Recovering the actual trajectory of economic cycles

Кармалита Вячеслав А.

doi:10.31857/S042473880024867-2

Русский

Главная>№ 2>Recovering the actual trajectory of economic cycles

Recovering the actual trajectory of economic cycles

Оглавление

Аннотация Оценить Содержание публикации

Библиография Комментарии

Recovering the actual trajectory of economic cycles

Аннотация

Код статьи

S042473880024867-2-1

DOI

10.31857/S042473880024867-2

Тип публикации

Статья

Статус публикации

Опубликовано

Авторы

Кармалита Вячеслав Алексеевич Связаться с автором

Должность: Частный консультант
Аффилиация: Private consultant
Адрес: Канада,

Выпуск

Том 59 № 2

Страницы

19-25

Аннотация

The paper deals with the development of a method for restoring the trajectory of economic cycles from estimates of the gross domestic product (GDP). The proposed approach to solve this problem is based on the interpretation of cycles in the form of random oscillations of the income with a certain natural frequency, also called a narrowband random process. The operators (Fourier transforms, filtering, etc.) used to recover the cycle trajectory are linear. Their inherent associativity property allows changing the sequence of implementation of the linear operators above. As a result, it is proposed to start the recovery with bandpass filtering of the GDP function, and after that to parry the influence of the inertia property of the GDP estimator. Taking the qualities of a narrowband random process into consideration made it possible to create a simplified procedure to recover the cycle trajectory. In the example of the Kuznets swing, the acceptability of this procedure is demonstrated for the practical econometrics. The developed method is applicable in problems that require knowledge of the trajectory of the considered cycle.

Ключевые слова

economic cycle, random oscillations, cycle trajectory, Fourier transforms, frequency response characteristics

Классификатор

Получено

15.03.2023

Дата публикации

02.07.2023

Всего подписок

Всего просмотров

190

Оценка читателей

0.0 (0 голосов)

Цитировать Скачать pdf

ГОСТ	Кармалита В. А. Recovering the actual trajectory of economic cycles // Экономика и математические методы. – 2023. – T. 59. – № 2 C. 19-25 . URL: https://emmras.ru/recovering-the-actual-trajectory-of-economic-cycles/?version_id=101974. DOI: 10.31857/S042473880024867-2
MLA	Karmalita, Viacheslav "Recovering the actual trajectory of economic cycles." Economics and the Mathematical Methods. 59.2 (2023).:19-25. DOI: 10.31857/S042473880024867-2
APA	Karmalita V. (2023). Recovering the actual trajectory of economic cycles. Economics and the Mathematical Methods. vol. 59, no. 2, pp.19-25 DOI: 10.31857/S042473880024867-2


1	1. INTRODUCTION	1. INTRODUCTION 1. INTRODUCTION

2	The model of economic cycles, as proposed in reference (Karmalita, 2020), is based on a probabilistic description of the investment function and the perception of the economic system as a material object with certain inherent properties. As for the system model itself, it represents the economic cycle as random oscillations generated by a linear elastic system with the natural frequency f₀= 1/T₀ and a damping coefficient h under the influence of the Gaussian white noise E(t) (Bolotin, 1984). Mathematically, the cycle model is represented by an ordinary differential equation of the second order:	The model of economic cycles, as proposed in reference (Karmalita, 2020), is based on a probabilistic description of the investment function and the perception of the economic system as a material object with certain inherent properties. As for the system model itself, it represents the economic cycle as random oscillations generated by a linear elastic system with the natural frequency <em>f</em><sub>0 </sub>= 1/<em>T</em><sub>0</sub> and a damping coefficient <em>h </em>under the influence of the Gaussian white noise <em>E</em>(<em>t</em>) (Bolotin, 1984). Mathematically, the cycle model is represented by an ordinary differential equation of the second order: The model of economic cycles, as proposed in reference (Karmalita, 2020), is based on a probabilistic description of the investment function and the perception of the economic system as a material object with certain inherent properties. As for the system model itself, it represents the economic cycle as random oscillations generated by a linear elastic system with the natural frequency <em>f</em><sub>0 </sub>= 1/<em>T</em><sub>0</sub> and a damping coefficient <em>h </em>under the influence of the Gaussian white noise <em>E</em>(<em>t</em>) (Bolotin, 1984). Mathematically, the cycle model is represented by an ordinary differential equation of the second order:

3	$\ddot{Ξ} (t) + 2 h \dot{Ξ} (t) + (2 π f_{0})^{2} Ξ (t) = E (t),$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="true"><mrow><mi mathvariant="normal">Ξ</mi></mrow><mo>¨</mo></mover><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced><mo>+</mo><mn>2</mn><mi>h</mi><mover accent="true"><mrow><mi mathvariant="normal">Ξ</mi></mrow><mo>˙</mo></mover><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced><mo>+</mo><mo>(</mo><mn>2</mn><mi mathvariant="normal">π</mi><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mi mathvariant="normal">Ξ</mi><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mi>E</mi><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="true"><mrow><mi mathvariant="normal">Ξ</mi></mrow><mo>¨</mo></mover><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced><mo>+</mo><mn>2</mn><mi>h</mi><mover accent="true"><mrow><mi mathvariant="normal">Ξ</mi></mrow><mo>˙</mo></mover><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced><mo>+</mo><mo>(</mo><mn>2</mn><mi mathvariant="normal">π</mi><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mi mathvariant="normal">Ξ</mi><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mi>E</mi><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo></math>

4	where Ξ(t) represents random oscillations (cycle under consideration) of the income function X(t), and E(t) — fluctuations of investments. The values of random oscillations Ξ(t) are determined mainly by harmonics in the frequency band f₁ = 0.7f₀ ≤ f ≤ f₂ = 1.4 f₀. This fact explains another name for random oscillations — a narrowband random process.	where Ξ(<em>t</em>) represents random oscillations (cycle under consideration) of the income function<em> X</em>(<em>t</em>), and <em>E</em>(<em>t</em>) — fluctuations of investments. The values of random oscillations Ξ(<em>t</em>) are determined mainly by harmonics in the frequency band <em>f</em><sub>1</sub> = 0.7<em>f</em><sub>0</sub> ≤ <em>f</em> ≤ <em>f</em><sub>2</sub> = 1.4<em> f</em><sub>0</sub>. This fact explains another name for random oscillations — a narrowband random process. where Ξ(<em>t</em>) represents random oscillations (cycle under consideration) of the income function<em> X</em>(<em>t</em>), and <em>E</em>(<em>t</em>) — fluctuations of investments. The values of random oscillations Ξ(<em>t</em>) are determined mainly by harmonics in the frequency band <em>f</em><sub>1</sub> = 0.7<em>f</em><sub>0</sub> ≤ <em>f</em> ≤ <em>f</em><sub>2</sub> = 1.4<em> f</em><sub>0</sub>. This fact explains another name for random oscillations — a narrowband random process.

