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This paper analytically investigates the performance of a wireless powered communication system, where a source powered by multiple dedicated power beacons communicates with a destination. Specifically, we consider a harvest-then-transmit multiple-input multiple-output communication system, where the source uses all harvested energy for information transmission in each time block. The random placement of terminals is characterized by stochastic geometry tools. Taking into account the effects of imperfect channel state information in wireless energy transfer and wireless information transmission, the ergodic achievable rate of information transmission is derived analytically by considering co-channel interference (CCI) and the noise. In addition, assuming the CCI can be accurately approximated by a Gamma distributed variable, the outage probability of information transmission is investigated. Numerical results validate the analytical results and show the impacts of the antenna number, distribution density, estimation error, and the transmit power of terminals on the system performance.