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

Optimized Training for Net Energy Maximization in Multi-Antenna Wireless Energy Transfer over Frequency-Selective Channel