The impact of the observation model on the Cramer-Rao bound for carrier phase and frequency synchronization

This contribution considers the Cramer-Rao bound (CRB) related to the joint estimation of the carrier phase and frequency of a noisy linearly modulated signal with random data symbols, using the correct continuous-time model of the received signal. We compare our results with the existing results obtained from a (common used) simplified discrete-time model of the matched filter output samples, that ignores useful signal reduction and ISI caused by nonzero frequency offset. We show that both models yield the same CRB for phase estimation. As far as frequency estimation is concerned, we point out that: (i) the correct observation models yield the smaller CRB, and (ii) the difference between the CRBs resulting from the two models is apparent only at very small SNR. This indicates that at practical SNR values, the frequency offset affects the likelihood function mainly through the signal rotation (rather than signal magnitude) at the matched filter output.