booklet format - inaf iasf bologna

More documents

Recommendations

Info

Temporal Data Analysis A.A. 2011/2012 convenient signals. Moreover, there are no simple, even linear stochastic differential equations generating signals with 1/f noise. The widespread occurrence of signals exhibiting such behavior suggests that a generic mathematical explanation might exist. Except for some formal mathematical descriptions like fractional Brownian motion (half-integral of a white noise signal), however, no generally recognized physical explanation of 1/f noise has been proposed. Consequently, the ubiquity of 1/f noise is one of the oldest puzzles of contemporary physics and science in general. The case of α = 1, or pink noise, is both the canonical case, and the one of most interest, but the more general form, where 0 < α ≤ 3, is sometimes referred to simply as 1/f . 1/f α noise is of interest because it occurs in many different systems, both in the natural world and in man-made processes (see Figure 6.8) from physics, biology, neuroscience, and psychology. Although 1/f noise appears in many natural systems and has been intensively studied for decades with many attempts to describe the phenomenon mathematically, researchers have not yet been able to agree on a unified explanation. Thus, there exist at present several formulations of systems that give rise to S(f ) = K /f α . 6.6.2 Shot Noise Process First, let t k be a Poisson point process. A shot noise process is obtained by attaching to each t k a relaxation function (unilateral exponential function) N (t) = N 0 e −λt , t ≥ 0 and summing on k (see Figure 6.9). The Fourier transform of the shot noise process is (see Page 25) 1 S(f ) = lim T →∞ T 〈| F (f ) |2 〉 = N 2 0 n λ 2 + f 2 where n is the average rate at which t k occur, and T is the interval over which the process is observed. As we have already seen, the power spectrum of an unilateral exponential function is a Lorentzian function. For an aggregation of shot noise processes with λ uniformly distributed on [λ 1 ,λ 2 ], the power spectrum is ⎧ N 2 0 n if 0 ≪ f ≪ λ 1 ≪ λ 2 ⎪⎨ S(f ) = ⎪⎩ N0 2nπ 2f (λ 2 − λ 1 ) · 1 f N 2 0 n · 1 f 2 if λ 1 ≪ f ≪ λ 2 if 0 ≪ λ 1 ≪ λ 2 ≪ f If the impulse response function is a power law, N 0 x −β , the process is called fractal shot noise, and the power spectrum is of the form 92 M.Orlandini
A.A. 2011/2012 Temporal Data Analysis Figure 6.7: Examples of 1/f noise: on the left the time series and on the right the corresponding power spectrum Figure 6.8: Examples of 1/f noise observed in both in the natural world and in man-made processes from physics, biology, neuroscience, and psychology M.Orlandini 93
Page 1 and 2:
Temporal Data Analysis: Theory and
Page 3 and 4:
Course Outline Table of Contents Li
Page 5 and 6:
A.A. 2011/2012 Temporal Data Analys
Page 7 and 8:
List of Figures 1.1 Classification
Page 9 and 10:
List of Definitions 2.1 Even and od
Page 11 and 12:
List of Theorems 2.1 Convolution th
Page 13:
Part I Theory 1
Page 16 and 17:
Temporal Data Analysis A.A. 2011/20
Page 18 and 19:
Page 20 and 21:
Page 22 and 23:
Page 24 and 25:
Page 26 and 27:
Page 28 and 29:
Page 30 and 31:
Page 32 and 33:
Page 34 and 35:
Page 36 and 37:
Page 38 and 39:
Page 40 and 41:
Page 42 and 43:
Page 44 and 45:
Page 46 and 47:
Page 48 and 49:
Page 50 and 51:
Page 52 and 53:
Page 54 and 55: Temporal Data Analysis A.A. 2011/20
Page 83 and 84: Chapter5 Procedures for Analyzing R
Page 85 and 86: A.A. 2011/2012 Temporal Data Analys
Page 91 and 92: Chapter6 Temporal Analysis in X-ray
Page 103: A.A. 2011/2012 Temporal Data Analys
Page 121 and 122: Chapter7 Bibliography Reference Tex
Page 123 and 124: AppendixA Examples Shown at the Bla
Page 145 and 146: AppendixB Practical Session on Timi
Page 155 and 156:
Page 157 and 158:
Page 159 and 160:
Page 161 and 162:
Page 163 and 164:
Page 165 and 166:
Page 167 and 168:
Page 169 and 170:
Page 171 and 172:
Page 173 and 174:
Page 175 and 176:
Page 177 and 178:
Page 179:
show all

booklet format - inaf iasf bologna

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?