I would like to understand the discrete Fourier transform (DFT) from first principles. I recently learned that the DFT can be derived as an instance of the change of basis formula. This was much more satisfying than seeing the formula simply stated in a definition.
But there is one piece of the puzzle that still seems to be taken for granted where ever the DFT is discussed. Where did these basis vectors come from? I understand that they are generated using the complex roots of unity. But who discovered that these were the discrete set of frequencies that would form an orthogonal basis? Is there some insight that makes the complex roots of unity an obvious choice for generating them?
[Edit] I think I read recently that the complex roots of unity generate a set of frequencies with an integer number of cycles in the signal length. But it wasn't obvious to me why this would be important.