  Wireless communications with extremely large-scale array (XL-array)
correspond to systems whose antenna sizes are so large that conventional
modelling assumptions, such as uniform plane wave (UPW) impingement, are longer
valid. This paper studies the mathematical modelling and performance analysis
of XL-array communications. By deviating from the conventional modelling
approach that treats the array elements as sizeless points, we explicitly model
their physical area/aperture, which enables a unified modelling for the
classical discrete antenna arrays and the emerging continuous surfaces. As
such, a generic array/surface model that accurately takes into account the
variations of signal phase, amplitude and projected aperture across array
elements is proposed. Based on the proposed model, a closed-form expression of
the resulting SNR with the optimal single-user MRC/MRT beamforming is derived.
The expression reveals that instead of scaling linearly with the antenna number
M as in conventional UPW modelling, the SNR with the more generic model
increases with M with diminishing return, which is governed by the collective
properties of the array, such as the array occupation ratio and the physical
sizes of the array along each dimension, while irrespective of the properties
of the individual array element. Additionally, we have derived an alternative
insightful expression for the optimal SNR in terms of the vertical and
horizontal angular spans. Furthermore, we also show that our derived results
include the far-field UPW modelling as a special case. One important finding
during the study of far-field approximation is the necessity to introduce a new
distance criterion to complement the classical Rayleigh distance, termed
uniform-power distance (UPD), which concerns the signal amplitude/power
variations across array elements, instead of phase variations as for Rayleigh
distance.
