New Possibilities of the Fourier Transformation: How to Describe an Arbitrary Frequency-Phase Modulated Signal?

Abstract

In this paper, the authors identify a transformation that is valid for any arbitrary signal. This transformation is strictly periodical and therefore it allows the ordinary F-transformation to be applied for the fitting of the transformed signal. The most interesting application (in accordance with the authors' opinion) is the fitting of the frequency-phase modulated signals that actually located inside the found transformation. This new transformation will be useful for the application of the responses of different complex systems when an ordinary model is absent. As available information we consider meteorological data corresponding to measurements of methane concentration (CH4) in atmosphere during 4 weeks of observation. For us, it is important to consider the integral (cumulative) data and find their amplitude-frequency response (AFR). If one considers each column as a frequency-phase modulated signal, then the AFR can be evaluated with the help of an F-transformation with a period equaling 2π that is valid for any analyzed random signal. This "universal" F-transformation allows us to fit a wide set of random signals and compare them with each other in terms of their AFRs. In conclusion, these new possibilities of traditional F-analysis will serve as a common tool in the armory of the methods used by researchers in data processing area.