Frequency Estimation Based on Hankel Matrices and the Alternating Direction Method of Multipliers

Fredrik Andersson (1) , Marcus Carlsson (1) , Jean-Yves Tourneret (2, 3) , Herwig Wendt (2, 4)

1 Centre for Mathematical Sciences [Lund]
2 IRIT-MINDS - CoMputational imagINg anD viSion
3 Toulouse INP - Institut National Polytechnique (Toulouse)
4 CNRS - Centre National de la Recherche Scientifique

We develop a parametric high-resolution method for the estimation of the frequency nodes of linear combinations of complex exponentials with exponential damping. We use Kronecker’s theorem to formulate the associated nonlinear least squares problem as an optimization problem in the space of vectors generating Hankel matrices of fixed rank. Approximate solutions to this problem are obtained by using the alternating direction method of multipliers. Finally, we extract the frequency estimates from the con-eigenvectors of the solution Hankel matrix. The resulting algorithm is simple, easy to implement and can be applied to data with equally spaced samples with approximation weights, which for instance allows cases of missing data samples. By means of numerical simulations, we analyze and illustrate the excellent performance of the method, attaining the Craḿer-Rao bound.

Mots clés: Frequency estimation; Nonlinear least squares; Hankel matrices; Kronecker’s theorem; Missing data; Alternating direction method of multipliers

Domaines: Synthèse d'image et réalité virtuelle [cs.GR]; Traitement du signal et de l'image [eess.SP]; Traitement des images [eess.IV]; Vision par ordinateur et reconnaissance de formes [cs.CV]; Intelligence artificielle [cs.AI]

Fredrik Andersson, Marcus Carlsson, Jean-Yves Tourneret, Herwig Wendt. Frequency Estimation Based on Hankel Matrices and the Alternating Direction Method of Multipliers. 21st European Signal and Image Processing Conference (EUSIPCO 2013), EURASIP : European Association For Signal Processing, Sep 2013, Marrakech, Morocco. pp.1-5. ⟨hal-03669328⟩