Mutual Coupling in Continuous Aperture Arrays: Physical Modeling and Beamforming Design

Giuseppe Thadeu Freitas de Abreu; Kuranage Roche Rayan Ranasinghe; Yuanwei Liu; Zhaolin Wang

REVIEW 4 major objections 4 minor 36 references

Mutual Coupling in Continuous Aperture Arrays: Physical Modeling and Beamforming Design

T0 review · 4 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read This paper establishes that mutual coupling in continuous aperture arrays is exactly captured by a kernel with polarization and surface dissipation, and the optimal beamformer is the kernel's Fredholm inverse, computed two ways.

desk verdict Useful polarization-aware coupling model for CAPA beamforming, but the printed Theorem 1 is mis-normalized and needs a condition-of-acceptance fix. read the letter →

arxiv 2511.11225 v4 pith:OIC7K33G submitted 2025-11-14 cs.IT math.IT

Zhaolin Wang , Kuranage Roche Rayan Ranasinghe , Giuseppe Thadeu Freitas de Abreu , Yuanwei Liu This is my paper · ORCID

classification cs.ITmath.IT MSC 78A5045B0594A12 PACS 84.40.Ba02.30.Rz

keywords continuousaperturearraysmutualcouplingpolarizationbeamformingFredholmintegralequationGauss-Legendrequadratureconjugategradientsuperdirectivity

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 sets out to give continuous aperture arrays (CAPAs) the same mutual-coupling rigor that discrete arrays have, including polarization and surface loss. It derives a scalar coupling kernel from the real part of the free-space dyadic Green's function plus a surface-resistance delta term, and shows that polarization makes the coupling anisotropic, so the textbook half-wavelength spacing rule no longer nulls coupling. The optimal beamforming problem is then posed as a functional optimization; its solution is the solution of a Fredholm integral equation, which the paper solves by a closed-form kernel approximation and by a conjugate-gradient iteration. The resulting analysis shows that uncoupled models of dense arrays are unphysical—they predict unbounded gain as spacing shrinks—while the coupled model converges correctly to the CAPA limit.

What carries the argument

The load-bearing object is the coupling kernel c(s) = Z_s δ(s) + c_rad(s), with c_rad(s)=κ0Z0(φ(s)+κ0^{-2}∂_y^2 φ(s)). Its wavenumber transform C_rad(κ)=Z0(1−κ_y^2/κ0²)/(2√(1−||κ||²/κ0²)) on the visible disk ||κ||≤κ0 is what makes the closed-form inverse possible; Gauss–Legendre quadrature turns the inverse Fourier integral into a sum of exponentials, giving an explicit inverse kernel in Proposition 2 and the closed-form beamformer (46). The same transform yields the large-aperture gain (68) and the beampattern ratio (79).

What would settle it

Place two small y-polarized probes on a conducting plane and measure the transmission coefficient versus separation in the y-direction. The paper predicts the first coupling null at 0.72λ (Eq. 21) rather than 0.5λ; observing a null at 0.5λ, or a null position that does not depend on orientation relative to polarization, would falsify the polarization-corrected kernel.

Watch

Extended reading notes

Core claim

The central discovery is that the effect of a CAPA's mutual coupling on transmit power is fully contained in a scalar kernel c(s)=Z_s δ(s)+c_rad(s), where c_rad is the polarization-projected real part of the free-space dyadic Green's function. Under a y-polarized current, c_rad takes the form κ0Z0(φ(s)+κ0^{-2}∂_y^2 φ(s)). With this kernel, the beamformer that maximizes received power subject to the physical power constraint is w_opt(s) ∝ v(s), where v solves the Fredholm equation ∫_S c(s−z)v(z)dz = h*(s). The paper proves this structure by calculus of variations, then gives two practical routes to compute v: a wavenumber-domain approximation of the kernel via Gauss–Legendre quadrature that y

Load-bearing premise

The model assumes the surface's loss is a local scalar resistance, Z_s δ(s), so all coupling is captured by the free-space Green's function plus a point-wise Ohmic term; if real surfaces have non-local impedance, surface waves, or other loss that cannot be written that way, the kernel and the derived beamformer and gains would change.

