On the physical relevance of the discrete Fourier transform

Abstract

This paper originated from the author's dissatisfaction with the way the discrete Fourier transform is usually presented in the literature. Although mathematically correct, the physical meaning of the common representation is unsatisfactory, and no direct relationship exists with the continuous Fourier transform and the Fourier series. The authors present the discrete Fourier transform in a form that is physically relevant and relates obviously to the continuous Fourier transformation and the Fourier series. They also discuss some consequences of this form for the smoothing and filtering of the Fourier expansion. In a brief section on applications the authors discuss the usefulness of this discrete Fourier transform for interpolation purposes and its limitations, in particular the problem of aliasing. They then comment on the application of the fast Fourier transform (FFT) methods in the new context.