Low-SNR limit of the Cramer-Rao bound for estimating the time delay of a PSK, QAM, or PAM waveform

In this letter we consider the Cramer-Rao bound (CRB) for the estimation of the time delay of a noisy linearly modulated signal with random data symbols and random carrier phase. Because of the presence of the nuisance parameters (i.e., data symbols and carrier phase), a closed-form expression of this CRB is very hard to obtain for arbitrary PSK, QAM or PAM constellations and a band-limited transmit pulse. Instead, we derive a simple expression for the limit of the CRB at low signal-to-noise ratio (SNR), which is a relevant benchmark for timing recovery algorithms operating at small E/sub s//N/sub 0/.