  Intelligent reflecting surface (IRS) is a promising technology for wireless
communications, thanks to its potential capability to engineer the radio
environment. However, in practice, such an envisaged benefit is attainable only
when the passive IRS is of a sufficiently large size, for which the
conventional uniform plane wave (UPW)-based channel model may become
inaccurate. In this paper, we pursue a new channel modelling and performance
analysis for wireless communications with extremely large-scale IRS (XL-IRS).
By taking into account the variations in signal's amplitude and projected
aperture across different reflecting elements, we derive both lower- and
upper-bounds of the received signal-to-noise ratio (SNR) for the general
uniform planar array (UPA)-based XL-IRS. Our results reveal that, instead of
scaling quadratically with the increased number of reflecting elements M as in
the conventional UPW model, the SNR under the more practically applicable
non-UPW model increases with M only with a diminishing return and gets
saturated eventually. To gain more insights, we further study the special case
of uniform linear array (ULA)-based XL-IRS, for which a closed-form SNR
expression in terms of the IRS size and transmitter/receiver location is
derived. This result shows that the SNR mainly depends on the two geometric
angles formed by the transmitter/receiver locations with the IRS, as well as
the boundary points of the IRS. Numerical results validate our analysis and
demonstrate the importance of proper channel modelling for wireless
communications aided by XL-IRS.
