{"id":"f969b97a-8ef8-4734-a71b-58570535bf34","arxiv_id":"2607.18736","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For non-square XL-MIMO planar arrays, the near field splits into fully near-field, anisotropic near-field, and far-field regions; distance multiplexing is set by the long axis, and an optimal aspect ratio minimizes 3D positioning error.","lead":"This paper studies extremely large antenna arrays shaped as non-square rectangles, showing that their focusing and positioning behavior splits into three distinct near-field zones governed by the long and short axes. The results give design rules for the array's width-to-height ratio, plus a cheaper way to estimate channels in such systems.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Three-region partition holds; the load-bearing soft spot is Theorem 3's uniform-in-1/r user distribution, which sets the EDoF 'fundamental limit'.","rationale":"After tracing the central partition from Eq. (4) through Appendix B, I find the three-region claim internally consistent: the Kronecker factorization of the gain is valid under the stated Fresnel/cross-term approximation, the half-power beam-depth threshold in Appendix A gives R_y and R_x consistently, and the asymptotic O(γ^{-4}) decay follows from the Maclaurin expansion of F. The reader's CONDITIONAL verdict should not be changed on the basis of the partition itself. The more consequential assumption is in Theorem 3: the correlation matrix in (18) assumes users are uniform in ζ=1/r. This is a modeling choice, not an error, but it is load-bearing because the closed-form EDoF (19) and the accompanying claim that distance-domain multiplexing scales with γN depend on the exact weight placed on small versus large distances. A uniform-r distribution puts more weight on large ζ (small distances) and changes the integrand in (45), altering the constant and possibly the logarithmic factor. The qualitative conclusion that the long-axis aperture governs is likely robust, but the exact 'fundamental limit' is then not universal. This does not warrant rejection; it warrants the same CONDITIONAL status, with the distribution assumption stated as a limitation. The CRB reciprocal typesetting issue in Eq. (21)/(22) is real but fixable and does not change the verdict.","tokens_in":21561,"tokens_out":23452,"duration_ms":203412,"concrete_test":"Symbolically recompute tr(R^2) in Appendix C with a uniform-distance user distribution, i.e., f(r)=1/(rmax-rmin) so f(ζ) ∝ ζ^{-2}, instead of f(ζ)=1/ζ_max, using the same steering-vector model. Compare the resulting EDoF with Eq. (19). If the prefactor or the argument of the logarithm changes, the exact closed-form EDoF and its 'governed by γN' claim are distribution-dependent.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Remark 1 and Theorem 2 are internally consistent: the gain factorization (5)-(7), the half-power beam-depth threshold (33)-(36), and the asymptotic expansion in Appendix B support the three-region partition, and I find no flaw there. The most load-bearing concern is in the paper's broader fundamental-limit claim, Theorem 3. Eq. (18) defines the spatial correlation by integrating over ζ=1/r with f(ζ)=1/ζ_max. This distribution is not derived from any deployment geometry; it is chosen to make the double integral in Appendix C tractable. Under uniform r or a path-loss-weighted user distribution, f(ζ) is different and the integral for tr(R^2) in (45)-(46) changes. The exact EDoF formula (19) and the statement that distance-domain multiplexing is governed by γN=N_x^2 are therefore distribution-dependent, not a geometry-only limit. The qualitative scaling may survive, but the closed-form constant and logarithmic term are not robust. Secondary: Eq. (21)/(22) as typeset omit the reciprocal of J_r from Eq. (53), contradicting the prose that CRB_r is maximal at γ=1; this is a typesetting error but should be corrected before use.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies XL-MIMO systems with non-square uniform planar arrays (UPA), where N_x >= N_y and the aspect ratio is gamma = N_x/N_y. It derives effective beamfocusing distances R_y and R_x for the short and long axes (Theorem 1), partitions the radiation space into fully near-field, anisotropic near-field, and far-field regions (Remark 1), and gives asymptotic expressions for the depth ratio and the normalized error of the effective beamfocusing distance (Theorem 2). It then derives an asymptotic effective-degree-of-freedom (EDoF) formula (Theorem 3), closed-form Cramer-Rao bounds for distance estimation (Theorem 4), and a 3D position error bound with an optimal aspect ratio (Theorem 5). Finally, it designs a 3D anisotropic near-field codebook based on the discrete fractional Fourier transform and a low-complexity OMP channel estimator, with simulation results showing performance comparable to the polar-domain codebook at reduced complexity.","tokens_in":21798,"tokens_out":11996,"duration_ms":106805,"significance":"If the results are established, the paper