Cyclic Point Processes with Limited Aftereffect for a Pulse Signal Analysis with Significant Pulse Rhythm Variability

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.

APPENDIX A

Information on Bessel Functions of the First Kind ${{\mathcal{J}}_{n}}\left( z \right)$, of the Integer Order of $n = 0, \pm ~1, \ldots$

Bessel functions of the first kind ${{\mathcal{J}}_{n}}\left( z \right)$ for integer values of $~n = 0, \pm ~1,~ \ldots$ ​​may be determined in several different ways [14]. In addition to the fact that they are defined as special solutions of the differential equation of the same name, ${{\mathcal{J}}_{n}}\left( z \right)$ may be defined by a number of hypergeometric types,

$$\begin{gathered} {{\mathcal{J}}_{n}}\left( z \right) = \frac{1}{{n!}}{{\left( {\frac{z}{2}} \right)}^{n}}\sum\limits_{m = 0}^{~ + \infty } {\frac{{ - {{1}^{m}}}}{{m!}}} \frac{{n!}}{{\left( {n + m} \right)!}}{{\left( {\frac{z}{2}} \right)}^{{2m}}} \\ = \frac{1}{{n!}}{{\left( {\frac{z}{2}} \right)}^{n}}\left[ {1 - \frac{1}{{\left( {n + 1} \right)}}{{{\left( {\frac{z}{2}} \right)}}^{2}} + \ldots } \right], \\ \end{gathered} $$

(A1)

or as coefficients $w$ at powers in the expansion of the generating function,

$$\exp \left( {\frac{z}{2}\left[ {w - \frac{1}{w}} \right]} \right) = \sum\limits_{ - \infty }^{ + \infty } {{{\mathcal{J}}_{n}}} \left( z \right){{w}^{n}}.$$

(A2)

Substitution of $w = \exp \left[ {j\varphi } \right]$ in (A2) leads to the Jacobi-Anger formula (to the expansion of $\exp \left[ {jz\sin \left( \varphi \right)} \right]$ in a Fourier series):

$$\exp \left[ {jz\sin \left( \varphi \right)} \right] = \sum\limits_{n = - \infty }^{~ + \infty } {{{\mathcal{J}}_{n}}\left( z \right)} \exp \left( {jn\varphi } \right).$$

(A3)

Formula (A3) implies, in particular, the relation

$$\begin{gathered} \frac{1}{K}\sum\limits_{l = 0}^{K - 1} {\exp } \left[ {jz\sin \left( {\frac{{2\pi }}{K}\left( {l - \frac{r}{2}} \right)} \right)} \right] \\ = \sum\limits_{ - \infty }^{ + \infty } {{{\mathcal{J}}_{n}}\left( z \right)} \exp \left[ { - \pi j\frac{n}{K}r)} \right]\frac{1}{K}\sum\limits_{l = 0}^{K - 1} {\exp } \left[ {2\pi j\frac{n}{K}l} \right] \\ = \sum\limits_{p = - \infty }^{~ + \infty } {{{{\left( { - 1} \right)}}^{{pr}}}} {{\mathcal{J}}_{{Kp}}}\left( z \right). \\ \end{gathered} $$

(A4)

Formula (A3) also implies the formula of the Bessel integral:

$$\begin{gathered} \int\limits_0^{2\pi } {\exp } \left[ {jz\sin \left( \varphi \right) - jm\varphi } \right]d\varphi \\ = \sum\limits_{ - \infty }^{ + \infty } {{{\mathcal{J}}_{n}}\left( z \right)} \int\limits_0^{2\pi } {\exp } \left( {j\left( {n - m} \right)\varphi } \right)d\varphi \\ = 2\pi {{\mathcal{J}}_{m}}\left( z \right). \\ \end{gathered} $$

(A5)

In turn, integral representations of the following form follow from (A5):

