[Paper Review] Applicable Regions of Spherical and Plane Wave Models for Extremely Large-Scale Array Communications

Renwang Li, Shu Sun+1 more|arXiv (Cornell University)|Jan 15, 2023

Antenna Design and Analysis4 citations

TL;DR

This paper proposes closed-form demarcation boundaries for the applicable regions of plane wave (PWM) and spherical wave (SWM) models in extremely large-scale array (XL-array) communications, based on channel gain and rank. It derives equi-power and equi-rank surfaces for various array configurations—point-to-ULA, point-to-UPA, ULA-to-ULA, and ULA-to-UPA—enabling accurate estimation of the transition between near-field (SWM) and far-field (PWM) regimes, with key results showing that the classical Rayleigh distance is insufficient for XL-arrays.

ABSTRACT

Extremely large-scale array (XL-array) communications can significantly improve the spectral efficiency and spatial resolution, and has great potential in next-generation mobile communication networks. A crucial problem in XL-array communications is to determine the boundary of applicable regions of the plane wave model (PWM) and spherical wave model (SWM). In this paper, we propose new PWM/SWM demarcations for XL-arrays from the viewpoint of channel gain and rank. Four sets of results are derived for four different array setups. First, an equi-power line is derived for a point-to-uniform linear array (ULA) scenario, where an inflection point is found at $\pm \fracπ{6}$ central incident angles. Second, an equi-power surface is derived for a point-to-uniform planar array (UPA) scenario, and it is proved that $\cos^2(ϕ) \cos^2(φ)=\frac{1}{2}$ is a dividing curve, where $ϕ$ and $φ$ denote the elevation and azimuth angles, respectively. Third, an accurate and explicit expression of the equi-rank surface is obtained for a ULA-to-ULA scenario. Finally, an approximated expression of the equi-rank surface is obtained for a ULA-to-UPA scenario. With the obtained closed-form expressions, the equi-rank surface for any antenna structure and any angle can be well estimated. Furthermore, the effect of scatterers is also investigated, from which some insights are drawn.

Motivation & Objective

• To address the lack of accurate demarcation between near-field (spherical wave model) and far-field (plane wave model) regions in extremely large-scale array (XL-array) communications.
• To overcome the limitations of the classical Rayleigh distance, which is based on phase difference and may not reflect practical performance metrics like channel gain and rank.
• To derive closed-form expressions for equi-power and equi-rank surfaces that define the boundary where PWM and SWM models are equally applicable.
• To investigate the impact of scatterers on model applicability and provide insights into practical system design.

Proposed method

• Derives an equi-power line for point-to-uniform linear array (ULA) by analyzing channel gain, identifying an inflection point at ±π/6 central incident angles.
• Establishes an equi-power surface for point-to-uniform planar array (UPA), proving that cos²(φ)cos²(θ) = 1/2 defines the dividing curve in azimuth and elevation angles.
• Develops an exact, explicit expression for the equi-rank surface in ULA-to-ULA configurations using eigenvalue analysis of the channel matrix.
• Derives an approximated expression for the equi-rank surface in ULA-to-UPA scenarios, enabling generalization to arbitrary array structures.
• Uses analytical methods including partial derivatives and root-finding techniques to prove monotonicity and behavior of the normalized channel gain ratio μ(r, φ, θ).
• Validates results through mathematical proofs involving higher-order derivatives and discriminant analysis of polynomial expressions in distance and angle parameters.

Experimental results

Research questions

• RQ1 What is the accurate boundary between the applicable regions of the plane wave model and spherical wave model in XL-array systems, based on channel gain?
• RQ2 How does the equi-power surface vary with elevation and azimuth angles in a point-to-UPA configuration?
• RQ3 What is the exact closed-form expression for the equi-rank surface in a ULA-to-ULA communication setup?
• RQ4 How can the equi-rank surface be approximated for a ULA-to-UPA configuration to enable generalization across array types?
• RQ5 How do scatterers affect the transition between PWM and SWM applicability, and what insights can be drawn from their influence?

Key findings

• For a point-to-ULA setup, the equi-power line exhibits an inflection point at central incident angles of ±π/6, indicating a critical transition in channel gain balance between models.
• In a point-to-UPA configuration, the equi-power surface is defined by the curve cos²(φ)cos²(θ) = 1/2, where φ and θ are elevation and azimuth angles, respectively.
• An exact, closed-form expression for the equi-rank surface is derived for ULA-to-ULA scenarios, enabling precise determination of the rank-equivalent boundary.
• An approximated expression for the equi-rank surface is obtained for ULA-to-UPA configurations, allowing estimation of the boundary for arbitrary antenna structures and angles.
• The normalized channel gain ratio μ(r, φ, θ) is always less than one and increases with distance, but for certain angle conditions (β < 1/2), it first increases then decreases, peaking at a finite distance.
• The analysis reveals that the classical Rayleigh distance is insufficient for XL-arrays, as it does not account for performance metrics like channel rank and gain, and thus fails to capture the true transition region.