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 signal-to-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.

翻译：本文推导了无导频污染与有导频污染条件下上行链路无小区大规模多输入多输出（CF-mMIMO）系统的近似中断概率（OP）表达式。系统中接入点（AP）被考虑为具有非理想信道状态信息（CSI）。通过两步矩匹配方法，将CF-mMIMO系统的信干噪比（SINR）近似为对数正态分布。基于该近似分布，推导了OP与遍历速率表达式。针对无导频污染场景，首先利用条件期望以多重积分形式推导精确表达式，随后采用一种新型降维方法将其近似为单变量积分之和。本文为CF-mMIMO系统推导的两种近似方法同样适用于单小区共置大规模MIMO（mMIMO）系统，并可得到闭式表达式。推导结果与OP及遍历速率的仿真数值高度吻合。