Editorial extensions

If this is right

Arrays designed with the common half-wavelength spacing rule will not null mutual coupling once polarization is accounted for; null positions become direction-dependent (e.g., ~0.72λ along the polarization axis).
Uncoupled discrete-array models overpredict gain at small spacing without bound, so any dense-array analysis should use a coupled kernel or state clearly that it is a non-physical upper bound.
The optimal beamformer under coupling can be computed in closed form by sampling the wavenumber domain with Gauss–Legendre points, making coupled CAPA beamforming computationally cheap.
At large apertures, array gain saturates at a value set by the surface resistance and the wavenumber-domain coupling kernel at the look direction, giving a design relation between material conductivity and achievable directivity.
The coupled beampattern is narrower than the uncoupled one in both broadside and end-fire, indicating that mutual coupling can be used to reach superdirectivity rather than being only a harmful effect.

Reading between the lines

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

If the kernel model survives measurement, the direction-dependent null positions (Eq. 21) could be used as a calibration test: measuring S-parameters of a small array along and across the polarization axis would directly verify the transcendental equation (17).
The paper's element-current-profile dependence in the SPDA coupling matrix hints at an unexplored design degree of freedom: shaping the unit-cell current to place the kernel nulls at the intended element spacings could replace the failed half-wavelength rule.
Because the kernel acts as a spatial-frequency low-pass filter on the beamformer spectrum, one could imagine a 'coupling-aware' codebook design that pre-distorts beams to compensate for the kernel, potentially improving sidelobe behavior beyond what the paper's matched structure provides.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague—

Bottom line: the paper is worth engaging with, and it deserves a real referee pass, but the printed Theorem 1 does not satisfy the paper's own power constraint. The stress-test note is right: the normalization constant in (28) should be sqrt(2Pt / ∫ h v), not sqrt(2Pt)/∫ h v. Appendix C's Eq. (106) has the correct form, and Eq. (47)'s gain expression is consistent with the corrected constant, so this is a typo rather than a wrong idea—but it sits in the theorem statement, so it has to be fixed.

What's actually new: the coupling kernel in Eq. (14) with the polarization-projected second derivative and surface dissipation goes beyond the isotropic treatment in [26]. The finding that nulls shift from half-wavelength spacing is concrete and useful. The kernel-approximation inversion, with the Gauss-Legendre wavenumber approximation and the closed-form expression (46), is a genuine attempt to solve the Fredholm inverse; the CG solver in Proposition 3 is a clean equivalent functional formulation. The SPDA extension is a natural bridge to discrete arrays, and the numerical cross-validation between KA and CG shows the machinery works. The large-aperture array-gain analysis in Proposition 4 is also coherent, despite a non-rigorous delta step in Appendix D.

Soft spots, in order of severity:

1. Theorem 1 mis-normalized. As printed, wopt = √(2Pt)/∫h v · v violates (27b). This is load-bearing because the theorem is the paper's central result, even though Appendix C shows the intended α. The same error appears in Eq. (46) (missing the square root over Zs(η − a^H DΛa)).

2. Proposition 2's proof is explicitly deferred to the authors' own [28]. The inversion formula may be right, but the paper should provide the key steps. Self-citation is not automatically a flaw, but here it's a missing load-bearing proof.

3. Eq. (17) and Appendix B have a typo: the last term should be 2(sin ε − ε cos ε), not 2(sin ε − ε sin ε). This is confirmed by Eq. (20)'s tan ε − ε = 0. Minor but confusing.

4. Numerical validation is self-consistent—KA vs CG—but no independent full-wave EM solver benchmark. That limits how much one can trust the physical kernel's fidelity, though it doesn't undermine the internal logic.

Who this is for: people working on holographic MIMO/CAPA beamforming, especially those who want a tractable way to include mutual coupling and polarization. The paper does not close the loop on real-antenna validation, but the model and inversion methods give a solid starting point.

My recommendation: send it to peer review, but make the normalization fix a condition of acceptance. The central model and methods are worth refereeing even with the current typos.

Referee Report

4 major / 4 minor

Summary. This paper studies mutual coupling in continuous aperture arrays (CAPAs). It models the transmitted EM power through a local surface-resistance dissipation term plus the real part of the free-space dyadic Green's function, then specializes to a scalar uni-polarized kernel. The authors show that polarization makes the coupling kernel anisotropic and hence that the conventional half-wavelength null-spacing rule fails. The beamforming problem of maximizing received signal power subject to a physical EM-power constraint is formulated as a functional optimization problem and solved by calculus of variations, giving an optimal beamformer proportional to the solution of a Fredholm equation. Since the inverse coupling kernel is not available in closed form, two methods are proposed: a wavenumber-domain Gauss-Legendre approximation leading to a separable kernel with an explicit inverse, and a conjugate-gradient method for an equivalent quadratic functional. Large-aperture array gain and beampatterns are analyzed, the model is extended to spatially discrete arrays, and numerical results demonstrate convergence of coupled SPDAs to the CAPA limit, anisotropy due to polarization, and superdirective behavior of the coupled beampattern.

