Theoretical Analysis of Near-Field MIMO Channel Capacity and Mid-Band Experimental Validation

Guangyi Liu; Haiyang Miao; Heng Wang; Jianhua Zhang; Lei Tian; Pan Tang

REVIEW 3 major objections 5 minor 22 references

Theoretical Analysis of Near-Field MIMO Channel Capacity and Mid-Band Experimental Validation

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Near-field UPA MIMO capacity can be written as a closed-form EDoF expression, and 13 GHz measurements confirm it decreases continuously with distance.

desk verdict Useful UPA EDoF expression and a 13 GHz measurement, but the capacity formula at the center is a heuristic without derivation, so the closed-form capacity claim does not hold as stated. read the letter →

arxiv 2506.15972 v1 pith:LHIOCEPH submitted 2025-06-19 eess.SP

Haiyang Miao , Jianhua Zhang , Pan Tang , Heng Wang , Lei Tian , Guangyi Liu This is my paper · ORCID

classification eess.SP

keywords near-fieldMIMOeffectivedegreeoffreedomchannelcapacityuniformplanararraymid-bandmeasurement13GHzclosed-formmodelXL-MIMO

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

This paper sets out to show that the capacity of a uniform planar array (UPA) MIMO link in the radiative near field can be written in closed form using the effective degree of freedom (EDoF) rather than the matrix rank. The central result, Eq. (22), expresses capacity as a ratio of two geometric sums that depend only on array positions, wavelength, and SNR, allowing capacity to be interpreted directly from distance and array size. A 13 GHz indoor measurement campaign with a 128-element transmitter and an 8-element receiver is used to argue that this closed-form model tracks the measured capacity as the link distance grows from 1 to 10 m, with capacity decreasing continuously and more slowly at longer range. The practical point the authors are pressing is that near-field capacity gain is substantial only when large arrays are used at both ends; with a small user-side array, the gain is limited.

What carries the argument

The load-bearing object is the effective degree of freedom $\beta=(\mathrm{tr}(R))^2/\mathrm{tr}(R^2)=|\sum_n R(n,n)|^2/\sum_{n_1,n_2}|R(n_1,n_2)|^2$, a continuous measure of how many independent single-input-single-output channels a MIMO matrix effectively provides. The paper derives closed forms for the numerator and denominator for a rectangular UPA by writing each channel element as a spherical-wave Green's function $h_{m,n}=\exp(-jk_0d_{m,n})/(4\pi d_{m,n})$, by approximating $d_{m,n}\approx x_R+((y_m-y_n)^2+(z_m-z_n)^2)/(2x_R)$, and by factoring the resulting phase differences into the function $f(d)$. The final capacity model, Eq. (22), is the ratio of those closed forms times $\log_2(1+P/(rN_0))$, which converts an eigenvalue problem into a geometric sum and makes the model easier to interpret and process.

What would settle it

Measure the complete 13 GHz channel matrix $H$ at distances from 1 to 10 m in the same indoor setup, compute the true capacity from its singular values with equal power allocation, and compare the result with Eq. (22); a systematic gap outside measurement error at near-field distances would disprove the closed-form claim.

Watch

Extended reading notes

Core claim

The paper's claim is that, for a UPA near-field MIMO channel under line-of-sight spherical-wave propagation, the EDoF $\beta=(\mathrm{tr}(R))^2/\mathrm{tr}(R^2)$ can be evaluated in closed form after a paraxial distance approximation, and that the channel capacity is $C=\beta\log_2(1+P/(rN_0))$. Substituting the closed forms for the trace terms gives Eq. (22), in which the capacity depends on the ratio of two squared sums: one over inverse distances that captures path loss and self-coupling, and one over exponential phase terms that captures the spatial coherence between transmit elements. The authors present Fig. 3(a) as evidence that this expression is basically consistent with measured capacity at 13 GHz over 1 to 10 m, and that capacity decreases continuously with distance.

