\documentclass[11pt,twoside]{article}\makeatletter

\IfFileExists{xcolor.sty}%
  {\RequirePackage{xcolor}}%
  {\RequirePackage{color}}
\usepackage{colortbl}
\usepackage{wrapfig}
\usepackage{ifxetex}
\ifxetex
  \usepackage{fontspec}
  \usepackage{xunicode}
  \catcode`⃥=\active \def⃥{\textbackslash}
  \catcode`❴=\active \def❴{\{}
  \catcode`❵=\active \def❵{\}}
  \def\textJapanese{\fontspec{Noto Sans CJK JP}}
  \def\textChinese{\fontspec{Noto Sans CJK SC}}
  \def\textKorean{\fontspec{Noto Sans CJK KR}}
  \setmonofont{DejaVu Sans Mono}
  
\else
  \IfFileExists{utf8x.def}%
   {\usepackage[utf8x]{inputenc}
      \PrerenderUnicode{–}
    }%
   {\usepackage[utf8]{inputenc}}
  \usepackage[english]{babel}
  \usepackage[T1]{fontenc}
  \usepackage{float}
  \usepackage[]{ucs}
  \uc@dclc{8421}{default}{\textbackslash }
  \uc@dclc{10100}{default}{\{}
  \uc@dclc{10101}{default}{\}}
  \uc@dclc{8491}{default}{\AA{}}
  \uc@dclc{8239}{default}{\,}
  \uc@dclc{20154}{default}{ }
  \uc@dclc{10148}{default}{>}
  \def\textschwa{\rotatebox{-90}{e}}
  \def\textJapanese{}
  \def\textChinese{}
  \IfFileExists{tipa.sty}{\usepackage{tipa}}{}
\fi
\def\exampleFont{\ttfamily\small}
\DeclareTextSymbol{\textpi}{OML}{25}
\usepackage{relsize}
\RequirePackage{array}
\def\@testpach{\@chclass
 \ifnum \@lastchclass=6 \@ne \@chnum \@ne \else
  \ifnum \@lastchclass=7 5 \else
   \ifnum \@lastchclass=8 \tw@ \else
    \ifnum \@lastchclass=9 \thr@@
   \else \z@
   \ifnum \@lastchclass = 10 \else
   \edef\@nextchar{\expandafter\string\@nextchar}%
   \@chnum
   \if \@nextchar c\z@ \else
    \if \@nextchar l\@ne \else
     \if \@nextchar r\tw@ \else
   \z@ \@chclass
   \if\@nextchar |\@ne \else
    \if \@nextchar !6 \else
     \if \@nextchar @7 \else
      \if \@nextchar (8 \else
       \if \@nextchar )9 \else
  10
  \@chnum
  \if \@nextchar m\thr@@\else
   \if \@nextchar p4 \else
    \if \@nextchar b5 \else
   \z@ \@chclass \z@ \@preamerr \z@ \fi \fi \fi \fi
   \fi \fi  \fi  \fi  \fi  \fi  \fi \fi \fi \fi \fi \fi}
\gdef\arraybackslash{\let\\=\@arraycr}
\def\@textsubscript#1{{\m@th\ensuremath{_{\mbox{\fontsize\sf@size\z@#1}}}}}
\def\Panel#1#2#3#4{\multicolumn{#3}{){\columncolor{#2}}#4}{#1}}
\def\abbr{}
\def\corr{}
\def\expan{}
\def\gap{}
\def\orig{}
\def\reg{}
\def\ref{}
\def\sic{}
\def\persName{}\def\name{}
\def\placeName{}
\def\orgName{}
\def\textcal#1{{\fontspec{Lucida Calligraphy}#1}}
\def\textgothic#1{{\fontspec{Lucida Blackletter}#1}}
\def\textlarge#1{{\large #1}}
\def\textoverbar#1{\ensuremath{\overline{#1}}}
\def\textquoted#1{‘#1’}
\def\textsmall#1{{\small #1}}
\def\textsubscript#1{\@textsubscript{\selectfont#1}}
\def\textxi{\ensuremath{\xi}}
\def\titlem{\itshape}
\newenvironment{biblfree}{}{\ifvmode\par\fi }
\newenvironment{bibl}{}{}
\newenvironment{byline}{\vskip6pt\itshape\fontsize{16pt}{18pt}\selectfont}{\par }
\newenvironment{citbibl}{}{\ifvmode\par\fi }
\newenvironment{docAuthor}{\ifvmode\vskip4pt\fontsize{16pt}{18pt}\selectfont\fi\itshape}{\ifvmode\par\fi }
\newenvironment{docDate}{}{\ifvmode\par\fi }
\newenvironment{docImprint}{\vskip 6pt}{\ifvmode\par\fi }
\newenvironment{docTitle}{\vskip6pt\bfseries\fontsize{22pt}{25pt}\selectfont}{\par }
\newenvironment{msHead}{\vskip 6pt}{\par}
\newenvironment{msItem}{\vskip 6pt}{\par}
\newenvironment{rubric}{}{}
\newenvironment{titlePart}{}{\par }

\newcolumntype{L}[1]{){\raggedright\arraybackslash}p{#1}}
\newcolumntype{C}[1]{){\centering\arraybackslash}p{#1}}
\newcolumntype{R}[1]{){\raggedleft\arraybackslash}p{#1}}
\newcolumntype{P}[1]{){\arraybackslash}p{#1}}
\newcolumntype{B}[1]{){\arraybackslash}b{#1}}
\newcolumntype{M}[1]{){\arraybackslash}m{#1}}
\definecolor{label}{gray}{0.75}
\def\unusedattribute#1{\sout{\textcolor{label}{#1}}}
\DeclareRobustCommand*{\xref}{\hyper@normalise\xref@}
\def\xref@#1#2{\hyper@linkurl{#2}{#1}}
\begingroup
\catcode`\_=\active
\gdef_#1{\ensuremath{\sb{\mathrm{#1}}}}
\endgroup
\mathcode`\_=\string"8000
\catcode`\_=12\relax

