Statistical Analysis of the Measurement Noise in Dynamic Impedance Spectra

Different strategies can be used to acquire dynamic impedance spectra during a cyclic voltammetry experiment. The spectra are then analyzed by fitting them with a model using a weighted non-linear least-squares minimization algorithm. The choice of the weighting factors is not trivial and influences the value of the extracted parameters. At variance with the classic electrochemical impedance spectroscopy, dynamic impedance measurements are performed under non-stationary conditions, making them typically more prone to errors arising from the voltage and current analog-to-digital conversion. Under the assumption that the noise in the voltage and current signals have a constant variance along the measurement and that it is uncorrelated between distinct samples, we calculate an expression for the expected variance of the error of the resulting immittances, which considers the specific procedure used to extract the spectra under the time-varying nature of the measurements. The calculated variance of the error is then used as a rigorous way to evaluate the weighting factors of the least-squares minimization, assuming that the fitted model is ideally exact and that there are no systematic errors. By exploring two classical electrochemical systems and fitting the measured spectra with a transfer function measurement model, namely with the Padé approximants, we show that the variance evaluated with our method captures the frequency dependence of the resulting residuals and can be used for reliably performing the complex non-linear least-squares fitting procedure.