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A New Frequency Estimation Method for Equally and Unequally Spaced Data

Andersson, Fredrik LU ; Carlsson, Marcus LU ; Tourneret, Jean-Yves and Wendt, Herwig (2014) In IEEE Transactions on Signal Processing 62(21). p.5761-5774
Abstract
Spectral estimation is an important classical problem that has received considerable attention in the signal processing literature. In this contribution, we propose a novel method for estimating the parameters of sums of complex exponentials embedded in additive noise from regularly or irregularly spaced samples. The method relies on Kronecker's theorem for Hankel operators, which enables us to formulate the nonlinear least squares problem associated with the spectral estimation problem in terms of a rank constraint on an appropriate Hankel matrix. This matrix is generated by sequences approximating the underlying sum of complex exponentials. Unequally spaced sampling is accounted for through a proper choice of interpolation matrices. The... (More)
Spectral estimation is an important classical problem that has received considerable attention in the signal processing literature. In this contribution, we propose a novel method for estimating the parameters of sums of complex exponentials embedded in additive noise from regularly or irregularly spaced samples. The method relies on Kronecker's theorem for Hankel operators, which enables us to formulate the nonlinear least squares problem associated with the spectral estimation problem in terms of a rank constraint on an appropriate Hankel matrix. This matrix is generated by sequences approximating the underlying sum of complex exponentials. Unequally spaced sampling is accounted for through a proper choice of interpolation matrices. The resulting optimization problem is then cast in a form that is suitable for using the alternating direction method of multipliers (ADMM). The method can easily include either a nuclear norm or a finite rank constraint for limiting the number of complex exponentials. The usage of a finite rank constraint makes, in contrast to the nuclear norm constraint, the method heuristic in the sense that the problem is non-convex and convergence to a global minimum can not be guaranteed. However, we provide a large set of numerical experiments that indicate that usage of the finite rank constraint nevertheless makes the method converge to minima close to the global minimum for reasonably high signal to noise ratios, hence essentially yielding maximum-likelihood parameter estimates. Moreover, the method does not seem to be particularly sensitive to initialization and performs substantially better than standard subspace-based methods. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Alternating direction method of multipliers, frequency estimation, Hankel matrix, irregular sampling, Kronecker's theorem, missing data, spectral estimation
in
IEEE Transactions on Signal Processing
volume
62
issue
21
pages
5761 - 5774
publisher
IEEE--Institute of Electrical and Electronics Engineers Inc.
external identifiers
  • wos:000344464500021
  • scopus:84908025123
ISSN
1053-587X
DOI
10.1109/TSP.2014.2358961
language
English
LU publication?
yes
id
b46e1cbd-888b-4c5a-b0dc-de8080e02886 (old id 4865303)
date added to LUP
2014-12-18 11:32:44
date last changed
2017-08-27 04:46:55
@article{b46e1cbd-888b-4c5a-b0dc-de8080e02886,
  abstract     = {Spectral estimation is an important classical problem that has received considerable attention in the signal processing literature. In this contribution, we propose a novel method for estimating the parameters of sums of complex exponentials embedded in additive noise from regularly or irregularly spaced samples. The method relies on Kronecker's theorem for Hankel operators, which enables us to formulate the nonlinear least squares problem associated with the spectral estimation problem in terms of a rank constraint on an appropriate Hankel matrix. This matrix is generated by sequences approximating the underlying sum of complex exponentials. Unequally spaced sampling is accounted for through a proper choice of interpolation matrices. The resulting optimization problem is then cast in a form that is suitable for using the alternating direction method of multipliers (ADMM). The method can easily include either a nuclear norm or a finite rank constraint for limiting the number of complex exponentials. The usage of a finite rank constraint makes, in contrast to the nuclear norm constraint, the method heuristic in the sense that the problem is non-convex and convergence to a global minimum can not be guaranteed. However, we provide a large set of numerical experiments that indicate that usage of the finite rank constraint nevertheless makes the method converge to minima close to the global minimum for reasonably high signal to noise ratios, hence essentially yielding maximum-likelihood parameter estimates. Moreover, the method does not seem to be particularly sensitive to initialization and performs substantially better than standard subspace-based methods.},
  author       = {Andersson, Fredrik and Carlsson, Marcus and Tourneret, Jean-Yves and Wendt, Herwig},
  issn         = {1053-587X},
  keyword      = {Alternating direction method of multipliers,frequency estimation,Hankel matrix,irregular sampling,Kronecker's theorem,missing data,spectral estimation},
  language     = {eng},
  number       = {21},
  pages        = {5761--5774},
  publisher    = {IEEE--Institute of Electrical and Electronics Engineers Inc.},
  series       = {IEEE Transactions on Signal Processing},
  title        = {A New Frequency Estimation Method for Equally and Unequally Spaced Data},
  url          = {http://dx.doi.org/10.1109/TSP.2014.2358961},
  volume       = {62},
  year         = {2014},
}