Pith. sign in

REVIEW 3 major objections 6 minor 27 references

Computational and Effective Degrees of Freedom for Spatially Stationary HMIMO Channel Modeling

T0 review · 3 major / 6 minor · reviewed 2026-07-30 · grok-4.5

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Gauss-Legendre Nyström sampling of continuous HMIMO channels needs a π/2 oversampling factor before quadrature error decays super-exponentially, and a Szegő–Widom formula then sets how many eigenmodes to keep.

desk verdict Solid methods paper that finally puts sampling and truncation numbers on the NGLQ HMIMO scheme; 1D theory is tight, 2D eDoF is useful but rests on an open Widom case they flag honestly. read the letter →

arxiv 2607.23487 v1 pith:KXSZ5J4Q submitted 2026-07-26 eess.SP

Hangsong Yan , Hong Yang , Shu Sun This is my paper · ORCID
classification eess.SP
keywords holographicMIMONyströmmethodGauss-LegendrequadraturecomputationaldegreesoffreedomeffectiveSzegő-Widomexpansionspatiallystationarychannels
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

Continuous holographic MIMO channels are random fields whose spatial correlation is an integral operator. To turn that operator into a stable discrete model the authors apply the Nyström method with Gauss-Legendre quadrature. They prove that the quadrature error stays large until the number of nodes exceeds a computational threshold they call cDoF—πL/λ in one dimension, a π/2 oversampling of the classical physical degrees of freedom—and then falls super-exponentially. In two dimensions the same separable grid wastes roughly 68 percent of its nodes on non-propagating spectral corners. Because the resulting eigenvalue problem is severely ill-conditioned, they further supply a semi-analytical count of the effective degrees of freedom (eDoF) that remain above any chosen numerical threshold; the count is taken from the multidimensional Szegő–Widom expansion and automatically incorporates the anisotropic edge corrections of a rectangle. Together the two thresholds define a numerically stable regime: sample at least at cDoF, keep only the first eDoF eigenmodes, and the continuous-to-discrete map converges to machine precision even for non-isotropic scattering.

What carries the argument

The computational DoF (cDoF) threshold together with the semi-analytical effective DoF (eDoF) obtained from the multidimensional Szegő–Widom trace expansion; cDoF fixes the minimal stable quadrature grid while eDoF fixes the safe truncation rank of the subsequent eigenvalue problem.

What would settle it

Compute the exact eigenvalue counting function of the isotropic sinc kernel on a large rectangle by high-order NGLQ and check whether the gap to the proposed asymptotic formula shrinks as o((ln c)/c) when the normalized aperture c=πL/λ grows.

Watch

Extended reading notes

Core claim

For a spatially stationary HMIMO aperture the Nyström–Gauss–Legendre quadrature error begins its super-exponential decay precisely when the number of nodes reaches the computational DoF Mc≈πL/λ (one dimension) or the corresponding tensor-product thresholds (two dimensions); once that grid is fixed, the number of eigenmodes that must be retained for a prescribed precision ε is given by the Szegő–Widom asymptotic that the authors write in closed form for rectangular apertures.

Load-bearing premise

The two-dimensional eDoF formula assumes that the still-unproven Widom conjecture continues to hold for a discontinuous step-function test on a rectangular spatial domain with disk-shaped wavenumber support.

Editorial extensions

If this is right

  • Any continuous-to-discrete HMIMO simulator must oversample physical DoF by at least π/2 (1-D) or roughly 3× (2-D separable) before spectral accuracy appears.
  • Partial eigensolvers can be seeded with the closed-form eDoF instead of a full O(M³) decomposition, cutting cost to roughly O(M²k).
  • Environment-aware angular sectors further shrink the required node count by the factors sin θ_max and sin ϕ_max.
  • Exact NUDFT evaluation of the spatial kernel removes the interpolation floor that previously masked super-exponential convergence under non-isotropic scattering.

