Author: Bastien Milani
The essential end-points of this course are:
- One line of code that serves to call the FFT algorithm in order to evaluate the discrete Fourier transform (DFT) of a sampled function with the correct scaling factors which take the sampling grids into accounts, and another line of code to do the same with the inverse DFT,
- A formula that gives the exact relation between the Fourier transform of a function on the one hand, and the estimate of that Fourier transform as given by the DFT on the other hand.
The second points requires some knowledge about convolutions and some carefully defined variants of the Dirichlet kernel, both explained in the course. The way we define the DFT, its inverse, and its adjoint, takes into account the step size of the sampling grids in X-space and K-space. These step sizes are introduced in the definition of the inner products on X-space and K-space so that the Parseval-Plancherel identity can be stated in an elegant way.