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Temporal Data Analysis A.A. 2011/2012 f (t) f (t) t t even odd f (t) f (t) f (t) t t t mixed = even + odd Figure 2.1: Examples of even, odd and mixed functions 12 M.Orlandini
A.A. 2011/2012 Temporal Data Analysis 2.1.1 Definition Any periodic function f (t) can be expanded into a series of trigonometric function, the so called Fourier series, as follows Definition 2.2 (Fourier series). f (t) = ∞∑ (A k cosω k t + B k sinω k t) with ω k = 2πk T , B 0 = 0 (2.1) k=0 T is the period of the function f (t). The amplitudes or Fourier coefficients A k and B k are determined in such a way that the infinite series is identical with the function f (t). Equation (2.1) therefore tells us that any periodic function can be represented as a superposition of sinefunction and cosine-function with appropriate amplitudes – with an infinite number of terms, if need be – yet using only precisely determined frequencies: ω = 0, 2π T , 4π T , 6π T ,... 2.1.2 Calculation of the Fourier Coefficients Before we compute the expressions of the Fourier coefficients, we need some tools. In all following integrals we integrate from −T /2 to +T /2, meaning over an interval with the period T that is symmetrical to t = 0. We could also pick any other interval, as long as the integrand is periodic with period T and gets integrated over a whole period. The letters n and m in the formulas below are natural numbers 0,1,2,... Let’s have a look at the following ∫ +T /2 −T /2 ∫ +T /2 −T /2 cosω n t dt = { 0 for n ≠ 0 T for n = 0 (2.2) sinω n t dt = 0 (2.3) This results from the fact that the areas on the positive half-plane and the ones on the negative one cancel out each other, provided we integrate over a whole number of periods. Cosine integral for n = 0 requires special treatment, as it lacks oscillations and therefore areas can’t cancel out each other: there the integrand is 1, and the area under the horizontal line is equal to the width of the interval T . Furthermore, we need the following trigonometric identities: cosαcosβ = 1 [ ] cos(α − β) + cos(α + β) 2 sinαsinβ = 1 [ ] cos(α − β) − cos(α + β) 2 sinαcosβ = 1 [ ] sin(α − β) + sin(α + β) 2 (2.4) M.Orlandini 13
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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
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Chapter5 Procedures for Analyzing R
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Chapter6 Temporal Analysis in X-ray
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Chapter7 Bibliography Reference Tex
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AppendixA Examples Shown at the Bla
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AppendixB Practical Session on Timi
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