Significance. If the technical errors are corrected, the paper makes a substantial contribution to CAPA beamforming under mutual coupling. The physics-based kernel, the explicit inclusion of polarization and surface dissipation, and the closed-form inversion method are valuable, as is the numerical cross-validation of the two solution algorithms. The claim that polarization invalidates the half-wavelength decoupling rule is concrete and testable, and the large-aperture gain formulas provide design insight. The paper is also transparent about its single-polarization and single-user limitations and suggests sensible extensions. However, several displayed equations that are load-bearing for the main results are mis-normalized or internally inconsistent, and one key proof is deferred rather than included. These issues must be fixed before the results are fully usable.

major comments (4)

[Theorem 1, Eq. (28); Eq. (46); Eq. (67)] The displayed optimal beamformer w_opt(s) = sqrt(2Pt)/[∫ h(z)v(z)dz] v(s) does not satisfy the power constraint (27b). Setting w=αv in (27b) gives α^2/2 ∫ h v = Pt, hence α = sqrt(2Pt/∫ h v). The printed expression omits the square root over the denominator. The same omission appears in Eq. (46): the denominator should be sqrt(Z_s(η−a^H D Λ a)), not Z_s(η−a^H D Λ a). Eq. (67) similarly needs sqrt(h^T Φ v_opt) in the denominator. Eq. (47), G_opt = 2(η−a^H D Λ a)/Z_s, is consistent with the corrected normalization and confirms the typo, but as printed Theorem 1 and Eq. (46) violate the power constraint.
[Proposition 1, Eq. (17); Appendix B, Eq. (101)] The null equation is printed as cos^2ψ((ε^2−3)sinε+3ε cosε)+2(sinε−ε sinε)=0. The final term should be 2(sinε−ε cosε). More importantly, the coordinate definition sx=r sinψ, sy=r cosψ is inconsistent with the text's claim that the x-axis is obtained by setting ψ=0 (i.e., sy=0); under the stated definition ψ=0 gives the y-axis. The axis equations (18) and (20) are correct for the x- and y-axes only if ψ is measured from the x-axis (sx=r cosψ, sy=r sinψ), or if the axis assignments in the text are swapped. Please correct Eq. (17), Eq. (101), and the coordinate conventions so that Proposition 1 is self-consistent and reproducible.
[Proposition 2, Eq. (42)] The inverse kernel in Eq. (42) is the foundation for the closed-form beamformer (46), but the proof states 'The proof assembles [28, Appendix B]. The details are thus omitted here.' Since this proposition is a central technical step and the present kernel includes both dissipation and a different structure, a self-contained verification of the inverse condition (30), or at least a complete derivation sketch, should be included. Deferring the proof entirely to a separate reference is not sufficient for a load-bearing result in this manuscript.
[Appendix D, Eq. (112)] The proof of Proposition 4 uses a Cauchy-Schwarz step with a Dirac delta and then writes ∫|δ(κ−κ̃_r)|^2 dκ=1. The square of a Dirac delta is not a well-defined distribution, and point evaluation W(κ̃_r) is not a bounded functional on L^2(C(κ)dκ). The large-aperture gain formula (68) may be correct, but the proof as written is not rigorous. Please replace this step with a limiting argument over finite apertures or a proper weak-convergence formulation, or justify the delta manipulation as a distributional limit with the necessary regularity conditions.

minor comments (4)

[Section III-B, first paragraph] 'the kernel approximation method and the conjugate method' should read 'the kernel approximation method and the conjugate gradient method'.
[Eq. (17) and Appendix B] In addition to the mathematical issue noted above, Appendix B contains a typo: 'we representation the coordinates' should be 'we represent the coordinates'.
[Section IV-B, Eq. (79)] The sentence 'which needs to be accurately evaluated using the numerical methods proposed in Section III' appears twice in close succession; please rephrase to avoid repetition.
[Figure 3 caption] The caption states the kernel is evaluated at 7.8 GHz while the default simulation setup in Section VI is 2.4 GHz. Please state the frequency and, if different, why this choice is made.

