On the true and the modified Cramer-Rao bounds for the estimation of a scalar parameter in the presence of nuisance parameters

We consider the Cramer-Rao bound (CRB) for the estimation of a scalar parameter in the presence of nuisance parameters. Whereas this true CRB is hard to evaluate in general, we present a simple analytical expression for its high SNR asymptote, i.e., the asymptotic CRB (ACRB). We show that this ACRB is related to the CRB for the joint estimation of the scalar parameter and the nuisance parameters. Previously, a modified CRB (MCRB) for the estimation of a scalar parameter in the presence of nuisance parameters has been derived. This MCRB is also simple to evaluate and is related to the CRB for the estimation of the scalar parameter assuming that the value of the nuisance parameters is a priori known. We show that the MCRB can be quite loose at high SNR, when the scalar parameter is coupled with the nuisance parameters. In the case of synchronization parameter estimation, we find that the ACRB equals the MCRB, whether or not (some of) the nuisance parameters are a priori known.