gives a concrete and useful design insight: a rectangular array should not be described by a single Rayleigh distance; it has two axis-dependent effective beamfocusing distances, and the anisotropic region can dominate the near-field space. The O(gamma^{-4}) convergence of the effective-distance error and the approximate 1 - 1/gamma^2 depth ratio in Theorem 2 are clean, falsifiable predictions, and the numerical validation of the three-region picture is convincing. The EDoF, CRB/PEB, and codebook formulations are also valuable, but the EDoF result depends on a specific user-distance distribution, and the CRB expression as typeset in Eq. (21)/(22) is inconsistent with its own derivation and with the prose. The paper does not provide machine-checked proofs or code, but the derivations are laid out in detail and the figures support the main qualitative claims.","major_comments":[{"comment":"The EDoF result is derived under the assumption that the user distance parameter zeta = 1/r is uniformly distributed on [0, zeta_max], f(zeta) = 1/zeta_max. This is a modeling choice made for tractability, not a consequence of the array geometry or of any deployment scenario. Under a uniform-distance model f(r) = const, the induced density is f(zeta) proportional to zeta^{-2}; under a path-loss-weighted user distribution it changes again. Consequently the exact constant and the logarithmic term in Eq. (19), and the statement that distance-domain multiplexing is governed by gamma N = N_x^2, are not geometry-only fundamental limits but are conditioned on this distribution. The qualitative dependence on the long-axis aperture may survive, but the closed-form EDoF should be presented as a model-dependent asymptotic, with the user distribution justified or its sensitivity quantified.","section":"Section IV, Eq. (18)-(19), Appendix C"},{"comment":"Eq. (53) in Appendix D is the Fisher information J_r for the distance r, and the CRB is the reciprocal, 1/J_r. As typeset, Eqs. (21) and (22) give J_r: along broadside the expression is proportional to gamma^2 + gamma^{-2}, which has a minimum at gamma = 1 and grows with gamma. This is exactly the opposite of what the text claims immediately after (22) (\"increasing gamma leads to a rapid decline in CRB_r\" and \"CRB_r reaches its global maximum if and only if gamma = 1\"), and it contradicts Eq. (24), where the distance component of the PEB correctly uses 1/(gamma^2 + gamma^{-2}). This appears to be a typesetting/reciprocal error, but it is load-bearing for the parameter-estimation contribution. Please correct (21) and (22) to the reciprocal form and re-check all statements, figures, and downstream formulas that depend on them.","section":"Theorem 4, Eq. (21)-(22), Appendix D"},{"comment":"The optimal aspect-ratio result is derived for the broadside direction, as stated before Eq. (24), but Theorem 5 and the conclusion present gamma_opt as the minimizer of the 3D PEB without a directional qualification. For off-broadside angles, Eq. (23) has additional terms through u_x, u_y, and the FIM off-diagonal structure, so the same gamma_opt is not automatically optimal. In addition, the proof in Appendix E requires kappa_2/kappa_1 = N d^2/(60 r^2) in (0,1), i.e. r > r_th; for r < r_th the minimizer is gamma = 1, not Eq. (26). The theorem should state explicitly that (25)-(26) hold along broadside and for r > r_th. This does not invalidate the practical regime used in Fig. 8, but it is needed for the stated scope.","section":"Theorem 5, Eq. (23)-(26), Appendix E"}],"minor_comments":[{"comment":"There are several spacing/formatting typos, e.g. \"3 dBbeam\" in Section III and \"Cram \\'er-Rao\" in the abstract. Please proofread the final version.","section":"Notation and typos"},{"comment":"The approximation (a) in Eq. (2) uses the Fresnel expansion and omits cross-terms; the condition under which the cross-term omission is negligible is only verified numerically in Fig. 2. A brief analytical condition (e.g., in terms of r relative to the array aperture) would make the approximation more rigorous.","section":"Eq. (2)"},{"comment":"The codebook sampling step is stated as Delta tau = 7/N_x^2 from setting the derivative of Xi(Delta tau) to zero, but the solution is not shown. Since the value of this constant affects the dictionary size in Table I and the quasi-orthogonality claim, a short derivation or a numerical justification for the constant 7 would improve reproducibility.","section":"Section VI, Eq. (32)"},{"comment":"The horizontal axis of Fig. 9(b) is labeled only with exponents 10^5 and 10^10; the unit or definition of computational complexity should be stated in the caption or text.","section":"Fig. 9(b)"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically competent and the three-region picture is a useful contribution. The two substantive issues are the user-distribution dependence of the EDoF theorem and the reciprocal error in the CRB formula; both are addressable by revision. The CRB error appears to be a typographical inversion rather than