Statistics of the MLE and Approximate Upper and Lower Bounds - Part 2: Threshold Computation and Optimal Signal Design

TL;DR

This paper derives closed-form expressions for ambiguity thresholds in time-of-arrival estimation using the method of interval estimation, and optimizes pulse design to minimize MSE based on SNR.

Contribution

It introduces a novel application of the method of interval estimation to derive thresholds and optimize signal design in time-of-arrival estimation.

Findings

Begin-ambiguity threshold depends only on the envelope shape of the ACR.

End-ambiguity and asymptotic thresholds depend only on the shape of the ACR.

Optimized pulse design minimizes MSE at given SNR levels.

Abstract

Threshold and ambiguity phenomena are studied in Part 1 of this work 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.