True Cramer–Rao Bound for Timing Recovery From a Bandlimited Linearly Modulated Waveform With Unknown Carrier Phase and Frequency

Abstract

This paper derives the Cramer-Rao bound (CRB) related to the estimation of the time delay of a linearly modulated bandpass signal with unknown carrier phase and frequency. We consider the following two scenarios: joint estimation of the time delay, the carrier phase, and the carrier frequency; and joint estimation of the time delay and the carrier frequency irrespective of the carrier phase. The transmit pulse is a bandlimited square-root Nyquist pulse. For each scenario, the transmitted symbols constitute either an a priori known training sequence or an unknown random data sequence. In spite of the presence of random data symbols and/or a random carrier phase, we obtain a relatively simple expression of the CRB, from which the effect of the constellation and the transmit pulse are easily derived. We show that the penalty resulting from estimating the time delay irrespective of the carrier phase decreases with increasing observation interval. However, the penalty, caused by not knowing the data symbols a priori, cannot be reduced by increasing the observation interval. Comparison of the true CRB to existing symbol synchronizer performance reveals that decision-directed timing recovery is close to optimum for moderate-to-large signal-to-noise ratios.