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arxiv: 2603.15311 · v2 · submitted 2026-03-16 · 📡 eess.SP

Near-field Boundary Distance in mmWave and THz Communications with Misaligned Antenna Arrays

Peng Zhang , Vitaly Petrov , Emil Bj\"ornson This is my paper

Pith reviewed 2026-05-15 10:04 UTC · model grok-4.3

classification 📡 eess.SP
keywords near-field boundarymmWaveTHzantenna arraysmisalignmentarray rotationclosed-form expressions
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The pith

Misaligned antenna arrays shift the radiative near-field boundary in mmWave and THz links, with new closed-form distances for rotated and offset setups.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a generalized framework to find the distance at which the near-field region ends when large antenna arrays at transmitter and receiver are rotated or laterally offset from each other. In mmWave and THz bands the near-field zone can extend tens or hundreds of meters, so an accurate boundary matters for deciding when plane-wave beamforming and channel models are safe to use. The authors derive compact closed-form expressions that cover array-to-array and array-to-point geometries for both linear and planar arrays under realistic misalignments. Numerical checks show that even modest rotations or offsets move the boundary by a noticeable fraction compared with the perfectly aligned case. Readers should care because real deployments rarely achieve perfect alignment, especially when nodes move.

Core claim

A generalized mathematical framework is presented to characterize the radiative near-field distance in directional mmWave and THz communication systems under various realistic array rotations and misalignments. With the developed framework, compact closed-form expressions are derived for the near-field boundary distance in a wide range of antenna configurations, including array-to-array and array-to-point setups, considering both linear and planar arrays. Numerical results indicate that the presence of antenna misalignment may significantly adjust the boundaries of the near-field region.

What carries the argument

Generalized mathematical framework that computes the radiative near-field boundary distance under array rotations and misalignments, yielding closed-form expressions for common linear and planar array geometries.

If this is right

  • Link budgets and beamforming codebooks can be adjusted using the corrected boundary distance for each expected misalignment angle and offset.
  • The same closed-form expressions apply to both linear and planar arrays in array-to-array and array-to-point scenarios.
  • Mobile systems must track changing near-field boundaries as relative array orientation varies over time.
  • Channel models used for capacity or coverage prediction need to switch from near-field to far-field formulas at the misalignment-adjusted distance.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The expressions could be embedded in real-time orientation control loops to keep the link inside or outside the near-field region as desired.
  • Extending the framework to full three-dimensional rotations and non-uniform array placements would likely reveal still larger variations in boundary distance.
  • Field trials could directly test the formulas by comparing measured phase progression against the predicted boundary under deliberate misalignment.

Load-bearing premise

Standard radiative near-field models remain valid under array rotations and misalignments without extra propagation effects caused by the misalignment geometry itself.

What would settle it

Measure the wavefront curvature or amplitude variation across the receive array at the predicted boundary distance under controlled misalignment; if the field matches far-field plane-wave behavior within a small tolerance, the boundary formula is supported; otherwise the formula needs revision.

Figures

Figures reproduced from arXiv: 2603.15311 by Emil Bj\"ornson, Peng Zhang, Vitaly Petrov.

Figure 1
Figure 1. Figure 1: Considered ULA-to-ULA scenario (analyzed in Sec. III). Section III-B [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Considered UPA-to-UPA scenario (analyzed in Sec. IV). Section IV-A [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: On-boresight ULA scenario (α = 0, θ ̸= 0, analyzed in Sec. III-B). Later, Sec. III-C extends the analysis to off-boresight ULA displacements (α ̸= 0). Finally, Sec. III-D addresses a specific extreme ULA-to-point setup with off-boresight misalignment. A. General Approach To characterize the near-field region for array-to-array links, we adopt a generalized phase-error-based criterion with a maximum allowab… view at source ↗
Figure 4
Figure 4. Figure 4: Selection region of expressions (14) and (15) under different angle pairs [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Off-boresight ULA-to-point system model (Sec. III-D). [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: On-boresight UPA-to-UPA setup with TxUPA rotation (Sec. IV-A). [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Dominance regions of the two closed-form expressions in the [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Mixed P2L on-boresight scenario [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Off-boresight UPA-to-point setup (analyzed in Sec. IV-E). [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of the near-field distance expressions for P2P and L2L [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: Comprehensive comparison of near-field distance characteristics for different antenna array configurations and size variations. [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗
Figure 14
Figure 14. Figure 14: Near-field distance of P2P scenario in (61) versus UPA-AP vertical [PITH_FULL_IMAGE:figures/full_fig_p014_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Statistical characterization of the near-field distance [PITH_FULL_IMAGE:figures/full_fig_p015_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Near-field distance distribution rF in the (θ ′ , ϕ) parameter space for different off-boresight angles α. Next, we consider the case where θ ′ and α have the same sign (i.e., θ ′α > 0), which implies sgn(sin θ ′ sin α) = 1. In this case, a sufficient condition for r L2L (a) (θ, α) > rL2L (b) (θ, α) is πD1D2 cos θ ′ cos α > λφ min (D1|sin θ ′ |, D2|sin α|). (83) To obtain a closed-form bound, we consider … view at source ↗
read the original abstract

