NAFASS: Fluctuation spectroscopy and the Prony spectrum for description of multi-frequency signals in complex systems

•Multi-frequency signal with beatings.•The generalized Prony spectrum.•Specific frequency distribution.•Fitting of economic and transcendental “noise” data (evaluation of meat prices and the π-number). In this paper, we essentially modernize the NAFASS (Non-orthogonal Amplitude Frequency Analysis of the Smoothed Signals) approach suggested earlier. Actually, we solved two important problems: (a) new and effective algorithm was proposed and (b) we proved that the segment of the Prony spectrum could be used as the fitting function for description of the desired frequency spectrum. These two basic elements open an alternative way for creation of the fluctuation spectroscopy when the segment of the Fourier series can fit any random signal with trend but the dispersion spectrum of the Fourier series ω0·k(ω0≡2π/T)⇒Ωk(k=0,1,2,...,K−1) is replaced by the specific dispersion law Ωk calculated with the help of original algorithm described below. It implies that any finite signal will have a compact amplitude-frequency response (AFR), where the number of the modes is much less in comparison with the number of data points (K