Infrared intensities of liquids. IX. The Kramers–Kronig transform, and its approximation by the finite Hilbert transform via fast Fourier transforms

Abstract

. Can. J . Chem. 70, 520 (1992). It is well known that the infinite Kramers-Kronig transform is equivalent to the infinite Hilbert transform, which is equivalent to the allied Fourier integrals. The Hilbert transform can thus be implemented using fast Fourier transform routines. Such implemcntation is usually some 60 times faster than the Kramers-Kronig transform for a data file containing about 7 points. This paper reports that, for transformations between the real and imaginary refractive indices, n(C) and k(C) in A = n(S) + ik(5), the FFT-based Hilbert transform can be much less accurate than, or as accurate as, the Kramers-Kronig transform, depending on the algorithm used. The Kramers-Kronig transform, incorporating Mclaurin's formula for finding the principal value of the integral, transforms k(G) spectra into n ( 5 ) spectra that are accurate to about 0.05%. Some Hilbert transform algorithms in the literature yield only about 470 accuracy. The BZ algorithm for the Hilbert transform is presented, for use on a laboratory computer running under DOS, that yields n(C) spectra accurate to 0.05%. For the transform from n(C) to k ( + ) , the BZ algorithm gives k ( 5 ) accurate to about -0.2% of the largest k value in the spectrum. This compares with an accuracy of 0.5% for the Kramers-Kronig transform. In cases where the k(D) spectrum is truncated at low wavenumbers, a simple method is presented that improves by a factor of -10 the accuracy at low wavenumber of the n(G) spectrum obtained by Hilbert or Kramers-Kronig transforms of the k(G) spectrum. Il est bien connu que la transformation infinie de Kramers-Kronig est 1'Cquivalent d'une transformation infinie de Hilbert qui est elle-m&me 1'Cquivalent des intCgrales allied Fourier. On peut donc rkaliser une transformation de Hilbert en utilisant des routines rapides de transformation de Fourier. Pour une base de donnees contenant 7000 points, une telle realisation est generalement 60 fois plus rapide qu'une transformation de Kramers-Kronig. Dans ce travail, on montre que, pour des transformations entre des indices de rkfraction reels et imaginaires, IT(:) , la transformation de Hilbert basCe sur FFT est beaucoup moins ou aussi prCcise que la transformation de Kramers-Kronig suivant I'algorithme utilise. La transformation de Kramers-Kronig, incorporant les formules de McLaurin pour trouver la valeur principale de I'integrale, transforme des spectres k ( 5 ) en spectres n(:) qui sont prCcis environ 0,05%. Quelques algorithmes rapportts dans la litterature pour la transformation de Hilbert ne conduisent qu'i une precision de 4%. On prCsente I'algorithme BZ qui peut &tre utilise sur un ordinateur de laboratoire, operant sous DOS, qui permet d'effectuer la transformation de Hilbert et qui donne des spectres n(:) precis h 0,05%. Pour la transformation n(C) en k(ij), I'algorithme BZ conduit i une precision d'environ -0,2% pour la valeur la plus importante du spectre. Cette valeur se compare h une prkcision de 0,5% pour la transformation de Kramers-Kronig. Pour les cas ou ...

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