\usepackage[a4paper,twoside,lmargin=1in,rmargin=1in,tmargin=1in,bmargin=1in,marginparwidth=0.75in]{geometry}
\usepackage{framed}

\definecolor{shadecolor}{gray}{0.95}
\usepackage{longtable}
\usepackage[normalem]{ulem}
\usepackage{fancyvrb}
\usepackage{fancyhdr}
\usepackage{graphicx}
\usepackage{marginnote}

\renewcommand{\@cite}[1]{#1}


\renewcommand*{\marginfont}{\itshape\footnotesize}

\def\Gin@extensions{.pdf,.png,.jpg,.mps,.tif}

  \pagestyle{fancy}

\usepackage[pdftitle={The Problem Solving Algorithm Time-Frequency Signals Analysis based on Behavior Functions and Arithmetic Series},
 pdfauthor={}]{hyperref}
\hyperbaseurl{}

	 \paperwidth210mm
	 \paperheight297mm
              
\def\@pnumwidth{1.55em}
\def\@tocrmarg {2.55em}
\def\@dotsep{4.5}
\setcounter{tocdepth}{3}
\clubpenalty=8000
\emergencystretch 3em
\hbadness=4000
\hyphenpenalty=400
\pretolerance=750
\tolerance=2000
\vbadness=4000
\widowpenalty=10000

\renewcommand\section{\@startsection {section}{1}{\z@}%
     {-1.75ex \@plus -0.5ex \@minus -.2ex}%
     {0.5ex \@plus .2ex}%
     {\reset@font\Large\bfseries}}
\renewcommand\subsection{\@startsection{subsection}{2}{\z@}%
     {-1.75ex\@plus -0.5ex \@minus- .2ex}%
     {0.5ex \@plus .2ex}%
     {\reset@font\Large}}
\renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}%
     {-1.5ex\@plus -0.35ex \@minus -.2ex}%
     {0.5ex \@plus .2ex}%
     {\reset@font\large}}
\renewcommand\paragraph{\@startsection{paragraph}{4}{\z@}%
     {-1ex \@plus-0.35ex \@minus -0.2ex}%
     {0.5ex \@plus .2ex}%
     {\reset@font\normalsize}}
\renewcommand\subparagraph{\@startsection{subparagraph}{5}{\parindent}%
     {1.5ex \@plus1ex \@minus .2ex}%
     {-1em}%
     {\reset@font\normalsize\bfseries}}


\def\l@section#1#2{\addpenalty{\@secpenalty} \addvspace{1.0em plus 1pt}
 \@tempdima 1.5em \begingroup
 \parindent \z@ \rightskip \@pnumwidth 
 \parfillskip -\@pnumwidth 
 \bfseries \leavevmode #1\hfil \hbox to\@pnumwidth{\hss #2}\par
 \endgroup}
\def\l@subsection{\@dottedtocline{2}{1.5em}{2.3em}}
\def\l@subsubsection{\@dottedtocline{3}{3.8em}{3.2em}}
\def\l@paragraph{\@dottedtocline{4}{7.0em}{4.1em}}
\def\l@subparagraph{\@dottedtocline{5}{10em}{5em}}
\@ifundefined{c@section}{\newcounter{section}}{}
\@ifundefined{c@chapter}{\newcounter{chapter}}{}
\newif\if@mainmatter 
\@mainmattertrue
\def\chaptername{Chapter}
\def\frontmatter{%
  \pagenumbering{roman}
  \def\thechapter{\@roman\c@chapter}
  \def\theHchapter{\roman{chapter}}
  \def\thesection{\@roman\c@section}
  \def\theHsection{\roman{section}}
  \def\@chapapp{}%
}
\def\mainmatter{%
  \cleardoublepage
  \def\thechapter{\@arabic\c@chapter}
  \setcounter{chapter}{0}
  \setcounter{section}{0}
  \pagenumbering{arabic}
  \setcounter{secnumdepth}{6}
  \def\@chapapp{\chaptername}%
  \def\theHchapter{\arabic{chapter}}
  \def\thesection{\@arabic\c@section}
  \def\theHsection{\arabic{section}}
}
\def\backmatter{%
  \cleardoublepage
  \setcounter{chapter}{0}
  \setcounter{section}{0}
  \setcounter{secnumdepth}{2}
  \def\@chapapp{\appendixname}%
  \def\thechapter{\@Alph\c@chapter}
  \def\theHchapter{\Alph{chapter}}
  \appendix
}
\newenvironment{bibitemlist}[1]{%
   \list{\@biblabel{\@arabic\c@enumiv}}%
       {\settowidth\labelwidth{\@biblabel{#1}}%
        \leftmargin\labelwidth
        \advance\leftmargin\labelsep
        \@openbib@code
        \usecounter{enumiv}%
        \let\p@enumiv\@empty
        \renewcommand\theenumiv{\@arabic\c@enumiv}%
	}%
  \sloppy
  \clubpenalty4000
  \@clubpenalty \clubpenalty
  \widowpenalty4000%
  \sfcode`\.\@m}%
  {\def\@noitemerr
    {\@latex@warning{Empty `bibitemlist' environment}}%
    \endlist}

\def\tableofcontents{\section*{\contentsname}\@starttoc{toc}}
\parskip0pt
\parindent1em
\def\Panel#1#2#3#4{\multicolumn{#3}{){\columncolor{#2}}#4}{#1}}
\newenvironment{reflist}{%
  \begin{raggedright}\begin{list}{}
  {%
   \setlength{\topsep}{0pt}%
   \setlength{\rightmargin}{0.25in}%
   \setlength{\itemsep}{0pt}%
   \setlength{\itemindent}{0pt}%
   \setlength{\parskip}{0pt}%
   \setlength{\parsep}{2pt}%
   \def\makelabel##1{\itshape ##1}}%
  }
  {\end{list}\end{raggedright}}
\newenvironment{sansreflist}{%
  \begin{raggedright}\begin{list}{}
  {%
   \setlength{\topsep}{0pt}%
   \setlength{\rightmargin}{0.25in}%
   \setlength{\itemindent}{0pt}%
   \setlength{\parskip}{0pt}%
   \setlength{\itemsep}{0pt}%
   \setlength{\parsep}{2pt}%
   \def\makelabel##1{\upshape ##1}}%
  }
  {\end{list}\end{raggedright}}
\newenvironment{specHead}[2]%
 {\vspace{20pt}\hrule\vspace{10pt}%
  \phantomsection\label{#1}\markright{#2}%

