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Normality of a non-linear transformation of AR parameters: application to reflection and cepstrum coefficients

Tourneret, Jean-Yves Normality of a non-linear transformation of AR parameters: application to reflection and cepstrum coefficients. (1997) Signal Processing, vol. 6 (n° 1). pp. 1-14. ISSN 0165-1684

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Official URL: http://dx.doi.org/10.1016/S0165-1684(97)00112-6

Abstract

Two sets of random vectors cannot both be Gaussian if they are nonlinearly related. Thus, Autoregressive (AR)parameters and reflection coefficient (resp. cepstrum coefficient) estimators cannot both be Gaussian for a finite number of samples. However, most estimators of AR parameters and reflection coefficients (resp. cepstrum coefficients) are Gaussian asymptotically. Thus, the distribution of AR parameter and reflection coefficient (resp. cepstrum coefficient) estimates are close to Gaussian for large samples. This paper studies the ``closeness'' between the Gaussian distribution and the ``non-linear transformation of Gaussian AR parameters'' distribution. A new distance is defined which is based on the Taylor expansion of the non-linear transformation. This ``Taylor'' distance called $M$-distance is used to measure the deviations from the Gaussian distribution of reflection coefficient and cepstrum coefficient statistics. A comparison is presented between this distance and Kullback's divergence. The main advantage of the M-distance with respect to other distances is a very simple closed form expression of the deviations from normality. This closed form expression shows that the convergence of the reflection and cepstrum coefficient distribution to its asymptotic Gaussian distribution (when the number of samples tends to infinity) depends on the position of AR model poles in the unit circle.

Item Type:Article
Additional Information:This publication is available at http://www.sciencedirect.com/science/journal/01651684
Audience (journal):International peer-reviewed journal
Uncontrolled Keywords:
Institution: Université de Toulouse > Institut National Polytechnique de Toulouse - INPT
Université de Toulouse > Université Paul Sabatier-Toulouse III - UPS
French research institutions > Centre National de la Recherche Scientifique - CNRS
Laboratory name:
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Deposited By: Jean-yves TOURNERET

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