Ziv-Zakai bound on time delay estimation in unknown convolutive random channels

Using the Ziv-Zakai bound (ZZB) methodology, we develop a Bayesian (MSE) bound on time delay estimation (TDE) in a wideband convolutive random channel. The channel is modeled as a tapped delay line, whose taps are Gaussian random variables that may be non-zero mean and correlated. The derivation does not assume channel knowledge at the receiver, and so realistically takes into account the effects of the unknown channel when estimating time delay. The channel model can represent wideband and ultra-wideband, as well as line of sight and non-line of sight cases, through selection of model parameters. The bound development involves the minimum probability of decision error under a hypothesis test, and we show this can be equivalently formulated as a probability of detection error for binary pulse position modulation (PPM) signals. An expression for the error probability, and thus the ZZB on TDE, is derived using a moment generating function approach. Effects of system design and channel distribution parameters are studied through the ZZB.