Reading between the lines

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

  • Non-separable quadrature rules matched to the circular wavenumber disk could eliminate the 68 percent corner redundancy the paper identifies.
  • The same cDoF/eDoF pair should remain meaningful for near-field spherical-wave kernels provided the field stays band-limited to the propagating disk.
  • Because the eDoF formula already encodes anisotropic edge corrections, it can be used as a cheap surrogate for capacity or mutual-information calculations on large rectangular apertures.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper develops a continuous-to-discrete modeling framework for spatially stationary HMIMO channels based on the Nyström method with Gauss–Legendre quadrature (NGLQ). Three main results are claimed: (i) a 1D analysis (Theorem 1, Appendix A) showing super-exponential decay of the GLQ error with the explicit bound |E_M(f)| ≤ C̃(M)(eπL/(2λM))^{2M}, and identifying a convergence-onset threshold ("cDoF") M_c ≈ πL/λ, a π/2 oversampling of the physical DoF 2L/λ, shown to be tight under dual end-fire illumination; (ii) a tensor-product extension to 2D rectangular apertures (Theorem 2, Appendix B) with per-axis thresholds M_x > πL_x/λ, M_y > πL_y/λ, yielding a 68% redundancy relative to the 2D physical DoF, plus an environment-aware variant for sectorized angular spectra; (iii) an effective-DoF (eDoF) analysis via the multidimensional Szegő–Widom expansion, recovering the known Landau–Widom form in 1D (Eq. (36)) and proposing a semi-analytical 2D formula (Eq. (37)) that is validated numerically against NGLQ eigenvalue counts. Simulations with an exact NUDFT kernel evaluation demonstrate machine-precision reconstruction under non-isotropic von Mises–Fisher scattering.

Significance. If the results hold, the paper provides the HMIMO literature with something it currently lacks: rigorous, parameter-free convergence guarantees and practically usable node-count and truncation thresholds for Nyström discretization of continuous-aperture channels. The 1D analysis is fully self-contained and chains classical tools (GLQ remainder, Robbins bounds, Paley–Wiener, Bernstein inequality) cleanly; I verified the algebra from (40)–(41) through (47) to (17), and the 1D eDoF in (36) reproduces the Landau–Widom second-order term exactly. The cDoF threshold carries no fitted constants, the end-fire tightness argument is constructive, and the eDoF predictions are checked against externally computed NGLQ eigenvalue distributions rather than fitted to them. The authors are commendably transparent that the 2D eDoF rests on an instance of Widom's conjecture that remains unproven, and they label the result "semi-analytical." The NUDFT demonstration of spectral convergence to machine precision and the partial-EVD complexity guidance (O(M²k) with k ≈ N_eDoF) are of direct practical value for large-aperture simulations.

