On the Performance of Bit-Synchronizers in an ISI Channel and a Related Lower Bound

The maximum likelihood (ML) criterion for symbol timing estimation is derived for a sequence of pulse amplitude modulated pulses in the presence of intersymbol interference (ISI) and Gaussian noise. The performance of this synchronizer is used as a benchmark to evaluate the performance of other synchronizers in a practical scenario. It is shown that a previous lower bound based on the ISI-free ML synchronizer cannot be used to lower bound the mean square error (MSE) of bit-synchronizers. A detection theory bound (DTB) is applied to the symbol timing recovery problem in an ISI channel and it is shown that this bound is a tight lower bound on the MSE of the ML synchronizer. A simple lower bound on this DTB is derived and it is shown that the simple bound is almost as tight as the well known modified Cramer-Rao bound (MCRB) at moderate values of SNR while it does not suffer from the shortcomings of the MCRB at small values of SNR.