Circularity Check

1 steps flagged · score 2.0 of 10

No significant circularity: the main beamforming derivation is self-contained; only the deferred inverse-kernel proof and the by-construction SPDA convergence check keep the score slightly above zero.

self citation load bearing [Section III-B, Proposition 2 proof]
"Proof: The proof assembles [28, Appendix B]. The details are thus omitted here."

The closed-form beamformer (46) depends on Proposition 2's inverse-kernel formula (42), but that formula is not derived in this paper. Instead, the proof is delegated to [28], a prior paper by two of the same authors. This makes the inversion step load-bearing on a self-citation. Mitigating factor: [28] is a published, parameter-free algebraic derivation, and the inverse can in principle be verified by substitution, so this is a minor circularity rather than a definitional reduction.

full rationale

The central derivation chain is not circular in any definitional sense. The coupling kernel (14) is built from the stated free-space Green's function and surface-resistance model; the beamforming problem (27) is a genuine functional optimization; Theorem 1 is obtained by the calculus of variations in Appendix C; and the closed-form array gain (47) is algebraically implied by the same kernel and power normalization, not by the target result. No free parameter is fitted to the quantity it later predicts. The large-aperture gain (68) follows from the same wavenumber-domain kernel C(kappa), and the polarization null analysis is a direct evaluation of the kernel. The numerical KA/CG cross-validation checks internal consistency rather than an external empirical benchmark, which is not circularity. Two small concerns prevent a score of zero: (i) Proposition 2's inverse-kernel proof is omitted and cited to the authors' own prior work [28], so the closed-form KA solution leans on a self-citation; (ii) the SPDA convergence claim in Fig. 6 is essentially a Riemann-sum consistency of the discretization in Eq. (82), so it confirms model consistency rather than an independent physical prediction. Separately, the normalization typo in Eq. (28)/(46) relative to Appendix C and Eq. (47) is a correctness issue, not a circularity issue.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

No free parameters are fitted to data; all constants (Z0, κ0, Zs from σs and µs) are physical inputs from the prior literature. The central claim rests on standard Fourier/Weyl identities, the far-field plane-wave approximation, and assumed smooth current profiles for the SPDA reduction.

assumptions (9)

standard math The radiated field is given by the tensor Green's function solution to the inhomogeneous Helmholtz equation (Eqs. 4–6).
Invoked directly in Eq. (4) as the starting model of mutual coupling.
domain assumption Surface dissipation is local and scalar: ediss(s)=Zs jt(s), with Zs computed from the good-conductor formula (Eqs. 7–8).
The entire coupling kernel depends on this local resistance model; non-local surface impedance would change the kernel.
domain assumption The receiver is in the far field, so the spherical wave is approximated as a plane wave across the aperture (Eq. 24).
Used to obtain h(s)=β e^{-jκr^T s}, which feeds the beamforming objective.
domain assumption The receiver polarization is perfectly matched to the y-polarized source (ur = uy).
Justifies projecting all fields onto uy in Section II-C.
domain assumption The continuous coupling operator is positive definite enough for the CG quadratic functional to have a unique minimum.
Proposition 3 and the CG algorithm rely on this; with Zs>0 and Crad≥0 the assumption is plausible but not proved.
domain assumption For large apertures the finite-aperture convolution can be replaced by the infinite-plane convolution theorem (Eq. 76).
Underpins the approximate coupled beampattern (78) and the large-aperture gain analysis.
domain assumption SPDA element current profiles are slowly varying and element apertures are small, giving Ψrad(n,m)≈Ad²|at(0)|²crad(Δpnm) (Eq. 87).
The discrete coupling matrix, and hence the claimed SPDA-to-CAPA convergence, depends on this approximation.
standard math Weyl identity and standard 2D Fourier transform relations for the Green's function (Eq. 33).
Used to derive the wavenumber-domain kernel Crad(κ).
standard math Gauss-Legendre quadrature converges rapidly for the smooth inverse-Fourier integrand (Eq. 35).
The kernel approximation method relies on this convergence, verified numerically but not bounded in the paper.

