  In this paper we consider a time-division duplex cell-free massive
multiple-input multiple-output (MIMO) system where many distributed access
points (APs) simultaneously serve many users. A normalized conjugate
beamforming scheme, which satisfies short-term average power constraints at the
APs, is proposed and analyzed taking into account the effect of imperfect
channel information. We derive an approximate closed-form expression for the
per-user achievable downlink rate of this scheme. We also provide, analytically
and numerically, a performance comparison between the normalized conjugate
beamforming and the conventional conjugate beamforming scheme in [1] (which
satisfies long-term average power constraints). Normalized conjugate
beamforming scheme reduces the beamforming uncertainty gain, which comes from
the users' lack of the channel state information knowledge, and hence, it
improves the achievable downlink rate compared to the conventional conjugate
beamforming scheme.
