The delay-lock discriminator is a nonlinear feedback system which employs a form of cross correlation to estimate the relative delay between a reference signal and a delayed version of that signal which is perturbed with additive noise. Previous treatments of delaylock tracking have assumed that the binary maximal-length code sequences used as the signal are detected in a coherent manner. The present paper is concerned with delay-lock discriminator tracking behavior when noncoherent detection of the received signal is required and when two-bit quantization is employed to reduce the cost of implementation of the cross-correlation network. It is found that the error signal generated by the cross-correlation network is characterized by "clustering" in the absence of noise, however, this mode of error signal behavior gradually vanishes as the noise level is increased and is replaced by an error signal generation characteristic that more closely resembles the behavior that obtains without quantization. The selection of an appropriate threshold value to achieve minimum steady-state errors for the delay-lock discriminator is reduced to the problem of determining the probability density function of the error signal under the condition of exact alignment.

The smoothing filter and smoothing error covariance matrix equations are developed for discrete linear systems whose observations may contain noise alone, where only the probability of occurrence of such cases is known to the estimator. An example of such a system arises in trajectory tracking, where the signal is first detected and then is processed by the estimator for tracking purposes. The results apply to any detection decision process, however, any such decision is associated with a false alarm probability, which is the probability that the detected signal contains only noise. The present results together with the earlier work of Nahi on prediction and filtering give a complete treatment of the discrete linear estimation problem for systems characterized by uncertain observations. These results, of course, reduce to well-known formulations for the classical estimation problem in the case where the observation is always assumed to contain the signal to be estimated.