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Cite this review

Pith. "Pith review of Mutual Coupling in Continuous Aperture Arrays: Physical Modeling and Beamforming Design." pith.science (2026). https://pith.science/paper/OIC7K33G

@misc{pith2026251111225,
  author       = {Pith},
  title        = {Pith review of: Mutual Coupling in Continuous Aperture Arrays: Physical Modeling and Beamforming Design},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OIC7K33G}},
  note         = {Machine review of arXiv:2511.11225}
}

read the original abstract

The phenomenon of mutual coupling in continuous aperture arrays (CAPAs) is studied. First, a general physical model for the phenomenon that accounts for both polarization and surface dissipation losses is developed. Then, the unipolarized coupling kernel is characterized, revealing that polarization induces anisotropic coupling and invalidates the conventional half-wavelength spacing rule for coupling elimination. Next, the beamforming design problem for CAPAs with coupling is formulated as a functional optimization problem, leading to the derivation of optimal beamforming structures via the calculus of variations. To address the challenge of inverting the coupling kernel in the optimal structure, two methods are proposed: 1) the kernel approximation method, which yields a closed-form solution via wavenumber-domain transformation and GaussLegendre quadrature, and 2) the conjugate gradient method, which addresses an equivalent quadratic functional optimization problem iteratively. Furthermore, the optimal array gain and beampattern are analyzed at the large-aperture limit. Finally, the proposed continuous mutual coupling model is extended to spatially discrete arrays (SPDAs), and comprehensive numerical results are provided, demonstrating that: 1) coupled SPDA performance correctly converges to the CAPA limit, while uncoupled models are shown to violate physics, 2) polarization results in anisotropic array gain behavior, and 3) the coupled beampattern exhibits higher directivity than the uncoupled beampattern.

Figures

Figures reproduced from arXiv: 2511.11225 by the authors.

**Figure 2.** Radiation mutual coupling kernel along y-axis when sx = 0 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Comparison of the true radiation mutual coupling ker [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: Convergence of the optimal array gain with respect to [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: Convergence of the conjugate gradient. A. Convergence of Proposed Methods We first evaluate the convergence performance of the proposed kernel approximation (KA) and conjugate gradient (CG) methods. First, the accuracy of both methods is highly dependent on the order …

**Figure 6.** Figure 6: Array gain versus the antenna spacing of SPDAs. [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 7.** Figure 7: Array gain versus the array aperture. model approximation; i.e., approximating the coupling kernel itself, whereas the CG method applies numerical approximation; i.e., approximating the final numerical implementation. Finally, [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

**Figure 10.** Figure 10: Normalized array gain in the H-plane (θ = 0) with different aperture size. CAPAs, the array gain grows linearly as the aperture size increases. However, the SPDA array gain exhibits a stepwise increase, since a larger aperture does not always accommodate more antenna …

**Figure 11.** Figure 11: Normalized beampatterns for coupled and uncoupled [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]

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Reference graph

Works this paper leans on

36 extracted references · 2 linked inside Pith

[26]

Mutual coupling in holographic MIMO: Physical modeling and information-theoretic analysis,

A. Pizzo and A. Lozano, “Mutual coupling in holographic MIMO: Physical modeling and information-theoretic analysis,” IEEE J. Sel. Areas Inf. Theory , vol. 6, pp. 111–126, May 2025

2025
[28]

Optimal beamforming for multi- user continuous aperture array (CAPA) systems,

Z. Wang, C. Ouyang, and Y . Liu, “Optimal beamforming for multi- user continuous aperture array (CAPA) systems,” IEEE Trans. Commun., early access, Mar. 2025. doi: 10.1109/TCOMM.2025.3554644

arXiv 2025
[1]

Enabling 6g performance in the upper mid-band by transitioning from massive to gigantic MIMO,

E. Bj¨ ornson, F. Kara, N. Kolomvakis, A. Kosasih, P . Rame zani, and M. B. Salman, “Enabling 6g performance in the upper mid-band by transitioning from massive to gigantic MIMO,” IEEE Open J. Commun. Soc., vol. 6, pp. 5450–5463, 2025

2025
[2]

Ne ar- ﬁeld communications: A tutorial review,

Y . Liu, Z. Wang, J. Xu, C. Ouyang, X. Mu, and R. Schober, “Ne ar- ﬁeld communications: A tutorial review,” IEEE Open J. Commun. Soc. , vol. 4, pp. 1999–2049, Aug. 2023

1999
[3]

A tutorial on extremely large-scal e mimo for 6g: Fundamentals, signal processing, and applications,

Z. Wang, J. Zhang, H. Du, D. Niyato, S. Cui, B. Ai, M. Debbah , K. B. Letaief, and H. V . Poor, “A tutorial on extremely large-scal e mimo for 6g: Fundamentals, signal processing, and applications,” IEEE Commun. Surv. Tut., vol. 26, no. 3, pp. 1560–1605, 3rd Quart. 2024

