Optimized Training Design for Wireless Energy Transfer

Radio-frequency (RF) enabled wireless energy transfer (WET), as a promising solution to provide cost-effective and reliable power supplies for energy-constrained wireless networks, has drawn growing interests recently. To overcome the significant propagation loss over distance, employing multi-antennas at the energy transmitter (ET) to more efficiently direct wireless energy to desired energy receivers (ERs), termed \emph{energy beamforming}, is an essential technique for enabling WET. However, the achievable gain of energy beamforming crucially depends on the available channel state information (CSI) at the ET, which needs to be acquired practically. In this paper, we study the design of an efficient channel acquisition method for a point-to-point multiple-input multiple-output (MIMO) WET system by exploiting the channel reciprocity, i.e., the ET estimates the CSI via dedicated reverse-link training from the ER. Considering the limited energy availability at the ER, the training strategy should be carefully designed so that the channel can be estimated with sufficient accuracy, and yet without consuming excessive energy at the ER. To this end, we propose to maximize the \emph{net} harvested energy at the ER, which is the average harvested energy offset by that used for channel training. An optimization problem is formulated for the training design over MIMO Rician fading channels, including the subset of ER antennas to be trained, as well as the training time and power allocated. Closed-form solutions are obtained for some special scenarios, based on which useful insights are drawn on when training should be employed to improve the net transferred energy in MIMO WET systems.Comment: 30 pages, 9 figures, to appear in IEEE Trans. on Communication

Analysis of contact Cauchy-Riemann maps II: canonical neighborhoods and exponential convergence for the Morse-Bott case

This is a sequel to the papers [OW1], [OW2]. In [OW1], the authors introduced a canonical affine connection on $M$ associated to the contact triad $(M,\lambda,J)$. In [OW2], they used the connection to establish a priori $W^{k,p}$-coercive estimates for maps $w: \dot \Sigma \to M$ satisfying $\overline{\partial}^\pi w= 0, \, d(w^*\lambda \circ j) = 0$ \emph{without involving symplectization}. We call such a pair $(w,j)$ a contact instanton. In this paper, we first prove a canonical neighborhood theorem of the locus $Q$ foliated by closed Reeb orbits of a Morse-Bott contact form. Then using a general framework of the three-interval method, we establish exponential decay estimates for contact instantons $(w,j)$ of the triad $(M,\lambda,J)$, with $\lambda$ a Morse-Bott contact form and $J$ a CR-almost complex structure adapted to $Q$, under the condition that the asymptotic charge of $(w,j)$ at the associated puncture vanishes. We also apply the three-interval method to the symplectization case and provide an alternative approach via tensorial calculations to exponential decay estimates in the Morse-Bott case for the pseudoholomorphic curves on the symplectization of contact manifolds. This was previously established by Bourgeois [Bou] (resp. by Bao [Ba]), by using special coordinates, for the cylindrical (resp. for the asymptotically cylindrical) ends. The exponential decay result for the Morse-Bott case is an essential ingredient in the set-up of the moduli space of pseudoholomorphic curves which plays a central role in contact homology and symplectic field theory (SFT).Comment: 69 pages, final version to appear in Nagoya Math J, improvement of overall presentatio