The fast Fourier transform algorithm: Programming considerations in the calculation of sine, cosine and Laplace transforms☆

Abstract In the organization of programming packages for computing Fourier and Laplace transforms, it is useful, both for conceptual understanding and for operational efficiency to consider the discrete complex Fourier transform as a kind of nucleus around which programming for special applications is performed. An advantage of these procedures is that the basic complex Fourier transform algorithm is systematic and can relatively easily be implemented in efficient subroutines, micro-programs and special hardware devices. Once this is done, programming for special properties of the data can efficiently be left to the user to implement on a general purpose computer. The problem of establishing the correspondence between the discrete transforms and the continuous functions with which one is usually dealing is described. The application of these results and the above-mentioned subroutines to the calculation and inversion of Laplace transforms is given with formulas and empirical results displaying the effect of optimal parameters on computational efficiency and accuracy.