5	In econometric studies, to quantify the income function X(t) the gross domestic product (GDP), hereinafter G(t), is usually used. Recall that the value of GDP is a monetary estimate of manufactured goods and provisioned services for a certain period ΔT. In the above reference, G(t) is mathematically described by the convolution equation:	In econometric studies, to quantify the income function <em>X</em>(<em>t</em>) the gross domestic product (GDP), hereinafter <em>G</em>(<em>t</em>), is usually used. Recall that the value<em> </em>of GDP is a monetary estimate of manufactured goods and provisioned services for a certain period Δ<em>T</em>. In the above reference, <em>G</em>(<em>t</em>) is mathematically described by the convolution equation: In econometric studies, to quantify the income function <em>X</em>(<em>t</em>) the gross domestic product (GDP), hereinafter <em>G</em>(<em>t</em>), is usually used. Recall that the value<em> </em>of GDP is a monetary estimate of manufactured goods and provisioned services for a certain period Δ<em>T</em>. In the above reference, <em>G</em>(<em>t</em>) is mathematically described by the convolution equation:

6	$G (t) = \int_{t - Δ t}^{t} X (t) d t = \int_{0}^{t} h (τ) X (t - τ) d τ .$ (1)	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>G</mi><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><msubsup><mrow><mo>∫</mo></mrow><mrow><mi>t</mi><mo>-</mo><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mi>X</mi><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced><mi>d</mi><mi>t</mi><mo>=</mo><msubsup><mrow><mo>∫</mo></mrow><mrow><mn>0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mi>h</mi><mfenced separators="\|"><mrow><mi>τ</mi></mrow></mfenced><mi>X</mi><mfenced separators="\|"><mrow><mi>t</mi><mo>-</mo><mi>τ</mi></mrow></mfenced><mi>d</mi><mi>τ</mi><mo>.</mo></math> (1) <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>G</mi><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><msubsup><mrow><mo>∫</mo></mrow><mrow><mi>t</mi><mo>-</mo><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow><mrow><mi>t</mi></mrow></msubsup><mi>X</mi><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced><mi>d</mi><mi>t</mi><mo>=</mo><msubsup><mrow><mo>∫</mo></mrow><mrow><mn>0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mi>h</mi><mfenced separators="\|"><mrow><mi>τ</mi></mrow></mfenced><mi>X</mi><mfenced separators="\|"><mrow><mi>t</mi><mo>-</mo><mi>τ</mi></mrow></mfenced><mi>d</mi><mi>τ</mi><mo>.</mo></math> (1)

7	In other words, the GDP function can be interpreted as the result of measuring the income function using an estimating means (estimator) whose inertial properties are described by the impulse response (IR) function h(τ):	In other words, the GDP function can be interpreted as the result of measuring the income function using an estimating means (estimator) whose inertial properties are described by the impulse response (<em>IR</em>) function<em> </em><em>h</em>(τ): In other words, the GDP function can be interpreted as the result of measuring the income function using an estimating means (estimator) whose inertial properties are described by the impulse response (<em>IR</em>) function<em> </em><em>h</em>(τ):

8	$h (τ) = \{\begin{array}{l} 1, 0 \leq τ \leq Δ t; \\ 0, τ < 0; τ > Δ t . \end{array}$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>h</mi><mfenced separators="\|"><mrow><mi>τ</mi></mrow></mfenced><mo>=</mo><mfenced open="{" close="" separators="\|"><mrow><mtable columnalign="left"><mtr><mtd><mn>1</mn><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mn>0</mn><mo>≤</mo><mi>τ</mi><mo>≤</mo><mi mathvariant="normal">Δ</mi><mi>t</mi><mo>;</mo></mtd></mtr><mtr><mtd><mn>0</mn><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi>τ</mi><mo><</mo><mn>0</mn><mo>;</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi>τ</mi><mo>></mo><mi mathvariant="normal">Δ</mi><mi>t</mi><mo>.</mo></mtd></mtr></mtable><mi mathvariant="normal"> </mi></mrow></mfenced></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>h</mi><mfenced separators="\|"><mrow><mi>τ</mi></mrow></mfenced><mo>=</mo><mfenced open="{" close="" separators="\|"><mrow><mtable columnalign="left"><mtr><mtd><mn>1</mn><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mn>0</mn><mo>≤</mo><mi>τ</mi><mo>≤</mo><mi mathvariant="normal">Δ</mi><mi>t</mi><mo>;</mo></mtd></mtr><mtr><mtd><mn>0</mn><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi>τ</mi><mo><</mo><mn>0</mn><mo>;</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi>τ</mi><mo>></mo><mi mathvariant="normal">Δ</mi><mi>t</mi><mo>.</mo></mtd></mtr></mtable><mi mathvariant="normal"> </mi></mrow></mfenced></math>

9	The reference above shows that the concept of the frequency (period) of the income function is quite applicable to the analysis of economic systems. If we consider equation (1) in the frequency domain, then it is transformed into the multiplication of the corresponding Fourier images:	The reference above shows that the concept of the frequency (period) of the income function is quite applicable to the analysis of economic systems. If we consider equation (1) in the frequency domain, then it is transformed into the multiplication of the corresponding Fourier images: The reference above shows that the concept of the frequency (period) of the income function is quite applicable to the analysis of economic systems. If we consider equation (1) in the frequency domain, then it is transformed into the multiplication of the corresponding Fourier images:

10	G(f) = H(f)X(f),(2)	<em>G</em>(<em>f</em>) = <em>H</em>(<em>f</em>)<em>X</em>(<em>f</em>),(2) <em>G</em>(<em>f</em>) = <em>H</em>(<em>f</em>)<em>X</em>(<em>f</em>),(2)

11	where G(f), H(f), and X(f) are the Fourier images of the GDP, impulse response, and income functions, respectively. Therefore, knowledge of h(t) and G(t) provides an opportunity to solve equation (2) with respect to X(f) = G(f)/H(f). The author (Karmalita, 2020) demonstrates that this problem is ill-posed and offers an approximate solution X_α(t) using the regularization method (Tikhonov, Arsenin, 1997). After recovering X_α(t), the trajectory of the concerned cycle Ξ(t) with a natural frequency f₀ is formed by a bandpass filter (Schlichtharle, 2011).	where <em>G</em>(<em>f</em>), <em>H</em>(<em>f</em>), and<em> X</em>(<em>f</em>) are the Fourier images of the GDP, impulse response, and income functions, respectively. Therefore, knowledge of <em>h</em>(<em>t</em>) and <em>G</em>(<em>t</em>) provides an opportunity to solve equation (2) with respect to <em>X</em>(<em>f</em>) = <em>G</em>(<em>f</em>)/<em>H</em>(<em>f</em>). The author (Karmalita, 2020) demonstrates that this problem is ill-posed and offers an approximate solution <em>X</em><sub>α</sub>(<em>t</em>) using the regularization method (Tikhonov, Arsenin, 1997). After recovering <em>X</em><sub>α</sub>(<em>t</em>), the trajectory of the concerned cycle Ξ(<em>t</em>) with a natural frequency <em>f</em><sub>0</sub> is formed by a bandpass filter (Schlichtharle, 2011). where <em>G</em>(<em>f</em>), <em>H</em>(<em>f</em>), and<em> X</em>(<em>f</em>) are the Fourier images of the GDP, impulse response, and income functions, respectively. Therefore, knowledge of <em>h</em>(<em>t</em>) and <em>G</em>(<em>t</em>) provides an opportunity to solve equation (2) with respect to <em>X</em>(<em>f</em>) = <em>G</em>(<em>f</em>)/<em>H</em>(<em>f</em>). The author (Karmalita, 2020) demonstrates that this problem is ill-posed and offers an approximate solution <em>X</em><sub>α</sub>(<em>t</em>) using the regularization method (Tikhonov, Arsenin, 1997). After recovering <em>X</em><sub>α</sub>(<em>t</em>), the trajectory of the concerned cycle Ξ(<em>t</em>) with a natural frequency <em>f</em><sub>0</sub> is formed by a bandpass filter (Schlichtharle, 2011).

12	Let us turn to the recovery procedure with a sample diagram represented in Fig. 1.	Let us turn to the recovery procedure with a sample diagram represented in Fig. 1. Let us turn to the recovery procedure with a sample diagram represented in Fig. 1.

