Similar basis function algorithm for numerical estimation of Fourier integrals

The methodological difficulties of estimating Fourier integrals using the fast Fourier transform (FFT) algorithm have intensified the interest in an alternative approach based on the Filon’s method of computing the trigonometric integrals. Following this approach, we introduce in this paper a similar basis function (SBF) algorithm that decomposes the function to be transformed into the sum of finite elements termed “similar basis functions”. Due to a simple analytical form of SBF, the reassignment of the SBFs’ similarity relationships into the transformation domain reduces the estimation of the Fourier integrals to a number of standard computational procedures. The SBF algorithm is capable to deal with both uniform and non-uniform samples of the function under analysis. Using this opportunity, we extend a general SBF algorithm by a fast SBF algorithm which deals with exponentially increasing sampling intervals. The efficiency and the accuracy of the method are illustrated by computer experiments with frequency characteristics and transient responses of a typical dynamic system.