Gaussian(f)

Gaussian(t)

Begin with

(2.1)

Complete the square in the integral

(2.2)

Define

 

(2.3)

Note that the width in time space is the 1/(πw) so that wide Gaussians in the frequency domain become narrow lines in the time domain.

            Note also that the only place the frequency remains is in the phase.

Gaussian(t)

Gaussian(x)

Begin with

    (2.1)

Complete the square in the integral

(2.2)

Define

 

(2.3)

Note that the width in time space is the 1/(πw) so that wide Gaussians in the frequency domain become narrow lines in the time domain.

            Note also that the only place the original time t₀ remains is in the phase.