13	Подпись к рисунку/медиа
14	Fig. 1. Forming the cycle trajectory	<strong>Fig. 1.</strong> Forming the cycle trajectory <strong>Fig. 1.</strong> Forming the cycle trajectory

15	This diagram shows the error Z(t) describing the imperfect performance of the estimation procedure. This error arises due to unreliable statistical data, its incompleteness, the performer’s skill, etc. Therefore, the available data related to the values of GDP are the estimates $\tilde{G} (t)$ determined as $\tilde{G} (t) = G (t) + Z (t) .$ The function Z(t) is called the measurement (instrumental) noise, which is usually wideband and is often modeled as “white” noise.	This diagram shows the error <em>Z</em>(<em>t</em>) describing the imperfect performance of the estimation procedure. This error arises due to unreliable statistical data, its incompleteness, the performer’s skill, etc. Therefore, the available data related to the values of GDP are the estimates <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="true"><mrow><mi>G</mi></mrow><mo>~</mo></mover><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced></math> determined as <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="true"><mrow><mi>G</mi></mrow><mo>~</mo></mover><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mi>G</mi><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced><mo>+</mo><mi>Z</mi><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced><mo>.</mo></math> The function <em>Z</em>(<em>t</em>) is called the measurement (instrumental) noise, which is usually wideband and is often modeled as “white” noise. This diagram shows the error <em>Z</em>(<em>t</em>) describing the imperfect performance of the estimation procedure. This error arises due to unreliable statistical data, its incompleteness, the performer’s skill, etc. Therefore, the available data related to the values of GDP are the estimates <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="true"><mrow><mi>G</mi></mrow><mo>~</mo></mover><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced></math> determined as <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="true"><mrow><mi>G</mi></mrow><mo>~</mo></mover><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mi>G</mi><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced><mo>+</mo><mi>Z</mi><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced><mo>.</mo></math> The function <em>Z</em>(<em>t</em>) is called the measurement (instrumental) noise, which is usually wideband and is often modeled as “white” noise.

16	The above approach has some features that complicate its application in econometric practice. In particular, the solution of the ill-posed problem of recovering the income function requires a few subjective decisions. The latter include, for example, the assumption about the nature of the smoothness of an unknown multicomponent income function. Another informal decision relates to establishing the range of the regularization parameter α, in which its optimum is sought. The absence of formal justifications of the above decisions requires a certain skill (a kind of art) to successfully solve the ill-posed problem. The specified influence of subjective factors on the recovery procedure determines the purpose of this article — the development of a formal method to obtain the trajectory of the cycle of interest while avoiding an ill-posed formulation of the problem. This method is presented in the second section of the article. Its third section illustrates the developed method with the example of recovering the Kuznets swing.	The above approach has some features that complicate its application in econometric practice. In particular, the solution of the ill-posed problem of recovering the income function requires a few subjective decisions. The latter include, for example, the assumption about the nature of the smoothness of an unknown multicomponent income function. Another informal decision relates to establishing the range of the regularization parameter <em>α</em>, in which its optimum is sought. The absence of formal justifications of the above decisions requires a certain skill (a kind of art) to successfully solve the ill-posed problem. The specified influence of subjective factors on the recovery procedure determines the purpose of this article — the development of a formal method to obtain the trajectory of the cycle of interest while avoiding an ill-posed formulation of the problem. This method is presented in the second section of the article. Its third section illustrates the developed method with the example of recovering the Kuznets swing. The above approach has some features that complicate its application in econometric practice. In particular, the solution of the ill-posed problem of recovering the income function requires a few subjective decisions. The latter include, for example, the assumption about the nature of the smoothness of an unknown multicomponent income function. Another informal decision relates to establishing the range of the regularization parameter <em>α</em>, in which its optimum is sought. The absence of formal justifications of the above decisions requires a certain skill (a kind of art) to successfully solve the ill-posed problem. The specified influence of subjective factors on the recovery procedure determines the purpose of this article — the development of a formal method to obtain the trajectory of the cycle of interest while avoiding an ill-posed formulation of the problem. This method is presented in the second section of the article. Its third section illustrates the developed method with the example of recovering the Kuznets swing.

17	2. FORMING CYCLE VALUES	2. FORMING CYCLE VALUES 2. FORMING CYCLE VALUES

18	In the frequency domain, the estimator is characterized by its frequency response (FR) function, denoted H(f), which is the Fourier transform of the IR function (Pavleino, Romadanov, 2007):	In the frequency domain, the estimator is characterized by its frequency response (<em>FR</em>) function, denoted <em>H</em>(<em>f</em>), which is the Fourier transform of the <em>IR</em> function (Pavleino, Romadanov, 2007): In the frequency domain, the estimator is characterized by its frequency response (<em>FR</em>) function, denoted <em>H</em>(<em>f</em>), which is the Fourier transform of the <em>IR</em> function (Pavleino, Romadanov, 2007):

19	$H (f) = \int_{- \infty}^{\infty} h (t) e^{- j 2 π f t} d t = Δ t s i n c (π Δ t f) e^{- j π Δ t f} = A_{h} (f) e^{j Φ_{h} (f)},$ (3)	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>H</mi><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced><mo>=</mo><msubsup><mrow><mo>∫</mo></mrow><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>h</mi><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>j</mi><mn>2</mn><mi>π</mi><mi>f</mi><mi>t</mi></mrow></msup><mi>d</mi><mi>t</mi><mo>=</mo><mi mathvariant="normal">Δ</mi><mi>t</mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">c</mi><mfenced separators="\|"><mrow><mi>π</mi><mi mathvariant="normal">Δ</mi><mi>t</mi><mi>f</mi></mrow></mfenced><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>j</mi><mi>π</mi><mi mathvariant="normal">Δ</mi><mi>t</mi><mi>f</mi></mrow></msup><mo>=</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>j</mi><msub><mrow><mi mathvariant="normal">Φ</mi></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced></mrow></msup><mo>,</mo></math> (3) <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>H</mi><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced><mo>=</mo><msubsup><mrow><mo>∫</mo></mrow><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>h</mi><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>j</mi><mn>2</mn><mi>π</mi><mi>f</mi><mi>t</mi></mrow></msup><mi>d</mi><mi>t</mi><mo>=</mo><mi mathvariant="normal">Δ</mi><mi>t</mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">c</mi><mfenced separators="\|"><mrow><mi>π</mi><mi mathvariant="normal">Δ</mi><mi>t</mi><mi>f</mi></mrow></mfenced><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>j</mi><mi>π</mi><mi mathvariant="normal">Δ</mi><mi>t</mi><mi>f</mi></mrow></msup><mo>=</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>j</mi><msub><mrow><mi mathvariant="normal">Φ</mi></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced></mrow></msup><mo>,</mo></math> (3)

20	where A_h(f) and Φ_h(f) are the amplitude and phase frequency (AF and PhF) characteristics of the estimator, while	where <em>A</em><sub><em>h</em></sub>(<em>f</em>) and Φ<sub><em>h</em></sub>(<em>f</em>) are the amplitude and phase frequency (<em>AF</em> and <em>PhF</em>) characteristics of the estimator, while where <em>A</em><sub><em>h</em></sub>(<em>f</em>) and Φ<sub><em>h</em></sub>(<em>f</em>) are the amplitude and phase frequency (<em>AF</em> and <em>PhF</em>) characteristics of the estimator, while

