re-derivation.","section":"§3.3, Eq. (17)"},{"comment":"The numerical validation should be clarified. If the 'Simulation' curves are generated from the same Karhunen–Loève truncation used to derive the analytical formulas, then Figure 3 validates the closed-form evaluation of the truncated sum but does not independently validate the truncation of the physical channel. After correcting Eq. (17), the simulations should be rerun against the physical channel model and the accuracy of the DOF truncation should be explicitly reported.","section":"§4, Figure 3"}],"minor_comments":[{"comment":"The symbol γ is used both for the random SNR and for the fixed transmit SNR P/σ²; introduce a distinct symbol such as η or γ̄ for the transmit SNR throughout the PDF and capacity expressions.","section":"§3.3, Eqs. (19)–(24)"},{"comment":"The caption contains a duplicated phrase: 'exhibit a step-like behavior: exhibit a step-like behavior:'.","section":"Figure 2 caption"},{"comment":"The application of Landau's theorem to the finite-interval sinc kernel would benefit from a sentence explaining why the theorem's hypotheses are satisfied for the kernel K(z,z′) defined on [−L/2,L/2]×[−L/2,L/2].","section":"§3.2, Eq. (15)"},{"comment":"The paper should state explicitly how the MIMO comparison is made in terms of total aperture length and power normalization, since the claimed capacity gain over conventional MIMO depends on this normalization.","section":"§4, parameter setup"}],"recommendation":"major_revision","confidential_remarks":"The matched-filter error is load-bearing, but it is also straightforward to repair: replacing |∫h|² by ∫|h|² changes only the scaling of the sum-of-exponentials representation and the resulting PDF/capacity formulas. If the authors make that correction and rerun the numerical validation, the central claim is defensible. I therefore see major revision rather than rejection as the appropriate outcome, but only if the revised derivation is clean and the notation is clarified."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper addresses a real gap—statistical characterization of fading in continuous-aperture arrays—and the overall structure is sensible: use Landau's eigenvalue theorem to show the step-like behavior, truncate to the first DOF terms, and apply Moschopoulos's gamma-sum formula for the PDF and capacity. The eigenvalue analysis and the Karhunen-Loève expansion are internally consistent, and the numerical eigenvalue plots look plausible. The idea itself is worth taking seriously.\n\nThe problem is in the matched-filter SNR derivation. Eq. (3) is correct, but the line immediately after it is not. With j = h*/sqrt(int |h|²), the matched-filter output is sqrt(int |h|²), so the SNR is (P/σ²) int |h|², not (P/σ²)|int h|². That error then poisons Eq. (17), which asserts |int g|² d= int |g|². That is false for the sinc-correlated Gaussian process in this paper: the left side is a single exponential random variable, the right side is a sum of many. Eq. (18) also drops the P/σ² factor, and Eq. (19) gives the PDF of the unscaled sum, not of the SNR as defined in Eqs. (1)–(3). So the closed-form results as stated do not follow from the stated system model.\n\nThe good news is that the error is repairable. If you start from the correct matched-filter output, the same KL expansion gives (P/σ²) sum σ_l |Φ_l|², which is exactly the weighted sum of exponentials they want. So the final expressions probably survive, but with a corrected scale factor and a corrected derivation. I cannot check the numerics because no simulation setup or code is provided, which is another weakness.\n\nThe Landau truncation is heuristic for finite L, but that is a standard approximation and not a fatal issue. The self-citations are background and not load-bearing. The paper is serious and the underlying mathematical machinery is sound, but the version I read is not correctly derived. A rigorous referee would catch this and ask for a substantial rewrite of the SNR derivation plus reproducibility details. I would send it to peer review, but I would not accept it in this form.","headline":"The paper's central SNR derivation contains a load-bearing error, but the intended result is likely repairable.","tokens_in":7619,"tokens_out":3819,"would_cite":false,"duration_ms":34344,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper shows that a CAPA's fading SNR is a finite weighted sum of exponentials whose term count is the electromagnetic degrees of freedom $2L/\\lambda$.","keywords":["continuous-aperture array","CAPA","electromagnetic channel","isotropic scattering","Landau eigenvalue