$\underline {\mathrm{SINR}} $ is given in at the top of the page.

The proof is given in Appendix B and is based on Jensen’s inequality, which leads to tractable expressions since only moments of the terms in need to be computed.

The tightness of the SE bound in Proposition \[prop:average-SE\] is demonstrated later on by means of simulations (see Fig. \[figureBSdensity\]). The numerator of $\underline{\mathrm{SINR}} $ in scales with $M$ due to the array gain from coherent processing, but the hardware impairments cause the multiplicative loss $(1-\epsilon^2)^2$. The last term in the denominator scales with $M$ and also with the sum of $\frac{1}{\beta (\alpha-1)}$ and $\epsilon^{2}$. The former term is called coherent pilot contamination since it accounts for interference received from UEs that use the same pilot symbol as the typical UE, while the latter is due to the distortion noise emitted from the typical UE itself. Many of the interference terms in increase with $K$ since having more UEs lead to more transmit power per cell. Note that is independent of the BS density $\lambda$, due to the power-control policy in . The average transmit power per UE in is, however, proportional to $\lambda^{-\alpha/2}$ so that less power is 