Performance bounds in synchronization for low signal-to-noise ratios

In this contribution we consider the Cramer-Rao bound (CRB) for the estimation of the synchronization parameters of a noisy linearly modulated signal with random data symbols. We explore three scenarios, i.e., (i) joint estimation of carrier phase, carrier frequency and time delay, irrespective of the data; (ii) joint estimation of carrier frequency and time delay, irrespective of the data and the carrier phase; and (iii) estimation of carrier frequency, irrespective of the data, the carrier phase and the timing. Because of the presence of the random data (and, in scenarios (ii) and (iii), also of random synchronization parameters), the exact computation of the corresponding CRBs is extremely difficult. Instead, here we derive a simple closed-form expression for the limit of these CRBs at low signal-to-noise ratio (SNR), which holds for arbitrary PAM, PSK and QAM constellations.