  In this paper, we consider the max-min signal-to-interference plus noise
ratio (SINR) problem for the uplink transmission of a cell-free Massive
multiple-input multiple-output (MIMO) system. Assuming that the central
processing unit (CPU) and the users exploit only the knowledge of the channel
statistics, we first derive a closed-form expression for uplink rate. In
particular, we enhance (or maximize) user fairness by solving the max-min
optimization problem for user rate, by power allocation and choice of receiver
coefficients, where the minimum uplink rate of the users is maximized with
available transmit power at the particular user. Based on the derived
closed-form expression for the uplink rate, we formulate the original user
max-min problem to design the optimal receiver coefficients and user power
allocations. However, this max-min SINR problem is not jointly convex in terms
of design variables and therefore we decompose this original problem into two
sub-problems, namely, receiver coefficient design and user power allocation. By
iteratively solving these sub-problems, we develop an iterative algorithm to
obtain the optimal receiver coefficient and user power allocations. In
particular, the receiver coefficients design for a fixed user power allocation
is formulated as generalized eigenvalue problem whereas a geometric programming
(GP) approach is utilized to solve the power allocation problem for a given set
of receiver coefficients. Numerical results confirm a three-fold increase in
system rate over existing schemes in the literature.
