Impact of Network Geometry on Large Networks with Intelligent Reflecting Surfaces

In wireless networks assisted by intelligent reflecting surfaces (IRSs), jointly modeling the signal received over the direct and indirect (reflected) paths is a difficult problem. In this work, we show that the network geometry (locations of serving base station, IRS, and user) can be captured using the so-called triangle parameter $\Delta$. We introduce a decomposition of the effect of the combined link into a signal amplification factor and an effective channel power coefficient $G$. The amplification factor is monotonically increasing with both the number of IRS elements $N$ and $\Delta$. For $G$, since an exact characterization of the distribution seems unfeasible, we propose three approximations depending on the value of the product $N\Delta$ for Nakagami fading and the special case of Rayleigh fading. For two relevant models of IRS placement, we prove that their performance is identical if $\Delta$ is the same given an $N$. We also show that no gains are achieved from IRS deployment if $N$ and $\Delta$ are both small. We further compute bounds on the diversity gain to quantify the channel hardening effect of IRSs. Hence only with a judicious selection of IRS placement and other network parameters, non-trivial gains can be obtained.