Fourier Transform of the Gaussian. Konstantinos G. Derpanis. October 20, 2005. In this note we consider the Fourier tran
Fourier Transform of the Gaussian Konstantinos G. Derpanis October 20, 2005 In this note we consider the Fourier transform1 of the Gaussian. The Gaussian function, g(x), is defined as, −x2 1 √ e 2σ2 , σ 2π
g(x) = where by:
R∞ −∞
(3)
g(x)dx = 1 (i.e., normalized). The Fourier transform of the Gaussian function is given G(ω) = e−
ω2 σ2 2
.
(4)
Proof: We begin with differentiating the Gaussian function: x dg(x) = − 2 g(x) dx σ Next, applying the Fourier transform to both sides of (5) yields, iωG(ω) = dG(ω) dω
1 dG(ω) iσ 2 dω
= −ωσ 2 .
G(ω)
(5)
(6) (7)
Integrating both sides of (7) yields, Zω
dG(ω 0 ) dω 0 G(ω 0 )
Zω
0
dω = −
0
ω 0 σ 2 dω 0
(8)
0 2 2
σ ω . 2 Since the Gaussian is normalized, the DC component G(0) = 0, thus (9) can be rewritten as, ln G(ω) − ln G(0) =
ln G(ω) = −
σ2 ω2 2
(9)
(10)
Finally, applying the exponent to each side yields, eln G(ω) = e−
