The Estimation and Tracking of Frequency
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.
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Truly a fantastic book. Everything is explained and proofed in a fantastic scientific manner and is recommended for anyone with an interest in any form of science, waves or anything statistical or mathematical.
Statistical and Probabilistic Methods
The Estimation of a Fixed Frequency
Techniques Derived from ARMA Modelling
Techniques Based on Phases and Autocovariances
Estimation using Fourier Coefficients
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algorithm amplitude assume asymptotic distribution asymptotic theory asymptotic variance asymptotically normally autocovariances autoregressive calculated central limit theorem Code function column complex compute converges almost surely cos2 covariance matrix derive Description The program eigenvalues eigenvectors ergodic estimators of frequency Figure filter fixed follows forward-backward equations Fourier coefficients Fourier frequencies frequency estimation Gaussian white noise Hannan Hence Hidden Markov Model identically distributed independent and identically initial estimates integers Iog2 iterative limsup Markov mean square mean zero minimising MUSIC estimators normal with mean number of sinusoids obtain parameters periodogram maximiser phase probability problem procedure quantities Quinn random variables sample Section sequence signal simulations sin2 spectral density spectrogram stationary process statistical sum of squares technique term vector Viterbi algorithm Viterbi track white noise zero and variance