And it’s defined by its nth component, so the nth component of the Fourier transform is the sum from say N equals zero to N minus one of the Nth component of F times E to the minus two pi I N M over N. All right? So everything is defined here in terms of the indices in the exponential, and these are the values of the discrete function at the index points, F of zero, F of one, F of two and so on. That’s the definition. They say you don’t see at all the fact that in our derivation this came from starting with a continuous signal, sampling it, sampling the Fourier transform, and then somehow ultimately leading to this definition. Here it’s just as an operation on one discrete signal producing another discrete signal. ...

See the whole transcript at The Fourier Transform and its Applications - Lecture 20