Channel Training for Analog FDD Repeaters: Optimal Estimators and   Cram\'er-Rao Bounds
  For frequency division duplex channels, a simple pilot loop-back procedure has been proposed that allows the estimation of the UL & DL channels at an antenna array without relying on any digital signal processing at the terminal side. For this scheme, we derive the maximum likelihood (ML) estimators for the UL & DL channel subspaces, formulate the corresponding Cram\'er-Rao bounds and show the asymptotic efficiency of both (SVD-based) estimators by means of Monte Carlo simulations. In addition, we illustrate how to compute the underlying (rank-1) SVD with quadratic time complexity by employing the power iteration method. To enable power control for the data transmission, knowledge of the channel gains is needed. Assuming that the UL & DL channels have on average the same gain, we formulate the ML estimator for the channel norm, and illustrate its robustness against strong noise by means of simulations. 
Stefan Wesemann and Thomas L. Marzetta
cs.IT math.IT