Optimal Multitaper Wigner Spectrum Estimation of a Class of Locally Stationary Processes Using Hermite Functions
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Research Article Optimal Multitaper Wigner Spectrum Estimation of a Class of Locally Stationary Processes Using Hermite Functions Maria Hansson-Sandsten (EURASIP Member) Mathematical Statistics, Centre for Mathematical Sciences, Lund University, P.O. Box 118, 221 00 Lund, Sweden Correspondence should be addressed to Maria Hansson-Sandsten, [email protected] Received 24 June 2010; Revised 27 October 2010; Accepted 3 December 2010 Academic Editor: Haldun emduh Ozaktas Copyright © 2011 Maria Hansson-Sandsten. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper investigates the time-discrete multitapers that give a mean square error optimal Wigner spectrum estimate for a class of locally stationary processes (LSPs). The accuracy in the estimation of the time-variable Wigner spectrum of the LSP is evaluated and compared with other frequently used methods. The optimal multitapers are also approximated by Hermite functions, which is computationally more efficient, and the errors introduced by this approximation are studied. Additionally, the number of windows included in a multitaper spectrum estimate is often crucial and an investigation of the error caused by limiting this number is made. Finally, the same optimal set of weights can be stored and utilized for different window lengths. As a result, the optimal multitapers are shown to be well approximated by Hermite functions, and a limited number of windows can be used for a mean square error optimal spectrogram estimate.
1. Introduction A locally stationary process (LSP) has a covariance function which is a multiplication of a covariance function of a stationary process and a time-variable function, [1]. The process is nonstationary with properties suitable for modeling measured signals that, for example, have a transient amplitude behavior, such as evoked or induced potentials arising in the electroencephalogram, [2, 3]. To statistically differentiate between responses from different types of stimuli, choosing a spectral estimator with small bias and low variance is important. Such estimators can be found but additionally important is that the estimators have discrete-time and discrete-frequency properties suitable from implementation aspects, such as few choices of parameters and computational efficiency. The mean square error optimal kernel for the class of Gaussian harmonizable processes has been obtained by Sayeed and Jones [4], and further, the optimal time-frequency kernel for LSPs, restricted to covariance functions defined by a multiplication of two variable Gaussian functions, is obtained in [5]. The calculation of the two-dimensional convolution between the kernel and the Wigner distribution
of a process realization can be simplified using kernel decomposition and calculating multitaper spectrograms, [6, 7]. The time-lag estimation kernel is rotated, and the corresponding eigenvectors and
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