  This paper investigates the performance of downlink simultaneous transmitting
and reflecting reconfigurable intelligent surface (STAR-RIS)-assisted cell-free
(CF) massive multiple-input multiple-output (mMIMO) systems, where user
equipments (UEs) are located on both sides of the RIS.
  We account for correlated Rayleigh fading and multiple antennas per access
point (AP), while the maximum ratio (MR) beamforming is applied for the design
of the active beamforming in terms of instantaneous channel state information
(CSI). Firstly, we rely on an aggregated channel estimation approach that
reduces the overhead required for channel estimation while providing sufficient
information for data processing. We obtain the normalized mean square error
(NMSE) of the channel estimate per AP, and design the passive beamforming (PB)
of the surface based on the long-time statistical CSI. Next, we derive the
received signal in the asymptotic regime of numbers of APs and surface
elements. Then, we obtain a closed-form expression of the downlink achievable
rate for arbitrary numbers of APs and STAR-RIS elements under statistical CSI.
Finally, based on the derived expressions, the numerical results show the
feasibility and the advantages of deploying a STAR-RIS into conventional CF
mMIMO systems. In particular, we theoretically analyze the properties of
STAR-RIS-assisted CF mMIMO systems and reveal explicit insights in terms of the
impact of channel correlation, the number of surface elements, and the pilot
contamination on the achievable rate.
