  Cell-free Massive MIMO (multiple-input multiple-output) refers to a
distributed Massive MIMO system where all the access points (APs) cooperate to
coherently serve all the user equipments (UEs), suppress inter-cell
interference and mitigate the multiuser interference. Recent works demonstrated
that, unlike co-located Massive MIMO, the \textit{channel hardening} is, in
general, less pronounced in cell-free Massive MIMO, thus there is much to
benefit from estimating the downlink channel. In this study, we investigate the
gain introduced by the downlink beamforming training, extending the previously
proposed analysis to non-orthogonal uplink and downlink pilots. Assuming
single-antenna APs, conjugate beamforming and independent Rayleigh fading
channel, we derive a closed-form expression for the per-user achievable
downlink rate that addresses channel estimation errors and pilot contamination
both at the AP and UE side. The performance evaluation includes max-min
fairness power control, greedy pilot assignment methods, and a comparison
between achievable rates obtained from different capacity-bounding techniques.
Numerical results show that downlink beamforming training, although increases
pilot overhead and introduces additional pilot contamination, improves
significantly the achievable downlink rate. Even for large number of APs, it is
not fully efficient for the UE relying on the statistical channel state
information for data decoding.
