On Frequency Estimation for Partially Observed System with Small Noises in State and Observation Equations

Oleg V Chernoyarov,Yury A Kutoyants,Mariana Marcokova

doi:10.26552/com.c.2018.1.67-72

Oleg V Chernoyarov, Yury A Kutoyants + Show 1 more

Open Access

https://doi.org/10.26552/com.c.2018.1.67-72

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Abstract

We consider the problem of frequency estimation of the periodic signal multiplied by a Gaussian process (Ornstein-Uhlenbeck) and observed in the presence of the white Gaussian noise. We demonstrate the consistency and asymptotic normality of the maximum likelihood and Bayesian estimators in the sense of the small noise asymptotics. The model of observations is a linear nonhomogeneous partially observed system and the construction of the estimators is based on the Kalman-Bucy filtration equations. For the study of the properties of the estimators, we apply the techniques introduced by Ibragimov and Has'minskii.

Highlights

The problem of the estimation of the frequency (Doppler shift) of the signal is of a great importance in the fields of radioand-hydroacoustic communications and positioning, radio-andhydrolocation, radio-and-hydroacoustic positioning, etc. [1,2,3]
It should be noted that all the results presented below for the O-U process can be directly extended to encompass the model presented in Example 2, but the much more complicated linear partially observed systems as well, though in this case the calculus and the expressions involved turn to be much more cumbersome
We study two estimators of the parameter j: jt T – the maximum likelihood estimator (MLE) and ju T – the Bayesian estimator (BE)

Summary

Introduction

The problem of the estimation of the frequency (Doppler shift) of the signal is of a great importance in the fields of radioand-hydroacoustic communications and positioning, radio-andhydrolocation, radio-and-hydroacoustic positioning, etc. [1,2,3]. The further extension of the specified problem is the use of non-stationary model of the useful random signal of the type (7) describing a practically important wide class of stochastic information processes and obtaining the properties of the estimate of unknown frequency parameter in wider range of asymptotics. In such context, the problem of statistical analysis of the estimate of random process band center is considered for the first time. The case T→ ∞can be treated following the study of similar problems in [11], where identification of the partially observed system is considered

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