major comments (3)
  1. [§IV, Theorem 1 and Remark 1; Appendix A, Eqs. (17), (48)–(52)] The claimed onset M_c ≈ πL/λ is not exhibited by the paper's own global bound. At M = πL/λ the base of (17) is eπL/(2λM) = e/2 ≈ 1.36 > 1, so (17) is vacuous (growing) at the claimed threshold and only certifies decay for M > eπL/(2λ) ≈ 1.36·M_c. The onset claim instead rests on the ratio test (51) applied to U_M, an upper bound on the Lagrange interpolation remainder (48). Two gaps follow: (a) decay of an upper-bound sequence does not by itself imply decay of the actual error; (b) (48) is the pointwise interpolation remainder, not the quadrature error (14) — the step connecting them (e.g., integrating the remainder over [−1,1], or a Lebesgue-constant argument) is never made explicit. The numerical evidence (Fig. 2) and the end-fire construction do support the threshold, so this appears fixable: either restate Theorem 1(1) as a bound-ratio prediction confirmed numerically, or supply the
  2. [§VI-B, Theorem 3; Appendix C, Lemma 4; Appendix E] Theorem 3 is conditional on the multidimensional Szegő–Widom expansion (Lemma 4) holding for the discontinuous step test function (63) on a piecewise-smooth spatial domain — an instance the authors themselves state is an open mathematical problem (Introduction; §VI-B). Presenting it as 'Theorem 3' with a 'proof' overstates its status; the conditionality should appear in the theorem statement itself (e.g., 'Conditional on Lemma 4 for f in (63)...') or the result should be relabeled a conjecture-based proposition. Additionally, the anisotropic decoupling in Appendix E inserts two different scaling parameters ln(c_x), ln(c_y) into an expansion (56) that is stated for a single isotropic scaling α; this is a second, unflagged heuristic step. Given that the numerical validation (Figs. 6–7) is the actual evidence base, the presentation should make the logical status — conjecture + heuristic dec
  3. [§VI-B, Theorem 3 and Appendix E vs. §VII, Eq. (39)] There is a symbol mismatch between Theorem 3 and the physical isotropic kernel. Theorem 3 posits the wavenumber symbol as the indicator of the disk K, and the boundary evaluation (71) uses that smooth indicator. But the actual isotropic HMIMO kernel — Eq. (39) with constant Ā²_h — has symbol ∝ (κ²−k_x²−k_y²)^{−1/2}·1_K, which is singular at ∂K (this is precisely the model whose 1D marginal is flat, yielding the sinc kernel used in §VI-A). An unbounded symbol at the wavenumber boundary can in principle modify the logarithmic correction term that constitutes the paper's edge-effect claim. The numerics may well be robust to this (the relative gap decreases with aperture size), but the manuscript should: (i) state exactly which closed-form 2D isotropic kernel generates Figs. 6–7 (the jinc from the disk indicator, or the sin(κr)/r-type kernel carrying the Jacobian singularity); (ii) state how
minor comments (6)
  1. [§IV and Appendix A] Notation E_M is overloaded for the quadrature error (14)/(15) and the pointwise interpolation remainder E_M(t) in (48); M_0 denotes both the Bernstein-inequality bound constant (Appendix A) and the extra GLQ nodes in Fig. 8; ε denotes both the eDoF eigenvalue threshold and generic machine precision in §VI. Please disambiguate.
  2. [§VI-B, Theorem 3 statement] ε is described as an 'energy containment threshold 1−ε', but the step test function (63) thresholds individual eigenvalues at ε, which is not the same as contained energy. Please align the wording with (63)–(64).
  3. [§VI-A, Fig. 5] The claimed agreement between N_eDoF^(1D)(10⁻¹⁶) ∈ {11,18,29,33,56} and the M values minimizing the RE curves should be quantified (e.g., mark the predicted eDoF on Fig. 5), and the dependence of the error floor on the reference point s and the test grid should be briefly discussed.
  4. [§VII, Eq. (39) and §VII-A] The kernel is defined only up to proportionality (∝); state the normalization. Also report the wavenumber grid size N_f used in the NUDFT and demonstrate that the MSRE floors in Fig. 8 are not limited by N_f rather than by M.
  5. [§V, Remark 2] The 'efficiency ratio 1/π ≈ 32%' should be defined explicitly as pDoF/cDoF; on first reading it is unclear which quantity is 32% of which.
  6. [References] Typo in [15]: 'survery'. Refs. [1], [2], [5] carry 2026 dates — please verify volume/page details are final. Consider citing Sobolev's survey or related Fisher–Hartwig-type literature on symbols with boundary singularities, which is directly relevant to the issue raised in Major Comment 3.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: cDoF and eDoF are derived from classical operator/quadrature analysis and Szegő–Widom asymptotics, then checked against independent numerical EVDs—not fitted or defined from the quantities they predict.

full rationale

The load-bearing claims are obtained by standard analytic steps that do not reduce to their own inputs. The 1D cDoF threshold Mc ≈ πL/λ follows from the Lagrange interpolation remainder plus Bernstein’s inequality on an entire function of exponential type 2πL/λ (Appendix A, ratio UM+1/UM < 1); the super-exponential bound is the classical GLQ 2M-th-derivative estimate after Robbins factorial bounds. Neither step fits a free parameter to the error curves it later plots. The 1D eDoF is the Landau–Widom/Szegő–Widom trace of a step test function on the band-limiting operator and is stated to recover the known 2L/λ + (1/π²)ln((1−ε)/ε)ln(πL/λ) form; the 2D formula is an explicit (and openly unproven for discontinuous tests) evaluation of the same volume-plus-anisotropic-edge expansion, then compared to eigenvalues obtained by an independent NGLQ matrix EVD. That comparison is an external numerical check, not a fit renamed as prediction. Prior self-citation [5] only supplies the NGLQ discretization under study; the thresholds, error rates, and eDoF expressions are derived in the present paper. No uniqueness theorem is imported from the authors to forbid alternatives, and no ansatz is smuggled in via self-citation. The acknowledged open status of Widom’s conjecture for discontinuous tests on rectangles is a rigor gap, not circularity.

