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Mixed Spectra Estimation for Stable P-adic Random Fields

Rachid SABRE1, and Walid HORRIGUE2
1.LE2I, Laboratory of University of Burgundy, Agrosup, 26 Av Docteur Petitjean 21000 Dijon, France
2.UMR Agroécologie Equipe Biocom, INRA Dijon, Agrosup, 26 Av Docteur Petitjean 21000 Dijon, France
Abstract—Alpha stable processes with p-adic time are considered. These signals whose variance is very large are often encountered in signal processing. P-adic time is chosen because it is a discrete space and infinitely small which is much requested when one discretizes an image with a high resolution. The problem encountered in the concrete case is that the spectral measurement of such a process is not always absolutely continuous with respect to the Lebesgue measure. In general, it is the sum of a continuous part and a discrete part (some jumps). In this work, we provided a solution to this problem by offering a new method to estimate the spectral density of the continuous part. Indeed, we selected two overlapping windows whose widths are well chosen to vanish the bias at jump points. We showed that the proposed estimator was asymptotically unbiased and consistent. The rate of convergence of our estimator has been studied in order to prove the effectiveness of the method used. 

Index Terms—spectral density, p-adic numbers, stable processes, periodogram, spectral windows

Cite: Rachid SABRE and Walid HORRIGUE, "Mixed Spectra Estimation for Stable P-adic Random Fields," International Journal of Signal Processing Systems, Vol. 7, No. 2, pp. 60-65, June 2019. doi: 10.18178/ijsps.7.2.60-65

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