is used to obtain the mutual information in the presence of imperfect CSI. Then, the statistical properties of the MMSE estimator and power-control policy are exploited to compute the average with respect to the channels in the presence of pilot contamination.

The expression in Lemma \[lemma:SINR-ideal\] holds for any $\beta\geq 1$ such that $\beta K \leq S$, since the pilot signals need to be contained in a coherence block. Observe also that $\beta K$ does not need to be an integer since an arbitrary $\beta K$ can be achieved by switching (for appropriate fractions of time) between the closest smaller integer, $\lfloor \beta K \rfloor$, and the closest larger integer, $\lceil \beta K \rceil$.

The average SE per UE can be obtained by taking the expectation of with respect to the PPP $\Psi_{\lambda}$ and the UE locations. This would require heavy numerical evaluations of integrals. As a key contribution of this paper, in the following we provide a tractable and tight lower bound on the average SE.

\[prop:average-SE\] If MRC is employed, a lower bound on the UL average SE \[bit/symbol/user\] is $$\underline {\mathrm{SE}} = \Big( 1 - \frac{\beta K}{S} \Big)\! \log_2 \! \left( 1 + \underline {\mathrm{SINR}} \right)\label{eq:average-SE}$$ where 