Modified Bhattacharyya bounds and their application to timing estimation

Recently, a modified Cramer-Rao bound has been proposed for the problem of estimating non-random parameters in the presence of random nuisance parameters. In its scalar or vector form, the modified Cramer-Rao bound (MCRB) is simple to evaluate and shown to be tight in some cases. Unfortunately, the MCRB is proven to be at most as tight as conventional Cramer-Rao bound (CRB). Another bound, the Miller-Chang bound (MCB), has also been evaluated in the literature. It applies to a more restricted class of estimators, namely, those are uniformly unbiased with respect to the random nuisance parameters. The MCB was shown to be much tighter than the CRB in several cases. In this article, we present a new bound, the modified Bhattacharyya bound (MBB). Like the MCRB, the MBB is easy to calculate. Taking the MCRB as a special case, the MBB is proven to be at least as tight as the MCRB. A case study on timing estimation shows that the MBB is not only tighter than the MCRB, but tighter than the CRB in low SNR range for the problem studied. It is also observed that significant improvement can be achieved by calculating bigger J matrix. Another bound, the modified Bhattacharyya bound 2 (MBB2) is also presented. As an extension of the MCB, the MBB2 is found to be at least as tight as the MCB.