Logarithmic Expectation of the Sum of Exponential Random Variables for Wireless Communication Performance Evaluation

Sums of exponentially distributed random variables (RVs) play important roles in performance analysis of various communication systems. Their logarithmic expectations can not only facilitate capacity analysis but also provide efficient analytical expressions of the system capacity. However, the analytical expressions for the logarithmic expectations have been rarely systematically provided in literature, resulting in inconvenience for related performance analysis. To overcome this issue, in this work, the analytical expressions for the logarithmic expectations of the sums of independent exponential RVs are summarized. Especially, for the case where the sum is composed of both independent non-identically distributed (i.n.i.d.) exponential RVs and independent identically distributed (i.i.d.) exponential RVs, a new closed-form probability density function (PDF) is derived. Compared to the previous PDF expressions, the derived PDF expression is much concise and easy to be determined. To demonstrate the effectiveness of the derived logarithmic expectations, case studies are performed by applying the derived logarithmic expectations to the analysis and derivations of the ergodic capacity of several multiple antenna systems. It is shown that the proposed approach can significantly facilitate the performance evaluation of multiple antenna communication systems.