A New Frequency Estimation Method for Equally and Unequally Spaced Data

Fredrik Andersson, Marcus Carlsson, Jean-Yves Tourneret, Herwig Wendt

Research output: Contribution to journalArticlepeer-review

37 Citations (SciVal)

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.
Original languageEnglish
Pages (from-to)5761-5774
JournalIEEE Transactions on Signal Processing
Volume62
Issue number21
DOIs
Publication statusPublished - 2014

Subject classification (UKÄ)

  • Mathematics

Keywords

  • Alternating direction method of multipliers
  • frequency estimation
  • Hankel matrix
  • irregular sampling
  • Kronecker's theorem
  • missing data
  • spectral estimation

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