Assumptions & free parameters 3 free parameters · 6 assumptions · 2 invented entities

Core claims rest on standard bandlimited entire-function analysis, classical quadrature remainder theory, and Widom-type operator asymptotics, plus standard far-field stationary HMIMO modeling assumptions. The load-bearing extra is the unproven extension of multidimensional Szegő–Widom to discontinuous eigenvalue-counting tests on rectangles; free parameters are essentially only the numerical precision threshold ε and optional angular sector limits, not fitted physics constants.

free parameters (3)
  • ε (eigenvalue / energy threshold) = examples: 1e-16, 1e-14, 1e-4
    User-chosen precision that defines eDoF via the step test function and Ntrunc cutoff (e.g. 10^{-16}, 10^{-14}, 10^{-4} in figures). Changes the logarithmic correction but not the leading area/length laws.
  • M0 (extra GLQ nodes above cDoF floor) = 0–3 in Fig. 8
    Integer refinement used in non-isotropic MSRE CDFs (M0 = 0..3); empirical cushion, not derived.
  • θmax, φmax (environment-aware angular sectors)
    Optional deployment-specific caps that shrink wavenumber support and thus cDoF; chosen from scenario, not fitted to the convergence theorems.
assumptions (6)
  • domain assumption Far-field monochromatic scalar field; evanescent waves neglected so wavenumber support is the propagating disk k_x²+k_y²≤κ².
    Section II; underpins bandlimited kernels and Paley–Wiener entire-function arguments.
  • domain assumption Spatial stationarity: kernel depends on difference r−r′ only, enabling Fourier/wavenumber representation and Toeplitz-like operator structure.
    Title and continuous model; required for the spectral factor representation and Szegő–Widom symbol.
  • standard math Integrand f is entire of exponential type set by twice the mapped bandwidth (product of bandlimited kernel and eigenfunction), hence Bernstein inequality applies to high derivatives.
    Lemmas 2–3, Appendix A; classical Paley–Wiener + Bernstein.
  • standard math M-point Gauss–Legendre rule has algebraic precision 2M−1 with standard remainder bound involving f^{(2M)}.
    Eq. (40) and classical numerical integration citations [24],[25].
  • ad hoc to paper Multidimensional Szegő–Widom trace expansion (Lemma 4) holds for piecewise-smooth spatial domains and the discontinuous step test f used for eigenvalue counting.
    Authors note the discontinuous-test rectangular case is still open; they invoke Widom’s conjecture and validate numerically (Intro; §VI-B; Appendices C–E).
  • domain assumption Isotropic scattering kernel (sinc / disk indicator) upper-bounds eDoF of non-isotropic kernels under fixed power.
    §VI-A citing concentration arguments [19]; used to treat isotropic eDoF as a safe ceiling.
invented entities (2)
  • computational degrees of freedom (cDoF) independent evidence
    purpose: Name the minimal GLQ node count at which super-exponential quadrature convergence onsets (Mc≈πL/λ in 1D).
    Definitional threshold from remainder-ratio analysis, not a new physical field; independent check via end-fire and PSWF experiments.
  • effective DoF (eDoF) as ε-numerical rank of the continuous kernel independent evidence
    purpose: Count eigenvalues above precision ε to truncate Nyström EVD safely and set partial-EVD target k.
    Operationalization of known rapid spectral decay; 2D formula is new packaging of Widom asymptotics for HMIMO rectangles.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Computational and Effective Degrees of Freedom for Spatially Stationary HMIMO Channel Modeling." pith.science (2026). https://pith.science/paper/KXSZ5J4Q

@misc{pith2026260723487,
  author       = {Pith},
  title        = {Pith review of: Computational and Effective Degrees of Freedom for Spatially Stationary HMIMO Channel Modeling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KXSZ5J4Q}},
  note         = {Machine review of arXiv:2607.23487}
}
read the original abstract

This paper establishes a comprehensive theoretical framework for the continuous-to-discrete modeling of spatially stationary holographic MIMO (HMIMO) channels utilizing the Nystrom method with Gauss-Legendre quadrature (NGLQ). Starting with an operator-theoretic analysis of the NGLQ method, we prove that its quadrature error exhibits a super-exponential decay. Furthermore, we derive a spatial sampling threshold, termed computational degrees of freedom (cDoF), which reveals a {\pi}/2 oversampling penalty over the physical DoF for 1D arrays, compounding to a 68% computational redundancy for 2D separable grids. To address the ill-conditioning of the eigenvalue decomposition (EVD) problem inherent to the Nystrom discretization, we invoke the multidimensional Szego-Widom asymptotic expansion. This analysis yields a physically grounded semi-analytical expression for the effective DoF (eDoF) of 2D rectangular apertures, capturing the anisotropic boundary truncation effects to guide partial EVD and reduce computational complexity. Numerical evaluations confirm the tightness of the cDoF threshold under worst-case end-fire conditions. Moreover, simulations utilizing closed-form kernels for isotropic scattering verify that the derived eDoF acts as an accurate asymptotic approximation. Finally, by deploying the exact non-uniform discrete Fourier transform to eliminate interpolation error floors, we demonstrate spectral convergence down to the machine-precision level for non-isotropic scattering environments.

