Accurate Numerical Fourier Transform in d-Dimensions

Abstract

The classical method of numerically computing Fourier transforms of digitized functions in one or in d-dimensions is the so-called Discrete Fourier Transform (DFT) efficiently implemented as Fast Fourier Transform (FFT) algorithms. In many cases, the DFT is not an adequate approximation of the continuous Fourier transform. Because the DFT is periodical, spectrum aliasing may occur. The method presented in this contribution provides accurate approximations of the continuous Fourier transform with similar time complexity. The assumption of signal periodicity is no longer posed and allows to compute numerical Fourier transforms in a broader domain of frequency than the usual half-period of the DFT. The aliasing introduced by periodicity can be reduced to a negligible level even with a relatively low number of sampled data points. In addition, this method yields accurate numerical derivatives of any order and polynomial splines of any odd order with their optimum boundary conditions. The numerical error on results is easily estimated. The method is developed in one and in d-dimensions and numerical examples are presented.

Normand Beaudoin is a post-doctoral fellow at the Department of Computer Science, Middlesex College, University of Western Ontario, London, Canada, N6A 5B7. Phone: (519) 661-2111.

Steven S. Beauchemin is with the Department of Computer Science, Middlesex College, University of Western Ontario, London, Canada, N6A 5B7. Phone: (519) 661-2111.