Discrete and integral fourier transforms: analytical examples.

Analytical examples of the discrete Fourier transform (DFT) help in understanding relations between the DFT and the Fourier integral transform (FIT). Such examples enable one to estimate the errors involved when one transform is approximated by the other, and they suggest how such approximation errors might be reduced. We present mathematical and numerical analyses of the time-to-frequency DFT of the complex exponential function and of the frequency-to-time inverse DFT of the relaxation function. The FITs of these functions are exact inverses, and so they serve to clarify the effects of aliasing and truncation on the DFT.