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Aliasing Free and Additive Error in Spectra for Alpha Stable Signals

Authors: R. Sabre


This work focuses on the symmetric alpha stable process with continuous time frequently used in modeling the signal with indefinitely growing variance, often observed with an unknown additive error. The objective of this paper is to estimate this error from discrete observations of the signal. For that, we propose a method based on the smoothing of the observations via Jackson polynomial kernel and taking into account the width of the interval where the spectral density is non-zero. This technique allows avoiding the “Aliasing phenomenon” encountered when the estimation is made from the discrete observations of a process with continuous time. We have studied the convergence rate of the estimator and have shown that the convergence rate improves in the case where the spectral density is zero at the origin. Thus, we set up an estimator of the additive error that can be subtracted for approaching the original signal without error.

Keywords: Spectral density, stable processes, aliasing, p-adic.

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[1] S. Cambanis (1983) “Complex symetric stable variables and processes” In P.K.SEN, ed, Contributions to Statistics”: Essays in Honour of Norman L. Johnson North-Holland. New York,(P. K. Sen. ed.), pp. 63-79
[2] S. Cambanis, and M. Maejima (1989). “Two classes of self-similar stable processes with stationary increments”. Stochastic Process. Appl. Vol. 32, pp. 305-329
[3] M.B. Marcus and K. Shen (1989). “Bounds for the expected number of level crossings of certain harmonizable infinitely divisible processes”. Stochastic Process. Appl., Vol. 76, no. 1 pp 1-32.
[4] E. Masry, S. Cambanis (1984). “Spectral density estimation for stationary stable processes”. Stochastic processes and their applications, Vol. 18, pp. 1-31 North-Holland.
[5] G. Samorodnitsky and M. Taqqu (1994). “Stable non Gaussian processes ». Chapman and Hall, New York.
[6] K., Panki and S. Renming (2014). “Stable process with singular drift”. Stochastic Process. Appl., Vol. 124, no. 7, pp. 2479-2516
[7] C. Zhen-Qing and W. Longmin (2016). “Uniqueness of stable processes with drift.” Proc. Amer. Math. Soc., Vol. 144, pp. 2661-2675
[8] K. Panki, K. Takumagai and W. Jiang (2017). “Laws of the iterated logarithm for symmetric jump processes”. Bernoulli, Vol. 23, n° 4 pp. 2330-2379.
[9] M. Schilder (1970). “Some Structure Theorems for the Symmetric Stable Laws”. Ann. Math. Statist., Vol. 41, no. 2, pp. 412-421.
[10] R. Sabre (2012b). “Missing Observations and Evolutionary Spectrum for Random”. International Journal of Future Generation Communication and Networking, Vol. 5, n° 4, pp. 55-64.
[11] E. Sousa (1992). “Performance of a spread spectrum packet radio network link in a Poisson of interferences”. IEEE Trans. Inform. Theory, Vol. 38, pp. 1743-1754
[12] M. Shao and C.L. Nikias (1993). “Signal processing with fractional lower order moments: Stable processes and their applications”, Proc. IEEE, Vol.81, pp. 986-1010
[13] C.L. Nikias and M. Shao (1995). “Signal Processing with Alpha-Stable Distributions and Applications”. Wiley, New York
[14] S. Kogon and D. Manolakis (1996). “Signal modeling with self-similar alpha- stable processes: The fractional levy motion model”. IEEE Trans. Signal Processing, Vol 44, pp. 1006-1010
[15] N. Azzaoui, L. Clavier, R. Sabre, (2002). “Path delay model based on stable distribution for the 60GHz indoor channel” IEEE GLOBECOM, IEEE, pp. 441-467
[16] J.P. Montillet and Yu. Kegen (2015). “Modeling Geodetic Processes with Levy alpha-Stable Distribution and FARIMA”, Mathematical Geosciences. Vol. 47, no. 6, pp. 627-646.
[17] M. Pereyra and H. Batalia (2012). “Modeling Ultrasound Echoes in Skin Tissues Using Symmetric alpha-Stable Processes”. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, Vol. 59, n°. 1, pp. 60-72.
[18] X. Zhong and A.B. Premkumar (2012). “Particle filtering for acoustic source tracking in impulsive noise with alpha-stable process”. IEEE Sensors Journal, Vol. 13, no. 2, pp. 589 - 600.
[19] Wu. Ligang and W. Zidong (2015). “Filtering and Control for Classes of Two-Dimensional Systems”. The series Studies in Systems of, Decision and Control, Vol.18, pp. 1-29.
[20] N. Demesh (1988). “Application of the polynomial kernels to the estimation of the spectra of discrete stable stationary processes”. (Russian) Akad.Nauk.ukrain. S.S.R. Inst.Mat. Preprint 64, pp. 12-36
[21] F. Brice, F. Pene, and M. Wendler, (2017) “Stable limit theorem for U-statistic processes indexed by a random walk”, Electron. Commun. Prob., Vol. 22, no. 9, pp.12-20.
[22] R. Sabre (2019). “Alpha Stable Filter and Distance for Multifocus Image Fusion”. IJSPS, Vol. 7, no. 2, pp. 66-72.
[23] JN. Chen, J.C. Coquille, J.P. Douzals, R. Sabre (1997). “Frequency composition of traction and tillage forces on a mole plough”. Soil and Tillage Research, Vol. 44, pp. 67-76.
[24] R. Sabre (1995). “Spectral density estimate for stationary symmetric stable random field”, Applicationes Mathematicaes, Vol. 23, n°. 2, pp. 107-133
[25] R. Sabre (2012a). “Spectral density estimate for alpha-stable p-adic processes”. Revisita Statistica, Vol. 72, n°. 4, pp. 432-448.
[26] R. Sabre (2017). “Estimation of additive error in mixed spectra for stable prossess”. Revisita Statistica, Vol. LXXVII, n°. 2, pp. 75-90.
[27] E. Masry, (1978). “Alias-free sampling: An alternative conceptualization and its applications”, IEEE Trans. Information theory, Vol. 24, pp.317-324.
[28] R. Sabre (1994). « Estimation de la densité de la mesure spectrale mixte pour un processus symétrique stable strictement stationnaire », C. R. Acad. Sci. Paris, Vol. 319, série I, pp. -1310.
[29] L. Clavier, M. Rachdi, Y. Delignon, V. Letuc, C. Garnier, P.A. Roland (2001). “Wide band 60GHz indoor channel: characterization and statistical modelling“. IEEE 54th VTC fall, Atantic City, NJ USA, 7-11October, 2001.
[30] A. Janicki and A. WERON (1993). “Simulation and Chaotic Behavior of Alpha-stable Stochastic Processes”. Series: Chapman and Hall/CRC Pure and Applied Mathematics, New York.