21	$s i n c (π Δ t f) = = \{\begin{array}{l} 1, f = 0; \\ (s i n π Δ t f) / π Δ t f, f \neq 0 . \end{array}$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">c</mi><mo>(</mo><mi>π</mi><mi mathvariant="normal">Δ</mi><mi>t</mi><mi>f</mi><mo>)</mo><mo>=</mo><mo>=</mo><mfenced open="{" close="" separators="\|"><mrow><mtable columnalign="left"><mtr><mtd><mn>1</mn><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi>f</mi><mo>=</mo><mn>0</mn><mo>;</mo></mtd></mtr><mtr><mtd><mfenced separators="\|"><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>π</mi><mi mathvariant="normal">Δ</mi><mi>t</mi><mi>f</mi></mrow></mfenced><mo>/</mo><mi>π</mi><mi mathvariant="normal">Δ</mi><mi>t</mi><mi>f</mi><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi>f</mi><mo>≠</mo><mn>0</mn><mo>.</mo></mtd></mtr></mtable><mi mathvariant="normal"> </mi></mrow></mfenced></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">c</mi><mo>(</mo><mi>π</mi><mi mathvariant="normal">Δ</mi><mi>t</mi><mi>f</mi><mo>)</mo><mo>=</mo><mo>=</mo><mfenced open="{" close="" separators="\|"><mrow><mtable columnalign="left"><mtr><mtd><mn>1</mn><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi>f</mi><mo>=</mo><mn>0</mn><mo>;</mo></mtd></mtr><mtr><mtd><mfenced separators="\|"><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>π</mi><mi mathvariant="normal">Δ</mi><mi>t</mi><mi>f</mi></mrow></mfenced><mo>/</mo><mi>π</mi><mi mathvariant="normal">Δ</mi><mi>t</mi><mi>f</mi><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi>f</mi><mo>≠</mo><mn>0</mn><mo>.</mo></mtd></mtr></mtable><mi mathvariant="normal"> </mi></mrow></mfenced></math>

22	Views of the above-mentioned AF and PhF characteristics are shown in Fig. 2.	Views of the above-mentioned <em>AF</em> and <em>PhF</em> characteristics are shown in Fig. 2. Views of the above-mentioned <em>AF</em> and <em>PhF</em> characteristics are shown in Fig. 2.

23	Подпись к рисунку/медиа
24	Fig. 2. The frequency characteristics of the estimator	<strong>Fig.</strong><strong> </strong><strong>2</strong><strong>.</strong> The frequency characteristics of the estimator <strong>Fig.</strong><strong> </strong><strong>2</strong><strong>.</strong> The frequency characteristics of the estimator

25	A_h(f) determines the ratio of the amplitudes of the input and output harmonics of the estimator. Φ_h(f) is the difference between their phases, which is equivalent to the time delay of the output process with respect to the input process. It follows from the expression (2) that H(f) only change the amplitude and phase of the harmonics that make up X(t). Despite this, the spectral composition of GDP and income functions is similar. Recall that the intensity (amplitude) of the income’s oscillations with the natural frequency f₀ is determined by the frequency range in the vicinity of f₀. This fact led to the use of the bandpass filtering to extract the data of the cycle of interest from the recovered income function. Such filtering allows signals to pass unaltered within the frequency range f₁ ≤ f ≤ f₂ while suppressing all the others. Amplitude spectrum A_c(f) of the income random oscillations and AF characteristics A_f(f) of the bandpass filter are shown in Fig. 3.	<em>A</em><sub><em>h</em></sub>(<em>f</em>) determines the ratio of the amplitudes of the input and output harmonics of the estimator. Φ<sub><em>h</em></sub>(<em>f</em>) is the difference between their phases, which is equivalent to the time delay of the output process with respect to the input process. It follows from the expression (2) that <em>H</em>(<em>f</em>) only change the amplitude and phase of the harmonics that make up <em>X</em>(<em>t</em>). Despite this, the spectral composition of GDP and income functions is similar. Recall that the intensity (amplitude) of the income’s oscillations with the natural frequency <em>f</em><sub>0</sub> is determined by the frequency range in the vicinity of <em>f</em><sub>0</sub>. This fact led to the use of the bandpass filtering to extract the data of the cycle of interest from the recovered income function. Such filtering allows signals to pass unaltered within the frequency range<em> </em><em>f</em><sub>1</sub> ≤ <em>f</em> ≤ <em>f</em><sub>2</sub> while suppressing all the others. Amplitude spectrum <em>A</em><sub><em>c</em></sub>(<em>f</em>) of the income random oscillations and<em> AF</em> characteristics <em>A</em><sub><em>f</em></sub>(<em>f</em>) of the bandpass filter are shown in Fig. 3. <em>A</em><sub><em>h</em></sub>(<em>f</em>) determines the ratio of the amplitudes of the input and output harmonics of the estimator. Φ<sub><em>h</em></sub>(<em>f</em>) is the difference between their phases, which is equivalent to the time delay of the output process with respect to the input process. It follows from the expression (2) that <em>H</em>(<em>f</em>) only change the amplitude and phase of the harmonics that make up <em>X</em>(<em>t</em>). Despite this, the spectral composition of GDP and income functions is similar. Recall that the intensity (amplitude) of the income’s oscillations with the natural frequency <em>f</em><sub>0</sub> is determined by the frequency range in the vicinity of <em>f</em><sub>0</sub>. This fact led to the use of the bandpass filtering to extract the data of the cycle of interest from the recovered income function. Such filtering allows signals to pass unaltered within the frequency range<em> </em><em>f</em><sub>1</sub> ≤ <em>f</em> ≤ <em>f</em><sub>2</sub> while suppressing all the others. Amplitude spectrum <em>A</em><sub><em>c</em></sub>(<em>f</em>) of the income random oscillations and<em> AF</em> characteristics <em>A</em><sub><em>f</em></sub>(<em>f</em>) of the bandpass filter are shown in Fig. 3.

26	Подпись к рисунку/медиа
27	Fig. 3. Amplitude spectrum of the cycle oscillations and AF characteristics of the bandpass filter	<strong>Fig. </strong><strong>3</strong><strong>.</strong> Amplitude spectrum of the cycle oscillations and<em> AF</em> characteristics of the bandpass filter <strong>Fig. </strong><strong>3</strong><strong>.</strong> Amplitude spectrum of the cycle oscillations and<em> AF</em> characteristics of the bandpass filter

28	In principle, extracting some cycle Ξ(t) with natural frequency f₀ does not require a complete recovery of the income function. The associativity inherent in linear convolution and filtering operators allows changing the sequence of operations. Namely, one can first form the GDP function corresponding to the considered cycle, and then recover only the values of the latter (Fig. 4).	In principle, extracting some cycle Ξ(<em>t</em>) with natural frequency <em>f</em><sub>0</sub> does not require a complete recovery of the income function. The associativity inherent in linear convolution and filtering operators allows changing the sequence of operations. Namely, one can first form the GDP function corresponding to the considered cycle, and then recover only the values of the latter (Fig. 4). In principle, extracting some cycle Ξ(<em>t</em>) with natural frequency <em>f</em><sub>0</sub> does not require a complete recovery of the income function. The associativity inherent in linear convolution and filtering operators allows changing the sequence of operations. Namely, one can first form the GDP function corresponding to the considered cycle, and then recover only the values of the latter (Fig. 4).

29	Подпись к рисунку/медиа
30	Fig. 4. Proposed recovery approach	<strong>Fig. </strong><strong>4</strong><strong>.</strong> Proposed recovery approach <strong>Fig. </strong><strong>4</strong><strong>.</strong> Proposed recovery approach

31	It should be noted that in addition to the extract function, this filtering will suppress the noise Z(t) that accompanies the realization $\tilde{G} (t)$ .	It should be noted that in addition to the extract function, this filtering will suppress the noise <em>Z</em>(<em>t</em>) that accompanies the realization <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="true"><mrow><mi>G</mi></mrow><mo>~</mo></mover><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced></math> . It should be noted that in addition to the extract function, this filtering will suppress the noise <em>Z</em>(<em>t</em>) that accompanies the realization <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="true"><mrow><mi>G</mi></mrow><mo>~</mo></mover><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced></math> .