Load-bearing premise

The whole closed-form capacity rests on replacing the rank $r$ in $C=r\log_2(1+P/(rN_0))$ with the EDoF $\beta$ for a channel whose singular values are unequal and path-loss scaled; if that substitution does not preserve capacity, Eq. (22) is not the capacity of the measured channel.

Editorial extensions

If this is right

If Eq. (22) is right, near-field UPA capacity is a deterministic function of array geometry, distance, and SNR, so coverage planning can be done without simulating eigenvalues.
Capacity will decrease continuously and monotonically with distance in the near field for fixed arrays, with the reduction rate leveling off at longer ranges.
Large-scale arrays at both ends are required to realize near-field capacity gain; an 8-element user array at 13 GHz leaves the gain small, which matches deployed user-device constraints.
The upper limit of near-field capacity is tied to the minimum of the transmit and receive antenna counts, so increasing only the base-station aperture has diminishing returns.

Reading between the lines

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

The paper leaves implicit that the same EDoF-based reasoning could be applied to other array geometries and frequency bands, because the closed form only requires recomputing the two geometric sums $q(d)$ and $f(d)$.
A testable extension the authors do not pursue is comparing Eq. (22) with capacity computed from full singular values at distances inside and beyond the near-field boundary; the model should fail gracefully where the paraxial approximation in Eq. (7) breaks down.
If the rank-substitution step is valid, the EDoF-to-capacity map generalizes the usual rank-based formula to any channel with unequal singular values, not only near-field UPA channels, which would imply a broader closed-form capacity principle.

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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

Quick take: the paper gives a specific closed-form EDoF for UPA near-field channels under a Fresnel approximation, and contributes a 13 GHz measurement with a 128-element Tx and 8-element Rx. That part is real and potentially useful. But the advertised closed-form capacity, Eq. (22), depends on Eq. (21), which simply replaces rank r with EDoF beta in the standard log formula. No derivation, citation, or error bound is given for that substitution, and it is not generally valid. For a channel with two singular values squared (1.5, 0.5), Eq. (21) gives about 0.94 bits/s/Hz while the actual equal-power capacity is 1.13 bits/s/Hz. This isn't a minor technicality; it's the load-bearing step of the paper.

What's new and good: the EDoF derivation itself is systematic. The paper traces the Green's function channel through the correlation matrix, works out the UPA index sums, and ends with an explicit expression in terms of array dimensions, element spacing, and distance. That is a legitimate extension of the EDoF framework in [14,20], and it may be handy for quick array sizing. The 13 GHz measurement is also a useful data point; mid-band near-field measurements are scarce, and the trend they report (capacity falls with distance, gain is capped by the smaller array) is plausible and consistent with physical intuition.

Soft spots, in order of importance. First, Eq. (21) is asserted, not derived. The definition of r is missing; presumably r = rank(H), but then the SNR term P/(rN0) in Eq. (21) should arguably be P/(beta N0) if beta is the equivalent number of SISO channels. Keeping r in the denominator while replacing the multiplier with beta makes the formula internally inconsistent. Second, the verification in Fig. 3(a) is qualitative: no error bars, no independent capacity benchmark, and the model and measurement are generated from the same EDoF framework, so agreement is partly tautological. Third, the paraxial approximation in Eq. (7) is borderline at 1 m with a 0.7 m aperture, though the error is probably a few percent and not the main issue.

Who is this for: readers working on near-field MIMO capacity approximations or mid-band channel measurements. The EDoF expression and the measurement deserve attention, but the capacity formula needs to be either derived properly, replaced by a known valid approximation with explicit conditions, or presented as a heuristic with numerical verification.

Recommendation: send it to peer review, but the referee should demand a rigorous basis for Eq. (21), a definition of r, and quantitative agreement metrics with error bars. As is, the closed-form capacity is not established.

