HILBERT TRANSFORMATION OF IMMITTANCE DATA USING THE FAST FOURIER-TRANSFORM

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 (pi-omega)-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 (pi-omega)-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.