32	The presence of filtered realization G_f(t) makes it possible to recover the values of income oscillations in the form of estimates $\tilde{Ξ} (t)$ . Using expression (2), the Fourier image $\tilde{Ξ} (t)$ can be represented in terms of AF and PhF characteristics in the following form:	The presence of filtered realization <em>G</em><sub><em>f</em></sub>(<em>t</em>)<em> </em>makes it possible to recover the values of income oscillations in the form of estimates <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="true"><mrow><mi mathvariant="normal">Ξ</mi></mrow><mo>~</mo></mover><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced></math> . Using expression (2), the Fourier image <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="true"><mrow><mi mathvariant="normal">Ξ</mi></mrow><mo>~</mo></mover><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced></math> can be represented in terms of <em>AF</em> and <em>PhF</em> characteristics in the following form: The presence of filtered realization <em>G</em><sub><em>f</em></sub>(<em>t</em>)<em> </em>makes it possible to recover the values of income oscillations in the form of estimates <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="true"><mrow><mi mathvariant="normal">Ξ</mi></mrow><mo>~</mo></mover><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced></math> . Using expression (2), the Fourier image <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="true"><mrow><mi mathvariant="normal">Ξ</mi></mrow><mo>~</mo></mover><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced></math> can be represented in terms of <em>AF</em> and <em>PhF</em> characteristics in the following form:

33	$\tilde{Ξ} (f) = G_{f} (f) / H (f) = \frac{A_{G_{f}} (f) e^{j Φ_{G_{f}} (f)}}{{\overset{⃡}{A}}_{h} (f) e^{j {\overset{⃡}{Φ}}_{h} (f)}} = \frac{A_{G_{f}} (f)}{{\overset{⃡}{A}}_{h} (f)} e^{j [Φ_{G_{f}} (f) - {\overset{⃡}{Φ}}_{h} (f)]} .$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="true"><mrow><mi mathvariant="normal">Ξ</mi></mrow><mo>~</mo></mover><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced><mo>=</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>f</mi></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced><mo>/</mo><mi>H</mi><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><msub><mrow><mi>A</mi></mrow><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>j</mi><msub><mrow><mi mathvariant="normal">Φ</mi></mrow><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced></mrow></msup></mrow><mrow><msub><mrow><mover accent="true"><mrow><mi>A</mi></mrow><mo>⃡</mo></mover></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>j</mi><msub><mrow><mover accent="true"><mrow><mi mathvariant="normal">Φ</mi></mrow><mo>⃡</mo></mover></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced></mrow></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mrow><mi>A</mi></mrow><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced></mrow><mrow><msub><mrow><mover accent="true"><mrow><mi>A</mi></mrow><mo>⃡</mo></mover></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced></mrow></mfrac><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>j</mi><mo>[</mo><msub><mrow><mi mathvariant="normal">Φ</mi></mrow><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced><mo>-</mo><msub><mrow><mover accent="true"><mrow><mi mathvariant="normal">Φ</mi></mrow><mo>⃡</mo></mover></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced><mo>]</mo></mrow></msup><mo>.</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="true"><mrow><mi mathvariant="normal">Ξ</mi></mrow><mo>~</mo></mover><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced><mo>=</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>f</mi></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced><mo>/</mo><mi>H</mi><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><msub><mrow><mi>A</mi></mrow><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>j</mi><msub><mrow><mi mathvariant="normal">Φ</mi></mrow><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced></mrow></msup></mrow><mrow><msub><mrow><mover accent="true"><mrow><mi>A</mi></mrow><mo>⃡</mo></mover></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>j</mi><msub><mrow><mover accent="true"><mrow><mi mathvariant="normal">Φ</mi></mrow><mo>⃡</mo></mover></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced></mrow></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mrow><mi>A</mi></mrow><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced></mrow><mrow><msub><mrow><mover accent="true"><mrow><mi>A</mi></mrow><mo>⃡</mo></mover></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced></mrow></mfrac><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>j</mi><mo>[</mo><msub><mrow><mi mathvariant="normal">Φ</mi></mrow><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>f</mi></mrow></msub></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced><mo>-</mo><msub><mrow><mover accent="true"><mrow><mi mathvariant="normal">Φ</mi></mrow><mo>⃡</mo></mover></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced><mo>]</mo></mrow></msup><mo>.</mo></math>

34	Here ${\overset{⃡}{A}}_{h} (f)$ and ${\overset{⃡}{Φ}}_{h} (f)$ include the fragments of corresponding AF and PhF characteristics in the range f₁ ≤ f ≤ f₂ (see Fig. 3), having the following values:	Here <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mover accent="true"><mrow><mi>A</mi></mrow><mo>⃡</mo></mover></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced></math> and <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mover accent="true"><mrow><mi mathvariant="normal">Φ</mi></mrow><mo>⃡</mo></mover></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced></math> include the fragments of corresponding <em>AF</em> and <em>PhF</em> characteristics in the range <em>f</em><sub>1</sub> ≤ <em>f</em> ≤ <em>f</em><sub>2</sub> (see Fig. 3), having the following values: Here <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mover accent="true"><mrow><mi>A</mi></mrow><mo>⃡</mo></mover></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced></math> and <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mover accent="true"><mrow><mi mathvariant="normal">Φ</mi></mrow><mo>⃡</mo></mover></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced></math> include the fragments of corresponding <em>AF</em> and <em>PhF</em> characteristics in the range <em>f</em><sub>1</sub> ≤ <em>f</em> ≤ <em>f</em><sub>2</sub> (see Fig. 3), having the following values:

35	${\overset{⃡}{A}}_{h} (f) = \{\begin{array}{l} A_{h} (f_{1}), f < f_{1}; \\ A_{h} (f), f_{1} \leq f \leq f_{2}; \\ A_{h} (f_{2}), f > f_{2}, \end{array}$ ${\overset{⃡}{Φ}}_{h} (f) = \{\begin{array}{l} Φ_{h} (f_{1}), f < f_{1}; \\ Φ_{h} (f), f_{1} \leq f \leq f_{2}; \\ Φ_{h} (f_{2}), f > f_{2} . \end{array}$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mover accent="true"><mrow><mi>A</mi></mrow><mo>⃡</mo></mover></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced><mo>=</mo><mfenced open="{" close="" separators="\|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>A</mi></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="\|"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfenced><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi>f</mi><mo><</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>;</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>A</mi></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≤</mo><mi>f</mi><mo>≤</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>;</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>A</mi></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="\|"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfenced><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi>f</mi><mo>></mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo></mtd></mtr></mtable></mrow></mfenced></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mover accent="true"><mrow><mi mathvariant="normal">Φ</mi></mrow><mo>⃡</mo></mover></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced><mo>=</mo><mfenced open="{" close="" separators="\|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi mathvariant="normal">Φ</mi></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="\|"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfenced><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi>f</mi><mo><</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>;</mo></mtd></mtr><mtr><mtd><mi mathvariant="normal"> </mi><msub><mrow><mi mathvariant="normal">Φ</mi></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≤</mo><mi>f</mi><mo>≤</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>;</mo></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="normal">Φ</mi></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="\|"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfenced><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi>f</mi><mo>></mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>.</mo></mtd></mtr></mtable></mrow></mfenced></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mover accent="true"><mrow><mi>A</mi></mrow><mo>⃡</mo></mover></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced><mo>=</mo><mfenced open="{" close="" separators="\|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>A</mi></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="\|"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfenced><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi>f</mi><mo><</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>;</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>A</mi></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≤</mo><mi>f</mi><mo>≤</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>;</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>A</mi></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="\|"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfenced><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi>f</mi><mo>></mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo></mtd></mtr></mtable></mrow></mfenced></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mover accent="true"><mrow><mi mathvariant="normal">Φ</mi></mrow><mo>⃡</mo></mover></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced><mo>=</mo><mfenced open="{" close="" separators="\|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi mathvariant="normal">Φ</mi></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="\|"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfenced><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi>f</mi><mo><</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>;</mo></mtd></mtr><mtr><mtd><mi mathvariant="normal"> </mi><msub><mrow><mi mathvariant="normal">Φ</mi></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≤</mo><mi>f</mi><mo>≤</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>;</mo></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="normal">Φ</mi></mrow><mrow><mi>h</mi></mrow></msub><mfenced separators="\|"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfenced><mo>,</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi>f</mi><mo>></mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>.</mo></mtd></mtr></mtable></mrow></mfenced></math>

