Abstract

The analysis of (approximately) periodic signals is an important
element in numerous applications. One generalization of standard
periodic signals often occurring in practice are harmonic chirp
signals where the instantaneous frequency increases/decreases linearly
as a function of time. A statistically efficient estimator for
extracting the parameters of the harmonic chirp model in additive
white Gaussian noise is the maximum likelihood (ML) estimator which
recently has been demonstrated to be robust to noise and accurate ---
even when the model order is unknown. The main drawback of the ML
estimator is that only very computationally demanding algorithms for
computing an estimate are known. In this paper, we give an algorithm
for computing an estimate to the ML estimator for a number of
candidate model orders with a much lower computational complexity than
previously reported in the literature. The lower computational
complexity is achieved by exploiting recursive matrix structures,
including a block Toeplitz-plus-Hankel structure, the fast Fourier
transform, and using a two-step approach composed of a grid and
refinement step to reduce the number of required function
evaluations. The proposed algorithms are assessed via Monte Carlo and
timing studies. The timing studies show that the proposed algorithm is
orders of magnitude faster than a recently proposed algorithm for
practical sizes of the number of harmonics and the length of the
signal.
Original languageEnglish
JournalI E E E Transactions on Signal Processing
Volume65
Issue number19
Pages (from-to)5137 - 5152
ISSN1053-587X
DOIs
Publication statusPublished - 2017

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Maximum likelihood estimation
Maximum likelihood
Computational complexity

Cite this

@article{1d0c35d90c404facada44465676cbe6e,
title = "A Fast Algorithm for Maximum Likelihood Estimation of Harmonic Chirp Parameters",
abstract = "The analysis of (approximately) periodic signals is an importantelement in numerous applications. One generalization of standardperiodic signals often occurring in practice are harmonic chirpsignals where the instantaneous frequency increases/decreases linearlyas a function of time. A statistically efficient estimator forextracting the parameters of the harmonic chirp model in additivewhite Gaussian noise is the maximum likelihood (ML) estimator whichrecently has been demonstrated to be robust to noise and accurate ---even when the model order is unknown. The main drawback of the MLestimator is that only very computationally demanding algorithms forcomputing an estimate are known. In this paper, we give an algorithmfor computing an estimate to the ML estimator for a number ofcandidate model orders with a much lower computational complexity thanpreviously reported in the literature. The lower computationalcomplexity is achieved by exploiting recursive matrix structures,including a block Toeplitz-plus-Hankel structure, the fast Fouriertransform, and using a two-step approach composed of a grid andrefinement step to reduce the number of required functionevaluations. The proposed algorithms are assessed via Monte Carlo andtiming studies. The timing studies show that the proposed algorithm isorders of magnitude faster than a recently proposed algorithm forpractical sizes of the number of harmonics and the length of thesignal.",
author = "Jensen, {Tobias Lindstr{\o}m} and Nielsen, {Jesper Kj{\ae}r} and Jensen, {Jesper Rindom} and Christensen, {Mads Gr{\ae}sb{\o}ll} and Jensen, {S{\o}ren Holdt}",
year = "2017",
doi = "10.1109/TSP.2017.2723342",
language = "English",
volume = "65",
pages = "5137 -- 5152",
journal = "I E E E Transactions on Signal Processing",
issn = "1053-587X",
publisher = "IEEE",
number = "19",

}

TY - JOUR

T1 - A Fast Algorithm for Maximum Likelihood Estimation of Harmonic Chirp Parameters

AU - Jensen, Tobias Lindstrøm

AU - Nielsen, Jesper Kjær

AU - Jensen, Jesper Rindom

AU - Christensen, Mads Græsbøll

AU - Jensen, Søren Holdt

PY - 2017

Y1 - 2017

N2 - The analysis of (approximately) periodic signals is an importantelement in numerous applications. One generalization of standardperiodic signals often occurring in practice are harmonic chirpsignals where the instantaneous frequency increases/decreases linearlyas a function of time. A statistically efficient estimator forextracting the parameters of the harmonic chirp model in additivewhite Gaussian noise is the maximum likelihood (ML) estimator whichrecently has been demonstrated to be robust to noise and accurate ---even when the model order is unknown. The main drawback of the MLestimator is that only very computationally demanding algorithms forcomputing an estimate are known. In this paper, we give an algorithmfor computing an estimate to the ML estimator for a number ofcandidate model orders with a much lower computational complexity thanpreviously reported in the literature. The lower computationalcomplexity is achieved by exploiting recursive matrix structures,including a block Toeplitz-plus-Hankel structure, the fast Fouriertransform, and using a two-step approach composed of a grid andrefinement step to reduce the number of required functionevaluations. The proposed algorithms are assessed via Monte Carlo andtiming studies. The timing studies show that the proposed algorithm isorders of magnitude faster than a recently proposed algorithm forpractical sizes of the number of harmonics and the length of thesignal.

AB - The analysis of (approximately) periodic signals is an importantelement in numerous applications. One generalization of standardperiodic signals often occurring in practice are harmonic chirpsignals where the instantaneous frequency increases/decreases linearlyas a function of time. A statistically efficient estimator forextracting the parameters of the harmonic chirp model in additivewhite Gaussian noise is the maximum likelihood (ML) estimator whichrecently has been demonstrated to be robust to noise and accurate ---even when the model order is unknown. The main drawback of the MLestimator is that only very computationally demanding algorithms forcomputing an estimate are known. In this paper, we give an algorithmfor computing an estimate to the ML estimator for a number ofcandidate model orders with a much lower computational complexity thanpreviously reported in the literature. The lower computationalcomplexity is achieved by exploiting recursive matrix structures,including a block Toeplitz-plus-Hankel structure, the fast Fouriertransform, and using a two-step approach composed of a grid andrefinement step to reduce the number of required functionevaluations. The proposed algorithms are assessed via Monte Carlo andtiming studies. The timing studies show that the proposed algorithm isorders of magnitude faster than a recently proposed algorithm forpractical sizes of the number of harmonics and the length of thesignal.

U2 - 10.1109/TSP.2017.2723342

DO - 10.1109/TSP.2017.2723342

M3 - Journal article

VL - 65

SP - 5137

EP - 5152

JO - I E E E Transactions on Signal Processing

JF - I E E E Transactions on Signal Processing

SN - 1053-587X

IS - 19

ER -