On the Ziv-Zakai Bound for Time Difference of Arrival Estimation in CP-OFDM Systems

We consider the pilot signal for time (difference) of arrival (ToA, TDoA) estimation in an orthogonal frequency division multiplexing system with a cyclic prefix (CP-OFDM). The Cramér-Rao and Ziv-Zakai lower bounds are used to assess the performance of different pilot signals, under the assumption that the delay is distributed uniformly within a certain range. We adapt the analysis from previous works that have analyzed the performance of waveforms in an AWGN channel, but have not considered CP-OFDM. We find that in a CP-OFDM system, the ZZLB of the estimate depends only on the power allocation on the subcarriers, but not on their phase, if the pilot symbol is detected with an uncertainty up to the duration of the CP prior to the T(D)oA estimation and the CP is dropped at the receiver. This simplifies pilot signal design. Numerical results for Dirac-rectangular and triangular power allocations show that the performance follows a similar trend as for (single carrier) waveforms with a Dirac-rectangular and triangular power spectral density. But for a small uncertainty in the delay and a small number of allocated subcarriers, there is no transition region and it is optimal to allocate the pilot signals to the edge subcarriers.