Time delay determination: Maximum likelihood and Kalman-Bucy type structures

Time delay determination is an important problem in numerous applications. The approach taken here models the signals via linear differential equations driven by white noise. The time delays are unknown parameters modulating the received signals. The maximum likelihood estimation of the delays requires the filtering in the minimum mean square error (MMSE) sense of the signals. The problem becomes that of the joint estimation of the signals with the identification of the delays. Due to the structure of the signal model, the signal MMSE estimate is obtained via a recursive structure of the Kalman-Bucy type. The class of signals considered includes the stationary signals, to which the cross-correlation receivers are restricted. In fact, it can be shown that the receiver studied in this paper is a generalization of the cross-correlation receiver. The paper presents the general receiver structure, discussing it in the context of a specific example. The Cramer-Rao bound associated with the delay estimation is also discussed.