The Estimation and Tracking of Frequency (Cambridge Series in Statistical and Probabilistic Mathematics, Series Number 9)

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 2001 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. Cover 1 Frontmatter 3 Contents 9 Preface 11 Introduction 15 1.1 Periodic functions 15 1.2 Sinusoidal regression and the periodogram 21 1.3 Testing for the presence of a sinusoid 27 1.4 Frequency estimation and tracking 30 Statistical and Probabilistic Methods 40 2.1 Introduction 40 2.2 Stationary processes, ergodicity and convergence concepts 41 2.3 The spectral theory for stationary processes 46 2.4 Maximum likelihood and the Cramér–Rao Theorem 53 2.5 Central limit theorem and law of the iterated logarithm 58 The Estimation of a Fixed Frequency 62 3.1 Introduction 62 3.2 The maximum likelihood method 63 3.3 Properties of the periodogram and the MLE 68 3.4 The Cramér–Rao Bound 75 3.5 Very low and closely adjacent frequencies 78 3.6 The estimation of the number of components 87 3.7 Likelihood ratio test for the number of frequencies 94 Techniques Derived from ARMA Modelling 116 4.1 ARMA representation 116 4.2 An iterative ARMA technique 117 4.3 Interpretation of the technique 118 4.4 Asymptotic behaviour of the procedure 122 4.5 More than one frequency 133 4.6 Asymptotic theory of the multi-frequency procedure 136 Techniques Based on Phases and Autocovariances 139 5.1 Introduction 139 5.2 An autoregressive technique 140 5.3 Pisarenko's technique 142 5.4 Kay's first estimator 148 5.5 Kay's second estimator 152 5.6 MUSIC 157 5.7 Complex MUSIC 176 Estimation using Fourier Coefficients 194 6.1 Introduction 194 6.2 Single series 194 6.3 More than one series 220 Tracking Frequency in Low SNR Conditions 229 7.1 Introduction 229 7.2 Maximum likelihood tracking 230 7.3 Hidden Markov models 232 7.4 HMM frequency tracking 241 7.5 Real data example 248 7.6 Simulated example 253 Appendix. MATLABTM programs 256 References 273 Author index 277 Subject index 279 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. B. G. Quinn, E. J. Hannan. Includes Bibliographical References (p. 259-262) And Indexes. Many electronic and acoustic signals can be modeled 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 characterize the periodicity or near-periodicity of a signal and consequently to identify its source. Quinn and Hannan present and analyze several practical techniques used for such estimation. The problem of tracking slow frequency changes of a very noisy sinusoid over time 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. This book presents and analyses practical techniques for estimating the frequencies and amplitudes of the sinusoids making up signals. Rigorous results and physical insight are both given, focusing on noisy signals and large sample sizes. Many applications are described and Matlab code is also included. An excellent introduction and reference for researchers.