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

A network of analog repeaters, each fed by a wireless fronthaul link and powered by e.g., solar energy, is a promising substitute for dense small cell deployments. A key challenge is the acquisition of accurate channel state information by the fronthaul hub (FH), which is needed for the spatial multiplexing of multiple fronthaul links over the same time/frequency resource. For frequency division duplex channels, we propose a simple pilot loop-back procedure that allows the estimation of the UL & DL channel subspaces at the FH, without relying on any digital signal processing at the repeater side. The UL & DL subspaces are extracted at the FH by computing the singular value decomposition (SVD) of the pilot reverse modulated receive signal. We show that this procedure coincides with the maximum likelihood (ML) subspace estimator for both, the UL & DL subspaces. Moreover, we provide the corresponding Cramér-Rao bounds (CRB) and illustrate their range of validity by means of Monte Carlo simulations. Finally, we show how to efficiently compute the ML estimates using the power iteration method, and empirically demonstrate a complexity reduction from O(n^2.95) down to O(n^2.25).