Applicable regions of spherical and plane wave models for extremely large-scale array communications

Extremely large-scale array (XL-array) communications can significantly improve the transmission rate, 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\pm\frac{\pi}{6}$</tex> central incident angles. Second, an equi-power surface is derived for a point-to-uniform planar array (UPA) scenario, and it is proved that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">${\rm cos}^{2}(\phi)\,{\rm cos}^{2}({\varphi})$</tex> 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.