Referee Report

3 major / 5 minor

Summary. The manuscript develops a near-field MIMO capacity analysis for uniform planar arrays operating at 13 GHz. It models the line-of-sight channel by scalar Green's functions, derives a closed-form effective degree of freedom (EDoF) for a UPA under a Fresnel approximation, and introduces a capacity expression C = β log2(1 + P/(r N0)). The paper then reports an indoor 13 GHz measurement campaign with a 128-element Tx and an 8-element Rx, and compares the closed-form capacity with the measured values as a function of distance. The main claims are that near-field capacity decreases with distance, that the closed-form model is basically consistent with the measurements, and that near-field capacity gain is significant only when both ends use large arrays.

Significance. The 13 GHz measurement data and the attempt to obtain a closed-form EDoF are useful, and the paper deserves credit for not fitting constants to the experiment. If the EDoF expression were rigorously established, it would provide a compact design formula for near-field UPA analysis. However, the central capacity step in Eq. (21) is not derived and is not generally correct, and the experimental comparison is only qualitative. The paper's broad qualitative conclusion that capacity decreases with distance and that gain is limited with small receive arrays is plausible, but the quantitative closed-form capacity claim and its validation are not currently supported.

major comments (3)

[II-C, Eq. (21)] The substitution of the EDoF β for the rank r in the Shannon formula is not derived and is not generally valid. For a channel with singular values σ_i, the equal-power capacity is C = Σ_i log2(1 + (P/(r N0)) σ_i^2), while β = (Σ_i σ_i^2)^2 / Σ_i σ_i^4. These two expressions coincide only when all nonzero σ_i^2 are equal. For σ^2 = (1.5, 0.5) and P/N0 = 1, Eq. (21) with r = 2 gives 0.94 bit/s/Hz, whereas the actual equal-power capacity is 1.13 bit/s/Hz. The rank r is also left undefined for the measured near-field channel; for a LOS UPA channel the rank can be as large as min(M,N) while β is much smaller. Since Eq. (22) is simply Eq. (18) multiplied by log2(1 + P/(r N0)), the closed-form capacity claim is not established. The authors should derive the capacity from the singular-value decomposition of H, or provide a rigorous error bound for the β-based interpolation, or explicitly restrict the paper's contribution to EDoF analysis.
[II-A/B, Eqs. (1), (7), (16)] The closed-form EDoF derivation rests on approximations that are not quantified and on a coordinate indexing that appears inconsistent. In Eq. (1), the Tx coordinates are (0,(i(n)-Nh)Δl,...), so for Nh=64 the Tx array center is at y = -32.5Δl rather than at the origin implied by setting yR=zR=0; thus xR is not the distance between array centers as claimed in Fig. 3(a). In addition, Eq. (15) retains the exact distance q(d) in the numerator while Eq. (16) replaces dm,n1 dm,n2 by xR^2 and uses only the linear phase term f(d); at xR=1 m and 13 GHz the amplitude error for edge elements of the 64×2 array can be tens of percent. The authors should correct the centering of the array coordinates, quantify the paraxial approximation error over the measured distance range, and show that Eq. (18) remains accurate, or restrict the claimed validity of the closed-form expression.
[III-A, Fig. 3(a)] The experimental validation is only qualitative and does not independently test the EDoF model. No error bars or repeated-measurement statistics are shown, and the text does not state how the measured channel matrix H and the capacity values are computed from the PN-sequence/TDM measurements, including phase calibration and noise subtraction. Because the theoretical capacity curve is built on the same EDoF interpolation whose validity is in question, the agreement in Fig. 3(a) does not verify the closed-form capacity model. The authors should compare the closed-form EDoF directly with the EDoF computed from the measured H, and report a quantitative error metric for the capacity comparison.

minor comments (5)

