The estimation and tracking of frequency (Q2711250)

This book is primarily concerned with the estimation of the fixed or varying frequencies \(\lambda_1, \dots, \lambda_r\) in the time series model NEWLINE\[NEWLINEy(t)= \mu+\sum^r_{j=1} A_j\cos (\lambda_j t+\varphi_j) +x(t),NEWLINE\]NEWLINE where \(x(t)\) is a noise associated with measurements of a periodic function. The aim of the book is to provide a comprehensive examination of various techniques for this problem; the text is accessible to both statisticians and engineers. NEWLINENEWLINENEWLINEChapter 1 (Introduction) motivates the problems by some physical examples. Chapter 2 (Statistical and probabilistic methods) lays the theoretical foundations for the rest of book. Chapter 3 (The estimation of a fixed frequency) examines in detail the statistical theory for the maximum likelihood estimators of the fixed frequencies. In particular, it looks at the cases where there are two frequencies very close together and where one estimates the number of frequencies. Chapter 4 (Techniques derived from ARMA modelling) presents the \textit{B.G. Quinn} and \textit{J.M. Fernandes} technique [Biometrika 78, No. 3, 489-497 (1991; Zbl 0737.62082)] for estimating frequencies using ARMA modelling. Chapter 5 (Techniques based on phases and autocovariances) examines a number of popular frequency estimation techniques which use sample autocovariances: autoregressive, Pisarenko and MUSIC (multiple signal characterization) techniques. The first and second Kay's estimator is also presented. Chapter 6 (Estimation using Fourier coefficients) obtains, e.g., closed form frequency estimators for a single time series using only three Fourier coefficients including suitable algorithms. Chapter 7 (Tracking frequency in low SUR conditions) treats the problem of changing frequency. One suggests hidden Markov model procedures for tracking frequency changing slowly in very low signal-to-noise environments.