$$\begin{gathered} {{\mathcal{J}}_{m}}\left( z \right) = \frac{1}{{2\pi }}\int\limits_0^{2\pi } {\exp \left[ {jz\sin \left( \varphi \right) - jm\varphi } \right]d\varphi } \\ = \frac{1}{{2\pi }}\int\limits_{ - {\pi \mathord{\left/ {\vphantom {\pi 2}} \right. \kern-0em} 2}}^{{\pi \mathord{\left/ {\vphantom {\pi 2}} \right. \kern-0em} 2}} {\exp \left[ {jz\sin \left( \varphi \right)} \right]\left\{ {\exp \left[ { - jm\varphi } \right] + {{{\left( { - 1} \right)}}^{m}}\exp \left[ {jm\varphi } \right]} \right\}d\varphi } \\ = \frac{1}{\pi }\int\limits_{ - {\pi \mathord{\left/ {\vphantom {\pi 2}} \right. \kern-0em} 2}}^{{\pi \mathord{\left/ {\vphantom {\pi 2}} \right. \kern-0em} 2}} {\exp \left[ {jz\sin \left( \varphi \right)} \right]{{{\left( j \right)}}^{m}}\cos \left[ {m\left( {\varphi + \frac{\pi }{2}} \right)} \right]d\varphi } \\ = \frac{{{{{\left( j \right)}}^{m}}}}{\pi }\int\limits_{ - 1}^1 {\frac{{\exp \left[ {jzy} \right]}}{{\sqrt {1 - {{y}^{2}}} }}} \cos \left[ {m\left( {\arcsin \left( y \right) + \frac{\pi }{2}} \right)} \right]dy = \frac{{{{{\left( { - j} \right)}}^{m}}}}{\pi }\int\limits_{ - 1}^1 {\frac{{\exp \left[ {jzy} \right]}}{{\sqrt {1 - {{y}^{2}}} }}{{T}_{m}}\left( y \right)dy} , \\ \end{gathered} $$

(A6)

where it is used that $\cos \left[ {m\arccos \left( y \right)} \right] = {{T}_{m}}\left( y \right)$ are the Chebyshev polynomials of the first kind. Taking the Fourier transform from both sides of (A6), we obtain a Fourier-image of ${{\mathcal{J}}_{m}}\left( z \right)$ (see also [15]):

$$\begin{gathered} F\left[ {{{\mathcal{J}}_{m}}\left( z \right)} \right]\left( x \right) = \int\limits_{ - \infty }^{ + \infty } {\exp } \left[ {jxz} \right]{{\mathcal{J}}_{m}}\left( z \right)dz = 2{{\left( { - j} \right)}^{m}}\int\limits_{ - 1}^1 {{\delta }\left( {x + y} \right)} \frac{1}{{\sqrt {1 - {{y}^{2}}} }}{{T}_{m}}\left( y \right)dy \\ = \left\{ \begin{gathered} \frac{{{{{\left( j \right)}}^{m}}2{{T}_{m}}\left( x \right)}}{{\sqrt {1 - {{x}^{2}}} }}\,\,~\,\,{\text{for}}~\,\,\,\,\left| x \right| < 1, \\ 0\,\,\,\,{\text{for}}\,\,\,\,~\left| x \right| > 1~. \\ \end{gathered} \right. \\ \end{gathered} $$

(A7)

For ${{\mathcal{J}}_{0}}\left( {2z\sin \left( \psi \right)} \right)$ it follows from (A5) that

$$\begin{gathered} {{\mathcal{J}}_{0}}\left( {2z\sin \left( \psi \right)} \right) = \frac{1}{{2\pi }}\int\limits_0^{2\pi } {\exp \left[ {jz2\sin \left( \psi \right)\sin \left( \varphi \right)} \right]d\varphi } \\ = \frac{1}{{2\pi }}\int\limits_0^{2\pi } {\exp \left[ {jz\cos \left( {\varphi - \psi } \right)} \right]\exp \left[ { - jz\cos \left( {\varphi + \psi } \right)} \right]d\varphi } \\ = \frac{1}{{2\pi }}\int\limits_0^{2\pi } {\exp \left[ {jz\sin \left( {\varphi - \psi + \frac{\pi }{2}} \right)} \right]\exp \left[ { - jz\sin \left( {\varphi + \psi + \frac{\pi }{2}} \right)} \right]d\varphi } \\ = \sum\limits_{p = - \infty }^{ + \infty } {\sum\limits_{q = - \infty }^{~ + \infty } {{{\mathcal{J}}_{p}}\left( z \right){{\mathcal{J}}_{q}}\left( z \right)\exp \left[ { - \psi \left( {p + q} \right)} \right]\frac{1}{{2\pi }}} } \int\limits_0^{2\pi } {\exp \left[ {j\left( {\varphi + \frac{\pi }{2}} \right)\left( {p - q} \right)} \right]d\varphi } \\ = \sum\limits_{p = - \infty }^{ + \infty } {\mathcal{J}_{p}^{2}\left( z \right)\exp \left[ { - 2p\psi } \right]} . \\ \end{gathered} $$

(A8)

