[math-fun] Really stupid physics question

We've all seen the classical analysis of a vibrating string, which leads to classical Fourier theory. In the 20th century, we all became converts to DFT's (Discrete Fourier Transforms). So.... I'd like to see the exact mechanical counterpart to the DFT. Given N, the _length_ of the DFT, how do I arrange for N masses & (?) Hookean springs so that the dynamics give me exactly the DFT ? In particular, for very small N (N<10, nowhere near the continuum case), what do such systems look like?