Optimal Placement of Training Symbols for Frequency Acquisition: A Cramér-Rao Bound Approach

We consider the problem of training symbol placement for timing acquisition in band-limited, baudrate sampled systems with inter-symbol interference and frequency offset. The conventional approach is to place known symbols at the start of the transmitted packet and use these at the receiver to run a trained phase-locked loop. We introduce an additional degree of freedom by allowing arbitrary locations for the training symbols. We first consider a simplified system model where only the training symbols are transmitted, or equivalently, the unknown data is assumed to be zero. We derive the modified Cramér-Rao bound (CRB) on the timing estimation error variance as a function of the trained symbol locations and then derive the optimal training symbol placement strategy to minimize the CRB. The optimal strategy, called the split-preamble strategy, is to split the known symbols into two halves and place these at the beginning and at the end of the packet. Simulations show that this arrangement also greatly reduces the occurrence of lost or added symbols, i.e., cycle slips. Finally, we present a simplified analysis of the problem when both known and unknown data are present, and show that the same arrangement also minimizes the CRB in this case.