More on the Fourier Transforms

Convolution

We have glimpsed this property in the previous section. In this section, we further explore the effect of convolution in the Fourier domain. Starting with two dots/dirac deltas in the Fourier domain (the inverse FT of which we know will be sinusoidal), we convolve different functions onto the dots and observe the effect on the inverse Fourier Transform.

Function 1: Circle

For the circle, we observe the effect of circle radius and dot spacing on the inverse Fourier Transform. (forgive the error in the plot titles)

Small circles 20 pixels apartpart

Medium circles 100 pixels apart

Large circles 100 pixels apart

The first noticeable thing is the anamorphism of the dots; the larger the circles, the smaller the appearance of the inverse Fourier Transform. In addition, the larger the spacing of the dots, the finer the fringes in the inverse Fourier Transform.

The second noticeable thing is that the inverse fourier transform is the product of the individual fourier signals. That is, knowing that the inverse of a circle results in an airy pattern, and that the inverse of two dots results in a sine, the multiplication of the two results in an airy pattern superimposed with fringes, which is observed. This is further proof that convolution in the fourier domain results in multiplication in real space.

Function 2: Gaussian

(Again forgive the plot titles).

Left: Gaussian convolved onto two dots. Right: inverse fourier transform of the left image.

Left: Gaussian with larger spread convolved onto two dots. Right: Inverse FT, which appears like a single gaussian due to the close proximity of the two gaussians.

Again, a convolution in Fourier space becomes multiplication in real space. The inverse FT of a gaussian is also a (anamorphic) gaussian, while the inverse FT of two dots is a sinusoid. Hence, the result is a gaussian superimposed by fringes, which is observed.

Function 3: Square

Lastly, the effect of convolving a square onto the two dots is investigated.

Small squares

Big square

Anamorphism is again evident, in that small squares result in larger inverse Fourier transform patterns. Also, the superposition of the square pattern and fringes is apparent.

Another thing of note is that the inverse Fourier transform of the larger squares close together resembles the inverse Fourier transform of a wide rectangle. This is because the spacing between the two squares is small enough such that it becomes almost negligible.