# Estimation of the time-of-arrival of a sampled noisy signal

## Abstract

ABSTRACT The focus of this work is the estimation of the time-of-arrival of the stepped sinusoid response of a system characterized by a linear ODE. The observed response is a linear combination of complex exponentials and a white Gaussian noise process which is recorded as a sampled sequence equally spaced in time. The maximum likelihood (ML) criteria is examined for the joint estimatation the of signal parameters: timeof- arrival {t ), damping and frequency {(a.,f.)}, and amplitude a 1 1 {(C.,S.)}. We show that this exponential model has a singular Fisher 1 1 information matrix or that the ML solution is not unique (i.e., for every t a there is a choice of {(C.,S.)} that maximizes the likelihood 1 1 function). An extended Cramer-Rao (ECR) lower bound on the parameter estimator variances is derived for this case of a singular information matrix. An ML algorithm is derived that takes advantage of the semilinear form of the error function (i.e., the amplitudes enter the model linearly while t a and {(a.,f.)} enter the model nonlinearly). 1 1 Introducing the natural constraint that the response is zero at t, a allows selecting unbiased estimates oft from the locus of equivalent a ML choices. The algorithm is a sequence of optimizations (1) estimate {(a.,f.)} by minimizing a variable projection functional (VPF), (2) 1 1 obtain unbiased estimates of the ideal t by optimizing another VPF, and a (3) maximize the likelihood funtion with linear least squares estimates of the amplitudes. Two distinct t estimators are developed by dividing a the domain of signals into two distinct sets: (1) sharply rising signals ( i.e., x'(t) ¢ 0), and (2) slowly rising signals (i.e., x'(t) = 0). a a For sharply rising signals, the error function is quadratic-like so that a combination of golden search and Newton's method is used (i.e., Brent's method). While for slowly rising signals, the error function is quartic-like for which we develop a novel approach which minimizes the curvature of the error surface to which Brent's method is applicable. Monte Carlo simulations demonstrate that the ML estimates closely A approach the extended Cramer-Rao lower bounds on the variance oft. a Two suboptimum time-of-arrival estimators are also examined: (1) the location of the maximum of the crosscorrelation function relating the excitation and the response, and (2) the estimation of the initial knot position of a linear combination of spline basis functions which approximate the data. The crosscorrelation approach is shown to be unacceptably biased except near resonance. Monte Carlo simulations show that the spline model t estimator has a mean-square-error {MSE) less a than ten times the ECR variance for sharply rising signals, but for slowly rising signals the MSE is more than ten times the ECR variance. However, a feature not exploited in this study is that the spline-based estimator can be used when there is no apriori knowledge of the functional form of the underlying signal.

## Notes

This item is only available in print in the UCF Libraries. If this is your thesis or dissertation, you can help us make it available online for use by researchers around the world by STARS for more information.

## Graduation Date

1990

## Semester

Spring

## Advisor

Mohapatra, Ram N.

## Degree

Master of Science (M.S.)

## College

College of Arts and Sciences

## Department

Mathematics

## Format

## Pages

100 p.

## Language

English

## Length of Campus-only Access

None

## Access Status

Masters Thesis (Open Access)

## Identifier

DP0027333

## Subjects

Arts and Sciences -- Dissertations, Academic; Dissertations, Academic -- Arts and Sciences

## STARS Citation

Muise, Robert Raymond, "Estimation of the time-of-arrival of a sampled noisy signal" (1990). *Retrospective Theses and Dissertations*. 4047.

https://stars.library.ucf.edu/rtd/4047

## Accessibility Status

Searchable text