  \pdfbookmark[2]{#2}{#1}%
  \hspace{-0.75in}{\bfseries\fontsize{16pt}{18pt}\selectfont#2}%
  }{}
      \def\TheFullDate{2019-01-15 (revised: 15 January 2019)}
\def\TheID{\makeatother }
\def\TheDate{2019-01-15}
\title{The Problem Solving Algorithm Time-Frequency Signals Analysis based on Behavior Functions and Arithmetic Series}
\author{}\makeatletter 
\makeatletter
\newcommand*{\cleartoleftpage}{%
  \clearpage
    \if@twoside
    \ifodd\c@page
      \hbox{}\newpage
      \if@twocolumn
        \hbox{}\newpage
      \fi
    \fi
  \fi
}
\makeatother
\makeatletter
\thispagestyle{empty}
\markright{\@title}\markboth{\@title}{\@author}
\renewcommand\small{\@setfontsize\small{9pt}{11pt}\abovedisplayskip 8.5\p@ plus3\p@ minus4\p@
\belowdisplayskip \abovedisplayskip
\abovedisplayshortskip \z@ plus2\p@
\belowdisplayshortskip 4\p@ plus2\p@ minus2\p@
\def\@listi{\leftmargin\leftmargini
               \topsep 2\p@ plus1\p@ minus1\p@
               \parsep 2\p@ plus\p@ minus\p@
               \itemsep 1pt}
}
\makeatother
\fvset{frame=single,numberblanklines=false,xleftmargin=5mm,xrightmargin=5mm}
\fancyhf{} 
\setlength{\headheight}{14pt}
\fancyhead[LE]{\bfseries\leftmark} 
\fancyhead[RO]{\bfseries\rightmark} 
\fancyfoot[RO]{}
\fancyfoot[CO]{\thepage}
\fancyfoot[LO]{\TheID}
\fancyfoot[LE]{}
\fancyfoot[CE]{\thepage}
\fancyfoot[RE]{\TheID}
\hypersetup{citebordercolor=0.75 0.75 0.75,linkbordercolor=0.75 0.75 0.75,urlbordercolor=0.75 0.75 0.75,bookmarksnumbered=true}
\fancypagestyle{plain}{\fancyhead{}\renewcommand{\headrulewidth}{0pt}}

\date{}
\usepackage{authblk}

\providecommand{\keywords}[1]
{
\footnotesize
  \textbf{\textit{Index terms---}} #1
}

\usepackage{graphicx,xcolor}
\definecolor{GJBlue}{HTML}{273B81}
\definecolor{GJLightBlue}{HTML}{0A9DD9}
\definecolor{GJMediumGrey}{HTML}{6D6E70}
\definecolor{GJLightGrey}{HTML}{929497} 

\renewenvironment{abstract}{%
   \setlength{\parindent}{0pt}\raggedright
   \textcolor{GJMediumGrey}{\rule{\textwidth}{2pt}}
   \vskip16pt
   \textcolor{GJBlue}{\large\bfseries\abstractname\space}
}{%   
   \vskip8pt
   \textcolor{GJMediumGrey}{\rule{\textwidth}{2pt}}
   \vskip16pt
}

\usepackage[absolute,overlay]{textpos}

\makeatother 
      \usepackage{lineno}
      \linenumbers
      
\begin{document}

             \author[1]{Victor  Bocharnikov}

\renewcommand\Authands{ and }

\date{\small \em Received: 14 December 2018 Accepted: 1 January 2019 Published: 15 January 2019}

\maketitle


\begin{abstract}
        


This article is devoted to time-frequency signals analysis algorithm.This algorithm introduce the approach based on behavior functions and arithmetic series. The basis of p-adic numbers will be used to describe the discrete signal values. It will allow to build system behavior functions as a distribution of possibility measure. The function data analysis allows to perform the meta systems identification and build impulse functions. These functions will be used for estimation of frequency spectrum of initial signal. The study results of the algorithm performance on non-stationary signals model are given.

\end{abstract}


\keywords{ime series, time-frequency analysis, p-adic numbers, system behavior functions, measure of possibility, fuzzy set, }

\begin{textblock*}{18cm}(1cm,1cm) % {block width} (coords) 
\textcolor{GJBlue}{\LARGE Global Journals \LaTeX\ JournalKaleidoscope\texttrademark}
\end{textblock*}

\begin{textblock*}{18cm}(1.4cm,1.5cm) % {block width} (coords) 
\textcolor{GJBlue}{\footnotesize \\ Artificial Intelligence formulated this projection for compatibility purposes from the original article published at Global Journals. However, this technology is currently in beta. \emph{Therefore, kindly ignore odd layouts, missed formulae, text, tables, or figures.}}
\end{textblock*}


