Optimal wireless power transfer with distributed transmit beamforming

This paper considers the performance of wireless power transfer (WPT) with distributed transmit beamforming (DTB) in a narrowband setting. One or more receive nodes, each equipped with energy harvesting and storage capabilities, provide periodic channel state feedback to a cluster of transmit nodes, each with an independent local oscillator, to facilitate beamforming and passband signal alignment for efficient WPT. Without channel state feedback, the transmit cluster can not align the passband signals at the receivers and the receivers can only harvest incoherent power. Since feedback improves the beamforming gain but requires the receivers to expend energy, there is a fundamental tradeoff between the feedback period and the energy harvesting efficiency. This paper analyzes the optimal feedback period to maximize the weighted mean energy harvesting rate as a function of the oscillator parameters. An optimization problem is formulated and an explicit method to numerically calculate the globally optimal feedback period is developed. Numerical results are provided to confirm the analysis and demonstrate the sensitivity of the weighted mean energy harvesting rate with respect to the feedback period and the DTB system parameters.