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

In this paper, the authors derive for the first time analytical expressions for the Cramér-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. Their new analytical expressions reveal that the CRLBs do not depend on the corresponding time delay parameter and that they change widely from one shaping pulse to another. They also corroborate previous works that computed them empirically and provide a meaningful tool for their quick and easy evaluation.