  This paper studies the training design problem for multiple-input
single-output (MISO) wireless energy transfer (WET) systems in
frequency-selective channels, where the frequency-diversity and
energy-beamforming gains can be both reaped to maximize the transferred energy
by efficiently learning the channel state information (CSI) at the energy
transmitter (ET). By exploiting channel reciprocity, a new two-phase channel
training scheme is proposed to achieve the diversity and beamforming gains,
respectively. In the first phase, pilot signals are sent from the energy
receiver (ER) over a selected subset of the available frequency sub-bands,
through which the ET determines a certain number of "strongest" sub-bands with
largest antenna sum-power gains and sends their indices to the ER. In the
second phase, the selected sub-bands are further trained by the ER, so that the
ET obtains a refined estimate of the corresponding MISO channels to implement
energy beamforming for WET. A training design problem is formulated and
optimally solved, which takes into account the channel training overhead by
maximizing the net harvested energy at the ER, defined as the average harvested
energy offset by that consumed in the two-phase training. Moreover, asymptotic
analysis is obtained for systems with a large number of antennas or a large
number of sub-bands to gain useful insights on the optimal training design.
Finally, numerical results are provided to corroborate our analysis and show
the effectiveness of the proposed scheme that optimally balances the diversity
and beamforming gains achieved in MISO WET systems with limited-energy
training.
