On this page, we'll make use of the
shifting property and the
scaling property of the Fourier Transform to
obtain the Fourier Transform of the scaled Gaussian function given by:
In Equation [1], we must assume
To start the process of finding the Fourier Transform of [1], let's recall the
fundamental Fourier Transform pair, the Gaussian:
Let's first define the function h(z):
Observe that we have defined the constant
In Equation [4], we have assumed
And thus, we have found the Fourier Transform of Equation [1]! Have a good day everybody.

[Equation 1]
*K*>0 or the function g(z) won't be a Gaussian function (rather, it will grow without
bound and therefore the Fourier Transform will not exist).
[Equation 2]
[Equation 3]
*c*=sqrt( 4**pi***K* ). We can relate the function
h(z) and n(z) by the simple relation: *h(z)=n(cz)*. Since we know the Fourier Transform of
*n(z)* (Equation [2]), we can use
the scaling property of the Fourier Transform
to get the Fourier Transform of *h(z)*:
[Equation 4]
*K* (and hence *c*) is positive. To find *G(f)*,
the Fourier Transform of *g(z)*, we note that *g(z) = h(z-a)*, and use
the shift property of the Fourier Transform:
[Equation 5]

Fourier Transform of a Gaussian

Fourier Transform Applications

Fourier Transform (Home)