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arxiv: 2507.01227 · v4 · submitted 2025-07-01 · 📡 eess.SP · cs.IT· math.IT

Distance-Domain Degrees of Freedom in Near-Field Region

Son T. Duong , Tho Le-Ngoc This is my paper

Pith reviewed 2026-05-19 06:00 UTC · model grok-4.3

classification 📡 eess.SP cs.ITmath.IT
keywords distance-domain degrees of freedomnear-field communicationslarge aperture arraysline-of-sight channelsintegral operatorsFourier transformmodular arrays
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The pith

Distance-domain degrees of freedom in near-field LoS links depend on the outer boundaries of the transmit and receive arrays rather than their internal details.

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

This paper studies the additional spatial resources unlocked when users share the same angle but occupy different distances from an extremely large transmit array in the near-field regime. By modeling the channel as an integral operator with a Hermitian convolution kernel, it derives a closed-form expression for the distance-domain DoF using Fourier analysis on the broadside case. The central result is that this DoF is set almost entirely by the farthest and closest points of the arrays. The framework is then extended through a projection method to angled geometries and to modular arrays, where splitting the transmit aperture yields higher DoF than a single contiguous array of the same total length.

Core claim

For a continuous-aperture transmit array and a linear receive array whose elements lie at varying distances, the distance-domain DoF equals the number of significant eigenvalues of the associated integral operator; these eigenvalues are obtained in closed form from the Fourier transform of the kernel. The resulting DoF value depends only on the extreme radial and lateral boundaries of the two arrays. The same count holds when the receive array is displaced from broadside via an equivalent projected geometry, and modular transmit arrays increase the count relative to a single-piece array under fixed total physical length.

What carries the argument

Reformulation of the line-of-sight channel as an integral operator whose Hermitian convolution kernel admits a closed-form Fourier-transform solution for the distance-domain eigenvalues.

If this is right

  • Users at the same angle but different distances can be spatially multiplexed using distance-domain resources alone.
  • Array design can prioritize overall span over precise interior element arrangement to achieve the distance-domain DoF bound.
  • Modular arrays deliver measurable DoF gains compared with a single contiguous aperture of identical total length.
  • The continuous-aperture result supplies a concrete upper bound that finite-element arrays approach as element density rises.

Where Pith is reading between the lines

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

  • Finite-element arrays with practical spacing should closely approach the derived DoF once element density is high enough to approximate the continuous limit.
  • Beamforming or precoding algorithms could be simplified by depending primarily on the positions of the outermost elements rather than the full array geometry.
  • The boundary-dominance finding may extend to three-dimensional user placements or multi-user scheduling that jointly exploits angle and distance.

Load-bearing premise

Both the transmit and receive arrays are modeled as continuous apertures containing infinitely many elements with infinitesimal spacing.

What would settle it

Build two transmit arrays that share identical outer boundaries but differ in internal element placement or density, then count how many independent distance-domain channels each supports to the same receive array at multiple distances.

Figures

Figures reproduced from arXiv: 2507.01227 by Son T. Duong, Tho Le-Ngoc.

Figure 1
Figure 1. Figure 1: Geometrical setup of the transmit CAP array [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Spectrum of g˜(ξ) for a squared CAP array P with a circular hole with radius = 60λ (Q has rmin = 200λ, rmax = 2000λ, and P has pmin = 60λ, pmax = 100λ) [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Normalized eigenvalues for three different shapes of [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Eigenvalue distribution for different rectangular CAP arrays [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Eigenvalue distribution and DoF versus rmin given fixed length of Q (P is a rectangular array with pmax = 100λ, pmin = 0λ.) rather than increasing its overall area of P. Remark: Although the number of dominant eigenvalues is determined solely by the extreme boundaries of the CAP array P and not on its detailed shape, the specific values of these eigenvalues are influenced by its detailed shape through the … view at source ↗
Figure 7
Figure 7. Figure 7: Eigenvalue distribution of two cases: 1) CAP array [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Eigenvalue distribution for different angle [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: Spectra of the array P and its sub-arrays P(1) and P(2) in the case where the main-lobes of g˜ (1)(ξ) and g˜ (2)(ξ) do not overlap. This result indicates that for sub-arrays with overlapping main lobes of g˜ (n) (ξ) the effective spatial DoF depends only on the extreme distances of the array, which is similar to the case of a single array derived in Section III. B. Spatial DoF Comparison Between Modular A… view at source ↗
Figure 12
Figure 12. Figure 12: Eigenvalue distributions of LoS channel between [PITH_FULL_IMAGE:figures/full_fig_p011_12.png] view at source ↗
Figure 11
Figure 11. Figure 11: Spectra of the array P and its sub-arrays P(1) and P(2) in the case where the main-lobes of g˜ (1)(ξ) and g˜ (2)(ξ) overlap. An important observation is that for a fixed sub-array length L, increasing the absolute positions (i.e., increasing both a and b) expands the effective spectrum bandwidth and hence the spatial DoF. This is demonstrated in [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
Figure 13
Figure 13. Figure 13: Eigenvalue distributions of LoS channel between [PITH_FULL_IMAGE:figures/full_fig_p011_13.png] view at source ↗
read the original abstract

