Calculating numerical derivatives using Fourier transform: some pitfalls and how to avoid them

Abstract

Numerical differentiation is commonly used by a number of science students and researchers for data analysis. The differentiation of vectors of data points representing discrete samples of some underlying signal can be implemented in a computer using the central differencing scheme or the fast Fourier transform (FFT)-based approach. We point out that a simple extension of the continuous Fourier transform derivative identity to the discrete case, however, gives rise to results that are inconsistent with the central differencing scheme. In particular, for functions with step-like discontinuities, the numerical derivative computed with the FFT-based approach is corrupted by ringing artifacts. We describe the idea of a modified wave number for FFT-based numerical differentiation which leads to results that are consistent with the central differencing scheme. Modified wave number identities for numerical computation of both first and second order derivatives are described. The pitfalls of using a simplistic extension of the derivative identity for continuous Fourier transform to the discrete case and the methodology to avoid them are illustrated with numerical examples.