Using the Rodrigues formula for the Chebyshev polynomials ${{T}_{m}}\left( y \right) = {{\left( { - 1} \right)}^{m}}\frac{{\sqrt \pi }}{{{{2}^{m}}\Gamma \left( {m + \frac{1}{2}} \right)}}\sqrt {1 - {{y}^{2}}}$ × $\frac{{{{\partial }^{m}}}}{{\partial {{y}^{m}}}}{{\left( {1 - {{y}^{2}}} \right)}^{{m - \frac{1}{2}}}}$ in (A6), we obtain the Poisson integral:

$$\begin{gathered} {{\mathcal{J}}_{m}}\left( z \right) = \frac{{{{{\left( { - j} \right)}}^{m}}}}{\pi }\mathop \smallint \limits_{ - 1}^1 \frac{{\exp \left[ {jzy} \right]}}{{\sqrt {1 - {{y}^{2}}} }}{{T}_{m}}\left( y \right)dy = \frac{{{{{\left( j \right)}}^{m}}}}{\pi }\frac{{\sqrt \pi }}{{{{2}^{m}}\Gamma \left( {m + \frac{1}{2}} \right)}}\mathop \smallint \limits_{ - 1}^1 \exp \left[ {jzy} \right]\frac{{{{\partial }^{m}}}}{{\partial {{y}^{m}}}}{{\left( {1 - {{y}^{2}}} \right)}^{{m - \frac{1}{2}}}}dy \\ = {{\left( j \right)}^{m}}\frac{{{{{\left( { - 1} \right)}}^{m}}}}{{{{2}^{m}}\sqrt \pi \Gamma \left( {m + \frac{1}{2}} \right)}}{{\left[ {jz} \right]}^{m}}\mathop \smallint \limits_{ - 1}^1 \exp \left[ {jzy} \right]{{\left( {1 - {{y}^{2}}} \right)}^{{m - \frac{1}{2}}}}dy = \frac{1}{{\sqrt \pi \Gamma \left( {m + \frac{1}{2}} \right)}}{{\left( {\frac{z}{2}} \right)}^{m}}\mathop \smallint \limits_{ - 1}^1 \exp \left[ {jzy} \right]{{\left( {1 - {{y}^{2}}} \right)}^{{m - \frac{1}{2}}}}dy \\ = \frac{2}{{\sqrt \pi \Gamma \left( {m + \frac{1}{2}} \right)}}{{\left( {\frac{z}{2}} \right)}^{m}}\mathop \smallint \limits_0^1 \cos \left[ {zy} \right]{{\left( {1 - {{y}^{2}}} \right)}^{{m - \frac{1}{2}}}}dy. \\ \end{gathered} $$

(A9)

From the Poisson integral (A9), the following estimate follows:

$$\begin{gathered} \left| {{{\mathcal{J}}_{m}}\left( z \right)} \right| < \frac{2}{{\sqrt \pi \Gamma \left( {m + \frac{1}{2}} \right)}}{{\left( {\frac{{\left| z \right|}}{2}} \right)}^{m}}\int\limits_0^1 {{{{\left( {1 - {{y}^{2}}} \right)}}^{{m - \frac{1}{2}}}}} dy = \frac{1}{{\sqrt \pi \Gamma \left( {m + \frac{1}{2}} \right)}}{{\left( {\frac{{\left| z \right|}}{2}} \right)}^{m}}\int\limits_0^1 {{{t}^{{m - \frac{1}{2}}}}} {{\left( {1 - t} \right)}^{{ - \frac{1}{2}}}}dt \\ = \frac{1}{{\sqrt \pi \Gamma \left( {m + \frac{1}{2}} \right)}}{{\left( {\frac{{\left| z \right|}}{2}} \right)}^{m}}B\left( {m + \frac{1}{2};\frac{1}{2}} \right) = \frac{1}{{\sqrt \pi \Gamma \left( {m + \frac{1}{2}} \right)}}{{\left( {\frac{{\left| z \right|}}{2}} \right)}^{m}}\frac{{\Gamma \left( {m + \frac{1}{2}} \right)\Gamma \left( {\frac{1}{2}} \right)}}{{\Gamma \left( {m + 1} \right)}} \\ = \frac{1}{{\Gamma \left( {m + 1} \right)}}{{\left( {\frac{{\left| z \right|}}{2}} \right)}^{m}} = \frac{1}{{m!}}{{\left( {\frac{{\left| z \right|}}{2}} \right)}^{m}}. \\ \end{gathered} $$

(A10)

Comparing (A10) and (A1), we can conclude that ${{\mathcal{J}}_{m}}\left( z \right)$ on the entire real axis does not exceed in absolute value the modulus of its first term in a series expansion in powers of argument z.