Extremely large aperture arrays operating in the near-field regime unlock additional spatial resources, which can be exploited to simultaneously serve multiple users even when they share the same angular direction. This work investigates the distance-domain degrees of freedom (DoF), defined as the DoF when a user varies only its distance to the base station and not the angle. To obtain the distance-domain DoF, we investigate a line-of-sight (LoS) channel between a base station (source) and observation region representing users. The base station is modeled as a large two-dimensional transmit (Tx) array with an arbitrary shape. The observation region is modeled as an arbitrarily long linear receive (Rx) array, where elements are collinearly aligned but located at varying distances from the Tx array. We assume that both the Tx and Rx arrays have continuous apertures with an infinite number of elements and infinitesimal spacing, which establishes an upper bound for the distance-domain DoF in the case of a finite number of elements. First, we analyze an ideal case where the Tx array is a single piece and the Rx array is on the broadside of the Tx array. By reformulating the channel as an integral operator with a Hermitian convolution kernel, we derive a closed-form expression for the distance-domain DoF via the Fourier transform. Our analysis shows that the distance-domain DoF is predominantly determined by the extreme boundaries of both the Tx and Rx arrays rather than their detailed interior structure. We further extend the framework to non-broadside configurations by employing a projection method that converts the problem to an equivalent broadside case. Finally, we extend the analytical framework to modular arrays and show the distance-domain DoF gain over a single-piece array under a fixed total physical length.

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 / 3 minor

Summary. The paper analyzes distance-domain degrees of freedom (DoF) for LoS near-field channels with extremely large aperture arrays. The BS is modeled as an arbitrary-shaped 2D continuous transmit aperture and users as a linear receive array at varying distances. For the ideal broadside case the channel is recast as an integral operator possessing a Hermitian convolution kernel; Fourier analysis then yields a closed-form expression showing that the effective spectral support (hence DoF) is fixed by the outermost radial extents of the two apertures rather than their interior geometry. The framework is extended to non-broadside geometries via a projection that reduces the problem to an equivalent broadside case and to modular arrays, where a DoF gain relative to a single-piece aperture of equal total length is reported. The continuous-aperture idealization is explicitly presented as an upper bound for finite-element realizations.

Significance. If the boundary-dominated result holds under the stated modeling assumptions, the work supplies a clean analytical handle on an additional spatial resource that can be exploited when users share the same angle but differ in range. The derivation is parameter-free once the aperture extents are fixed, the continuous-aperture limit is correctly labeled as an upper bound, and the modular-array extension offers a concrete, falsifiable prediction of DoF improvement. These features make the contribution potentially useful for both theoretical capacity studies and practical array partitioning in XL-MIMO systems.

major comments (1)
  1. [§3.2] §3.2, after Eq. (12): the projection step that maps a non-broadside geometry onto an equivalent broadside kernel is asserted to preserve the convolution structure required for the Fourier-transform argument, yet the transformed kernel is not explicitly recomputed; without this step the claim that the same closed-form DoF expression applies is not fully load-bearing.
minor comments (3)
  1. [Abstract] Abstract and §2.1: the phrase 'infinitesimal spacing' should be accompanied by a brief statement of the limiting process (e.g., element density → ∞ while total aperture length fixed) to avoid ambiguity for readers unfamiliar with continuous-aperture models.
  2. [Figure 3] Figure 3: the modular-array illustration would benefit from explicit annotation of the outermost radial coordinates that, according to the main result, determine the DoF.
  3. [§2] Notation: the symbol for the radial distance variable is reused in both the Tx and Rx domains; a subscript (e.g., r_T, r_R) would improve readability in the integral-operator definition.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive review and positive recommendation for minor revision. The single major comment concerns the clarity of the projection argument in §3.2, which we address below by committing to an explicit addition in the revised manuscript.

read point-by-point responses

  1. Referee: [§3.2] §3.2, after Eq. (12): the projection step that maps a non-broadside geometry onto an equivalent broadside kernel is asserted to preserve the convolution structure required for the Fourier-transform argument, yet the transformed kernel is not explicitly recomputed; without this step the claim that the same closed-form DoF expression applies is not fully load-bearing.

    Authors: We agree that an explicit recomputation of the transformed kernel would make the argument fully rigorous. The projection maps each Rx position to an equivalent broadside coordinate by replacing the slant-range distance with its projected component along the array axis; the resulting kernel then depends only on the difference of these projected coordinates and remains Hermitian. Consequently the integral operator stays convolutional in the new variable, so the same Fourier-support analysis and closed-form DoF expression apply directly. In the revised §3.2 we will insert the explicit expression for the projected kernel immediately after Eq. (12) and verify that it is a function of the difference variable only. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained via standard Fourier analysis on first-principles operator

full rationale

The paper defines the LoS channel from first principles as an integral operator between continuous Tx and Rx apertures, reformulates it for the broadside case as a Hermitian convolution kernel, and applies the standard Fourier transform property to extract the spectral support (hence distance-domain DoF) directly from the kernel's frequency-domain representation. The conclusion that DoF is fixed by outermost radial boundaries follows immediately from the support of that transform without any fitted parameters, self-referential definitions, or load-bearing self-citations. The projection method for non-broadside geometries and the modular-array extension are presented as algebraic consequences of the same boundary-determined kernel. No step reduces by construction to its own inputs or to an unverified prior result by the same authors.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the continuous-aperture modeling choice and standard properties of integral operators and Fourier transforms; no new physical entities are introduced.

axioms (2)
  • domain assumption LoS channel between Tx array and linear Rx observation region
    Stated as the channel model under investigation.
  • domain assumption Continuous apertures with infinite elements and infinitesimal spacing
    Explicitly invoked to establish an upper bound on finite-element DoF.

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Reference graph

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