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Rational approximations of spectral densities based on the Alpha divergence

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Abstract

We approximate a given rational spectral density by one that is consistent with prescribed second-order statistics. Such an approximation is obtained by selecting the spectral density having minimum “distance” from under the constraint corresponding to imposing the given second-order statistics. We analyze the structure of the optimal solutions as the minimized “distance” varies in the Alpha divergence family. We show that the corresponding approximation problem leads to a family of rational solutions. Secondly, such a family contains the solution which generalizes the Kullback–Leibler solution proposed by Georgiou and Lindquist in 2003. Finally, numerical simulations suggest that this family contains solutions close to the non-rational solution given by the principle of minimum discrimination information.

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Acknowledgments

This work was supported by University of Padova under the project “A Unifying Framework for Spectral Estimation and Matrix Completion: A New Paradigm for Identification, Estimation, and Signal Processing”.

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  1. Department of Electrical Engineering and Computer Science, University of Liège, 4000,  Liège, Belgium

    Mattia Zorzi

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  1. Mattia Zorzi
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Correspondence to Mattia Zorzi.

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Zorzi, M. Rational approximations of spectral densities based on the Alpha divergence. Math. Control Signals Syst. 26, 259–278 (2014). https://doi.org/10.1007/s00498-013-0118-2

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  • DOI: https://doi.org/10.1007/s00498-013-0118-2

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