# Understanding the 2D discrete Fourier transform

I just learned about the 1-dimensional discrete Fourier transform which got introduced to me by means of circulant matrices (in the context of convolutions) and noticing that they all have the same eigenvectors. Now I want to learn about the 2D Fourier transform but I couldn't find an explanation so far that satisfied me (most of them solely state the formula without deriving it), which is why I'm asking whether someone knows a good source where the 2D discrete Fourier transform gets properly derived/developed.

• You could try to review a book about Optics, like "Fourier Optics" by J. Goodman.. the continuous version of the 2D Fourier Transform is important in Optics since the Far Field profile of a light beam is the 2D Fourier Transform of the incoming beam, as is also the profile that happens in the focal plane of a aspherical lens, so you could in practice to calculate and apply filters in the Fourier Plane doing calculations at speedlight because of this, also taking cross-correlations. In the same book, I believe is constructed the discrete case from the continuous case, to use it in simulations. Nov 7, 2021 at 21:51
• One can consider 2D Fourier transform as a sequence of two 1-dimensional discrete Fourier transforms: applied to the first variable and then to the second. The properties follow immediately. Not much new in the multidimensional case. Nov 8, 2021 at 5:38