Taylor–Fourier spectra to study fractional order systems

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In measurement science mathematical models are often used as an indirect measurement of physical properties which are mapped to measurands through the mathematical model. Dynamical systems describing a physical process with a dominant diffusion or dispersion phenomenon requires a large dimensional model due to its long memory. Ignoring a dominant difussion or dispersion component acts as a confounder which may introduce a bias in the estimated quantities of interest. For linear systems it has been observed that fractional order models outperform classical rational forms in terms of the number of parameters for the same fitting error. However it is not straightforward to deal with a fractional order system or long memory effects without prior knowledge. Since the parametric modeling of a fractional system is very involved, we put forward the question whether fractional insight can be gathered in a non-parametric way. In this paper we show that classical Fourier basis leading to the frequency response function lacks fractional insight. To circumvent this problem, we introduce a fractional Taylor–Fourier basis to obtain non-parametric insight in the fractional system. This analysis proposes a novel type of spectrum to visualize the spectral content of a fractional system: Taylor–Fourier spectrum. This spectrum is fully measurement driven which can be used as a first to explore the fractional dynamics of a measured diffusion or dispersion system.