A Survey of Time Delay Estimation Performance Bounds

In this paper we review performance bounds, as well as some current trends, in time delay estimation (TDE). Research over several decades reveals that a few key parameters determine TDE performance. The most basic are the signal-to-noise ratio (SNR), and the signal time-bandwidth (TB) product; larger values for each are desirable. The Cramer-Rao bound (CRB) reveals asymptotic maximum-likelihood estimation (MLE) behavior with respect to TB and SNR. At moderate to lower SNR, TDEs generally break down as ambiguities arise due to increased noise and the cross-correlation of the signal, causing the TDE to deviate (often quite sharply) away from the CRB. Because it is a local bound, the CRB does not indicate the threshold behavior, and Ziv-Zakai and other bounds have been developed to handle this. When TD is measured between multiple sensors, the coherence between them can fundamentally limit the result, an effect that occurs in acoustics due to the turbulent atmosphere. We discuss modifications to the classical bounds that accommodate the coherence loss, and reveal a threshold coherence phenomenon. When communications and other signals are utilized for TDE, they may have significant nuisance parameters, including carrier uncertainty, unknown symbols, as well as effects due to an unknown channel. Recent TDE performance limits reveal the effect of these parameters for various signal models, including the impact of diversity channels on TDE