Achievable Rate Analysis and Max-Min SINR Optimization in Intelligent Reflecting Surface Assisted Cell-Free MIMO Uplink

In this paper, we study the uplink transmission in an intelligent reflecting surface (IRS) assisted cell-free multiple-input multiple-output (MIMO) system where the central processing unit (CPU) only has statistical channel state information (CSI) to detect symbols, and to design the receiver filter coefficients, the power allocations, and the IRS phase shifts. The access points (APs) estimate only their local end-to-end channels with the users using minimum mean squared error (MMSE) estimation to implement matched filtering, thereby avoiding the large overhead associated with estimating individual IRS-assisted channels. Under this framework, we derive a closed-form expression for the achievable uplink net rate that only depends on the channel statistics. Using this expression, we formulate the problem of maximizing the minimum (max-min) signal-to-interference plus noise ratio (SINR) to design the receiver filter coefficients at the CPU, the power allocations at the users, and the phase shifts at the IRS, subject to per user power constraints as well as IRS phase shift resolution constraints. The resulting problem is jointly non-convex in the three design variables and is solved using an alternating optimization algorithm. In particular, the receiver filter design is formulated as a generalized eigenvalue problem leading to a closedform solution, the power allocation problem is solved using a geometric programming (GP) approach, and the IRS phase shifts are designed using an alternating maximization algorithm. For comparison, we also formulate and solve the max-min SINR problem for the scenario where the instantaneous imperfect CSI of all individual direct and IRS-assisted channels is available at the CPU. Numerical results show that the scheme designed using statistical CSI has the potential to outperform the scheme based on instantaneous CSI for moderate to large number of IRS elements, due to savings in the channel estimation overhead.