36	The inverse Fourier transform of $\tilde{Ξ} (f)$ will provide the actual trajectory of the considered cycle:	The inverse Fourier transform of <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="true"><mrow><mi mathvariant="normal">Ξ</mi></mrow><mo>~</mo></mover><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced></math> will provide the actual trajectory of the considered cycle: The inverse Fourier transform of <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="true"><mrow><mi mathvariant="normal">Ξ</mi></mrow><mo>~</mo></mover><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced></math> will provide the actual trajectory of the considered cycle:

37	$\tilde{Ξ} (t) = \frac{1}{2 π} \int_{- \infty}^{\infty} \tilde{Ξ} (f) e^{j 2 π f t} d f .$	<math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="true"><mrow><mi mathvariant="normal">Ξ</mi></mrow><mo>~</mo></mover><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mi>π</mi></mrow></mfrac><msubsup><mrow><mo>∫</mo></mrow><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mover accent="true"><mrow><mi mathvariant="normal">Ξ</mi></mrow><mo>~</mo></mover><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>j</mi><mn>2</mn><mi>π</mi><mi>f</mi><mi>t</mi></mrow></msup><mi>d</mi><mi>f</mi><mo>.</mo></math> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="true"><mrow><mi mathvariant="normal">Ξ</mi></mrow><mo>~</mo></mover><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mi>π</mi></mrow></mfrac><msubsup><mrow><mo>∫</mo></mrow><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msubsup><mover accent="true"><mrow><mi mathvariant="normal">Ξ</mi></mrow><mo>~</mo></mover><mfenced separators="\|"><mrow><mi>f</mi></mrow></mfenced><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>j</mi><mn>2</mn><mi>π</mi><mi>f</mi><mi>t</mi></mrow></msup><mi>d</mi><mi>f</mi><mo>.</mo></math>

38	The reconstruction of the time domain processes from its Fourier images has its own specifics. As a rule, the initial and final values of the filtered GDP realization, presented on the time interval t₁_,…, t_c, are different. The Fourier transform of G_f(t) assumes the periodic nature of this fragment. The specified inequality G_f(t₁) ≠ G_f(t_c) leads to the appearance of a gap (discontinuity) between the last value G_f(t_c) and the first one of the realization G_f(t) with its hypothetical continuation. Therefore, the values of the recovered $\tilde{Ξ} (t)$ will oscillate in the regions adjacent to break points (t₁ and t_c) of the recovered trajectory. In (Karmalita, 2020) the following technique is proposed to eliminate the gap in the periodicity of the realization G_f(t). Its endpoint is complemented by a cubic parabola Q(t) in the time interval t_c,…, t₂, as shown in Fig. 5.	The reconstruction of the time domain processes from its Fourier images has its own specifics. As a rule, the initial and final values of the filtered GDP realization, presented on the time interval <em>t</em><sub>1</sub><sub>,</sub>…, <em>t</em><sub><em>c</em></sub>,<em> </em>are different. The Fourier transform of <em>G</em><sub><em>f</em></sub>(<em>t</em>)<em> </em>assumes the periodic nature of this fragment. The specified inequality <em>G</em><sub><em>f</em></sub>(<em>t</em><sub>1</sub>) ≠ <em>G</em><sub><em>f</em></sub>(<em>t</em><sub><em>c</em></sub>) leads to the appearance of a gap (discontinuity) between the last value <em>G</em><sub><em>f</em></sub>(<em>t</em><sub>c</sub>)<em> </em>and the first one of the realization <em>G</em><sub><em>f</em></sub>(<em>t</em>) with its hypothetical continuation. Therefore, the values of the recovered <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="true"><mrow><mi mathvariant="normal">Ξ</mi></mrow><mo>~</mo></mover><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced></math> will oscillate in the regions adjacent to break points (<em>t</em><sub>1</sub> and <em>t</em><sub>c</sub>) of the recovered trajectory. In (Karmalita, 2020) the following technique is proposed to eliminate the gap in the periodicity of the realization <em>G</em><sub><em>f</em></sub>(<em>t</em>). Its endpoint is complemented by a cubic parabola <em>Q</em>(<em>t</em>) in the time interval <em>t</em><sub><em>c</em></sub>,…, <em>t</em><sub>2</sub>, as shown in Fig. 5<em>.</em> The reconstruction of the time domain processes from its Fourier images has its own specifics. As a rule, the initial and final values of the filtered GDP realization, presented on the time interval <em>t</em><sub>1</sub><sub>,</sub>…, <em>t</em><sub><em>c</em></sub>,<em> </em>are different. The Fourier transform of <em>G</em><sub><em>f</em></sub>(<em>t</em>)<em> </em>assumes the periodic nature of this fragment. The specified inequality <em>G</em><sub><em>f</em></sub>(<em>t</em><sub>1</sub>) ≠ <em>G</em><sub><em>f</em></sub>(<em>t</em><sub><em>c</em></sub>) leads to the appearance of a gap (discontinuity) between the last value <em>G</em><sub><em>f</em></sub>(<em>t</em><sub>c</sub>)<em> </em>and the first one of the realization <em>G</em><sub><em>f</em></sub>(<em>t</em>) with its hypothetical continuation. Therefore, the values of the recovered <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mover accent="true"><mrow><mi mathvariant="normal">Ξ</mi></mrow><mo>~</mo></mover><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced></math> will oscillate in the regions adjacent to break points (<em>t</em><sub>1</sub> and <em>t</em><sub>c</sub>) of the recovered trajectory. In (Karmalita, 2020) the following technique is proposed to eliminate the gap in the periodicity of the realization <em>G</em><sub><em>f</em></sub>(<em>t</em>). Its endpoint is complemented by a cubic parabola <em>Q</em>(<em>t</em>) in the time interval <em>t</em><sub><em>c</em></sub>,…, <em>t</em><sub>2</sub>, as shown in Fig. 5<em>.</em>

39	Подпись к рисунку/медиа
40	Fig. 5. Fragment of filtered realization G_f(t)	<strong>Fig. 5.</strong> Fragment of filtered realization <em>G</em><sub><em>f</em></sub>(<em>t</em>) <strong>Fig. 5.</strong> Fragment of filtered realization <em>G</em><sub><em>f</em></sub>(<em>t</em>)

41	Q(t) is a polynomial of the form: Q(t) = q₀ + q₁t + q₂t² + q₃t³. Its coefficients q_l (l = 0, 3) are completely determined by the following boundary requirements:	<em>Q</em>(<em>t</em>) is a polynomial of the form: <em>Q</em>(<em>t</em>) = <em>q</em><sub>0</sub> + <em>q</em><sub>1</sub><em>t</em> + <em>q</em><sub>2</sub><em>t</em><sup>2</sup> + <em>q</em><sub>3</sub><em>t</em><sup>3</sup>. Its coefficients <em>q</em><sub><em>l</em></sub> (<em>l</em> = 0, 3) are completely determined by the following boundary requirements:<strong> </strong> <em>Q</em>(<em>t</em>) is a polynomial of the form: <em>Q</em>(<em>t</em>) = <em>q</em><sub>0</sub> + <em>q</em><sub>1</sub><em>t</em> + <em>q</em><sub>2</sub><em>t</em><sup>2</sup> + <em>q</em><sub>3</sub><em>t</em><sup>3</sup>. Its coefficients <em>q</em><sub><em>l</em></sub> (<em>l</em> = 0, 3) are completely determined by the following boundary requirements:<strong> </strong>