Figures

Figures reproduced from arXiv: 2607.23487 by the authors.

Figure 1
Figure 1. Illustration of end-fire and broadside incidence scenarios. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 3
Figure 3. PSWF reconstruction against the number of GLQ nodes, [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 2
Figure 2. Illustration of quadrature errors for cos(2π L λ t) with respect to the number of GLQ nodes. To numerically validate this, [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Illustration of the effective wavenumber domain and the [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: RE of sin(c(s−t)) π(s−t) against NGLQ nodes under various normalized linear aperture sizes. linear aperture sizes, the number of NGLQ nodes M starts from the physical DoF 2L/λ. As expected, the RE initially decays rapidly as M increases. We first note that the [PITH_F…
Figure 6
Figure 6. Figure 6: Eigenvalue distribution comparison between the NGLQ re [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: Eigenvalue distribution comparison between the NGLQ re [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: CDF of NGLQ based MSRE under non-isotropic scattering [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

27 extracted references · 2 linked inside Pith

  1. [5]

    A spectrally convergent discretization of holographic MIMO channels,

    H. Yan et al., “A spectrally convergent discretization of holographic MIMO channels,” IEEE Communications Letters, vol. 30, pp. 1964–1968, 2026

    1964

  2. [1]

    Electromagnetic information theory for holo- graphic MIMO communications,

    L. Wei et al., “Electromagnetic information theory for holo- graphic MIMO communications,” IEEE Communications Sur- veys & Tutorials, vol. 28, pp. 6211–6240, 2026

    2026

  3. [2]

    Wavenumber-domain signal processing for holographic MIMO: Foundations, methods, and future direc- tions,

    Z. Zhang et al., “Wavenumber-domain signal processing for holographic MIMO: Foundations, methods, and future direc- tions,” IEEE Communications Standards Magazine, vol. 10, no. 2, pp. 127–133, 2026

    2026

  4. [3]

    Towards 6G MIMO: Massive spatial mul- tiplexing, dense arrays, and interplay between electromagnetics and processing,

    E. Bjornson et al., “Towards 6G MIMO: Massive spatial mul- tiplexing, dense arrays, and interplay between electromagnetics and processing,” arXiv:2401.02844, 2024

    arXiv 2024

  5. [4]

    Spatially-stationary model for holographic MIMO small-scale fading,

    A. Pizzo et al., “Spatially-stationary model for holographic MIMO small-scale fading,” IEEE Journal on Selected Areas in Communications, vol. 38, no. 9, pp. 1964–1979, 2020

    1964

  6. [6]

    Fourier plane-wave series expansion for holo- graphic MIMO communications,

    A. Pizzo et al., “Fourier plane-wave series expansion for holo- graphic MIMO communications,” IEEE Transactions on Wire- less Communications, vol. 21, no. 9, pp. 6890–6905, 2022

    2022

  7. [7]

    Eigenvalue distribution of time and frequency limiting,

    H. J. Landau et al., “Eigenvalue distribution of time and frequency limiting,” Journal of Mathematical Analysis and Applications, vol. 77, no. 2, pp. 469–481, 1980

    1980

  8. [8]

    Widom, On a Class of Integral Operators with Discontinuous Symbol

    H. Widom, On a Class of Integral Operators with Discontinuous Symbol. Basel: Birkhäuser Basel, 1982, pp. 477–500

    1982

Show all 27 references
  1. [9]

    Entanglement entropy of fermions in any dimension and the widom conjecture,

    D. Gioev et al., “Entanglement entropy of fermions in any dimension and the widom conjecture,” Phys. Rev. Lett., vol. 96, p. 100503, Mar 2006

    2006

  2. [10]

    A. V. Sobolev, Pseudo-differential operators with discontinuous symbols: Widom’s conjecture. Memoirs of the American Mathematical Society, 2013, vol. 222, no. 1043

