Fourier series of periodic functions with variable period and evaluation of the variable period for determination of heart rhythm variability

The article draws attention to the fact that in addition to periodic empirical signals, whose model is a periodic function, there are signals that behave like periodic, but the period of their values repetition is no longer constant and changes in some way. An illustrative example is electrocardiograms (ECGs) obtained during or after an impact of some “exciter of calm”, for example, physical exertion, on the patient. How to study periodic signals with a variable period (PSVP)? The literature review shows that until recently there has been no scientifically substantiated answer to this question. Therefore, the problem of developing information technologies (IT) for doing research into PSVP is relevant both from theoretical and applied point of view. To solve the problem, we propose to use an approach, whose essence is the triad «model-algorithm-program». Certain results in this direction have already been achieved in our previous works. Particularly, we give a definition of periodic functions with a variable period (PFVP), consider examples of trigonometric FVP (TFVP) and record their variable periods, develop a method for the formation of orthogonal TFVP system, and determine a scalar product for the functions of the system. In this paper, a Fourier series for PFVP is written, and formulas for finding its coefficients are obtained. As an example, a finite Fourier series is constructed for the analytically given PFVP, and it is shown that with number of coefficients increasing, the series approaches the function itself, which confirms the correctness of the theoretical results obtained.

Taking into account that for the vast majority of empirical PFVP their variable period is unknown, the question of its evaluation is raised. For the case of an ECG, obtained after physical activity, evaluations of its variable frequency (VF) and variable period (VP) are derived. The evaluation of a VF turned out to have the form of exponential function, which is determined by three parameters. The IT developed for the study of PFVP provide the opportunity to explore real PSVP, in particular, ECGs with VP, and the obtained numerical values of the parameters can be used in diagnostic tasks and decision making support.