42	Q(t_c) = G_f (t_c); Q(t₂) = G_f (t₁); ${\frac{d Q (t)}{d t}\|}_{t_{c}} = {\frac{d G_{f} (t)}{d t}\|}_{t_{c}}$ ; ${\frac{d Q (t)}{d t}\|}_{t_{2}} = {\frac{d G_{f} (t)}{d t}\|}_{t_{1}} .$	<em>Q</em>(<em>t</em><sub>c</sub>) =<em> </em><em>G</em><sub><em>f</em></sub> (<em>t</em><sub>c</sub>);<strong> </strong><em>Q</em>(<em>t</em><sub>2</sub>) = <em>G</em><sub><em>f</em></sub> (<em>t</em><sub>1</sub>); <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mfenced open="" close="\|" separators="\|"><mrow><mfrac><mrow><mi>d</mi><mi>Q</mi><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></msub><mo>=</mo><msub><mrow><mfenced open="" close="\|" separators="\|"><mrow><mfrac><mrow><mi>d</mi><msub><mrow><mi>G</mi></mrow><mrow><mi>f</mi></mrow></msub><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></msub></math> ;<strong> </strong> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mfenced open="" close="\|" separators="\|"><mrow><mfrac><mrow><mi>d</mi><mi>Q</mi><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><mo>=</mo><msub><mrow><mfenced open="" close="\|" separators="\|"><mrow><mfrac><mrow><mi>d</mi><msub><mrow><mi>G</mi></mrow><mrow><mi>f</mi></mrow></msub><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>.</mo></math> <em>Q</em>(<em>t</em><sub>c</sub>) =<em> </em><em>G</em><sub><em>f</em></sub> (<em>t</em><sub>c</sub>);<strong> </strong><em>Q</em>(<em>t</em><sub>2</sub>) = <em>G</em><sub><em>f</em></sub> (<em>t</em><sub>1</sub>); <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mfenced open="" close="\|" separators="\|"><mrow><mfrac><mrow><mi>d</mi><mi>Q</mi><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></msub><mo>=</mo><msub><mrow><mfenced open="" close="\|" separators="\|"><mrow><mfrac><mrow><mi>d</mi><msub><mrow><mi>G</mi></mrow><mrow><mi>f</mi></mrow></msub><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></msub></math> ;<strong> </strong> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mfenced open="" close="\|" separators="\|"><mrow><mfrac><mrow><mi>d</mi><mi>Q</mi><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><mo>=</mo><msub><mrow><mfenced open="" close="\|" separators="\|"><mrow><mfrac><mrow><mi>d</mi><msub><mrow><mi>G</mi></mrow><mrow><mi>f</mi></mrow></msub><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>.</mo></math>

43	Such a “loopback” of realization of G_f(t) practically excludes the Gibbs’ phenomenon under the inverse Fourier transform.	Such a “loopback” of realization of <em>G</em><sub><em>f</em></sub>(<em>t</em>) practically excludes the Gibbs’ phenomenon under the inverse Fourier transform.<strong> </strong> Such a “loopback” of realization of <em>G</em><sub><em>f</em></sub>(<em>t</em>) practically excludes the Gibbs’ phenomenon under the inverse Fourier transform.<strong> </strong>

44	Let us recall that accepted model of economic cycles considers them to be the processes which values are determined by harmonics in the vicinity of the natural frequency f₀. This fact allows the simplification of the above recovery procedure based on considering the influence of the estimator only on the following harmonic — ξ(t) = A sin(2πf₀t). The influence of the estimator lies in its transformation to the following form:	Let us recall that accepted model of economic cycles considers them to be the processes which values are determined by harmonics in the vicinity of the natural frequency <em>f</em><sub>0</sub>. This fact allows the simplification of the above recovery procedure based on considering the influence of the estimator only on the following harmonic — ξ(<em>t</em>) = <em>A</em><em> </em>sin(2π<em>f</em><sub>0</sub><em>t</em>). The influence of the estimator lies in its transformation to the following form: Let us recall that accepted model of economic cycles considers them to be the processes which values are determined by harmonics in the vicinity of the natural frequency <em>f</em><sub>0</sub>. This fact allows the simplification of the above recovery procedure based on considering the influence of the estimator only on the following harmonic — ξ(<em>t</em>) = <em>A</em><em> </em>sin(2π<em>f</em><sub>0</sub><em>t</em>). The influence of the estimator lies in its transformation to the following form:

45	g(t) = А_h(f₀) A sin[2πf₀t + Φ_h(f₀)] = А_h(f₀) A sin(2πf₀t – πf₀ $Δ T$ ) =	<em>g</em>(<em>t</em>)<em> = </em><em>А</em><sub><em>h</em></sub>(<em>f</em><sub>0</sub>) <em>A</em><em> </em>sin[2π<em>f</em><sub>0 </sub><em>t</em><em> </em>+<em> </em>Φ<sub><em>h</em></sub>(<em>f</em><sub><em>0</em></sub>)] = <em>А</em><sub><em>h</em></sub>(<em>f</em><sub>0</sub>)<em> </em><em>A</em><em> </em>sin(2π<em>f</em><sub>0</sub><em>t</em><em> </em><em>–</em><em> </em>π<em>f</em><sub>0 </sub> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi mathvariant="normal">Δ</mi><mi>T</mi></math> ) = <em>g</em>(<em>t</em>)<em> = </em><em>А</em><sub><em>h</em></sub>(<em>f</em><sub>0</sub>) <em>A</em><em> </em>sin[2π<em>f</em><sub>0 </sub><em>t</em><em> </em>+<em> </em>Φ<sub><em>h</em></sub>(<em>f</em><sub><em>0</em></sub>)] = <em>А</em><sub><em>h</em></sub>(<em>f</em><sub>0</sub>)<em> </em><em>A</em><em> </em>sin(2π<em>f</em><sub>0</sub><em>t</em><em> </em><em>–</em><em> </em>π<em>f</em><sub>0 </sub> <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi mathvariant="normal">Δ</mi><mi>T</mi></math> ) =

46	=А_h(f₀) A sin[2πf₀(t — $Δ T$ /2)] = А_h(f₀) ξ(t – $Δ T$ /2).	<em>=</em><em>А</em><sub><em>h</em></sub>(<em>f</em><sub>0</sub>)<em> </em><em>A</em><em> </em>sin[2π<em>f</em><sub>0</sub>(<em>t</em> — <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi mathvariant="normal">Δ</mi><mi>T</mi></math> /2)] = <em>А</em><sub><em>h</em></sub>(<em>f</em><sub>0</sub>)<em> </em><em> </em>ξ(<em>t</em> – <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi mathvariant="normal">Δ</mi><mi>T</mi></math> /2). <em>=</em><em>А</em><sub><em>h</em></sub>(<em>f</em><sub>0</sub>)<em> </em><em>A</em><em> </em>sin[2π<em>f</em><sub>0</sub>(<em>t</em> — <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi mathvariant="normal">Δ</mi><mi>T</mi></math> /2)] = <em>А</em><sub><em>h</em></sub>(<em>f</em><sub>0</sub>)<em> </em><em> </em>ξ(<em>t</em> – <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi mathvariant="normal">Δ</mi><mi>T</mi></math> /2).

