Determination of relaxation-time distributions from dielectric spectra of complex impedance or dielectric permittivity remains a challenge. This problem is one of a wider class of ill-posed inverse problems where the measurement is a superposition or convolution of functions containing the sought-after information. An expectation-maximization (EM) algorithm is shown to be useful for obtaining dielectric relaxation-time distributions from impedance data. This algorithm is stable and converges to realistic relaxation-time spectra without the need for constraints or initial values. The implementation used herein updates expectations in an iterative multiplication step. The models and basic assumptions of impedance spectroscopy are outlined in the first part of this paper. Frequency-dependent impedance measurements are obtained for calibration samples and saturated montmorillonite clays. The EM algorithm is subsequently used to determine the dielectric relaxation times. The dielectric relaxation-time spectra allow facile interpretation of otherwise complicated impedance.
Stephen E. Bialkowski, Lynn Dudley, and Dani Or, "Using an Expectation–Maximization Algorithm to Obtain Dielectric Relaxation-Time Spectra of Aqueous Montmorillonite Clay Suspensions," Appl. Spectrosc. 56, 1470-1474 (2002)