Closed-Form Expressions for the Exact Cramer-Rao Bounds of Timing Recovery Estimators from BPSK and Square-QAM Transmissions

In this paper, we derive for the first time analytical expressions for the Cramer-Rao lower bounds (CRLBs) of timing recovery estimators from binary phase shift keying (BPSK) and square quadrature amplitude modulation (QAM) transmissions. The bounds are derived in the presence of additive white Gaussian noise (AWGN). Moreover the carrier phase and frequency are considered as unknown nuisance parameters. Our new analytical expressions reveal that the CRLBs do not depend on the corresponding time delay parameter and that they do not change widely from one modulation order to another. They also corroborate previous works that computed them empirically and provide a meaningful tool for their quick and easy evaluation.