This paper derives approximate outage probability (OP) expressions for uplink cell-free massive multiple-input-multiple-output (CF-mMIMO) systems with and without pilot contamination. The system's access points (APs) are considered to have imperfect channel state information (CSI). The signalto-interference-plus-noise ratio (SINR) of the CF-mMIMO system is approximated via a Log-normal distribution using a two-step moment matching method. OP and ergodic rate expressions are derived with the help of the approximated Log-normal distribution. For the no-pilot contamination scenario, an exact expression is first derived using conditional expectations in terms of a multi-fold integral. Then, a novel dimension reduction method is used to approximate it by the sum of single-variable integrations. Both the approximations derived for the CF-mMIMO systems are also useful for single-cell collocated massive MIMO (mMIMO) systems and lead to closed-form expression. The derived expressions closely match the simulated numerical values for OP and ergodic rate.
INDEX TERMS Cell-free massive MIMO, outage probability, univariate dimension reduction
I. INTRODUCTIONC ELL-free massive multiple-input multiple-output (CF-mMIMO) system is envisioned as a key enabler for sixth-generation (6G) communication systems. [1]-[4]. CF-mMIMO system contains many access points (APs) that are connected to a central processing unit (CPU) and jointly serve all the user equipment (UE) by coherent joint transmission and reception [5], [6]. The name cell-free comes from the fact that there are no boundaries, and each AP serves all the UEs. This differs from a conventional small-cell system where each AP serves only a particular set of UEs. Early works in [7], [8] compare the CF-mMIMO system with a conventional small-cell system and show a multifold improvement in 95%-likely throughput can be expected from CF-mMIMO.