Ziv–Zakai Bounds on Time Delay Estimation in Unknown Convolutive Random Channels

Using the Ziv-Zakai bound (ZZB) methodology, we develop a Bayesian mean-square error bound on time delay estimation (TDE) in convolutive random channels, and compare it with time delay estimator performance and a Crame??r-Rao bound. The channel is modeled as a tapped delay line, whose taps are Gaussian random variables that may be nonzero mean and correlated, a model widely adopted in many applications such as wideband fading in a multipath channel. The time delay has a uniform prior distribution. A ZZB is developed that incorporates the prior on the random time delay, as well as the convolutive random Gaussian channel, and does not assume the receiver has knowledge of the channel realization. The ZZB provides good performance prediction for maximum a posteriori (MAP) time delay estimation, tracking the low, medium, and high signal-to-noise ratio (SNR) regimes. The convolutive channel model includes important special cases, such as narrowband Gaussian channels corresponding to Rayleigh/Rician fading, wideband multipath channels with a power decay profile (such as exponential decay), and known channels. We show that the associated Crame??r-Rao bound is tight only at high SNR, whereas the ZZB predicts threshold behavior and TDE breakdown as the SNR decreases. When compared to a ZZB that assumes knowledge of the channel realization, the ZZB developed here provides a more realistic and tighter bound, revealing the performance loss due to lack of channel knowledge. The MAP estimator incorporates knowledge of the channel statistics, and so performs much better than a maximum likelihood estimator that minimizes mean square error but does not use knowledge of the random channel statistics. Several examples illustrate the estimator and bound behaviors.