Determination of the Relaxation and Retardation Spectra – a View from the Up-to-date Signal Processing Perspective

The problem of determination of relaxation and retardation spectra (RRS) is considered from the viewpoint of up-to-date signal processing. It is shown that the recovery of RRS represents the Mellin deconvolution problem, which transforms into the Fourier deconvolution problem for data on a logarithmic time or frequency scale, where it can also be treated as the inverse filtering problem. On this basis, discrete deconvolution (inverse) filters operating with geometrically sampled data are proposed to use as RRS estimators. Appropriate frequency responses and algorithms are derived for estimating RRS from eight different material functions. The noise amplification coefficient is suggested to use as a measure for quantifying the degree of ill-posedness and ill-conditioness of the RRS recovery problem and algorithms. A methodology is developed for designing RRS estimators with a desired noise amplification, producing maximum accurate spectra for available limited input data. Practical algorithms for determining RRS are proposed, and their performance is studied. The algorithms suggested are compared with the so-called moving-average formulae. It is demonstrated that the minimum frequency range for recovering the point estimate of a relaxation spectrum depends on the allowable noise amplification (the degree of ill-conditioness) and is in no way limited by 1.36 decades, as it is stated by the sampling localization theorem.