In this paper we introduce projection-based precoders for secure transmission in cell-free massive multiple-input multipleoutput (MIMO) systems. Assuming that the channel state information (CSI) of both legitimate users (referred to as Bob) and passive eavesdroppers (Eve) is known, we derive three secure precoder designs using well-known linear precoders, such as maximum ratio transmission (MRT), zero-forcing (ZF) and minimum mean square error (MMSE). The first method projects these linear precoders onto the orthogonal complement space of Eve. The second method processes Bob's channel with the same projection operator before applying linear precoding to this newly formed "secure" channel matrix. Additionally, we propose a novel MMSE-based precoder with added security constraints, designed to reduce energy leaked towards Eve. By reformulating the proposed precoders, we provide several interpretations of the latter, which result in simpler expressions and equivalent projected precoding schemes. Analytical results are supported by simulations in realistic environments, and demonstrate that the proposed methods can completely eliminate Eve's received power. It results in improved secrecy performance compared to their non-secrecy aware counterparts.

This paper deals with the minimum mean square error (MMSE) digital beamforming techniques at both transmitter (precoding) and receiver (combining) sides in multiuser multiple input multiple output (MU-MIMO) systems, under errors of channel state information (CSI). We consider very general system models in both downlink and uplink cases, where the number of transmit and receive antennas can take any values, and the CSI errors are random samples. Based on this model, we analyze the performance of the MMSE beamforming methods considering imperfect CSI in the construction of the beamforming matrices, reflecting possible errors of channel estimation. We theoretically show that the additional inter-user interference power due to the CSI errors is quadratically proportional to the square of a parameter α quantifying the CSI errors. This analytic result is supported by simulations, and we also show the effect of the CSI errors on the achievable bit error rate (BER), considering multipath and free space line-of-sight channel models. In both cases, simulations and theoretical results match, and highlight that the MMSE beamforming methods experience few dB loss in presence of potentially high CSI errors compared to perfect CSI.