Estimation of Spectral Exponent Parameter of Process in Additive White Background Noise
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Research Article Estimation of Spectral Exponent Parameter of 1/ f Process in Additive White Background Noise ¨ ¨ 1, 2 and Semih Ergintav2 Suleyman Baykut,1 Tayfun Akgul, 1 Department 2 TUB ¨ ITAK ˙
of Electronics and Communications Engineering, Istanbul Technical University, 34469 Maslak, Istanbul, Turkey Marmara Research Center, Earth and Marine Sciences Institute, 41470 Gebze, Kocaeli, Turkey
Received 29 September 2006; Revised 5 February 2007; Accepted 29 April 2007 Recommended by Abdelhak M. Zoubir An extension to the wavelet-based method for the estimation of the spectral exponent, γ, in a 1/ f γ process and in the presence of additive white noise is proposed. The approach is based on eliminating the effect of white noise by a simple difference operation constructed on the wavelet spectrum. The γ parameter is estimated as the slope of a linear function. It is shown by simulations that the proposed method gives reliable results. Global positioning system (GPS) time-series noise is analyzed and the results provide experimental verification of the proposed method. Copyright © 2007 S¨uleyman Baykut et al. 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.
1.
INTRODUCTION
1/ f γ processes, also referred to as self-similar processes, are observed in many diverse fields and have gained importance in various signal processing applications from geophysical records to biomedical signals, from economical indicators to internet network traffic [1–6]. 1/ f γ processes are generally characterized by a power-law relationship in the frequency domain, that is, the empirical (or measured) power spectra of such processes are considered to be of the form [1] σ2 Sx (ω) ∼ x γ |ω |
(1)
over some decades of frequency ω, where σx2 is a finite nonzero constant and γ is the so-called spectral exponent (or sometimes it is called the self-similarity parameter). In general, 1/ f γ processes can be modeled by fractional Gaussian noise (fGn) and fractional Brownian motion (fBm). fBms are zero-mean, normally distributed, nonstationary random processes with 1 < γ < 3, whereas fGns are zero-mean, normally distributed, stationary incremental processes of fBms with −1 < γ < 1 [1, 7]. 1/ f γ processes are also named as colored noise. White noise having a flat spectrum is the special case of colored noise, where the spectral exponent γ = 0. For
γ = 1, it is called flicker noise and for γ = 2, it is known as classical Brownian motion (random walk process). The importance of such processes is due to the fact that they can be modeled by a single parameter γ which can be used for diagnosis, prediction, and control purposes in many applications. Therefore, an accurate estimation of γ is needed. However, estimation of this parameter is not often straightforward, especially when the data is considered to be corrupted by additive white noise. For this case, the measured power spectrum Sx (ω
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