# Optimal Channel Estimation for Hybrid Energy Beamforming under Phase Shifter Impairments

## TL;DR

This paper develops an optimal channel estimation and energy beamforming strategy for hybrid energy transmission systems with phase shifter impairments, maximizing harvested energy while considering hardware imperfections.

## Contribution

It derives the optimal least-squares channel estimator and joint resource allocation scheme for hybrid energy beamforming with phase shifter impairments, providing a closed-form solution.

## Key findings

- Optimized resource allocation improves energy harvesting by 37% over fixed schemes.
- Analytical approximation of harvested power considering phase shifter impairments.
- Impact of API and estimation errors on beamforming performance is quantified.

## Abstract

Smart multiantenna wireless power transmission can enable perpetual operation of energy harvesting (EH) nodes in the internet-of-things. Moreover, to overcome the increased hardware cost and space constraints associated with having large antenna arrays at the radio frequency (RF) energy source, the hybrid energy beamforming (EBF) architecture with single RF chain can be adopted. Using the recently proposed hybrid EBF architecture modeling the practical analog phase shifter impairments (API), we derive the optimal least-squares estimator for the energy source to EH user channel. Next, the average harvested power at the user is derived while considering the nonlinear RF EH model and a tight analytical approximation for it is also presented by exploring the practical limits on the API. Using these developments, the jointly global optimal transmit power and time allocation for channel estimation (CE) and EBF phases, that maximizes the average energy stored at the EH user is derived in closed form. Numerical results validate the proposed analysis and present nontrivial design insights on the impact of API and CE errors on the achievable EBF performance. It is shown that the optimized hybrid EBF protocol with joint resource allocation yields an average performance improvement of $37\%$ over benchmark fixed allocation scheme.