Wireless communications in the millimeter wave (mmWave) and terahertz (THz) spectrum allow harnessing large frequency bands, thus achieving ultra-high data rates. However, the inherently short wavelengths of mmWave and THz signals lead to an extended radiative near-field region, where certain canonical far-field assumptions fail. Most prior works aimed to characterize this radiative near-field region either do not consider antenna arrays on both communicating nodes or, if they do, assume perfect alignment between the arrays. However, such assumptions break down in many realistic deployments, where both sides must employ large-scale mmWave/THz antenna arrays to maintain the desired communication range, while perfect antenna alignment cannot be guaranteed particularly under nodes mobility. In this work, a generalized mathematical framework is presented to characterize the radiative near-field distance in directional mmWave and THz communication systems under various realistic array rotations and misalignments. With the use of the developed framework, compact closed-form expressions are derived for the near-field boundary distance in a wide range of antenna configurations, including array-to-array and array-to-point setups, considering both linear and planar arrays. Our numerical study reveals that the presence of antenna misalignment may significantly adjust the boundaries of the near-field region in mmWave and THz communication systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper develops a generalized mathematical framework to characterize the radiative near-field boundary distance for directional mmWave and THz communications when antenna arrays at both ends are subject to rotations and misalignments. It derives compact closed-form expressions for the boundary in array-to-array and array-to-point configurations using both linear and planar arrays, and presents a numerical study demonstrating that misalignment can substantially alter the extent of the near-field region compared to the perfectly aligned case.

Significance. If the closed-form expressions remain accurate, the work fills a practical gap in near-field modeling for realistic mmWave/THz deployments where perfect alignment is difficult to maintain due to mobility or installation tolerances. The emphasis on compact analytical expressions (rather than purely numerical or simulation-based results) is a strength that could facilitate integration into link-budget tools and system-level analysis. The numerical results provide concrete illustration of the misalignment effect across frequencies and array sizes.

major comments (1)
  1. [Section 3] Section 3: The derivations extend the conventional phase-error criterion via coordinate transformations on the array geometry, but it is not shown that higher-order terms in the exact path-length expression remain negligible once rotations are introduced. For large misalignment angles or THz carrier frequencies, these terms can alter the effective curvature and invalidate the Fresnel-region approximation underlying the reported closed-form boundary. A direct comparison against the exact spherical-wave distance formula (or full-wave validation) for representative extreme-angle cases is needed to confirm the expressions remain load-bearing.
minor comments (2)
  1. The abstract and introduction should explicitly list the key assumptions retained from the standard radiative near-field model (e.g., that polarization mismatch and higher-order propagation effects remain negligible).
  2. [Numerical study] In the numerical study, include a sensitivity plot or table showing how the boundary distance varies with small perturbations around the reported misalignment angles to illustrate robustness.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive and detailed review of our manuscript. The feedback highlights an important aspect of the approximation validity, which we address below. We will incorporate the suggested validation to strengthen the paper.

read point-by-point responses

  1. Referee: [Section 3] Section 3: The derivations extend the conventional phase-error criterion via coordinate transformations on the array geometry, but it is not shown that higher-order terms in the exact path-length expression remain negligible once rotations are introduced. For large misalignment angles or THz carrier frequencies, these terms can alter the effective curvature and invalidate the Fresnel-region approximation underlying the reported closed-form boundary. A direct comparison against the exact spherical-wave distance formula (or full-wave validation) for representative extreme-angle cases is needed to confirm the expressions remain load-bearing.

    Authors: We agree that confirming the negligibility of higher-order terms under large misalignments and THz frequencies is essential for the robustness of the Fresnel approximation. In the revised manuscript, we will add a direct numerical comparison in Section 4 between the proposed closed-form boundary distances and the exact spherical-wave path-length formula. This will cover representative extreme cases, including misalignment angles up to 45 degrees and carrier frequencies up to 300 GHz (and beyond where relevant). The comparison will either validate the approximation within practical ranges or explicitly delineate its limitations, thereby addressing the concern without altering the core derivations. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained from electromagnetic principles

full rationale

The paper presents a generalized framework deriving compact closed-form expressions for radiative near-field boundaries under array rotations and misalignments by applying geometric coordinate transformations to standard spherical-wave path-length differences and Fresnel-region phase-error criteria. These steps begin from the canonical electromagnetic model of wavefront curvature across the array aperture and do not reduce to fitted parameters, self-referential definitions, or load-bearing self-citations; the resulting boundary distances are direct algebraic consequences of the input geometry and frequency parameters rather than circular restatements of them.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on standard electromagnetic near-field propagation models from prior literature; no new free parameters, ad-hoc axioms, or invented entities are indicated in the abstract.

axioms (1)
  • domain assumption Standard radiative near-field electromagnetic propagation models remain applicable under array misalignment
    Invoked to extend prior aligned-array results to rotated cases

pith-pipeline@v0.9.0 · 5522 in / 1177 out tokens · 45201 ms · 2026-05-15T10:04:53.771933+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. User Mobility Demands Near-Field Communications in Terahertz Band Wireless Networks Beyond 6G

    eess.SP 2026-04 unverdicted novelty 6.0

    Mobile THz wireless links cannot achieve high bandwidths while staying in the far field without unrealistic transmit power, unlike stationary THz or sub-6 GHz/mmWave systems.

Reference graph

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