On proportional fairness of uplink spectral efficiency in cell-free massive MIMO systems

This paper is concerned with the proportional fairness (PF) of the spectral efficiency (SE) maximization of uplinks in a cell-free (CF) massive multiple-input multiple-output (MIMO) system in which a large number of single-antenna access points (APs) connected to a central processing unit (CPU) serve many single-antenna users. To detect the user signals, the APs use matched filters based on the local channel state information while the CPU deploys receiver filters based on knowledge of channel statistics. We devise the maximization problem of the SE PF, which maximizes the sum of the logarithm of the achievable user rates, as a jointly nonconvex optimization problem of receiver filter coefficients and user power allocation subject to user power constraints. To handle the challenges associated with the nonconvexity of the formulated design problem, we develop an iterative algorithm by alternatively finding optimal filter coefficients at the CPU and transmit powers at the users. While the filter coefficient design is formulated as a generalized eigenvalue problem, the power allocation problem is addressed by a gradient projection (GP) approach. Simulation results show that the SE PF maximization not only offers approximately the achievable sum rates as compared to the sum-rate maximization but also provides an improved trade-off between the user rate fairness and the achievable sum rate.