booklet format - inaf iasf bologna

A.A. 2011/2012
Temporal Data Analysis
Figure A.3: The bilateral exponential function (left) and its Fourier transform (right)
Its Fourier transform is
F (ω) =
∫ +∞
−∞
e −|t|/τ e −iωt dt = 2
∫ +∞
0
e −t/τ 2τ
cosωt dt =
1 + ω 2 τ 2
As f (t) is even, the imaginary part of its Fourier transform is null. The Fourier transform of the
bilateral exponential function is a Lorentzian. Both functions are shown in Figure A.3.
Unilateral exponential function
Its Fourier transform is
f (t) =
{ e
−λt
for t ≥ 0
0 else
F (ω) =
=
=
=
∫ +∞
0
e −λt e −iωt dt
e −(λ+iω)t
∣
−(λ + iω)
∣
1
+∞
λ + iω
λ
λ 2 + ω 2 − i ω
λ 2 + ω 2
0
F (ω) is complex, as f (t) is neither even nor odd. We now can write the real and the imaginary
parts separately. The real part has a Lorentzian shape we are familiar with by now, and the
imaginary part has a dispersion shape.
M.Orlandini 121