a conceptual flaw, since the PEB formula uses the correct reciprocal. I do not see a circularity or novelty problem; the self-citations to prior framework papers are appropriate. I would support acceptance after a careful revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague—\n\nThis paper is worth your time if you care about near-field XL-MIMO with rectangular arrays. The genuinely new piece is the three-region partition: the radiation space splits at two effective beamfocusing distances R_y and R_x, with an anisotropic middle region that asymptotically dominates for large aspect ratio. Theorems 1 and 2 are well supported; the Fresnel-integral machinery is standard, but the O(γ^{-4}) decay of K(γ) and the ~1-1/γ² depth ratio are new and I don't see a flaw in the derivation. The codebook section is also practical: DFrFT along the long axis and DFT along the short axis, with a solid complexity reduction in the simulations.\n\nWhere the paper gets softer is Theorem 3. The EDoF formula (19) follows from assuming users are uniform in inverse distance ζ=1/r. That choice is analytically convenient but is not derived from any deployment geometry. Under uniform r or a path-loss-weighted distribution, the integral giving tr(R²) changes, so the exact constant and the logarithmic term are not robust. The qualitative statement that distance-domain multiplexing is governed by the long-axis aperture γN will probably survive, but calling this a fundamental limit overstates it. The authors should either redo this under a physical user distribution or clearly state the sensitivity.\n\nAlso flag Eq. (21)/(22). As typeset, it looks like the Fisher information J_r (proportional to γ²+γ^{-2}) rather than CRB_r, which would be its reciprocal. That contradicts the prose that CRB_r is maximal at γ=1. I assume this is a typesetting error, but it needs correcting.\n\nMinor: no code and no error bars on the NMSE curves. That's acceptable for a theory paper, but I'd ask.\n\nOverall, the central partition claim is solid and the CRB/PEB analysis is coherent. This deserves a serious referee. Send it out, with the EDoF section requiring revision and the CRB equation fixed.","headline":"Solid new analysis of non-square UPA near-field behavior; the three-region partition and codebook are worth publishing, but the EDoF theorem rests on an unmotivated user distribution and the CRB equation has a typesetting slip.","tokens_in":22380,"tokens_out":2819,"would_cite":true,"duration_ms":24135,"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":"This paper establishes that a non-square uniform planar array possesses two effective focusing distances—one per axis—so its radiation space splits into fully near-field, anisotropic near-field, and far-field regions, with the anisotropic r","keywords":["XL-MIMO","near-field communications","non-square UPA","anisotropic beamfocusing","effective Rayleigh distance","effective degrees of freedom","Cramér-Rao bound","channel estimation"],"falsifier":"Measure the normalized array gain of a rectangular UPA (e.g., 128×16 elements at 28 GHz) focused at a distance r_F inside the claimed anisotropic region (R_y ≤ r_F < R_x) and check whether the short-axis gain still shows a finite 3-dB beam-depth peak; a resolved peak would falsify the R_y boundary. Similarly, recomputing EDoF for users uniformly distributed in physical distance rather than inverse distance should break the close match to equation (19) if the assumed prior is load-bearing.","tokens_in":21382,"feed_emoji":"📡","tokens_out":3809,"duration_ms":36170,"temperature":0.7,"pith_summary":"This paper argues that a rectangular (non-square) extremely large antenna array has two distinct effective focusing distances—one for its long axis and one for its short axis—so its radiation space splits into three regions: a fully near-field zone, an anisotropic near-field zone where only the long axis can focus, and a far-field zone. The anisotropic zone is not a small correction: as the aspect ratio grows, it asymptotically occupies almost the entire near-field space. From that geometric fact the paper derives closed-form limits—effective degrees of freedom, distance-estimation error bound, and 3D positioning error bound—and shows how to choose the aspect ratio that minimizes positioning error. It then builds a low-complexity channel-estimation codebook that matches a polar-domain codebook's accuracy while cutting runtime by roughly 96.6%.","feed_headline":"Two focusing distances replace one for rectangular XL arrays","feed_subtitle":"Long and short axes focus at different distances, reshaping multiplexing, positioning, and channel estimation.","key_machinery":"The load-bearing objects are the two effective beamfocusing distances R_x and R_y, defined as the distances at which the 3-dB beam depth of the long or short axis becomes infinite, derived from the half-power equation of the normalized array gain expressed through Fresnel integrals. Together with the aspect ratio γ = N_x/N_y, these boundaries partition the radiation