  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.
