**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**30309

##### Discrete Estimation of Spectral Density for Alpha Stable Signals Observed with an Additive Error

**Authors:**
R. Sabre,
W. Horrigue,
J. C. Simon

**Abstract:**

This paper is interested in two difficulties encountered in practice when observing a continuous time process. The first is that we cannot observe a process over a time interval; we only take discrete observations. The second is the process frequently observed with a constant additive error. It is important to give an estimator of the spectral density of such a process taking into account the additive observation error and the choice of the discrete observation times. In this work, we propose an estimator based on the spectral smoothing of the periodogram by the polynomial Jackson kernel reducing the additive error. In order to solve the aliasing phenomenon, this estimator is constructed from observations taken at well-chosen times so as to reduce the estimator to the field where the spectral density is not zero. We show that the proposed estimator is asymptotically unbiased and consistent. Thus we obtain an estimate solving the two difficulties concerning the choice of the instants of observations of a continuous time process and the observations affected by a constant error.

**Keywords:**
periodogram,
spectral density,
stable processes,
aliasing

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.3607870

**References:**

[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] R. Sabre (2019) “Aliasing Free and Additive Error in Spectra for Alpha Stable Signals”, International Journal of Electrical and Computer Engineering, Vol. 13, No. 10, pp. 668-673.

[28] E. Masry, (1978). “Alias-free sampling: An alternative conceptualization and its applications”, IEEE Trans. Information theory, Vol. 24, pp.317-324.

[29] 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.