A class of rational systems for modelling 1/f power spectra [of biological signals]

A class of rational systems for modelling the power spectra of 1/f signals are introduced. This class is characterized by the cascade of frequency scaled versions of an arbitrary prototype rational function which has the same number of zeros and poles. The novelty of the approach lies in the fact that the scale-invariant nature of the power-law response is intentionally imposed on the rational system via the frequency scaling operation, resulting in γ-homogeneous rational functions. For finite bandwidth systems, as the number of elements in the cascade increases, the deviations from exact 1/f behavior decrease so that arbitrarily good approximations to an exact power-law behavior can be achieved. The rational models previously derived using other approaches can be shown to be special cases of the class of rational models introduced here. While these results have been presented in the context of the rational modelling of 1/f power spectra, the solution can be used to achieve a power-law magnitude response where the scaling exponent λ satisfies -∞ < λ < ∞