2024
[4]

CAPA: Continuous-aperture arrays for revolutionizing 6G wirele ss communica- tions,

Y . Liu, C. Ouyang, Z. Wang, J. Xu, X. Mu, and Z. Ding, “CAPA: Continuous-aperture arrays for revolutionizing 6G wirele ss communica- tions,” IEEE Wireless Commun. , vol. 32, no. 4, pp. 38–45, Aug. 2025

2025
[5]

Electrom agnetic information theory: Fundamentals, modeling, application s, and open problems,

J. Zhu, Z. Wan, L. Dai, M. Debbah, and H. V . Poor, “Electrom agnetic information theory: Fundamentals, modeling, application s, and open problems,” IEEE Wireless Communications, vol. 31, no. 3, pp. 156–162, Jun. 2024

2024
[6]

Towards 6G MIMO: Massive spatial multiplexing, dense arra ys, and interplay between electromagnetics and processing,

E. Bj¨ ornson, C.-B. Chae, R. W. Heath Jr, T. L. Marzetta, A . Mezghani, L. Sanguinetti, F. Rusek, M. R. Castellanos, D. Jun, and ¨O. T. Demir, “Towards 6G MIMO: Massive spatial multiplexing, dense arra ys, and interplay between electromagnetics and processing,” arXiv preprint arXiv:2401.02844, 2024

arXiv 2024

Show all 36 references

[7]

Mutual coupling in MIMO wirele ss systems: a rigorous network theory analysis,

J. Wallace and M. Jensen, “Mutual coupling in MIMO wirele ss systems: a rigorous network theory analysis,” IEEE Wireless Commun. , vol. 3, no. 4, pp. 1317–1325, Jul. 2004

2004
[8]

Impact of antenna coupling on 2 × 2 MIMO communications,

B. Clerckx, C. Craeye, D. V anhoenacker-Janvier, and C. O estges, “Impact of antenna coupling on 2 × 2 MIMO communications,” IEEE Trans. V eh. Technol., vol. 56, no. 3, pp. 1009–1018, May 2007

2007
[9]

Toward a circuit theory of communi- cation,

M. T. Ivrlaˇ c and J. A. Nossek, “Toward a circuit theory of communi- cation,” IEEE Trans. Circuits Syst. I: Regul. Pap. , vol. 57, no. 7, pp. 1663–1683, Jul. 2010

2010
[10]

Large -scale MIMO transmitters in ﬁxed physical spaces: The effect of transmi t correlation and mutual coupling,

C. Masouros, M. Sellathurai, and T. Ratnarajah, “Large -scale MIMO transmitters in ﬁxed physical spaces: The effect of transmi t correlation and mutual coupling,” IEEE Trans. Commun. , vol. 61, no. 7, pp. 2794– 2804, Jul. 2013

2013
[11]

Super-wideband massive MIMO,

M. Akrout, V . Shyianov, F. Bellili, A. Mezghani, and R. W . Heath, “Super-wideband massive MIMO,” IEEE J. Sel. Areas Commun., vol. 41, no. 8, pp. 2414–2430, Aug. 2023

2023
[12]

Effects of mutual coupling on degree of freedom and antenna efﬁciency in holographic MIMO communications,

S. S. Y uan et al., “Effects of mutual coupling on degree of freedom and antenna efﬁciency in holographic MIMO communications,” IEEE Open J. Antennas Propag. , vol. 4, pp. 237–244, Feb. 2023

2023
[13]

Beamforming per- formances of holographic surfaces,

P . Wang, M. N. Khormuji, and B. M. Popovic, “Beamforming per- formances of holographic surfaces,” IEEE Trans. Wireless Commun. , vol. 23, no. 6, pp. 5816–5831, Jun. 2024

2024
[14]

Holographic MIMO com munica- tions: What is the beneﬁt of closely spaced antennas?

