Closed Form Sum Rate of Random Beamforming

In this letter, we carry out analysis to derive the closed form sum rate for a general multi-antenna random beamforming system with an arbitrary but finite number of users. A simple and analytical useful expansion with general utility is first developed to decompose and interpret the probability density function of the selected user´s signal-to-interference-plus-noise ratio. By using this decomposition, along with standard integration techniques, the closed form result for the sum rate is obtained. The established result explains the interplay between the system parameters for arbitrary but finite number of users.