🔬 applied physics

Inversion of Electrochemical Immittance Spectra based on the Mellin Transform

This paper demonstrates that electrochemical immittance spectra inversion problems share a common Mellin convolution structure, enabling the derivation of an exact algebraic inversion formula that unifies impedance- and admittance-based analyses for characterizing electrochemical systems.

Original authors: Anis Allagui, Sohaib Majzoub, Ahmed Elwakil

Published 2026-06-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Anis Allagui, Sohaib Majzoub, Ahmed Elwakil

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a complex, mysterious machine—like a battery or a supercapacitor—and you want to understand exactly how it works inside. You can’t open it up, so instead, you tap on it with different rhythms (frequencies) and listen to how it responds. This is called Electrochemical Impedance Spectroscopy (EIS).

The problem is, the response you hear is a jumbled mix of many different internal processes happening at different speeds. It’s like trying to identify every instrument in an orchestra just by listening to the final, mixed recording. Traditionally, scientists have tried to guess which instruments are playing by fitting the sound to pre-made templates (like saying, "This sounds like a violin and a drum"). But this is subjective and often misses the subtle details.

This paper introduces a new, more precise mathematical "decoder ring" to separate those sounds perfectly. Here is how it works, explained with some everyday analogies.

The Core Idea: The "Mellin" Shortcut

The authors discovered that the mathematical relationship between the "tap" (frequency) and the "response" (impedance) has a special structure. In math terms, the response depends on the product of the frequency and the time constant of the internal process, not on them separately.

Think of it like this: If you are mixing paint, the final color depends on the ratio of red to blue, not just how much red you have or how much blue you have individually. This specific type of relationship is called a Mellin Convolution.

For decades, scientists have been trying to untangle this mix using complicated, messy methods that often distort the result (like trying to un-mix paint by shaking it harder). The authors realized that if you use a specific mathematical tool called the Mellin Transform, you can turn this messy mixing problem into a simple division problem.

The Analogy: The Spectral Equalizer

Imagine the internal processes of the battery are different musical notes. The "Mellin Transform" is like a super-powered equalizer that doesn’t just show you the volume of each note, but translates the entire song into a different language where the mixing rules are simple.

Translate: You take the jumbled response from the battery and translate it into "Mellin Space."
Divide: In this space, the complex mixing becomes simple multiplication. So, to find the original "notes" (the distribution of relaxation times), you just divide the translated response by the known "mixing rule" (the kernel).
Translate Back: You translate the result back into normal time, and voilà—you have a clear map of exactly which processes are happening and how long they take.

What They Tested

The authors didn’t just claim it works; they proved it with three different types of "machines" (models):

The Constant Phase Element (CPE): Think of this as a battery that doesn’t have one single speed, but a smooth spread of speeds. The new method perfectly recovered this spread, even when the data was noisy (like listening to the orchestra in a windy room).
The Davidson-Cole Model: A more complex model where the response changes shape. The method accurately identified the sharp cutoff point where the process stops.
The Warburg Model (Diffusion): This models how ions move through a material, like sugar dissolving in water. The method correctly identified that this process happens at one specific timescale, acting like a single, pure note.

Why This Matters

No More Guessing: Instead of guessing the circuit diagram beforehand, this method lets the data speak for itself. It unifies different ways of looking at the data (impedance vs. admittance) into one single framework.
Handles Noise: Real-world data is always a bit messy. The authors showed that their method can filter out this noise effectively, giving a clean result even when the input is up to 5% noisy.
Exact Math: Unlike older methods that require "regularization" (a fancy way of saying "making educated guesses to smooth out errors"), this method is mathematically exact. The only "smoothing" needed is to handle the practical limits of computer calculations.

The Limitations

The authors are honest about the catch:

You need a wide range: You need to tap the machine at a very wide range of frequencies (from very slow to very fast) to get a good picture. If you only listen to a small part of the song, you can’t identify all the instruments.
Choosing the "Lens": You have to choose the right mathematical "lens" (the abscissa $\sigma$ ) to look through. If you pick the wrong one, the picture gets blurry. However, the authors note that this can often be determined automatically from the data itself.

