Timing Recovery With Frequency Offset and Random Walk: Cramer–Rao Bound and a Phase-Locked Loop Postprocessor

We consider the problem of timing recovery for bandlimited, baud-rate sampled systems with intersymbol interference and a timing offset that can be modeled as a combination of a frequency offset and a random walk. We first derive the Cramer–Rao bound (CRB), which is a lower bound on the estimation-error variance for any timing estimator. Conventional timing recovery is based on a phase-locked loop (PLL). We compare the conventional timing recovery method with the CRB for realistic timing parameters for the magnetic recording channel, and observe a 7 dB signal-to-noise ratio gap between the two. Next, we propose a PLL postprocessor based on the maximum a posteriori estimation principle that performs to within 1.5 dB of the CRB. This postprocessor performs time-invariant filtering and time-varying scaling of the PLL timing estimates. The refined timing estimates from the postprocessor are then used to get refined samples by interpolating the samples taken at the PLL's timing estimates. Finally, we present suboptimal implementations that allow a performance-complexity tradeoff.