New Digital Signal Processing Methods pp 289-341 | Cite as

# The Non-orthogonal Amplitude Frequency Analysis of Smoothed Signals Approach and Its Application for Describing Multi-Frequency Signals

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## Abstract

This chapter deals with non-reproducible experiments and contains the answer to the following question: is it possible to find a “universal” fitting function for some non-reproducible and random results from an experiment/observation? There are a lot of examples of such kind of experiments producing various types of data: economic data, medical data, geophysical data, etc. The approach abbreviated as NAFASS – non-orthogonal amplitude-frequency analysis of the smoothed signals – can answer the posed question. This approach is an alternative to the Fourier-transform method, which is used only for periodic signals and in case of an “ideal experiment”, and can be applied to a wide class of smoothed signals representing the behaviour of different complex systems. If one assumes the reasonable hypothesis that a given signal presents itself as the combination of different beatings, then one can find the dispersion law (i.e. the dependence of a set of frequencies Ω_{k} from the wave vector *k*) and fit a broad set of complex signals. The effectiveness and the power of the NAFASS-approach in the calculation of the amplitude-frequency response are demonstrated by considering two types of complex signals: (a) one is associated with economic data and (b) the second is determined by the integral envelopes of the well-known numbers π and * E* (Euler constant). Based on the NAFASS-approach, the authors try to outline a new type of spectroscopy. Moreover, this chapter serves as an invitation for ambitious readers and researchers to support and develop this spectroscopy by the analysis of different kinds of fluctuations. The chapter also establishes an interesting link between the NAFASS-approach and the chaotic behaviour of a large class of complex systems.

## Keywords

NAFASS-approach Beatings and dispersion law Economic data and their fit Fluctuation spectroscopy Control by chaotic processes## References

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