[II, before Eq. (1)] The phrase 'mode operation' should be 'modulo operation'.
[Figs. 3(a) and 4(a)] The label 'near-field range' is used without a definition; the authors should state the Rayleigh distance or another near-field boundary for the 13 GHz array parameters.
[Fig. 4(a)] The annotation 'More than 6 times' should identify which two curves are being compared.
[Table I and Fig. 3(b)] Table I reports SNR = 70 dB for the measurement, but Fig. 3(b) is labeled SNR = 30 dB; the text should clarify which setting applies to each result.
[Abstract and Section II-C] The abstract and introduction state that closed-form capacity expressions are 'derived in detail', but Section II-C contains no derivation for the key step in Eq. (21); the wording should be aligned with the actual content.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the closed-form capacity model is derived from the channel Green's function and validated against measurements without fitted parameters.

full rationale

The central derivation is self-contained: Eq. (18) for the EDoF is obtained from the Green's-function channel model H_{m,n}=exp(-jk0 d_{m,n})/(4π d_{m,n}) through Eqs. (8)–(17), with only the explicitly stated paraxial approximation in Eq. (7) and the denominator approximation in Eq. (16). Eq. (22) is then the product of Eq. (18) with log2(1+P/(rN0)), and no constant is fitted to the measured capacity. The 13 GHz validation uses independently fixed system parameters from the measurement campaign (frequency, array dimensions, element spacing, distance, and SNR set to 70 dB), so the theoretical capacity is a genuine prediction rather than a fitted reproduction. The self-citations in the paper ([6], [7], [15], [17], [21], [22]) are contextual (surveys, mid-band motivation, measurement-platform references) and do not supply the load-bearing EDoF or capacity expressions, which are attributed to external prior work [14], [20]. A separate correctness concern is that Eq. (21) replaces the rank r by the EDoF beta inside the log-capacity expression with no derivation or error bound; for unequal singular values this substitution is not generally the equal-power Shannon capacity. That is an unproven modeling assumption, however, not a circular step, because beta is computed from the channel matrix independently of the claimed capacity. No circularity by construction, fitted-input renaming, or self-citation loop is present.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The model rests on four unproven or weakly checked premises: a LOS free-space channel, a Fresnel distance approximation, a capacity formula that replaces rank by EDoF, and a denominator simplification. No constants are fitted to the measurement, so the circularity burden is modest, but the capacity formula itself is imported rather than derived.

free parameters (1)

rank r in the capacity formula = not specified
Appears in Eqs. (20)-(22) as the number of active modes in log2(1 + P/(r N0)), but the paper never states its value for the UPA; it only notes 1 ≤ β ≤ r. The closed-form model therefore has an unspecified parameter.

assumptions (4)

domain assumption The channel is LOS free-space propagation described by the scalar Green's function h_{m,n} = exp(-jk0 d_{m,n})/(4π d_{m,n}) in Eqs. (3)-(5).
The paper models the indoor 13 GHz channel as pure LOS spherical waves with no reflections, polarization, or antenna patterns. This is invoked in Section II-A and used for all theoretical and numerical capacity values.
domain assumption The paraxial distance approximation sqrt(1+x) ≈ 1 + x/2 in Eq. (7), giving d ≈ x_R + (offset^2)/(2x_R).
Used to turn the distance into a quadratic form. It requires the Fresnel condition with negligible higher-order terms, which is not checked; at 1 m range with a roughly 0.72 m horizontal transmit aperture, higher-order terms are likely non-negligible.
ad hoc to paper The capacity formula C = β log2(1 + P/(r N0)) in Eq. (21) is a valid approximation of MIMO capacity for the near-field UPA channel.
The paper introduces this formula by analogy with a unit-rank channel and cites the EDoF concept from [14,20], but does not derive it from the channel's singular value distribution. It is the bridge between EDoF and the claimed capacity model.
domain assumption The denominator simplification d_{m,n1} d_{m,n2} ≈ x_R^2 and the drop of m-independent quadratic phases in Eq. (16) are justified.
The paper says the transmit array size is smaller than the distance d, but does not quantify the error; close to the array this is not obviously satisfied.

