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

Abstract 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 ω 0 · k ( ω 0 ≡ 2 π / T ) ⇒ Ω k ( k = 0 , 1 , 2 , . . . , K − 1 ) is replaced by the specific dispersion law Ω k 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 N ). The NAFASS approach can be applicable for quantitative description of a wide set of random signals/fluctuations and allows one to compare them with each other based on one general platform. As the first example, we considered economic data and compare 30-years world prices for meat (beef, chicken, lamb and pork) entering as the basic components to every-day food consumption. These data were taken from the official site http://www.indexmundi.com/commodities/ . We fitted these random functions with the high accuracy and calculated the desired “amplitude-frequency” response for these random price fluctuations. The calculated distribution of the amplitudes ( Ac k , As k ) and frequency spectrum Ω k ( k = 0, 1,…, K −1) allows one to compress initial data ( K (number of modes)  N (number of data points), N / K ≅ 20–40) and receive an additional information for their comparison with each other. As the second example, we considered the transcendental/irrational numbers description in the frame of the proposed NAFASS approach, as well. This possibility was demonstrated on the quantitative description of the transcendental number π = 3.1415926535897932…, containing initially 6⋅10 4 digits. The results obtained for the second type of data can be useful for cryptography purposes. We do believe that the NAFASS approach can be widely used for creation of the new metrological standards based on comparison of different test fluctuations with the fluctuations registered from the pattern equipment. Apart from this obvious application, the NAFASS approach can be applicable for description of different nonlinear random signals containing the hidden beatings in radioelectronics and acoustics.