Bounds on bearing and symbol estimation with side information

We develop Cramer-Rao bounds (CRBs) for bearing, symbol, and channel estimation of communications signals in flat-fading channels. We do this using the constrained CRB formulation of German and Hero (1990), and Stoica and Ng (see IEEE Signal Processing Lett., vol.5, p.177-79, 1998), with the unknown parameters treated as deterministic constants. The equality constraints may be combined arbitrarily, e.g., we may develop CRBs for bearing estimation of constant modulus (CM) signals where a subset of the symbols are known (semi-blind, CM case). The results establish the value of side information in a large variety of communications scenarios. We focus on the CM and semi-blind properties and develop closed-form CRBs for these cases. Examples are presented indicating the relative value of the training and CIM property. These show the significant amount of signal processing information provided under these two conditions. In addition, we consider the performance of the maximum-likelihood beamformer for the semi-blind case, assuming the bearings are known. This semi-blind beamformer achieves the appropriate (constrained) CRB with finite data at finite SNR. Analysis also reveals that in a semi-blind scenario with two closely spaced sources, ten or more training symbols are sufficient to achieve the asymptotic training regime. Together with previous results on angle estimation for known sources, these results indicate that relatively few training samples enable both angle estimation and closely spaced co-channel source separation that approaches the CRB with finite data and finite SNR.