Distance-Domain Degrees of Freedom in Near-Field Region

TL;DR

The paper defines distance-domain DoF for near-field LoS channels with continuous Tx and Rx apertures and derives a closed-form bound showing DoF scales with the Tx-edge span and the inverse-distance span: $DoF \approx \frac{(p_{\max}^2 - p_{\min}^2)}{2\lambda} \left( \frac{1}{r_{\min}} - \frac{1}{r_{\max}} \right) + O(1)$. It uncovers that the DoF is governed by array-edge extents rather than interior shapes and extends the analysis to non-broadside configurations via a projection that preserves DoF, as well as to modular Tx arrays where gaps can be advantageous under fixed total length. The findings provide insights into distance-domain multiplexing and distance-domain channel estimation, highlighting finite DoF even as $r_{\max} \to \infty$ and suggesting practical benefits of modular aperture designs for spatial multiplexing in dense user scenarios. Overall, the work offers a rigorous, geometry-driven limit on distance-based spatial resources and actionable guidance for near-field array design and training strategies.

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 study a line-of-sight (LoS) channel with a source representing a base station and an observation region representing users, where the source is a large two-dimensional transmit (Tx) array with arbitrary shape and the observation region is an arbitrarily long linear receive (Rx) array with collinearly aligned elements located at different 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.

Figures (15)

• Figure 1: Geometrical setup of the Tx array $\mathcal{P}$ (with an example of a gap inside $\mathcal{P}$) and the Rx broadside linear array $\mathcal{Q}$.
• Figure 2: Implication of distance-domain DoF for users distributed within the distance span $[r_{\min}, r_{\max}]$
• Figure 3: Spectrum of $\tilde{g}(\xi)$ for a squared Tx array $\mathcal{P}$ with a circular gap (radius = $60\lambda$) (Rx array spans over $[r_{\min}, r_{\max}] = [200\lambda, 2000\lambda]$, and Tx array is bounded by $[{p}_{\min},{p}_{\max}] = [60\lambda,100\lambda]$).
• Figure 4: Normalized eigenvalues for three different shapes of Tx array with the same ${p}_{\max}=100\lambda,{p}_{\min}=60\lambda$.
• Figure 5: Eigenvalue distribution for different rectangular Tx arrays with the same area: CAP array 1 ($50\lambda \times 30\lambda$), CAP array 2 ($100\lambda \times 15\lambda$), CAP array 3 ($150\lambda \times 10\lambda$). (The broadside linear array $\mathcal{Q}$ has $r_{min}=400\lambda$ and $r_{max}=4000\lambda$.)

Theorems & Definitions (3)

• Lemma 1
• Lemma 2
• Theorem 1: Distance-Domain DoF Expression for Broadside Case