theorem","SNR distribution","average capacity","degrees of freedom"],"falsifier":"A direct check is to measure the SNR of a CAPA of length $L$ with a single-antenna receiver in an anechoic chamber built for isotropic Rayleigh fading, and compare the empirical PDF with Eq. (19). Equivalently, sample the spatial response $g(z)$ across the aperture, estimate its autocorrelation, and compute the eigenvalue spectrum: the prediction is that roughly $2L/\\lambda$ eigenvalues sit near one and the rest near zero. A failure to see that step, or a systematic deviation of the SNR distribution from (19) beyond Monte Carlo error, would invalidate the central approximation. A cheaper boundary test is to set $L=\\lambda$: the step is then weak, so the truncation in (18) should be visibly inaccurate, delimiting how large the aperture must be.","tokens_in":6594,"feed_emoji":"📡","tokens_out":11945,"duration_ms":89986,"temperature":0.7,"pith_summary":"The paper establishes that the received SNR of a single-user continuous-aperture array (CAPA) under isotropic scattering is statistically a weighted sum of independent exponential random variables, and that for an electromagnetically large aperture the sum is effectively finite with $2L/\\lambda$ terms. This reduces the fading channel to a tractable object: the eigenvalues of a sinc autocorrelation kernel, whose step-like spectrum is controlled by Landau's theorem. From that representation the paper derives closed-form expressions for the SNR probability density, the average capacity, and the high-SNR slope. If correct, these results let a designer predict a CAPA link's average capacity from the aperture length, wavelength, and transmit power without Monte Carlo simulation.","feed_headline":"CAPA fading SNR is a finite weighted sum of exponentials","feed_subtitle":"Landau's theorem sets the term count at 2L/λ, enabling closed-form SNR PDF and capacity.","key_machinery":"The load-bearing object is the autocorrelation kernel $K(z,z') = \\frac{1}{2\\pi}\\int_{-k_0}^{k_0} e^{j(z-z')\\kappa_z}\\,d\\kappa_z$ acting on $L^2([-L/2,L/2])$; this is a sinc kernel, and its eigenvalues are the $\\sigma_\\ell$ scaled by the constant in (10). Landau's eigenvalue theorem for bandlimited kernels shows that, as $L\\to\\infty$, the fraction of eigenvalues exceeding any fixed positive threshold tends to the electromagnetic degrees of freedom $DOF = 2L/\\lambda$, with a transition band whose width grows like $\\log DOF$. That step spectrum is what truncates the infinite Karhunen--Loève expansion of the Gaussian field to a finite weighted sum of exponentials, and it is what makes the PDF and capacity integrals evaluable in closed form.","core_discovery":"Under the isotropic Gaussian scattering model of Eqs. (4)--(5), the normalized EM spatial response $g(z)$ is a zero-mean stationary complex Gaussian process with autocorrelation $R_g(z,z') = \\frac{1}{2k_0}\\int_{-k_0}^{k_0} e^{j(z-z')\\kappa_z}\\,d\\kappa_z$. Its Karhunen--Loève expansion gives $\\gamma \\stackrel{d}{=} \\bar{\\gamma}\\sum_{\\ell=1}^{\\infty}\\sigma_\\ell|\\Phi_\\ell|^2$, where the $\\Phi_\\ell$ are independent $\\mathcal{CN}(0,1)$ variables and the $\\sigma_\\ell$ are the eigenvalues of that sinc kernel. By Landau's eigenvalue theorem these eigenvalues polarize: for $L\\gg\\lambda$, the leading $DOF = 2L/\\lambda$ eigenvalues are near one and the rest are near zero, with a transition band of width proportional to $\\log DOF$. The paper therefore approximates the infinite sum by the first $DOF$ terms and, using the series representation of a sum of independent exponentials, obtains the SNR PDF (19) and the average capacity (22), including the high-SNR asymptotics (23) with unit slope and the explicit power offset (24).","pith_inferences":["For non-isotropic but spatially stationary scattering, the kernel would be a filtered version of the sinc kernel, and Landau-type estimates would still predict a step spectrum; the effective number of branches would equal the measure of the angular support times the aperture length, so the paper's framework extends in spirit to directive scattering once the kernel is known.","The truncation at $DOF$ is a convenience, not a necessity: Eq. (19) is the exact distribution of any finite weighted sum of exponentials, and the infinite sum is the limit as $DOF\\to\\infty$, so one could retain small tail eigenvalues if a particular aperture length requires it.","Because the SNR is a sum of $DOF$ independent exponentials with generally distinct scales, the outage CDF near zero behaves like $x^{DOF}/(DOF!