Downlink Spectral Efficiency of Distributed Massive MIMO Systems with Linear Beamforming under Pilot Contamination

In this paper, the downlink spectral efficiency of multi-cell multi-user distributed massive MIMO systems with linear beamforming is studied in the presence of pilot contamination. According to the levels of effective channel gain information at user side, we provide the lower bound and upper bound on user ergodic achievable downlink rate. Due to the different access distance from each user to different remote antenna units, the entries of user channel vectors are no longer identically distributed in distributed massive MIMO systems, which makes the spectral efficiency analysis challenging. Using the properties of Gamma distributions together with the approximate methods for non-isotropic vectors, we derive tractable but accurate closed-form expressions for the rate bounds with maximum ratio transmission (MRT) and zero-forcing (ZF) beamforming in distributed massive MIMO systems. Based on these expressions, user ultimate achievable rates are also given when the ratio of the total number of transmit antennas to the number of users goes to infinity. It is shown that MRT and ZF beamforming achieve the same ultimate rate no matter what levels of effective channel gain information at user side. Numerical results show that ZF achieves better performance gain and faster convergence speed than MRT. When the coherence interval is large, the downlink beamforming training scheme is more preferable for the distributed massive MIMO systems.