\let\tabcellsep& 	 	 		 \par
Any space-time signal can be described by a set of basic functions \hyperref[b0]{[1]}. This allows us to obtain a signal spectrum that reflects the basis functions proportion of the content in the original signal. Such a decomposition is often useful for signal analysis. For example, the spectral representation is effective for analyzing the compressibility of signals, the synthesis of compression algorithms with minimal losses, the signals filtering problems solving, the synthesis of optimal regulators, etc.\par
The transition to the spectrum can be carried out using orthogonal and unitary transformations. Most often, to obtain a spectrum, the decomposition in orthogonal functions is used \hyperref[b1]{[2]}. For example, the spectra obtained on the basis of decomposition into a Fourier series \hyperref[b2]{[3]} (with harmonic basis), Walsh series \hyperref[b3]{[4]} (using a non-harmonic orthogonal system of rectangular functions with values of ± 1), wavelet transform \hyperref[b4]{[5,}\hyperref[b5]{6]}, etc. In \hyperref[b6]{[7]}, a new approach to the timefrequency analysis of non-stationary signals was proposed. In this paper, we consider an algorithm implementing the approach developed in \hyperref[b6]{[7]} and present the results of its application to the timefrequency analysis of non-stationary signals.\par
A detailed analysis of existing approaches to the spectral-temporal analysis of signals is given in \hyperref[b6]{[7]}. From the analysis it follows that the following elements are common to all the main approaches:\par
1. Arbitrary of the function ( ) tend to be decomposed into a set of basis functions \{?( , )\}. Such functions form, as a rule, an orthogonal basis. Each function from this system conditionally plays the role of a coordinates axis.\par
2. To determine the projection on such an coordinates axis, the integral convolution is used = ? [ ( ) ? ?( , )] , where are decomposition coefficients determining the degree of coincidence of the original function ( ) and ?( , ). The values of are perceived as coordinates on the respective axes of the basis orthogonal functions. Thus, the convolution integral or its discrete analogue is perceived as a measure of the similarity of the original function ( ) and the function ?( , ). 3. To study non-stationary signals and obtain the time-frequency decomposition of a signal a "window" is used, which is "cutting out" a portion of the function ( ) over a certain time interval. For each time interval, a measure of similarity with the basis functions ?( , ) is found.\par
4. Window accounting when analyzing the original function can be carried out explicitly or implicitly. In the first case, the window function ( ) is explicitly introduced into the expression for integral convolution. This function may have various forms, for example, in the form of a Gaussian function. In the second case, the window is accounted for by direct selection of the basis orthogonal functions, which are called bursts or wavelets. The use of wavelets allows to further determine the frequency localization.\par
5. Depending on the selected basic orthogonal functions and window functions, different time-frequency localization is obtained (Fig. 1). In any case, such localization is determined in advance before the beginning of the analysis and affects its result. 1. The "grid" of the time-frequency localization superimposed on the original signal may not "coincide" with the characteristics of the original function ( ), which leads to distortions of the time-frequency signal evaluation. That is why in the existing approaches special attention is paid to the selection of basic functions and window modeling functions.\par
2. In some cases, such as when using wavelet transforms, the time window has fuzzy boundaries, which leads to distortions of the time-frequency estimate due to the windows overlapping.\par
3. The choice of functions integral convolution instrument as a measure of their similarity affects the assessment of initial and basic functions coincidence degree. The measure of similarity can be constructed in other ways and be more effective. In addition, in practice, the use of integral convolution introduces its own errors, which are associated with numerical integration methods.\par
4. The rigid "grid" of time-frequency localization used in existing approaches is not adaptive. It does not take into account the behavior of the non-stationary function of the signal. At present, localization is more effective, which is used in wavelet transforms (Fig. 1d). However, it is also rigid, which leads to the need for a more careful selection of wavelet basis functions. Selection of basic functions adapted to a signal is used in the method of signal analysis proposed by N. Huang  {\ref (Huang-Hilbert Transform)}  \hyperref[b9]{[10]}. However, studies have shown that this approach also has several disadvantages \hyperref[b10]{[11]}.\par
5. It can be assumed that if the function ( ) behaves not stationary, then the time-frequency localization should be adaptive and adapt to the behavior of the signal under study (Fig.  {\ref 2}). Adaptive time-frequency localization Thus, to ensure the adaptation of the time-frequency signal localization, it is necessary to determine the local areas of stationary behavior for the system that generates the signal. These areas are defined as a locally-invariant constraints on system behavior.\par
Earlier, theoretical studies were presented in \hyperref[b6]{[7]}, which allow us to obtain estimates of the frequency spectrum of a signal in an adaptive version. The approach is based on the fact that the behavior of the system has locally invariant sections, in which the signal spectrum is relatively stable. To identify these sections, it was proposed to use a behavior functions change in the system \hyperref[b11]{[12]}. This behavior function is proposed to build using the mathematical basis of the -adic calculus in the form of possibility measure distribution. The moments of the behavior function change allow us to determine the coordinates of the impulse function that models the original signal. It is noteworthy that periodic functions (for example, harmonic) are described by impulse functions, in which the coordinates of the pulses correspond to the members of arithmetic progressions In the above presented work, an approach is proposed to determine the current signal spectrum estimates based on the coincidence degree of the impulse functions of this signal and the harmonic signals of the basis functions. To ensure the effectiveness of the approach, it is necessary to develop an algorithm for solving the problem and test its workability for non-stationary signals.\par
The purpose of the study, the results of which are presented in the article, was to develop an algorithm for solving the problem of frequency-time analysis of signals, which implements the approach proposed in \hyperref[b6]{[7]}, as well as testing the efficiency of the algorithm based on a model example of a non-stationary signal. To achieve the goal, three tasks were solved: To solve these problems the following methods were used: the theory of fuzzy sets and fuzzy measures; system analysis based on changes in system behavior; -adic analysis; linear programming; discrete signal modeling; spectral analysis Based on the approach proposed in \hyperref[b6]{[7]}, an algorithm for time-frequency analysis of nonstationary signals was developed. The algorithm allows to obtain the spectrum of the signal at each point in time when observing the signal. The delay in estimating the spectrum is determined by the time for the formation of the observation window. Below is the synthesized algorithm for solving the problem:\par
Step. 1. Initial data set:\par
1. The observation time W and the sampling interval of the signal (for example, in seconds). 2. The set of basis functions of impulse sines \hyperref[b6]{[7]} ? = \{ ( )\}, = 1, , ? . Pulse sines are called impulse functions, in which the coordinates of the pulses are determined by the formula of an arithmetic progression:= + ( ? 1) ? ,\par
where ? , -index -th sinusoid, = -is a step of arithmetic progression. In fact, the functions ( ) are given by two parameters and , which determine the phase and frequency of the harmonic signal, respectively.\par
