Hilbert Transform: A Brief Overview

This article is pedagogical. It briefly outlines the historical background of the Hilbert transform and then elaborates on some of its attributes, including the convolution and spectral characteristics associated with an analytic function having a compact support. The article covers three algorithms: (1) the Fourier spectral method, (2) the method of algebraic convolution, and (3) the quadrature method to estimate the Hilbert transform of a discrete data set. An important transform that is closely related to the Hilbert transform, namely the Kramers–Kronig transform (KKT), is also discussed. Examples of some Hilbert transform applications are mentioned, and the results of numerical experiments on three specific algorithms are presented.