  This letter studies a new array architecture, termed as modular extremely
large-scale array (XL-array), for which a large number of array elements are
arranged in a modular manner. Each module consists of a moderate number of
array elements and the modules are regularly arranged with the inter-module
space typically much larger than signal wavelength to cater to the actual
mounting structure. We study the mathematical modelling and conduct the
performance analysis for modular XL-array communications, by considering the
non-uniform spherical wave (NUSW) characteristic that is more suitable than the
conventional uniform plane wave (UPW) assumption for physically large arrays. A
closed-form expression is derived for the maximum signal-to-noise ratio (SNR)
in terms of the geometries of the modular XL-array, including the total array
size and module separation, as well as the user's location. The asymptotic SNR
scaling law is revealed as the size of modular array goes to infinity.
Furthermore, we show that the developed modelling and performance analysis
include the existing results for collocated XL-array or far-field UPW
assumption as special cases. Numerical results demonstrate the importance of
near-field modelling for modular XL-array communications since it leads to
significantly different results from the conventional far-field UPW modelling.
