The modified Cramer-Rao bound and its application to synchronization problems

We introduce the modified Cramer-Rao bound (CRB) which, like the true CRB, is a lower bound to the error variance of any parameter estimator. The modified CRB proves useful when, in addition to the parameter to be estimated, the observed data also depend on other unwanted parameters. The relationship between the modified and true CRB is established and applications are discussed regarding the estimation of carrier-frequency offset, carrier phase, and timing epoch in linearly modulated signals. Modified CRBs for phase and timing estimation have been already discussed in previous works where it is shown that several practical carrier-phase and clock recovery circuits do attain such bounds. Frequency discrimination, instead, is not so well-represented in the literature and a significant contribution of this paper is the calculation of the modified CRB for frequency estimation. This bound is compared with the performance of some frequency detectors and it is concluded that further work is needed in search of more efficient frequency discrimination methods