Hilbert transforms using fast Fourier transforms

Abstract

Fourier transforms are unreliable near discontinuities because of the Gibbs phenomenon. Before using the fast Fourier transform technique to evaluate Hilbert transforms, it is desirable to remove any discontinuities by smoothing, as suggested by Papoulis (1962). This is especially true if one wishes to use Fourier transforms to find the Hilbert transform h(t) of a function f(t) which has an infinite discontinuity: it is then necessary to smooth f(t) as well as the Hilbert transform kernel. An alternative to smoothing f(t) is to remove a discontinuity at a point t=d by multiplying f(t) by the factor (t-d): this has the advantage of having better asymptotic behaviour. A numerical example is given and here the two methods perform about equally.