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Beamfocusing Capabilities of a Uniform Linear Array in the Holographic Regime

Xavier Mestre, Adrian Agustin

TL;DR

This work develops a formal near-field beamfocusing analysis for Uniform Linear Arrays (ULAs) using a Green dyadic channel model that captures amplitude variations across the aperture. It derives a second-order Taylor expansion of SNR around the intended receiver to define a beamfocusing feasibility region, showing that a minimum total array length of $4.4\lambda$ is required for any focusing; in the holographic regime this region collapses to an ellipsoid whose axes and center are given in closed form in terms of $\varrho=L/D$, elevation $\theta$, and $L/\lambda$. The authors provide a closed-form asymptotic description of the ellipsoid radii $l_k$ and the ellipsoid center, and show the maximum feasible distance as a function of angle. Numerical results demonstrate accurate holographic approximations even for moderate inter-element spacing and highlight the practical relevance for coverage planning.

Abstract

The use of multiantenna technologies in the near field offers the possibility of focusing the energy in spatial regions rather than just in angle. The objective of this paper is to provide a formal framework that allows to establish the region in space where this effect can take place and how efficient this focusing can be, assuming that the transmit architecture is a uniform linear array (ULA). A dyadic Green's channel model is adopted, and the amplitude differences between the receiver and each transmit antenna are effectively incorporated in the model. By considering a second-order expansion of the SNR around the intended receiver, a formal criterion is derived in order to establish whether beamfocusing is feasible or not. An analytic description is provided that determines the shape and position of the asymptotic ellipsoid where a minimum SNR is achieved. Further insights are provided by considering the holographic regime, whereby the number of elements of the ULA increase without bound while the distance between adjacent elements converges to zero. This asymptotic framework allows to simplify the analytical form of the beamfocusing feasibility region, which in turn provides some further insights into the shape of the coverage regions depending on the position of the intended receiver. In particular, it is shown that beamfocusing is only possible if the size of the ULA is at least $4.4λ$ where $λ$ is the transmission wavelength. Furthermore, a closed form analytical expression is provided that asymptotically determines the maximum distance where beamfocusing is feasible as a function of the elevation angle. In particular, beamfocusing is only feasible when the receiver is located between a minimum and a maximum distance from the array, where these upper and lower distance limits effectively depend on the angle of elevation

Beamfocusing Capabilities of a Uniform Linear Array in the Holographic Regime

TL;DR

This work develops a formal near-field beamfocusing analysis for Uniform Linear Arrays (ULAs) using a Green dyadic channel model that captures amplitude variations across the aperture. It derives a second-order Taylor expansion of SNR around the intended receiver to define a beamfocusing feasibility region, showing that a minimum total array length of is required for any focusing; in the holographic regime this region collapses to an ellipsoid whose axes and center are given in closed form in terms of , elevation , and . The authors provide a closed-form asymptotic description of the ellipsoid radii and the ellipsoid center, and show the maximum feasible distance as a function of angle. Numerical results demonstrate accurate holographic approximations even for moderate inter-element spacing and highlight the practical relevance for coverage planning.

Abstract

The use of multiantenna technologies in the near field offers the possibility of focusing the energy in spatial regions rather than just in angle. The objective of this paper is to provide a formal framework that allows to establish the region in space where this effect can take place and how efficient this focusing can be, assuming that the transmit architecture is a uniform linear array (ULA). A dyadic Green's channel model is adopted, and the amplitude differences between the receiver and each transmit antenna are effectively incorporated in the model. By considering a second-order expansion of the SNR around the intended receiver, a formal criterion is derived in order to establish whether beamfocusing is feasible or not. An analytic description is provided that determines the shape and position of the asymptotic ellipsoid where a minimum SNR is achieved. Further insights are provided by considering the holographic regime, whereby the number of elements of the ULA increase without bound while the distance between adjacent elements converges to zero. This asymptotic framework allows to simplify the analytical form of the beamfocusing feasibility region, which in turn provides some further insights into the shape of the coverage regions depending on the position of the intended receiver. In particular, it is shown that beamfocusing is only possible if the size of the ULA is at least where is the transmission wavelength. Furthermore, a closed form analytical expression is provided that asymptotically determines the maximum distance where beamfocusing is feasible as a function of the elevation angle. In particular, beamfocusing is only feasible when the receiver is located between a minimum and a maximum distance from the array, where these upper and lower distance limits effectively depend on the angle of elevation

Paper Structure

This paper contains 13 sections, 4 theorems, 78 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Scenario and Signal Model
  3. The holographic regime
  4. Case $y_{0}=0$ (intended receiver on the broadside)
  5. Approximation for high $D/L$
  6. Numerical Analysis
  7. Conclusions
  8. Proof of Proposition \ref{['prop:Taylor']}
  9. Computation of the gradient $\mathcal{G}_{m}\left( \mathbf{\Delta }_{r}\right)$
  10. Computation of the Hessian $\mathcal{H}_{m}\left( \mathbf{\Delta }_{r}\right)$
  11. Analysis of the reminder $\mathcal{R}_{m}(\mathbf{\bar{r}})$
  12. Proof of Corollary \ref{['cor:TaylorSNR']}
  13. Proof of Proposition \ref{['prop:quadric']}

Key Result

Proposition 1

The channel matrix $\mathbf{H}_{m}\left( \mathbf{r} \right)$ accepts the following Taylor expansion around $\mathbf{r} =\mathbf{r}_{0}$ where $\mathbf{\Delta}_{r}=\left( \mathbf{r}-\mathbf{r}_{0}\right) =\left[ \Delta_{x},\Delta_{y},\Delta_{z}\right] ^{T}$ and where $\mathbf{\bar{r}}$ is in the segment joining $\mathbf{r}$ and $\mathbf{r}_{0}$. The three matrices $\mathcal{G}_{m}\left( \mathb

Figures (10)

  • Figure 1: Scenario configuration. The transmitter is a ULA consisting of $2M+1$ elements, each incorporating 3 orthogonal infinitesimal dipoles. It is assumed that at least the polarization in the $x$-axis (red dipoles) is employed at the transmitter. Here, $D$ is the distance between the receiver and the center of the array, $\theta$ is the elevation angle.
  • Figure 2: Spatial regions where beamfocusing is feasible for different values of the array size relative to the wavelength in the holographic regime ($L/\lambda$).
  • Figure 3: Volume of the asymptotical ellipsoid that characterizes the 3dB loss with respect to the maximum SNR ($\kappa = 0.5$) for different values of $L/\lambda$.
  • Figure 4: Representation of the function on the left hand side of (\ref{['eq:beamfocusBroadside']}) as a function of $\varrho^{-1} = D/L$. The feasibility segment for beamfocusing on the broadside is obtained by selecting the interval where this curve is lower than $L/\lambda$. Additionally, we represent the two lines that determine the Fraunhofer distance by the same procedure (red dash-dotted line) as well as the beamfocusing radius approximation in Bjornson2021primer, consisting of one tenth of the Fraunhofer distance (magenta dashed line).
  • Figure 5: Comparision of the actual feasibility region for beamfocusing and the large $D/L$ approximation for different values of $L/\lambda$.
  • ...and 5 more figures

Theorems & Definitions (9)

  • Remark 1
  • Proposition 1
  • proof
  • Corollary 1
  • proof
  • Proposition 2
  • proof
  • Remark 2
  • Lemma 1