Time Delay Estimation Bounds in Convolutive Random Channels

We develop bounds on time delay estimation (TDE) in wideband convolutive random channels. The improved Ziv-Zakai bound (ZZB) is adapted to this case, resulting in mean-square error (MSE) bounds that are independent of estimator bias, and avoid regularity conditions. We assume a known transmitted signal, a tapped delay line random channel model, and a uniform prior on the delay. The taps are modeled as Gaussian, including non-zero mean and possible correlation between taps. The channel model may be tuned to a variety of environments through the choice of mean and covariance, and models wideband and ultra-wideband line of sight (LOS) and non-LOS (NLOS) cases. Flat fading is a simple special case with one tap. The channel power delay profile is incorporated through the tap covariance matrix, and results for an exponential power delay profile are given. We derive the ZZB for a given channel realization, and then average over the random channel model. The resulting bound reflects the impact of the random channel, providing an average MSE performance bound, that assumes perfect channel estimates are available to the receiver. Specific cases and examples illustrate the impact of the key system, channel, and signal parameters.