2 research outputs found
Statistics of the MLE and Approximate Upper and Lower Bounds - Part 2: Threshold Computation and Optimal Signal Design
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
Statistics of the MLE and Approximate Upper and Lower Bounds - Part 1: Application to TOA Estimation
In nonlinear deterministic parameter estimation, the maximum likelihood
estimator (MLE) is unable to attain the Cramer-Rao lower bound at low and
medium signal-to-noise ratios (SNR) due the threshold and ambiguity phenomena.
In order to evaluate the achieved mean-squared-error (MSE) at those SNR levels,
we propose new MSE approximations (MSEA) and an approximate upper bound by
using the method of interval estimation (MIE). The mean and the distribution of
the MLE are approximated as well. The MIE consists in splitting the a priori
domain of the unknown parameter into intervals and computing the statistics of
the estimator in each interval. Also, we derive an approximate lower bound
(ALB) based on the Taylor series expansion of noise and an ALB family by
employing the binary detection principle. The accurateness of the proposed
MSEAs and the tightness of the derived approximate bounds are validated by
considering the example of time-of-arrival estimation