The Estimation and Tracking of Frequency

by B. G. Quinn , E. J. Hannan

• ISBN13: 9780521804462
• ISBN10: 0521804469
• Format: Hardcover
• Copyright: 2001-02-05
• Publisher: Cambridge University Press

Summary

Many electronic and acoustic signals can be modelled as sums of sinusoids and noise. However, the amplitudes, phases and frequencies of the sinusoids are often unknown and must be estimated in order to characterise the periodicity or near-periodicity of a signal and consequently to identify its source. This book presents and analyses several practical techniques used for such estimation. The problem of tracking slow frequency changes over time of a very noisy sinusoid is also considered. Rigorous analyses are presented via asymptotic or large sample theory, together with physical insight. The book focuses on achieving extremely accurate estimates when the signal to noise ratio is low but the sample size is large. Each chapter begins with a detailed overview, and many applications are given. Matlab code for the estimation techniques is also included. The book will thus serve as an excellent introduction and reference for researchers analysing such signals.

Table of Contents

Preface	ix
Introduction	1	(25)
Periodic functions	1	(6)
Sinusoidal regression and the periodogram	7	(6)
Testing for the presence of a sinusoid	13	(3)
Frequency estimation and tracking	16	(10)
Statistical and Probabilistic Methods	26	(22)
Introduction	26	(1)
Stationary processes, ergodicity and convergence concepts	27	(5)
The spectral theory for stationary processes	32	(7)
Maximum likelihood and the Cramer-Rao Theorem	39	(5)
Central limit theorem and law of the iterated logarithm	44	(4)
The Estimation of a Fixed Frequency	48	(54)
Introduction	48	(1)
The maximum likelihood method	49	(5)
Properties of the periodogram and the MLE	54	(7)
The Cramer-Rao Bound	61	(3)
Very low and closely adjacent frequencies	64	(9)
The estimation of the number of components	73	(7)
Likelihood ratio test for the number of frequencies	80	(22)
Techniques Derived from ARMA Modelling	102	(23)
ARMA representation	102	(1)
An iterative ARMA technique	103	(1)
Interpretation of the technique	104	(4)
Asymptotic behaviour of the procedure	108	(11)
More than one frequency	119	(3)
Asymptotic theory of the multi-frequency procedure	122	(3)
Techniques Based on Phases and Autocovariances	125	(55)
Introduction	125	(1)
An autoregressive technique	126	(2)
Pisarenko's technique	128	(6)
Kay's first estimator	134	(4)
Kay's second estimator	138	(5)
MUSIC	143	(19)
Complex MUSIC	162	(18)
Estimation using Fourier Coefficients	180	(35)
Introduction	180	(1)
Single series	180	(26)
Likelihood method	180	(5)
Three coefficient techniques	185	(14)
Techniques involving only the moduli of Fourier coefficients	199	(7)
More than one series	206	(9)
Tracking Frequency in Low SNR Conditions	215	(27)
Introduction	215	(1)
Maximum likelihood tracking	216	(2)
Hidden Markov models	218	(9)
The Viterbi track and the Viterbi algorithm	218	(2)
The EM algorithm	220	(4)
The scaled forward-backward equations	224	(2)
Other trackers	226	(1)
HMM frequency tracking	227	(7)
Real data example	234	(5)
Simulated example	239	(3)
Appendix. MATLAB™ programs	242	(17)
References	259	(4)
Author index	263	(2)
Subject index	265