A. A. D’Amico and L. Sanguinetti, “Holographic MIMO com munica- tions: What is the beneﬁt of closely spaced antennas?” IEEE Trans. Wireless Commun., vol. 23, no. 10, pp. 13 826–13 840, Oct. 2024

2024
[15]

Communicating with waves between volume s: evaluating orthogonal spatial channels and limits on coupling strengt hs,

D. A. Miller, “Communicating with waves between volume s: evaluating orthogonal spatial channels and limits on coupling strengt hs,” Appl. Opt., vol. 39, no. 11, pp. 1681–1699, 2000

2000
[16]

Communicating with large intelligent sur faces: Fundamen- tal limits and models,

D. Dardari, “Communicating with large intelligent sur faces: Fundamen- tal limits and models,” IEEE J. Sel. Areas Commun. , vol. 38, no. 11, pp. 2526–2537, Nov. 2020

2020
[17]

On Landau’s eigenvalue theorem for line-of- sight MIMO channels,

A. Pizzo and A. Lozano, “On Landau’s eigenvalue theorem for line-of- sight MIMO channels,” IEEE Wireless Commun. Lett. , vol. 11, no. 12, pp. 2565–2569, Dec. 2022

2022
[18]

Capacity of the continuo us-space electromagnetic channel,

M. A. Jensen and J. W. Wallace, “Capacity of the continuo us-space electromagnetic channel,” IEEE Trans. Antennas Propag., vol. 56, no. 2, pp. 524–531, Feb. 2008

2008
[19]

Horse (electromagnetics) is more impo rtant than horse- man (information) for wireless transmission,

M. D. Migliore, “Horse (electromagnetics) is more impo rtant than horse- man (information) for wireless transmission,” IEEE Trans. Antennas Propag., vol. 67, no. 4, pp. 2046–2055, Apr. 2019
[20]

Mutual i nformation for electromagnetic information theory based on random ﬁel ds,

Z. Wan, J. Zhu, Z. Zhang, L. Dai, and C.-B. Chae, “Mutual i nformation for electromagnetic information theory based on random ﬁel ds,” IEEE Trans. Commun., vol. 71, no. 4, pp. 1982–1996, Apr. 2023

1982
[21]

Wavenumb er-division multiplexing in line-of-sight holographic MIMO communica tions,

L. Sanguinetti, A. A. D’Amico, and M. Debbah, “Wavenumb er-division multiplexing in line-of-sight holographic MIMO communica tions,” IEEE Trans. Wireless Commun. , vol. 22, no. 4, pp. 2186–2201, Apr. 2023

2023
[22]

Pattern-division multiplexing fo r multi-user continuous-aperture MIMO,

Z. Zhang and L. Dai, “Pattern-division multiplexing fo r multi-user continuous-aperture MIMO,” IEEE J. Sel. Areas Commun. , vol. 41, no. 8, pp. 2350–2366, Aug. 2023

2023
[23]

Beamforming optimizati on for continuous aperture array (CAPA)-based communications,

Z. Wang, C. Ouyang, and Y . Liu, “Beamforming optimizati on for continuous aperture array (CAPA)-based communications,” IEEE Trans. Wireless Commun., vol. 24, no. 6, pp. 5099–5113, Jun. 2025

2025
[24]

Beamforming design for continuous aperture array (CAPA)-based MIMO systems,

——, “Beamforming design for continuous aperture array (CAPA)-based MIMO systems,” IEEE Transactions on Wireless Communications, Aug. 2025, early access. doi: 10.1109/TWC.2025.3595157

2025
[25]

Deep learni ng for beamforming in multi-user continuous aperture array (CAPA ) systems,

J. Guo, Y . Liu, H. Shin, and A. Nallanathan, “Deep learni ng for beamforming in multi-user continuous aperture array (CAPA ) systems,” arXiv preprint arXiv:2411.09104 , 2024

2024 arXiv
[27]

D. M. Pozar, Microwave Engineering. Hoboken, NJ, USA: Wiley, 2012

2012
[29]

A rrays of isotropic radiators-a ﬁeld-theoretic justiﬁcation,

H. Y ordanov, M. T. Ivrlac, P . Russer, and J. A. Nossek, “A rrays of isotropic radiators-a ﬁeld-theoretic justiﬁcation,” in Proc. ITG/IEEE W orkshop on Smart Antennas , 2009

2009
[30]

On channel capacity of communication via antenna arrays with r eceiver noise matching,

R. R. M¨ uller, B. E. Godana, M. A. Sedaghat, and J. B. Hube r, “On channel capacity of communication via antenna arrays with r eceiver noise matching,” in Proc. IEEE Inf. Theory W orkshop (ITW) , 2012, pp. 396–400

2012
[31]

The extended manifold for antenna arr ays,

B. Friedlander, “The extended manifold for antenna arr ays,” IEEE Trans. Signal Process. , vol. 68, pp. 493–502, Jan. 2020

2020
[32]

