Signal Processing

Transform theory
Example: Fourier transform of the Gaussian pulse
{ √
F e −at2} π ω2
= e− 4a
a
Proof: Notice that x(t) = e −at2 is a solution to
∂x(t)
∂t
+ 2atx(t) = 0
Take the Fourier transform and use F{ ∂x(t) } = iωX(ω) and F{tx(t)} = i ∂X(ω)
∂t
∂ω
iωX(ω) + 2ai ∂X(ω)
∂ω
Parseval’s theorem
= 0 ⇒ ∂X(ω) ( ) 1
∂ω + 2 ωX(ω) = 0 ⇒ X(ω) = Ce − 4a 1 ω2
4a
∫ ∞
−∞
∫ ∞
x 2 (t) dt = e −2at2 dt = 1
−∞
2π
= {ω = 2at, dω = 2a dt} = C2
2π
∫ ∞
−∞
∫ ∞
−∞
∫
X 2 (ω) dω = C2 ∞
e − ω2
2a dω
2π −∞
e −2at2 2a dt ⇒ 1 = C2
2π 2a ⇒ C = √ π
a
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 26(28)