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

In this paper, the authors found a transformation that is valid for any arbitrary signal. This transformation is strictly periodical and, therefore, it allows to apply the ordinary $$F$$ -trans- formation for the fitting of the transformed signal. The most interesting application (in accordance with the author’s opinion) is the fitting of the frequency-phase modulated signals that located actually inside the found transformation. This new transformation will be useful for application of the responses of different complex systems when a particular model is absent. As available data, we consider cosmic microwave background data (CMB) associated with the background temperature fluctuations near $$T=2.725$$ K. These electro-magnetic (EM) fluctuations of the early Universe were measured at the wide frequency range 30–857 GHz. In this paper, we analyzed the measured data at 353 GHz corresponding to the taken zero pixels. Other details are described in the second section of the paper. This squared matrix corresponding to the measured data contains 2047 lines $$\times$$ 2047 columns. If one considers each column as frequency-phase modulated signal, then amplitude-frequency response can be evaluated with the help of $$F$$ -transformation that has the period equals $$2\pi$$ that is valid for any analyzed random signal. These ‘‘universal’’ behavior allows to fit a wide set of random signals and compare them with each other in terms of their amplitude-frequency responses (AFR). Concluding the abstract, one can say that these new possibilities of the traditional $$F$$ -analysis will serve as a common tool in the armory of the methods used by researchers in the data processing area.