Cyclic Point Processes with Limited Aftereffect for a Pulse Signal Analysis with Significant Pulse Rhythm Variability
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RY AND METHODS OF SIGNAL PROCESSING
Cyclic Point Processes with Limited Aftereffect for a Pulse Signal Analysis with Significant Pulse Rhythm Variability V. E. Antsiperov* Kotelnikov Institute of Radioengineering and Electronics, Russian Academy of Sciences, Moscow, 125009 Russia *e-mail: [email protected] Received February 2, 2020; revised February 2, 2020; accepted February 12, 2020
Abstract—In this paper, we present the results of applying the model of cyclic point processes with a limited aftereffect for the analysis of the rhythmic characteristics of pulsed signals. Being a generalization of recurrent and alternating point processes, we show that cyclic processes can describe event flows with more than two states. The latter circumstance significantly expands the scope of their application, in particular, to biomedical signals. Here, a complete (local) statistical description of the cyclic processes is derived, the asymptotics of their behavior are studied, and a simplified statistical description for the case of stationary modes is given. In the latter case, analytical expressions for the average and second mixed moments of the cyclic process are obtained based on the local statistics. In the most important particular case, the dependence of the features of their structure on the time scales of the signal dynamics and on the ratios between the scales was clarified. DOI: 10.1134/S1064226920070013
INTRODUCTION Many important pulse signals in nature and in technology are the result of the interaction of periodic processes and related random events. Although the implementations of such signals are not strictly periodic functions, their average statistical characteristics possess the property of periodicity. Such signals are often called cyclostationary. They are met in telecommunication systems, radars, telemetry, astronomy, mechanics, econometrics, and biology. An extensive list of models, algorithms, and applications of cyclostationary signals is presented in [1]. To simulate pulsed signals of a cyclostationary type, random point processes filtered with some impulse transfer function are often used. The wideranging class of point processes used for these purposes consists of the so-called point processes with a limited aftereffect (in the terminology of Khinchin [2]). Processes with a limited aftereffect are far-reaching generalizations of a simple Poisson process that inherit from it the property of the statistical independence of the durations of intervals between pulses. One of the popular families of point processes of this class are recurrent processes [3] characterized by the same distribution of interval durations. In the theory of queuing systems, such processes are also called recovery processes [4]. Recurrent processes are widely used in theories of communication, queues, resource allocation, etc. However, they are not entirely adequate for modeling biomedical signals. This is because the pulse flows described by recurrent processes have a constant aver-
age rhythm, while biomedical signals are c
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