Hilbert transformation of immittance data using the Fast Fourier Transform

Abstract

The study of electrochemical systems and dielectric materials using immittance methods relies upon the fact that they may be considered to be causal, stable, finite and linear-time-invariant. Under these conditions, the components of immittance are related to one another by Hilbert Transforms (or the Kramers-Kronig relations). The Hilbert Transform is a convolution of a given immittance component with (πω) −1. This convolution may be conveniently performed using the Fourier Transform (FT), since a convolution corresponds to a multiplication in the Fourier domain. Because the FT of (πω) −1 is the signum function, the FT of the data is multiplied by ± 1. The Fourier Transform requires data which is equi-spaced on a linear scale. Unfortunately, immittance data are usually collected on a log-spaced scale with no measurement corresponding to zero frequency. Immittance data in this form will not Hilbert Transform correctly. However, a non-linear transformation of the log-spaced data to give a new “frequency” variable enables the use of the convenient Fourier Transform approach. The complete numerical procedure is described and illustrated. The numerical accuracy and precision are better than 1%. The implications of bounded/unbounded immittances and Constant Phase Elements are also discussed. The Hilbert Transformation of data spanning very wide frequency ranges is described.