This paper presents a tutorial review of work in coherence and time delay estimation. A review of coherence research and development is presented. A derivation of the ML estimator for time delay is presented together with an interpretation of that estimator as a special member of a class of generalized cross correlators. The performance of the estimator is given for both high and low signal-to-noise ratio cases. The proposed correlator is implemented and stimulated with synthetic data. The results are compared with performance predictions and found to be in good agreement.

Coherence and time delay estimation

An overview of applied research in passive sonar signal processing estimation techniques for naval systems is presented. The naval problem that motivates time delay estimation is the source state estimation problem. A discussion of this problem in terms of estimating the position and velocity of a moving acoustic source is presented. Optimum bearing and range estimators are presented for the planar problem and related to the optimum time delay vector estimator. Suboptimum realizations are considered together with the effects of source motion and receiver positional uncertainty.

Time delay estimation for passive sonar signal processing

The purpose of this paper is to apply the renewal theory approach for analyzing aperiodic finite S -curve code-tracking loops developed by Meyr [9] to the case of a noncoherent, arbitrary offset, early-late, delay-locked loop ("noncoherent δ-DLL") The exact (renewal) approach is compared to the approximate (periodic-extension) approach of using the periodic S -curve or phase-locked loop theory, as well as to the linear theory developed herein for the aforementioned codetracking loop Finally, loop optimization with respect to the offset δ is carried out according to certain performance criteria The results indicate that, for low SNR, the exact and approximate theories could deviate significantly while, for high SNR, all three theories yield identical performance, as expected Furthermore, it is shown that the optimal δ for both low and high SNR could differ from the commonly accepted choice \delta = 1/2 

Analysis and Optimization of Correlative Code-Tracking Loops in Spread-Spectrum Systems

Cycle slips in phase-locked loops are statistical, nonlinear phenomena. This makes a mathematical analysis extremely difficult. As a consequence, the results of such an analysis are not easily accessible to the practicing engineer. It is the purpose of this survey paper to present a self-contained discussion of cycle slips in phase-locked loop avoiding advanced mathematical tools. Based on the results of an extensive experimental study we explain the underlying principle of the complex interaction between nonlinearity and noise. The results are complemented by simple, approximate analysis which agrees well with the experimental findings. In addition, we present a new and complete set of diagrams on cycle slip statistics not presently available in the literature.

Cycle Slips in Phase-Locked Loops: A Tutorial Survey

We show that the steady-state entropy production rate of a stochastic process is inversely proportional to the minimal time needed to decide on the direction of the arrow of time. Here we apply Wald's sequential probability ratio test to optimally decide on the direction of time's arrow in stationary Markov processes. Furthermore, the steady-state entropy production rate can be estimated using mean first-passage times of suitable physical variables. We derive a first-passage time fluctuation theorem which implies that the decision time distributions for correct and wrong decisions are equal. Our results are illustrated by numerical simulations of two simple examples of nonequilibrium processes.

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Decision Making in the Arrow of Time

The complete statistical description of a first-order correlative tracking system with periodic nonlinearity is shown to be embedded in a renewal process. The time-dependent probability density function of the phase error, as well as the distribution of the cycle slips, is computed. The use of the renewal process approach makes it possible for the first time to compute the distribution of the positive and negative number of cycle slips within a given time interval. This information is sufficient to determine the probability density function of the absolute phase error.

H. Meyr

Papers

Complete statistical description of the phase-error process generated by correlative tracking systems