The Estimation and Tracking of Frequency ( Cambridge Series in Statistical and Probabilistic Mathematics )

Publication series ：Cambridge Series in Statistical and Probabilistic Mathematics

Author： B. G. Quinn; E. J. Hannan  

Publisher： Cambridge University Press‎

Publication year： 2001

E-ISBN: 9781316047880

P-ISBN(Paperback): 9780521804462

Subject： O29 applied mathematics

Keyword： 应用数学

Language： ENG

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The Estimation and Tracking of Frequency

Description

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.

Chapter

Cover

Frontmatter

Contents

Preface

Introduction

1.1 Periodic functions

1.2 Sinusoidal regression and the periodogram

1.3 Testing for the presence of a sinusoid

1.4 Frequency estimation and tracking

Statistical and Probabilistic Methods

2.1 Introduction

2.2 Stationary processes, ergodicity and convergence concepts

2.3 The spectral theory for stationary processes

2.4 Maximum likelihood and the Cramér–Rao Theorem

2.5 Central limit theorem and law of the iterated logarithm

The Estimation of a Fixed Frequency

3.1 Introduction

3.2 The maximum likelihood method

3.3 Properties of the periodogram and the MLE

3.4 The Cramér–Rao Bound

3.5 Very low and closely adjacent frequencies

3.6 The estimation of the number of components

3.7 Likelihood ratio test for the number of frequencies

Techniques Derived from ARMA Modelling

4.1 ARMA representation

4.2 An iterative ARMA technique

4.3 Interpretation of the technique

4.4 Asymptotic behaviour of the procedure

4.5 More than one frequency

4.6 Asymptotic theory of the multi-frequency procedure

Techniques Based on Phases and Autocovariances

5.1 Introduction

5.2 An autoregressive technique

5.3 Pisarenko's technique

5.4 Kay's first estimator

5.5 Kay's second estimator

5.6 MUSIC

5.7 Complex MUSIC

Estimation using Fourier Coefficients

6.1 Introduction

6.2 Single series

6.3 More than one series

Tracking Frequency in Low SNR Conditions

7.1 Introduction

7.2 Maximum likelihood tracking

7.3 Hidden Markov models

7.4 HMM frequency tracking

7.5 Real data example

7.6 Simulated example

Appendix. MATLAB™ programs

References

Author index

Subject index

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