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Available Degrees of Spatial Multiplexing of a Uniform Linear Array with Multiple Polarizations: A Holographic Perspective

Xavier Mestre, Adrian Agustin, David Sarda

TL;DR

This work addresses the problem of quantifying spatial multiplexing capabilities in near-field holographic MIMO using a uniform linear array with three orthogonal dipoles per element. It adopts a holographic regime where $M\to\infty$, $\Delta_T\to0$ with fixed aperture $2L$, and employs full CSIT to derive a limiting eigenstructure and SNR thresholds that determine how many spatial streams can be activated. For both fully polarized ($t_{pol}=r_{pol}=3$) and double-polarized ($t_{pol}=2$, $r_{pol}=3$) transmitters, the authors obtain closed-form eigenvalues $\gamma_i$ of the limiting channel matrix and associated activation thresholds $\overline{\mathsf{SNR}}^{(1)}$, $\overline{\mathsf{SNR}}^{(2)}$, which define regions supporting 1–3 streams. They further provide simplified large-$D/L$ expressions to characterize the support regions in geometry ($D$, $\theta$) and demonstrate that three polarizations guarantee at least two streams everywhere under reasonable transmit power, while two polarizations yield a more limited coverage. The results offer practical insights for designing holographic surfaces and near-field MIMO systems by linking geometry, power, and multiplexing feasibility in closed-form terms.

Abstract

The capabilities of multi-antenna technology have recently been significantly enhanced by the proliferation of extra large array architectures. The high dimensionality of these systems implies that communications take place in the nearfield regime, which poses some questions as to their effective perfomrance even under simple line of sight configurations. In order to study these limitations, a uniform linear array (ULA) is considered here, the elements of which are three infinitesimal dipoles transmitting different signals in the three spatial dimensions. The receiver consists of a single element with three orthogonal infinitesimal dipoles and full channel state information is assumed to be available at both ends. A capacity analysis is presented when the number of elements of the ULA increases without bound while the interelement distance converges to zero, so that the total aperture length is kept asymptotically fixed. In particular, the total number of available spatial eigenmodes is shown to depend crucially on the receiver position in space, and closed form expressions are provided for the different achievability regions. From the analysis it can be concluded that the use of three orthogonal polarizations at the transmitter guarantees the almost universal availability of two spatial streams, whereas the use of only two polarizations results in a more extensive region where maximum multiplexing gain is available.

Available Degrees of Spatial Multiplexing of a Uniform Linear Array with Multiple Polarizations: A Holographic Perspective

TL;DR

This work addresses the problem of quantifying spatial multiplexing capabilities in near-field holographic MIMO using a uniform linear array with three orthogonal dipoles per element. It adopts a holographic regime where , with fixed aperture , and employs full CSIT to derive a limiting eigenstructure and SNR thresholds that determine how many spatial streams can be activated. For both fully polarized () and double-polarized (, ) transmitters, the authors obtain closed-form eigenvalues of the limiting channel matrix and associated activation thresholds , , which define regions supporting 1–3 streams. They further provide simplified large- expressions to characterize the support regions in geometry (, ) and demonstrate that three polarizations guarantee at least two streams everywhere under reasonable transmit power, while two polarizations yield a more limited coverage. The results offer practical insights for designing holographic surfaces and near-field MIMO systems by linking geometry, power, and multiplexing feasibility in closed-form terms.

Abstract

The capabilities of multi-antenna technology have recently been significantly enhanced by the proliferation of extra large array architectures. The high dimensionality of these systems implies that communications take place in the nearfield regime, which poses some questions as to their effective perfomrance even under simple line of sight configurations. In order to study these limitations, a uniform linear array (ULA) is considered here, the elements of which are three infinitesimal dipoles transmitting different signals in the three spatial dimensions. The receiver consists of a single element with three orthogonal infinitesimal dipoles and full channel state information is assumed to be available at both ends. A capacity analysis is presented when the number of elements of the ULA increases without bound while the interelement distance converges to zero, so that the total aperture length is kept asymptotically fixed. In particular, the total number of available spatial eigenmodes is shown to depend crucially on the receiver position in space, and closed form expressions are provided for the different achievability regions. From the analysis it can be concluded that the use of three orthogonal polarizations at the transmitter guarantees the almost universal availability of two spatial streams, whereas the use of only two polarizations results in a more extensive region where maximum multiplexing gain is available.

Paper Structure

This paper contains 10 sections, 4 theorems, 43 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Scenario and Signal Model
  3. Holographic regime
  4. Fully polarized transceiver ($t_\mathrm{pol}=r_\mathrm{pol}=3$)
  5. Double polarized transmitter ($t_\mathrm{pol}=2, r_\mathrm{pol}=3$)
  6. Support regions of spatial multiplexing
  7. Simplified expressions for the multiplexing boundaries
  8. Geometric representation and discussion
  9. Conclusions
  10. Sketch of the Proof of Lemmas \ref{['lem:highD3x3']} and \ref{['lem:highD2x3']}

Key Result

Lemma 3.1

The three eigenvalues of $\overline{\mathcal{W}}^{3\times3}$ are given by $\gamma_1 = \psi_2$, $\gamma_2 = (\psi_2+\Delta^{-1})/2$ and $\gamma_3 = (\psi_2-\Delta^{-1})/2$, where Furthermore we always have $\gamma_1>\gamma_2>\gamma_3>0$.

Figures (6)

  • Figure 1: Scenario configuration. The transmitter is a ULA consisting of $2M+1$ elements, each incorporating 3 orthogonal infinitesimal dipoles. The receiver is assumed to lie on the $yz$-plane. Here, $D$ is the distance between the receiver and the center of the array, $\theta$ is the elevation angle and $\Gamma$ is the angle of view of the ULA from the receiver.
  • Figure 2: Representation of the right hand side of (\ref{['eq:normalization']}) as a function of $D/L$ for fixed elevations $\theta$ when $t_\mathrm{pol}=r_\mathrm{pol}=3$.
  • Figure 3: Representation of the right hand side of (\ref{['eq:normalization']}) as a function of $D/L$ for fixed elevations $\theta$ when $t_\mathrm{pol}=2, r_\mathrm{pol}=3$.
  • Figure 4: Relative error between the true distance activation boundaries and the holographic ones (denoted as $D_{th}^{(i)}$ for $i=1,2$) as a function of $M$ for fixed $L=\Delta_T M =1$ and some selected values of $\theta$ and $\overline{\mathsf{SNR}}_0$.
  • Figure 5: Boundary between $n_+=2$ and $n_+=1$ when $t_\mathrm{pol}=2, r_\mathrm{pol}=3$ for different values of $\overline{\mathsf{SNR}}_0$. Solid lines represent the actual asymptotic boundaries and dash-dotted lines represent the high $D/L$ approximations.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Remark 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof