  In this paper, we study the uplink (UL) and downlink (DL) spectral efficiency
(SE) of a cell-free massive multiple-input-multiple-output (MIMO) system with
Rician fading channels. The phase of the line-of-sight (LoS) path is modeled as
a uniformly distributed random variable to take the phase-shifts due to
mobility and phase noise into account. Considering the availability of prior
information at the access points (APs), the phase-aware minimum mean square
error (MMSE), non-aware linear MMSE (LMMSE), and least-square (LS) estimators
are derived. The MMSE estimator requires perfectly estimated phase knowledge
whereas the LMMSE and LS are derived without it.
  In the UL, a two-layer decoding method is investigated in order to mitigate
both coherent and non-coherent interference. Closed-form UL SE expressions with
phase-aware MMSE, LMMSE, and LS estimators are derived for maximum-ratio (MR)
combining in the first layer and optimal large-scale fading decoding (LSFD) in
the second layer. In the DL, two different transmission modes are studied:
coherent and non-coherent. Closed-form DL SE expressions for both transmission
modes with MR precoding are derived for the three estimators. Numerical results
show that the LSFD improves the UL SE performance and coherent transmission
mode performs much better than non-coherent transmission in the DL. Besides,
the performance loss due to the lack of phase information depends on the pilot
length and it is small when the pilot contamination is low.
