booklet format - inaf iasf bologna

Temporal Data Analysis A.A. 2011/2012
Please note that |F (ω)| is no Lorentzian! If you want to “stick” to this property, you better represent
the square of the magnitude: |F (ω)| 2 = 1/(λ 2 +ω 2 ), that is a Lorentzian. This representation
is often also called the power representation: |F (ω)| 2 = (real part) 2 +(imaginary part) 2 . The phase
goes to 0 at the maximum of |F (ω)|, i.e. when “in resonance”.
Warning: The representation of the magnitude as well as of the squared magnitude does away
with the linearity of the Fourier transformation!
Finally, let us try out the inverse transformation and find out how we return to the “unilateral”
exponential function (the Fourier transform did not look all that “unilateral”!):
∫ +∞
f (t) = 1 λ − iω
2π −∞ λ 2 + ω 2 e+iωt dω
= 1 { ∫ +∞
∫
cosωt
+∞
}
2λ
2π 0 λ 2 + ω 2 dω + 2 ωsinωt
0 λ 2 + ω 2 dω
= 1 { π
π 2 e−|λt| ± π 2 e−|λt|} where + for t ≥ 0
− else
{ e
−λt
for t ≥ 0
=
0 else
is valid
122 M.Orlandini