Channel Training for Analog FDD Repeaters: Optimal Estimators and Cramér–Rao Bounds

For frequency division duplex channels, a simple pilot loop-back procedure has been proposed that allows the estimation of the uplink (UL) and downlink (DL) channel subspaces between an antenna array and a fully analog repeater. For this scheme, we derive the maximum likelihood (ML) estimators for the UL and DL subspaces, formulate the corresponding Cramér–Rao bounds, and show the asymptotic efficiency of both (singular value decomposition (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 and DL channels have on average the same gain, we formulate the ML estimator for the uplink channel vector norm, and illustrate its robustness against strong noise perturbations by means of simulations.