Inversion of Electrochemical Immittance Spectra based on the Mellin Transform

Abstract

In this work, we show that the Fredholm integral equations underlying the distribution of relaxation times (DRT), the distribution of capacitive times (DCT), and related frameworks share a common mathematical structure, namely that of a Mellin convolution. This comes from the fact that all standard immittance (impedance or admittance) kernels depend on the product $\omega\tau$ rather than on $\omega$ and $\tau$ independently. Exploiting this structure, we derive an exact algebraic inversion formula in Mellin space that converts the deconvolution problem into a closed-form relation between the Mellin transform of the measured immittance and that of the unknown distribution function. The framework is validated analytically on a set of examples including the constant phase element (CPE), the Davidson-Cole (DC) model, and the finite-length Warburg model with blocking boundary conditions. It is also validated numerically using the fast Mellin transform via the fast Fourier transform algorithm for both the CPE and the DC model, including their DRT and DCT recovery under clean and noisy conditions. The approach unifies the impedance- and admittance-based inversions under a single spectral framework, and provides a new approach for the characterization of electrochemical systems from immittance data.