W. C. Chew, W aves and ﬁelds in inhomogenous media . John Wiley & Sons, 1999

1999
[33]

F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions . Cambridge, U.K.: Cambridge Univ. Press, 2010

2010
[34]

An introduction to the conjugate gradi ent method without the agonizing pain,

J. R. Shewchuk, “An introduction to the conjugate gradi ent method without the agonizing pain,” Carnegie-Mellon University, Tech. Rep., 1994

1994
[35]

D esign of unit cells and demonstration of methods for synthesizing hu ygens metasurfaces,

J. P . Wong, M. Selvanayagam, and G. V . Eleftheriades, “D esign of unit cells and demonstration of methods for synthesizing hu ygens metasurfaces,” Photonics Nanostructures-Fundam. Appl. , vol. 12, no. 4, pp. 360–375, 2014

2014
[36]

Degrees of freedom in multiple- antenna channels: A signal space approach,

A. Poon, R. Brodersen, and D. Tse, “Degrees of freedom in multiple- antenna channels: A signal space approach,” IEEE Trans. Inf. Theory , vol. 51, no. 2, pp. 523–536, Feb. 2005

2005

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [26]

Mutual coupling in holographic MIMO: Physical modeling and information-theoretic analysis,

A. Pizzo and A. Lozano, “Mutual coupling in holographic MIMO: Physical modeling and information-theoretic analysis,” IEEE J. Sel. Areas Inf. Theory , vol. 6, pp. 111–126, May 2025

2025

[2] [28]

Optimal beamforming for multi- user continuous aperture array (CAPA) systems,

Z. Wang, C. Ouyang, and Y . Liu, “Optimal beamforming for multi- user continuous aperture array (CAPA) systems,” IEEE Trans. Commun., early access, Mar. 2025. doi: 10.1109/TCOMM.2025.3554644

arXiv 2025

[3] [1]

Enabling 6g performance in the upper mid-band by transitioning from massive to gigantic MIMO,

E. Bj¨ ornson, F. Kara, N. Kolomvakis, A. Kosasih, P . Rame zani, and M. B. Salman, “Enabling 6g performance in the upper mid-band by transitioning from massive to gigantic MIMO,” IEEE Open J. Commun. Soc., vol. 6, pp. 5450–5463, 2025

2025

[4] [2]

Ne ar- ﬁeld communications: A tutorial review,

Y . Liu, Z. Wang, J. Xu, C. Ouyang, X. Mu, and R. Schober, “Ne ar- ﬁeld communications: A tutorial review,” IEEE Open J. Commun. Soc. , vol. 4, pp. 1999–2049, Aug. 2023

1999

[5] [3]

A tutorial on extremely large-scal e mimo for 6g: Fundamentals, signal processing, and applications,

Z. Wang, J. Zhang, H. Du, D. Niyato, S. Cui, B. Ai, M. Debbah , K. B. Letaief, and H. V . Poor, “A tutorial on extremely large-scal e mimo for 6g: Fundamentals, signal processing, and applications,” IEEE Commun. Surv. Tut., vol. 26, no. 3, pp. 1560–1605, 3rd Quart. 2024

2024

[6] [4]

CAPA: Continuous-aperture arrays for revolutionizing 6G wirele ss communica- tions,

Y . Liu, C. Ouyang, Z. Wang, J. Xu, X. Mu, and Z. Ding, “CAPA: Continuous-aperture arrays for revolutionizing 6G wirele ss communica- tions,” IEEE Wireless Commun. , vol. 32, no. 4, pp. 38–45, Aug. 2025

2025

[7] [5]

Electrom agnetic information theory: Fundamentals, modeling, application s, and open problems,

J. Zhu, Z. Wan, L. Dai, M. Debbah, and H. V . Poor, “Electrom agnetic information theory: Fundamentals, modeling, application s, and open problems,” IEEE Wireless Communications, vol. 31, no. 3, pp. 156–162, Jun. 2024

2024

[8] [6]

Towards 6G MIMO: Massive spatial multiplexing, dense arra ys, and interplay between electromagnetics and processing,

E. Bj¨ ornson, C.-B. Chae, R. W. Heath Jr, T. L. Marzetta, A . Mezghani, L. Sanguinetti, F. Rusek, M. R. Castellanos, D. Jun, and ¨O. T. Demir, “Towards 6G MIMO: Massive spatial multiplexing, dense arra ys, and interplay between electromagnetics and processing,” arXiv preprint arXiv:2401.02844, 2024

arXiv 2024