# 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

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## 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