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Wireless communications with extremely large-scale array (XL-array)correspond to systems whose antenna sizes are so large that conventionalmodelling assumptions, such as uniform plane wave (UPW) impingement, are longervalid. This paper studies the mathematical modelling and performance analysisof XL-array communications. By deviating from the conventional modellingapproach that treats the array elements as sizeless points, we explicitly modeltheir physical area/aperture, which enables a unified modelling for theclassical discrete antenna arrays and the emerging continuous surfaces. Assuch, a generic array/surface model that accurately takes into account thevariations of signal phase, amplitude and projected aperture across arrayelements is proposed. Based on the proposed model, a closed-form expression ofthe resulting SNR with the optimal single-user MRC/MRT beamforming is derived.The expression reveals that instead of scaling linearly with the antenna numberM as in conventional UPW modelling, the SNR with the more generic modelincreases with M with diminishing return, which is governed by the collectiveproperties of the array, such as the array occupation ratio and the physicalsizes of the array along each dimension, while irrespective of the propertiesof the individual array element. Additionally, we have derived an alternativeinsightful expression for the optimal SNR in terms of the vertical andhorizontal angular spans. Furthermore, we also show that our derived resultsinclude the far-field UPW modelling as a special case. One important findingduring the study of far-field approximation is the necessity to introduce a newdistance criterion to complement the classical Rayleigh distance, termeduniform-power distance (UPD), which concerns the signal amplitude/powervariations across array elements, instead of phase variations as for Rayleighdistance.

Communicating with Extremely Large-Scale Array/Surface: Unified Modelling and Performance Analysis