Evaluation of Fourier Transforms

Abstract

This paper deals with numerical methods as well as with the use of digital and analog computer methods for the evaluation of Fourier integrals. Graphical methods of integration are described which enable one to obtain arbitrarily good accuracy with a minimum amount of computational effort through the use of the Fourier integral property that permits a representation in terms of an appropriate higher derivative of the real part of the transform. This method, which makes use of impulse trains, can be applied in such a way as to take proper account of the pertinent asymptotic behavior of the transform, even though the integration for practical purposes is extended over only finite frequency intervals. A simple error criterion is given for estimating the accuracy of a contemplated calculation. Computational economy is achieved through letting the nonuniform spacing of impulses, in the impulse train representation, be dictated by the nature of the function dealt with, rather than choosing an arbitrary uniform interval as is done in many conventional methods of graphical procedure. If digital computers are used in this kind of numerical evaluation, then economy of impulses is of no consequence, and one achieves a simplification of the programing through using an arbitrary uniform spacing. Other than this, the digital computer process is fundamentally like the numerical method already discussed. Analog methods belong in a different category. Here circuit analyzers, as well as the potential analog provided by the electrolytic tank methods, are presented and fields of usefulness discussed.