In Summary

This paper provides a new, elegant mathematical tool that turns the difficult problem of interpreting battery data into a straightforward algebraic calculation. By using the Mellin Transform, scientists can now "un-mix" the complex signals from electrochemical systems with high precision, revealing the hidden internal processes without relying on subjective guesses. It’s like going from guessing the ingredients of a cake by taste to having a perfect recipe printed out from the crumbs.

Technical Summary: Inversion of Electrochemical Immittance Spectra based on the Mellin Transform

Problem Statement
Electrochemical Impedance Spectroscopy (EIS) is a standard technique for characterizing electrochemical systems, yet interpreting the resulting frequency-resolved data to identify underlying physical processes and their timescales remains challenging. The conventional approach relies on fitting data to equivalent circuit models, which requires a priori assumptions about topology and introduces subjectivity. Alternatively, the Distribution of Relaxation Times (DRT) framework maps impedance to the time domain via a Fredholm integral equation of the first kind. However, recovering the distribution function from finite, noisy measurements is an ill-posed inverse problem typically requiring regularization, which can distort results. Furthermore, standard DRT models based on the Voigt kernel are physically limited for systems with blocking electrodes (e.g., batteries, supercapacitors), where impedance diverges at low frequencies. To address this, alternative frameworks such as the Distribution of Capacitive Times (DCT) and Distribution of Diffusion Times (DDT) have been proposed for admittance-based analysis. Despite these variations, the mathematical structure underlying these inversions has not been fully exploited.

Methodology
The authors identify that the integral equations governing DRT, DCT, and DDT share a common mathematical structure: they are all Mellin convolutions. This arises because standard immittance (impedance or admittance) kernels depend on the product $\omega\tau$ rather than on frequency $\omega$ and time constant $\tau$ independently.

By exploiting this structure, the authors derive an exact algebraic inversion formula in Mellin space. The procedure converts the deconvolution problem into a closed-form relation:

The measured immittance $X(j\omega)$ is transformed into Mellin space.
The Mellin transform of the immittance is divided by the Mellin transform of the specific kernel function.
An argument shift ( $s \to 1-s$ ) is applied.
The inverse Mellin transform yields the unknown distribution function $h(\tau)$ .

This approach unifies impedance- and admittance-based inversions under a single spectral framework. For numerical implementation, the authors utilize the Fast Mellin Transform (FMT) via the Fast Fourier Transform (FFT) algorithm. To handle the ill-posed nature of numerical division and noise, the method employs Tikhonov regularization and Hann windowing to act as a low-pass denoising filter.

Key Contributions

Mathematical Unification: The paper demonstrates that DRT, DCT, and DDT problems are mathematically equivalent Mellin convolution integrals, allowing for a unified inversion strategy.
Exact Analytical Inversion: An exact algebraic inversion formula is derived that does not require regularization at the analytical level, relating the Mellin transform of the distribution function directly to the ratio of the immittance and kernel spectra.
Numerical Framework: A practical numerical algorithm (Mellin-FFT) is presented, including strategies for stabilization (Tikhonov regularization) and spectral truncation to handle noisy data.

Results
The framework is validated both analytically and numerically:

Analytical Validation: The method is applied to known models, including the Constant Phase Element (CPE), the Davidson-Cole (DC) model, and the finite-length Warburg model with blocking boundary conditions. The derived distribution functions match known analytical solutions (e.g., power-law distributions for CPE and DC models, and a delta function for the Warburg element).
Numerical Validation: The Mellin-FFT algorithm is tested on CPE and DC models under clean and noisy conditions (noise levels up to 5%).
- For noise-free data, the relative error reaches the floating-point limit ( $\sim 10^{-14}$ ).
- For noisy data, the pointwise relative errors remain below the input noise level across the timescale window, demonstrating that the regularization and windowing effectively suppress noise without significantly distorting the recovered distributions.

Significance
The paper claims that this approach provides a new, unified method for characterizing electrochemical systems from immittance data. By recognizing the Mellin convolution structure, the inversion problem is reduced to simple algebraic operations in the spectral domain. This eliminates the need for a priori circuit topology assumptions and offers a robust tool for recovering distribution functions (DRT, DCT, DDT) from experimental data, handling both bounded and unbounded low-frequency behaviors within a consistent mathematical framework.