3. The basis of the -adic number ( ? 2), as well as the number of observation channel blocks \hyperref[b12]{[13]} to represent the data that describe the signal. The number of blocks is usually from 3 to 9. 4. The set of shift parameters ? , = 1, | | is determined for the whole set of sample variables \hyperref[b13]{[14]} describing the signal. 5. The set of cut-off thresholds \{? \}, ? ? [0; 1] to determine the impulse function.\par
Thresholds are chosen at regular intervals in the range 0.12 -0.25 in the amount of 10 -15. 6. The algorithm parameter for determining the pulses coordinates for the impulse function of the studied series: an integer ? 1. 7. Select the type of window. There are four options:\par
a. Asymmetrical window. The boundaries of the window lie in the middle between the coordinates of nearby pulses of the signal that is being studied. Calculation of the window using the formula:= [ ? ( ); + ( )] ? .\par
where ( ) = 0. 8. A method is chosen for determining the influence degrees of the -th impulse sine function from a set of basis functions ? for building the signal spectrum: method 1based on optimization (linear programming); method 2 -on the basis of an approximate approach.\par
Step 2. The maximum normalizing integer for the time series is determined on the basis of the -adic number formula of the form \hyperref[b14]{[15]}:= ? ,\par
where ? ? \{0, ? , \} = ? 1.\par
Step 3. All values of the time series of the signal under study are normalized so that the minimum value of the series is 0, and the maximum value of the series . The resulting values are rounded to integers.\par
Step 4. Each value of the series is decomposed into -adic number and is represented in canonical form \hyperref[b15]{[16]}: ( ) = \{ , , ? , , ? , \}. In fact, this is a representation of a number in the -adic number system. If = 2, then these are binary numbers. The value of determines the value of theth variable.\par
Step 5. For each point in time, the confidence distribution is calculated for each system variable describing the signal. Let the value of the system variable ? , be determined on the set of states = , ? , , , where = , = + 1, = 0, . Then the state of the system for this variable for ? is determined by the distribution function of the possibility , : × ? [0,1], which is given on the basis of -adic number ( ) in the form:= ( ) ? max , ( ) ,\par
where ( ) -value of -th element of canonical form -adic number with ? .\par
Step 6. A set of sample variables that determine the current state of the signal is defined. Sample variables are given by the relation , = , ( ) ? ? ,, where , is the state of the -th sample variable with the parameter ? , , ( ) ? -the state of the variable ? when the value of the parameter ( ) = + , ? . The full set of signal states will be defined as Step 7. The pulses coordinate of the time series of the signal under study are determined. The algorithm is calculated for all thresholds \{? \}. It is carried out iteratively on the basis of the metasystem identification subalgorithm described below:= × × ? × | | ,\par
Step. 7.1. At the first step, the initial conditions for the parameter = 1 and the coefficient of the algorithm = 1 are accepted. Step 7.2. For the data subset [ , + ], the behavior function ( ) is defined as the distribution of the possibility by the formula:( ) = ( ) ?? ? max ? ( ) ?? .\par
where( ) = min ,| | \{ ( [ ])\},\par
where ? is the system state. The specific sample variable in the state ? takes the value [ ] ? ? . ( [ ]) -the possibility to observe the state ? with the sample variable at the time ? .\par
Step 7.3. The index value of generating fuzziness of the system \hyperref[b16]{[17]} ( ) is calculated by the formula:( ) = ? ? log ( ) | |\par
, where ( ) is ordered by descending of ? , ? behavior function with a fictitious element |?| = 0, |?| -power set of states.\par
Step 7.4. = + 1. is given. + ? ? , then go to step 7.7.\par
Step 7.5. The behavior function ( ) is determined for the data subset [ , + ? ] ? and the generating fuzziness ( ) is determined.\par
Step 7.6. If ( ) ? ( ) max ( ) , ( ) < ? , then go to Step 7.4. If the condition is not satisfied, then the point + ( ? 1) ? ? is taken as an approximation of the replacement point of the metasystem elements. For this point, the value = 1 is assumed and the transition to Step 7.2 is performed.\par
Step 7.7. Stop. At the moments of change in the behavior of the system, single impulses form. For a fixed threshold ? ? [0,1], = 1, ? for discrete moments ? , we obtain a twodimensional impulse function:( , ) = 1, = ; 0, ? .\par
Step 8. There is a generalized impulse function of the signal under study by the formula:( ) = ? ? ( , ) ? ?\{?\} ,\par
where ( , ) is the impulse function for the threshold ? of the metasystem identification algorithm. Year 2019F © 2019 Global Journals\par
The time coordinates ? of the approximating impulse function ( ) for the signal under study are defined as the local maxima of the function ( ). The coordinates will specify a numeric sequence with variable step ( ). The total number of pulses of the function under study . Thus, there will be local intervals, where the original signal has relatively stable frequency characteristics.\par
Step.9. For each pulse , a window ? is determined based on the selected window type in accordance with Step 1.\par
Step 10. For each pulse of the signal under investigation with the coordinate , for each basis function of the pulse sines, the coefficients of the balance equation \hyperref[b6]{[7]} are determined by the formula:? ( ) = \{ ( ) ? ? ( )\},\par
where the coefficient ( ) = ? (?1) ? ? \{?1,0,1\}. ( ) -is the partial sum of the sign-variable number series obtained from the arithmetic progression for the pulse sine ( ):(?1) ? ? = ( ),\par
where ? is the time window near the pulse coordinate of the function under study. The partial sum ( ) takes into account the influence of all the pulses of the function ( ) falling into the window .\par
Step 9. For each -th impulse sine function, the degree of its influence on the formation of the resulting impulse ( ) is determined. For each pulse ( ) the determination of the coefficients of the influence degrees ( ) ? [0,1] is found either by the optimization method (method 1) or by the approximate method (method 2).\par
According to method 1, the coefficients ( ) ? [0,1] are found from the solution of an optimization linear programming problem based on the simplex method \hyperref[b17]{[18]}:? ? ? ? ? ( ) ? ? ( ) ? min \{ ( )\} ( ) = 1, ? , ( ) ? 0,\par
According to method 2, the coefficients ( ) ? [0,1] are found by the formula:( ) = | ( )| ? 1 ? 2 ? |? ( )| ( ) + ( ) , ( ) ? 0, ( ) = ? ( ), ( ) = 0, ? [0,1].\par
where ( ) and ( ) are the right and left deviations from the coordinate of the pulse . Coefficient ? [0,1] is determined from the correct consideration condition of the low-frequency components of the signal spectrum: For all pulses of the signal under study, in the vicinity of the coordinates , a function of the conditional instantaneous spectral density ( ): ? ? [0,1] is formed, where ( ) = ( ).\par
Step 10. Spectrum estimates are determined based on fuzzy filtering. The estimated spectrum is determined by the fuzzy filter formula \hyperref[b21]{[19,}\hyperref[b18]{20]}. In the simplest case, a filter is used in the form( ) = ( ) + ? \{ ( ) ? ( )\},\par
where ? [0,1] is the gain of a fuzzy filter \hyperref[b19]{[21]}.\par
Step 11. Stop.\par
The function ( ) is taken as the basis for the time-frequency analysis of the signal under study. It is also used to restore the original signal. In this case, the reconstructed signal is found as the sum of sinusoidal functions, which determine the set of basis functions ? with amplitudes multiplied by the value ( ). The accuracy of the algorithm is checked by the error criterion, which is calculated, for example, by the degree of correlation \hyperref[b20]{[22]} between two functions. To test the performance of the algorithm, studies were performed on a model example of a non-stationary signal.\par
