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Temporal Data Analysis A.A. 2011/2012 This means: the convolution of a Gaussian distribution in the frequency domain results in exponential “damping” of the cosine term, where the damping happens to be the Fourier transform of the frequency distribution. This, of course, is due to the fact that we have chosen to use a cosine function (i.e. a basis function) for f (t). P(ω) makes sure that oscillations for ω ≠ ω 0 are slightly shifted with respect to each other, and will more and more superimpose each other destructively in the long run, averaging out to 0. Lorentzian frequency distribution If we convolute our signal f (t) = cosω 0 t with a Lorentzian distribution P(ω) = σ 1 π ω 2 + σ 2 then, by following what we have done previously with the Gaussian, we have This is a damped wave. ¯ f (t) = ∫ +∞ −∞ h(t 0 ) = FT −1 ( ¯ f (t)) = e −σt 0 ¯ f (t) = e −σt 0 cosω 0 t Gaussian convoluted with a Gaussian σ 1 π ω 2 + σ 2 cos(ω − ω 0)t dω [ δ(t0 − t) + δ(t ] 0 + t) 2 2 We perform a convolution of a Gaussian with σ 1 with another Gaussian with σ 2 . As the Fourier transforms are Gaussians again — yet with σ 2 1 and σ2 2 in the numerator of the exponent — it’s immediately obvious that σ 2 total = σ2 1 + σ2 2 . Therefore, we get another Gaussian with geometric addition of the widths σ 1 and σ 2 . 124 M.Orlandini
A.A. 2011/2012 Temporal Data Analysis 3.1 Discrete Fourier Transformation — Page 46 Constant function with N = 4 f k = 1 for k = 0,1,2,3 For the continuous Fourier transformation we expect a δ-function with the frequency ω = 0. The discrete Fourier transformation therefore will only result in F 0 ≠ 0. Indeed, by using (3.11) — or even a lot smarter using (3.8): F 0 = 1 4 4 = 1 F 1 = 0 F 2 = 0 F 3 = 0 As {f k } is an even series, {F j } contains no imaginary part. Cosine function with N = 4 f 0 = 1 f 1 = 0 f 2 = −1 f 3 = 0 We get, using (3.11) and W 4 = i F 0 = 0 (this is the average) F 1 = 1 4 (1 + (−1)(−1)) = 1 2 F 2 = 1 (1 + (−1)(1)) = 0 4 F 3 = 1 4 (1 + (−1)(−1)) = 1 2 Sine function with N = 4 M.Orlandini 125
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Temporal Data Analysis: Theory and
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Course Outline Table of Contents Li
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A.A. 2011/2012 Temporal Data Analys
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List of Figures 1.1 Classification
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List of Definitions 2.1 Even and od
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List of Theorems 2.1 Convolution th
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Part I Theory 1
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