\\prod \\sigma_\\ell)$, implying a diversity order of $DOF$ for fixed-rate transmission, even though the high-SNR capacity slope is capped at 1."],"forward_implications":["The high-SNR capacity slope of a single-user CAPA link is exactly $1$ bit/s/Hz per 3 dB regardless of aperture size; a larger aperture only improves the power offset $\\mathcal{L}$ in (24).","For design purposes, the average capacity of a CAPA under isotropic fading can be computed in closed form from $L$, $\\lambda$, and transmit power, replacing Monte Carlo simulations.","The same eigenvalue expansion gives other performance metrics such as outage probability directly, since the SNR PDF (19) contains all the statistical information.","A CAPA of physical length $L$ under isotropic scattering has $2L/\\lambda$ effective fading branches, so increasing the aperture beyond that does not add independent signal components; capacity gains come from better weighting of the existing branches."],"supporting_citations":[{"why":"Supply the electromagnetic multipath channel model of Eqs. (4)-(5) that makes $g(z)$ a stationary complex Gaussian field with a sinc autocorrelation.","marker":"[4, 5]"},{"why":"Landau's eigenvalue theorem, the basis for the step-like eigenvalue spectrum and the degrees-of-freedom count.","marker":"[6]"},{"why":"Series representation of the distribution of a sum of independent gamma variables, used to write the SNR PDF in closed form.","marker":"[8]"},{"why":"Numerical algorithm for the Fredholm eigenvalue problem, used to compute the eigenvalues in Figure 2 and validate the step behavior.","marker":"[9]"}],"fun_headline_variants":["Landau's theorem simplifies CAPA fading SNR to closed form","Polarized eigenvalues yield closed-form CAPA SNR PDF and capacity","CAPA channel stats: finite exponential sum from Landau's theorem","Eigenvalue polarization gives exact CAPA SNR distribution","How Landau's theorem makes CAPA fading SNR tractable"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The derivation requires the scattering to be isotropic and the angular response $W(k,\\kappa)$ in Eq. (5) to be a zero-mean unit-variance complex Gaussian random field; without that, $g(z)$ is not a stationary Gaussian process with a sinc autocorrelation, and the Landau step spectrum with $2L/\\lambda$ significant eigenvalues does not apply.","fun_headline_variants_meta":{"raw":{"variants":["Landau's theorem simplifies CAPA fading SNR to closed form","Polarized eigenvalues yield closed-form CAPA SNR PDF and capacity","CAPA channel stats: finite exponential sum from Landau's theorem","Eigenvalue polarization gives exact CAPA SNR distribution","How Landau's theorem makes CAPA fading SNR tractable"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000374,"raw_usage":{"total_tokens":1969,"prompt_tokens":891,"completion_tokens":1078,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":507,"completion_tokens_details":{"reasoning_tokens":992}},"tokens_in":507,"tokens_out":1078,"duration_ms":8953,"temperature":1.0,"reasoning_tokens":992,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T14:10:47.976546+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct check is to measure the SNR of a CAPA of length $L$ with a single-antenna receiver in an anechoic chamber built for isotropic Rayleigh fading, and compare the empirical PDF with Eq. (19). Equivalently, sample the spatial response $g(z)$ across the aperture, estimate its autocorrelation, and compute the eigenvalue spectrum: the prediction is that roughly $2L/\\lambda$ eigenvalues sit near one and the rest near zero. A failure to see that step, or a systematic deviation of the SNR distribution from (19) beyond Monte Carlo error, would invalidate the central approximation. A cheaper boundary test is to set $L=\\lambda$: the step is then weak, so the truncation in (18) should be visibly inaccurate, delimiting how large the aperture must be.","supporting_citations":[{"cited_title":"On Szegö’s eingenvalue distribution theorem and non-Hermitian kernels,","cited_arxiv_id":null,"evidence_quote":"Landau's eigenvalue theorem, the basis for the step-like eigenvalue spectrum and the degrees-of-freedom count."},{"cited_title":"The distribution of the sum of independent gamma random variables,","cited_arxiv_id":null,"evidence_quote":"Series representation of the distribution of a sum of independent gamma variables, used to write the SNR PDF in closed form."},{"cited_title":"Solving Fredholm integral equations of the second kind in MA TLAB,","cited_arxiv_id":null,"evidence_quote":"Numerical algorithm for the Fredholm eigenvalue problem, used to compute the eigenvalues in Figure 2 and validate the step behavior."}],"review_version":1}