To substantiate the performance of the proposed algorithm for time-frequency analysis of non-stationary signals, it is necessary to solve two classical interrelated tasks: 1. To determine the spectrum of the investigated signal. Compare the results obtained with the actual signal spectrum. Estimate the errors of the proposed algorithm. 2. Based on the obtained spectra, restore the signal and compare it with the original signal.\par
Rate the error.\par
For the study of the algorithm, a non-stationary signal with a sampling interval of 0.  
\section[{2.}]{2.}\par
We will assume that quantization by level of the signal under study provides the power of the set of signal values ( ) = 128. To represent the signal value, we will use -adic numbers with = 2. In this case, to represent one data in the data system, the number of blocks of the observation channel will be = 7. 
\section[{3.}]{3.}\par
To determine the set of sample variables and build the function of the system behavior, we will use the simplest mask with the shift parameter = 0 for all variables of the system , ( ) ? . In the metasystem identification algorithm, the set of cut-off threshold values \{? \} is presented in Table  {\ref 1}.\par
The set of cut-off thresholds taken in the study In the balance equations, we will use a symmetric window LR with the averaged deviation ( ) = 0.5 ? ( ) + ( ) . 5.\par
To test the algorithm performance in determining the influence degrees of the -th impulse sine function ( ) from the set of basis functions ?, we will use an approximate approach.\par
Based on the initial data, as a result of applying the algorithm proposed above, the data system was obtained for the signal under study. Data system is a matrix × with a dimension of 7 × 100. In Table \hyperref[tab_1]{2} shows a fragment of this matrix for a time subset up to? [0; 1] .\par
Fragment of the data matrix for the time series under study 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 1 0 0 0 1 0 1 0 0 0 0 1 0 1 0 1 1 1 0 0 1\par
As a result of using the metasystem identification algorithm, the full two-dimensional impulse function ( , ) was obtained (Fig.  {\ref 5}). Based on this function, a generalized impulse function ( ) was obtained (Fig.  {\ref 6}).\par
Full t wo-dimensional pulse function ( , ) for the signal  The time series of the signal under study and the corresponding generalized impulse function ( ) (normalized to the maximum value of the series under study)\par
The function ( ) allows to obtain the impulse function of the signal under study ( ). The coordinates of the pulses were determined as the local extrema coordinates of the function ( ) (Fig.  {\ref 7}). 
\section[{Pulse function of the signal under study ( )}]{Pulse function of the signal under study ( )}\par
Based on the obtained impulse function ( ) for a given set of impulse functions ? = \{ ( )\}, = 1,4 , we obtained the coefficients of the balance equation for all coordinates of the impulses , = 1, . For example, for the coordinate = 19 and for the average deviation, the window will be defined as: Then the coefficient ? ( ) for the balance equation will take the value:? ( ) = \{ ( ) ? ? ( )\} = ?1 ? 19 ? (?21) = 2.\par
Similarly, the coefficients of the balance equation are calculated for all basis functions and coordinates , = 1, . (Table  {\ref 3}). In Table  {\ref 4} estimates of the coefficients ( ) (normalized to unity) obtained using the direct approximate approach to their definition are presented. Table  {\ref 4a} presents the directly obtained estimates of the coefficients ( ), and in Table  {\ref 4b} are estimates of the coefficients ( ) after filtering with the gain = 0.53.\par
Estimates of the coefficients ( ) of the signal spectrum, obtained by the approximate method Table  {\ref 4a} Table 4b The change in the ( ) coefficients of the signal spectrum over time is shown below (Fig.  {\ref 8}).\par
Change dynamics of the ( ) coefficients of the signal spectrum From Fig.  {\ref 8}, it follows that the algorithm provides the correct spectrum estimate for a nonstationary signal. In Fig.  {\ref 9} a comparison of the true and estimated signal spectrum at the initial and final measurement interval is shown. 
\section[{Comparison of the true and estimated spectrum}]{Comparison of the true and estimated spectrum}\par
At the same time, over the entire observation interval, the degree of correlation between the spectra at each frequency remains high (more than 0.7, Fig. 10).\par
The correlation coefficient between the spectra for each frequency over the entire time interval\par
The change in the value of the correlation coefficient between the true and estimated spectra over the entire time interval is shown in Fig. 11.  The correlation degree of the spectra at the stationary parts of the signal refers to a high one (the correlation coefficient is more than 0.7). At the moment of changing the spectra of a true signal, the algorithm provides fast adaptation to a new signal spectrum (within 1 -2 pulses of an impulse function simulating a true signal). Thus, with a large number of measurements, the algorithm will provide an increase in the accuracy of estimation of the current signal spectrum.\par
Based on the estimated spectra in each windows, the estimated signal is easily restored. In this case, the coefficients ( ) as amplitudes of the sinusoidal signals normalized to one are used. These sinusoidal signals correspond to the impulse sines , from the set of basis functions ?. Fig. 12 shows the true and reconstructed signals from the estimated spectrum.\par
True and reconstructed signals from the estimated spectrum.\par
The correlation coefficient of the signals is 0.933, which corresponds to a high level of correlation between the true and reconstructed signals. The average error for the Hamming distance is about 0.1 for the entire observation interval of the signal, taking into account the sharp change in its spectrum. From the graph it is seen that after a sharp change of the signal spectrum, there is an error in its recovery. However, by the 9th second the signals almost coincide. Thus, a relatively high degree of adaptation of the spectrum estimation algorithm for a non-stationary signal can be made.\par
The developed algorithm based on the results given in article \hyperref[b6]{[7]} allows to reduce the problem of estimating the signal spectrum to the solution of a system of linear equations. These equations are based on the use of arithmetic progressions. In this case, the task of calculating the integral convolutions of functions that introduce additional errors in the definition of the signal spectrum is excluded. Additionally algorithm provides adaptive time-frequency localization which is linked to the measured signal. This reduces errors at the borders of the windows when determining the current spectra. These results were made possible due to the properties of the system behavior functions. These functions are the distribution of the possibility measure on a set of system states.\par
Studies of the algorithm have shown that the accuracy of determining the current signal spectrum will depend on the set of its parameters. In particular, the accuracy will depend on the signal discretization conditions, on parameters of the construction algorithm of behavioral functions, on algorithm for identifying the metasystem, on the choice of window, on the gain of the fuzzy filter, and other parameters. These parameters are the algorithm settings. The study of the algorithm on the example of a non-stationary signal with a sharp change in the spectrum showed the efficiency of using the algorithm and a high degree of correlation between the estimated and true signal spectra. Estimates of the signal spectra, which are obtained using the algorithm, make it possible to recover the true signal with a high degree of correlation (with a correlation coefficient higher than 0.9).\begin{figure}[htbp]
\noindent\textbf{1201918}\includegraphics[]{image-2.png}
\caption{\label{fig_0}1 2019 F 18 ©}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-3.png}
\caption{\label{fig_1}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{13}\includegraphics[]{image-4.png}
\caption{\label{fig_2}1 . 3 .}\end{figure}
            \begin{figure}[htbp]
\noindent\textbf{2} \par 
\begin{longtable}{P{0.85\textwidth}}
Year 2019\\
27\\
of Researches in Engineering ( ) Volume XIX Issue I Version I F\\
Global Journal\end{longtable} \par
 