space into the three regions. The normalized array gain factorizes as g_x(r_F,u_x)·g_y(r_F,u_y) into long-axis and short-axis contributions, and this factorization is what makes the EDoF, CRB, and codebook derivations tractable.","core_discovery":"The central discovery is that a non-square UPA has two effective beamfocusing distances, one per axis: the short axis loses its focusing capability at R_y = N_y² d² (1−u_y²)/(2λη₀²), while the long axis keeps focusing until R_x = N_x² d² (1−u_x²)/(2λη₀²), where η₀ is the primary root of the half-power Fresnel equation. Between these distances lies the anisotropic near-field region, where the long-axis steering vector remains spherical but the short-axis steering vector becomes planar. Theorem 2 shows the error in treating R_x as the array's focusing distance decays as O(γ^{-4}) and the anisotropic region's depth ratio approaches 1−1/γ². The paper then derives an asymptotic effective-degree-o","pith_inferences":["If the three-region picture holds, the common practice of using a single diagonal-based Rayleigh distance overestimates the near-field range for rectangular arrays in most directions; designers should treat the long-axis distance as the operative boundary for beamfocusing and treat the short axis as negligible except near broadside.","The optimal-aspect-ratio law γ_opt ∝ r^{2/3} N^{−1/3} suggests a testable dynamic sub-array strategy: a base station could activate progressively more elongated sub-arrays as a user moves farther away to keep positioning error low, without requiring new hardware.","Theorem 3's EDoF formula assumes users are uniformly spread in inverse distance (ζ=1/r), a convenience prior rather than a deployment-derived distribution; under uniform-distance or path-loss-weighted user distributions the constant changes, so the result should be read as a scaling law rather than an exact capacity prediction."],"forward_implications":["Near-field taxonomy for rectangular arrays becomes three-region instead of a single Rayleigh-distance dichotomy; the anisotropic region dominates for elongated arrays, so the long-axis focusing distance is the operative boundary for beamfocusing in most directions.","Effective degrees of freedom in the large-aspect-ratio regime scale with γN = N_x², meaning distance-domain multiplexing is governed by the long-axis physical aperture, not the total antenna count; elongating an array adds multiplexing without adding antennas.","Distance-estimation CRB decreases with increasing aspect ratio and is maximized at the square configuration γ=1, while elevation angular resolution degrades, creating a quantified trade-off; the optimal aspect ratio minimizing 3D positioning error is approximately (120 r²/(N d²))^{1/3}.","The DFrFT/DFT anisotropic codebook enables OMP-based channel estimation with accuracy comparable to a 3D polar-domain codebook at substantially lower computational cost, about a 96.6% reduction in average runtime in the simulated configuration."],"fun_headline_variants":["Rectangular XL arrays get two focusing distances, not one","Two focusing distances for rectangular XL-MIMO arrays","Anisotropic near-field: long and short axes focus differently","Why rectangular arrays need two beamfocusing distances","Non-square UPA: two focusing distances reshape MIMO"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"Theorem 3's asymptotic EDoF formula assumes users are uniformly distributed in the inverse-distance domain ζ=1/r over [0, ζ_max], a mathematical convenience not derived from any deployment geometry; if real user distances follow a different distribution, the exact EDoF constant and logarithmic term change, although the N_x² scaling is likely to survive.","fun_headline_variants_meta":{"raw":{"variants":["Rectangular XL arrays get two focusing distances, not one","Two focusing distances for rectangular XL-MIMO arrays","Anisotropic near-field: long and short axes focus differently","Why rectangular arrays need two beamfocusing distances","Non-square UPA: two focusing distances reshape MIMO"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000172,"raw_usage":{"total_tokens":1185,"prompt_tokens":893,"completion_tokens":292,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":637,"completion_tokens_details":{"reasoning_tokens":228}},"tokens_in":637,"tokens_out":292,"duration_ms":46209,"temperature":1.0,"reasoning_tokens":228,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T14:30:51.129906+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the normalized array gain of a rectangular UPA (e.g., 128×16 elements at 28 GHz) focused at a distance r_F inside the claimed anisotropic region (R_y ≤ r_F < R_x) and check whether the short-axis gain still shows a finite 3-dB beam-depth peak; a resolved peak would falsify the R_y boundary. Similarly, recomputing EDoF for users uniformly distributed in physical distance rather than inverse distance should break the close match to equation (19) if the assumed prior is load-bearing.","supporting_citations":[],"review_version":1}