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Temporal Data Analysis A.A. 2011/2012
Figure A.2: The normalized Gaussian function (left) and its Fourier transform (right), that is
another Gaussian. Note that the Fourier transform is not normalized.
2.5 Fourier Transformation of Relevant Functions — Page 24
Normalized Gaussian function
The prefactor is chosen in such a way that the area under the function is normalized to unity.
Its Fourier transform is
f (t) = 1
σ 1 2π e− 2
t 2
σ 2
∫ +∞
e − 1 t 2
2
−∞
∫ +∞
e − 1 t 2
2 σ 2
0
1
F (ω) =
σ 2π
2
=
σ 2π
= exp
(− 1 )
2 σ2 ω 2
σ 2 e −iωt dt
cosωt dt
Again, the imaginary part is null because the function is even. The Fourier transform of a
Gaussian results to be another Gaussian. Note that the Fourier transform is not normalized to
unit area.
f (t) has σ in the exponent denominator, while F (ω) has it in the exponent numerator: the
slimmer f (t), the wider F (ω) and vice versa, as shown in Figure A.2.
The bilateral exponential function
f (t) = e −|t|/τ
120 M.Orlandini