\caption{\label{tab_1}Table 2 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{} \par 
\begin{longtable}{P{0.038636363636363635\textwidth}P{0.09659090909090909\textwidth}P{0.09659090909090909\textwidth}P{0.09659090909090909\textwidth}P{0.09659090909090909\textwidth}P{0.038636363636363635\textwidth}P{0.09659090909090909\textwidth}P{0.09659090909090909\textwidth}P{0.09659090909090909\textwidth}P{0.09659090909090909\textwidth}}
19\tabcellsep 0.167\tabcellsep 0.667\tabcellsep 0.667\tabcellsep 0.000\tabcellsep 19\tabcellsep 0.111\tabcellsep 0.444\tabcellsep 0.444\tabcellsep 0.000\\
30\tabcellsep 0.167\tabcellsep 0.667\tabcellsep 0.667\tabcellsep 0.333\tabcellsep 30\tabcellsep 0.100\tabcellsep 0.402\tabcellsep 0.402\tabcellsep 0.096\\
43\tabcellsep 0.000\tabcellsep 0.200\tabcellsep 0.667\tabcellsep 0.400\tabcellsep 43\tabcellsep 0.047\tabcellsep 0.272\tabcellsep 0.468\tabcellsep 0.213\\
49\tabcellsep 0.333\tabcellsep 0.667\tabcellsep 0.333\tabcellsep 0.400\tabcellsep 49\tabcellsep 0.124\tabcellsep 0.332\tabcellsep 0.322\tabcellsep 0.222\\
56\tabcellsep 1.000\tabcellsep 0.333\tabcellsep 0.167\tabcellsep 0.200\tabcellsep 56\tabcellsep 0.370\tabcellsep 0.260\tabcellsep 0.203\tabcellsep 0.167\\
67\tabcellsep 0.000\tabcellsep 0.000\tabcellsep 0.800\tabcellsep 0.800\tabcellsep 67\tabcellsep 0.174\tabcellsep 0.122\tabcellsep 0.360\tabcellsep 0.343\\
76\tabcellsep 1.000\tabcellsep 0.000\tabcellsep 0.400\tabcellsep 0.400\tabcellsep 76\tabcellsep 0.376\tabcellsep 0.057\tabcellsep 0.287\tabcellsep 0.279\\
85\tabcellsep 0.750\tabcellsep 0.500\tabcellsep 0.000\tabcellsep 0.750\tabcellsep 85\tabcellsep 0.376\tabcellsep 0.159\tabcellsep 0.135\tabcellsep 0.330\\
92\tabcellsep 0.667\tabcellsep 0.333\tabcellsep 0.000\tabcellsep 0.750\tabcellsep 92\tabcellsep 0.378\tabcellsep 0.176\tabcellsep 0.063\tabcellsep 0.382\end{longtable} \par
 
