Statistics of the MLE and approximate upper and lower bounds-part II: Threshold computation and optimal pulse design for TOA estimation

Threshold and ambiguity phenomena are studied in Part I of this paper where approximations for the mean-squared error (MSE) of the maximum-likelihood estimator are proposed using the method of interval estimation (MIE), and where approximate upper and lower bounds are derived. In this part, we consider time-of-arrival estimation and we employ the MIE to derive closed-form expressions of the begin-ambiguity, end-ambiguity and asymptotic signal-to-noise ratio (SNR) thresholds with respect to some features of the transmitted signal. Both baseband and passband pulses are considered. We prove that the begin-ambiguity threshold depends only on the shape of the envelope of the ACR, whereas the end-ambiguity and asymptotic thresholds only on the shape of the ACR. We exploit the results on the begin-ambiguity and asymptotic thresholds to optimize, with respect to the available SNR, the pulse that achieves the minimum attainable MSE. The results of this paper are valid for various estimation problems.}, keywords={maximum likelihood estimation;time-of-arrival estimation;MLE;TOA estimation;approximate lower bound;approximate upper bound;asymptotic signal-to-noise ratio threshold;closed form expressions;interval estimation;maximum likelihood estimation;mean squared error;optimal pulse design;threshold computation;time-of-arrival estimation;Approximation methods;Bandwidth;Maximum likelihood estimation;Shape;Signal to noise ratio;Time of arrival estimation;Maximum likelihood estimator;mean-squared-error;nonlinear estimation;optimal signal design;signal-to-noise ratio;threshold and ambiguity phenomena;time-of-arrival