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Cite this review

Pith. "Pith review of Theoretical Analysis of Near-Field MIMO Channel Capacity and Mid-Band Experimental Validation." pith.science (2026). https://pith.science/paper/LHIOCEPH

@misc{pith2026250615972,
  author       = {Pith},
  title        = {Pith review of: Theoretical Analysis of Near-Field MIMO Channel Capacity and Mid-Band Experimental Validation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LHIOCEPH}},
  note         = {Machine review of arXiv:2506.15972}
}

read the original abstract

With the increase of multiple-input-multiple-output (MIMO) array size and carrier frequency, near-field MIMO communications will become crucial in 6G wireless networks. Due to the increase of MIMO near-field range, the research of near-field MIMO capacity has aroused wide interest. In this paper, we focus on the theoretical analysis and empirical study of near-field MIMO capacity. First, the near-field channel model is characterized from the electromagnetic information perspective. Second, with the uniform planar array (UPA), the channel capacity based on effective degree of freedom (EDoF) is analyzed theoretically, and the closed-form analytical expressions are derived in detail. Finally, based on the numerical verification of near-field channel measurement experiment at 13 GHz band, we reveal that the channel capacity of UPA-type MIMO systems decreases continuously with the communication distance increasing. It can be observed that the near-field channel capacity gain is relatively obvious when large-scale MIMO is adopted at both receiving and transmitter ends, but the near-field channel capacity gain may be limited in the actual communication system with the small antenna array at receiving end. This work will give some reference to the near-field communication systems.

Figures

Figures reproduced from arXiv: 2506.15972 by the authors.

**Figure 2.** Mid-band near-field MIMO channel measurement experiment. (a) [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Measurement results and verification of channel capacity. (a) Versus [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Numerical results of channel capacity. (a) Versus the communication [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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3-D MIMO: How much does it meet our expectations observed from channel measurements?,

J. Zhang, Y . Zhang, Y . Yu et al., “3-D MIMO: How much does it meet our expectations observed from channel measurements?,” IEEE Journal on Selected Areas in Communications. , vol. 35, no. 8, pp. 1887-1903, Aug. 2017

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Y . Yu, J. Zhang, P. J. Smith et al., “Theoretical analysis of 3-D channel spatial correlation and capacity,” IEEE Communications Letters. , vol. 22, no. 2, pp. 420-423, Feb. 2018

work page 2018

[3] [3]

A vision of 6G wireless systems: applications, trends, technologies, and open research problems,

W. Saad, M. Bennis, and M. Chen, “A vision of 6G wireless systems: applications, trends, technologies, and open research problems,” IEEE Network., vol. 34, no. 3, pp. 134-142, Jun. 2020

work page 2020

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ITU, “Framework and overall objectives of the future development of IMT for 2030 and beyond,” in Recommendation ITU-R M.2160. , 2023

work page 2023

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Y . Zhao, L. Dai, J. Zhang et al., “Near-field communications: char- acteristics, technologies, and engineering,” Frontiers of Information Technology & Electronic Engineering., vol. 25, no. 12, pp. 1580-1626, Feb. 2025

work page 2025

[6] [6]

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J. Zhang, J. Lin, P. Tang et al., “Channel measurement, modeling, and simulation for 6G: a survey and tutorial,” arXiv preprint arXiv: 2305.16616, 2023

arXiv 2023

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P. Tang, J. Zhang, H. Miao et al., “XL-MIMO channel measurement, characterization, and modeling for 6G: a survey,” Frontiers of Informa- tion Technology & Electronic Engineering. , vol. 25, no. 12, pp. 1627- 1650, 2024

work page 2024

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M. Cui, Z. Wu, Y . Lu et al., “Near-field MIMO communications for 6G: fundamentals, challenges, potentials, and future directions,” IEEE Communications Magazine., vol. 61, no. 1, pp. 40-46, Jan. 2023

work page 2023