\caption{\label{tab_2}}\end{figure}
 			\footnote{The Problem Solving Algorithm time-Frequency Signals Analysis based on Behavior Functions and Arithmetic Series} 			\footnote{The Problem Solving Algorithm time-Frequency Signals Analysis based on Behavior Functions and Arithmetic Series © 2019 Global Journals} 			\footnote{© 2019 Global Journals} 		 		\backmatter  			  				\begin{bibitemlist}{1}
\bibitem[Gabor ()]{b7}\label{b7} 	 		\textit{},  		 			D Gabor 		.  	 	 		\textit{Journal IEE (London)}  		1946. 93 p. .  	 
\bibitem[????? (ed.) ()]{b15}\label{b15} 	 		\textit{},  		 			? ????? 		.  		????. ?.?. ??????????. -?. ????? (ed.)  		2004. 112.  	 
\bibitem[?????????? ()]{b6}\label{b6} 	 		\textit{????????-??????? ?????? ?? ?????? ??????? ????????? ? ???????????? ?????. ??????? 1. ?????? ???????? ?? ???? ??????. ??????? ???????? ????? ?????? ??Ñ?"???-???????????? ?????????? ???? ????? ????? ??????????????},  		 			? ? ?????????? 		.  		2018. 3 p. .  	 
\bibitem[Taylor ()]{b20}\label{b20} 	 		\textit{An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements},  		 			John R Taylor 		.  		1997. Sausalito, CA: University Science Books. p. 217.  	 	 (2nd ed.) 
\bibitem[Beauchamp ()]{b3}\label{b3} 	 		\textit{Applications of Walsh and Related Functions},  		 			K G Beauchamp 		.  		1984. Orlando: Academic Press. p. 308.  	 
\bibitem[Klir and Elias ()]{b11}\label{b11} 	 		\textit{Architecture of Systems Problem Solving},  		 			G Klir 		,  		 			D Elias 		.  		1985. New York: Plenum Press. 354.  	 
\bibitem[Heil and Walnut ()]{b4}\label{b4} 	 		‘Continuous and discrete wavelet transforms’.  		 			C Heil 		,  		 			D Walnut 		.  	 	 		\textit{SIAM Review}  		1989. 31  (4)  p. .  	 
\bibitem[Bocharnikov and Bocharnikov]{b21}\label{b21} 	 		‘Discrete fuzzy filter of UAV's flight parameters’.  		 			V Bocharnikov 		,  		 			I Bocharnikov 		.  	 	 		\textit{Proceedings of the NAU. 2010. ?3},  				 (the NAU. 2010. ?3)  		p. .  	 
\bibitem[Wu and Huang ()]{b10}\label{b10} 	 		\textit{Ensemble Empirical Mode Decomposition: a noise assisted data analysis method // Advances in Adaptive Data Analysis},  		 			Z Wu 		,  		 			N E Huang 		.  		2008. 1 p. .  	 
\bibitem[Rioul and Duhamel ()]{b5}\label{b5} 	 		‘Fast algorithms for discrete and continuous wavelet transforms’.  		 			O Rioul 		,  		 			P Duhamel 		.  	 	 		\textit{IEEE Trans. Inform. Th., sp. iss. Wavelet Transforms and Multiresolution Signal Analysis}  		1992. 38  (2)  p. .  	 
\bibitem[Kova?cevi?c et al. ()]{b2}\label{b2} 	 		\textit{Fourier and Wavelet Signal Processing},  		 			J Kova?cevi?c 		,  		 			V K Goyal 		,  		 			M Vetterli 		.  		 ?. 272.  		2013.  	 
\bibitem[Bocharnikov et al. ()]{b12}\label{b12} 	 		\textit{Fundamentals of the systemic organizations management. Theory and Practice},  		 			V Bocharnikov 		,  		 			I Bocharnikov 		,  		 			S Sveshnikov 		.  		2012. Berlin: LAP LAMBERT Academic Publishing.  	 
\bibitem[?????????? ()]{b13}\label{b13} 	 		\textit{Fuzzy-??????????. ??????????? ? ???????? ??????? ??? ????????????? ?????????????. -????: ????-?????, ?????},  		 			? ? ?????????? 		.  		2002. 224.  	 
\bibitem[Huang ()]{b9}\label{b9} 	 		 			N E Huang 		.  	 	 		\textit{The Hilbert-Huang transform and its applications / Ed. By S.S.Shen. Interdisciplinary mathematical sciences. 5 Toh Tuck Link},  				 (Singapore)  		2005. World Scienti?c Publishing Company Co. 596224 p. 311.  	 
\bibitem[Higashi and Klir ()]{b16}\label{b16} 	 		‘Measure of uncertainty and information based of possibility distribution’.  		 			M Higashi 		,  		 			J Klir 		.  	 	 		\textit{International Journal of General System}  		1983. 9  (1)  p. .  	 
\bibitem[Bocharnikov and Bocharnikov]{b19}\label{b19} 	 		‘Optimal discrete fuzzy filter of UAV's flight parameters’.  		 			V Bocharnikov 		,  		 			I Bocharnikov 		.  	 	 		\textit{Proceedings of the NAU. 2012. ?3. -p},  				 (the NAU. 2012. ?3. -p)  		p. .  	 
\bibitem[Daubechies ()]{b0}\label{b0} 	 		‘Orthonormal bases of compactly supported wavelets’.  		 			I Daubechies 		.  	 	 		\textit{XLI}  		1988. 7 p. .  	 
\bibitem[Daubechies ()]{b8}\label{b8} 	 		‘Orthonormal bases of compactly supported wavelets’.  		 			I Daubechies 		.  	 	 		\textit{//Commun. On Pure and Appl}  		1998. 41 p. .  	 
\bibitem[Bocharnikov and Bocharnikov (2013)]{b18}\label{b18} 	 		‘Simplified and adopted to the MatLab fuzzy filter of UAV's flight parameters’.  		 			V Bocharnikov 		,  		 			I Bocharnikov 		.  	 	 		\textit{IEEE 2nd International Conference Actual Problems of Unmanned Air Vehicles Developments Proceedings},  				 (Kiev, Ukraine)  		2013. October 2013. p. .  	 
\bibitem[Borgwardt ()]{b17}\label{b17} 	 		\textit{The simplex algorithm takes on average D steps for a cube. The simplex method: A probabilistic analysis},  		 			Karl-Heinz Borgwardt 		.  		 P.268.  		1987. Berlin: Springer-Verlag. 1.  	 
\bibitem[Harmuth ()]{b1}\label{b1} 	 		\textit{Transmission of Information by Orthogonal Functions},  		 			H F Harmuth 		.  		1969. New York: Springer-Verlag.  	 
\bibitem[Hensel]{b14}\label{b14} 	 		\textit{Untersuchung der Fundamentalglelchung einer Gattung fur eine reelle Prlmzahl als Modul und Bestimmung der Theiler Ihrer Discriminante // J. Reine Angew. Math. 1894},  		 			K Hensel 		.  		p. .  	 
\end{bibitemlist}
 			 		 	 
\end{document}