    2013

  3. [11]

    Wiener–hopf operators in higher dimensions: The widom conjecture for piece-wise smooth domains,

    ——, “Wiener–hopf operators in higher dimensions: The widom conjecture for piece-wise smooth domains,” Integral Equations and Operator Theory, vol. 81, no. 3, pp. 435–449, Mar 2015

    2015

  4. [12]

    Functions of self-adjoint operators in ideals of com- pact operators,

    ——, “Functions of self-adjoint operators in ideals of com- pact operators,” Journal of the London Mathematical Society, vol. 95, no. 1, pp. 157–176, 2017

    2017

  5. [13]

    Hildebrand, Advanced Calculus for Applications

    F. Hildebrand, Advanced Calculus for Applications. Upper Saddle River, NJ, USA, Prentice-Hall, 1962

    1962

  6. [14]

    Riesz et al., Functional Analysis

    F. Riesz et al., Functional Analysis. New York: Courier Corporation, 1990

    1990

  7. [15]

    Reproducing kernel Hilbert space, Mercer’s theorem, eigenfunctions, Nystrom method, and use of kernels in machine learning: Tutorial and survery,

    B. Ghojogh et al., “Reproducing kernel Hilbert space, Mercer’s theorem, eigenfunctions, Nystrom method, and use of kernels in machine learning: Tutorial and survery,” arXiv preprint arXiv:2106.08443, 2021

    2021 arXiv

  8. [16]

    Prolate spheroidal wave functions, fourier analysis and uncertainty — I,

    D. Slepian et al., “Prolate spheroidal wave functions, fourier analysis and uncertainty — I,” The Bell System Technical Journal, vol. 40, no. 1, pp. 43–63, 1961

    1961

  9. [17]

    S. M. Nikol’skii, Approximation of Functions of Several Vari- ables and Imbedding Theorems. New York: Springer-Verlag, 1975

    1975

  10. [18]

    On Bernstein’s inequality for entire functions of exponential type,

    Q. Rahman et al., “On Bernstein’s inequality for entire functions of exponential type,” Journal of Mathematical Analysis and Applications, vol. 359, no. 1, pp. 168–180, 2009

    2009

  11. [19]

    Degrees of freedom in multiple-antenna chan- nels: a signal space approach,

    A. Poon et al., “Degrees of freedom in multiple-antenna chan- nels: a signal space approach,” IEEE Transactions on Informa- tion Theory, vol. 51, no. 2, pp. 523–536, 2005

    2005

  12. [20]

    An iteration method for the solution of the eigenvalue problem of linear differential and integral operators,

    C. Lanczos, “An iteration method for the solution of the eigenvalue problem of linear differential and integral operators,” Journal of research of the National Bureau of Standards, vol. 45, no. 4, pp. 255–282, 1950

    1950

  13. [21]

    Spatial characterization of electromagnetic random channels,

    A. Pizzo et al., “Spatial characterization of electromagnetic random channels,” IEEE Open Journal of the Communications Society, vol. 3, pp. 847–866, 2022

    2022

  14. [22]

    Electromagnetic-compliant channel modeling and performance evaluation for holographic MIMO,

    T. Wang et al., “Electromagnetic-compliant channel modeling and performance evaluation for holographic MIMO,” in 2022 IEEE Globecom Workshops (GC Wkshps), 2022, pp. 747–752

    2022

  15. [23]

    How to differentiate between near field and far field: Revisiting the Rayleigh distance,

    S. Sun et al., “How to differentiate between near field and far field: Revisiting the Rayleigh distance,” IEEE Communications Magazine, vol. 63, no. 1, pp. 22–28, Jan. 2025

    2025

  16. [24]

    P. J. Davis et al., Methods of numerical integration. Courier Corporation, 2007

    2007

  17. [25]

    F. B. Hildebrand, Introduction to Numerical Analysis, 2nd ed. New York: Dover Publications, 1987

    1987

  18. [26]

    A remark on Stirling’s formula,

    H. Robbins, “A remark on Stirling’s formula,” The American mathematical monthly, vol. 62, no. 1, pp. 26–29, 1955

    1955

  19. [27]

    Rudin, Real and Complex Analysis

    W. Rudin, Real and Complex Analysis. McGraw-Hill, 1974

    1974

Pith tools

Reviewed July 30, 2026 · model on record in the stance chip above.