47	In other words, the estimator changes the amplitude of the harmonic and shifts it in time towards the delay. This fact simplifies the procedure for recovering the business cycle, since only bandpass filtering of the GDP function can provide estimates of the cycle trajectory with an accuracy acceptable for practical econometrics.	In other words, the estimator changes the amplitude of the harmonic and shifts it in time towards the delay. This fact simplifies the procedure for recovering the business cycle, since only bandpass filtering of the GDP function can provide estimates of the cycle trajectory with an accuracy acceptable for practical econometrics. In other words, the estimator changes the amplitude of the harmonic and shifts it in time towards the delay. This fact simplifies the procedure for recovering the business cycle, since only bandpass filtering of the GDP function can provide estimates of the cycle trajectory with an accuracy acceptable for practical econometrics.

48	3. RECOVERING A FRAGMENT OF THE KUZINETS SWING	3. RECOVERING A FRAGMENT OF THE KUZINETS SWING 3. RECOVERING A FRAGMENT OF THE KUZINETS SWING

49	Now consider the approach above in the case of recovering the Kuznets swing from quarterly GDP estimates of the USA economy. Such estimates for the time period 2000–2021 are shown in Fig. 6 (Federal Reserve Economic Data, 2022).	Now consider the approach above in the case of recovering the Kuznets swing from quarterly GDP estimates of the USA economy. Such estimates for the time period 2000<em>–</em>2021 are shown in Fig. 6 (Federal Reserve Economic Data, 2022). Now consider the approach above in the case of recovering the Kuznets swing from quarterly GDP estimates of the USA economy. Such estimates for the time period 2000<em>–</em>2021 are shown in Fig. 6 (Federal Reserve Economic Data, 2022).

50	Подпись к рисунку/медиа
51	Fig. 6. Real GDP estimates of the US economy	<strong>Fig. 6.</strong> Real <em>GDP</em> estimates of the US economy <strong>Fig. 6.</strong> Real <em>GDP</em> estimates of the US economy

52	In the case of quarterly GDP estimates, the sampling interval is Δt = ∆T ≈ 0.25 year. The non-equidistance of G_i = G(iΔt) readings is related to the different duration of quarters during the year. For the first quarter, ∆t₁ is equal to 90 or 91 (leap year) days, ∆t₂= 91, ∆t₃= 92, and ∆t₄= 92. Hence, the additional contribution to the measurement noise Z(t) due to fluctuations in the sampling interval yield an additional randomization of considered processes. Such a randomization leads to a bias (increase) in the estimates of the damping coefficient h. For example, it transforms a harmonic process (h = 0) into a narrowband random process.	In the case of quarterly GDP estimates, the sampling interval is Δ<em>t</em> = ∆<em>T</em> ≈ 0.25 year. The non-equidistance of <em>G</em><sub><em>i</em></sub><em> = </em><em>G</em>(<em>i</em>Δ<em>t</em>) readings is related to the different duration of quarters during the year. For the first quarter, ∆<em>t</em><sub>1</sub> is equal to 90 or 91 (leap year) days, ∆<em>t</em><sub>2 </sub>= 91, ∆<em>t</em><sub>3 </sub>= 92, and ∆<em>t</em><sub>4 </sub>= 92. Hence, the additional contribution to the measurement noise <em>Z</em>(<em>t</em>) due to fluctuations in the sampling interval yield an additional randomization of considered processes. Such a randomization leads to a bias (increase) in the estimates of the damping coefficient <em>h</em>. For example, it transforms a harmonic process (<em>h </em>= 0) into a narrowband random process. In the case of quarterly GDP estimates, the sampling interval is Δ<em>t</em> = ∆<em>T</em> ≈ 0.25 year. The non-equidistance of <em>G</em><sub><em>i</em></sub><em> = </em><em>G</em>(<em>i</em>Δ<em>t</em>) readings is related to the different duration of quarters during the year. For the first quarter, ∆<em>t</em><sub>1</sub> is equal to 90 or 91 (leap year) days, ∆<em>t</em><sub>2 </sub>= 91, ∆<em>t</em><sub>3 </sub>= 92, and ∆<em>t</em><sub>4 </sub>= 92. Hence, the additional contribution to the measurement noise <em>Z</em>(<em>t</em>) due to fluctuations in the sampling interval yield an additional randomization of considered processes. Such a randomization leads to a bias (increase) in the estimates of the damping coefficient <em>h</em>. For example, it transforms a harmonic process (<em>h </em>= 0) into a narrowband random process.

53	To increase the representativeness of the empirical data (number of samples), additional terms were calculated via linear interpolation of quarterly GDP estimates. Thus, the sampling interval was reduced to a value of 0.125 years, and the samples number increased to n = 175. Subsequently, oscillations ${\tilde{g}}_{i}$ of GDP estimates (Fig. 7) were evaluated relative to the regression ${\tilde{G}}_{i} = 71.242 i + 9609.753$ , obtained by the least squares method (Brandt, 2014).	To increase the representativeness of the empirical data (number of samples), additional terms were calculated via linear interpolation of quarterly GDP estimates. Thus, the sampling interval was reduced to a value of 0.125 years, and the samples number increased to <em>n</em> = 175. Subsequently, oscillations <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mover accent="true"><mrow><mi>g</mi></mrow><mo>~</mo></mover></mrow><mrow><mi>i</mi></mrow></msub></math> of GDP estimates (Fig. 7) were evaluated relative to the regression <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mover accent="true"><mrow><mi>G</mi></mrow><mo>~</mo></mover></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mn>71.242</mn><mi>i</mi><mo>+</mo><mn>9609.753</mn></math> , obtained by the least squares method (Brandt, 2014). To increase the representativeness of the empirical data (number of samples), additional terms were calculated via linear interpolation of quarterly GDP estimates. Thus, the sampling interval was reduced to a value of 0.125 years, and the samples number increased to <em>n</em> = 175. Subsequently, oscillations <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mover accent="true"><mrow><mi>g</mi></mrow><mo>~</mo></mover></mrow><mrow><mi>i</mi></mrow></msub></math> of GDP estimates (Fig. 7) were evaluated relative to the regression <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mover accent="true"><mrow><mi>G</mi></mrow><mo>~</mo></mover></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mn>71.242</mn><mi>i</mi><mo>+</mo><mn>9609.753</mn></math> , obtained by the least squares method (Brandt, 2014).

54	Подпись к рисунку/медиа
55	Fig. 7. Fragment of GDP oscillations	<strong>Fig. 7.</strong> Fragment of <em>GDP</em> oscillations <strong>Fig. 7.</strong> Fragment of <em>GDP</em> oscillations

56	If we proceed to the dimensionless interval of sampling Δt = 1, then it will correspond to the range of the relative natural frequency 0 ≤ θ₀ ≤ 0.5. The value of θ₀of the cycleof interestcan be determined through Fourier analysis (Cho, 2018) of oscillations ${\tilde{g}}_{i}$ (Fig. 8).	If we proceed to the dimensionless interval of sampling Δ<em>t</em> = 1, then it will correspond to the range of the relative natural frequency 0 ≤ θ<sub>0</sub> ≤ 0.5. The value of θ<sub>0 </sub>of the cycle<sub> </sub>of interest<sub> </sub>can be determined through Fourier analysis (Cho, 2018) of oscillations <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mover accent="true"><mrow><mi>g</mi></mrow><mo>~</mo></mover></mrow><mrow><mi>i</mi></mrow></msub></math> (Fig. 8). If we proceed to the dimensionless interval of sampling Δ<em>t</em> = 1, then it will correspond to the range of the relative natural frequency 0 ≤ θ<sub>0</sub> ≤ 0.5. The value of θ<sub>0 </sub>of the cycle<sub> </sub>of interest<sub> </sub>can be determined through Fourier analysis (Cho, 2018) of oscillations <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mover accent="true"><mrow><mi>g</mi></mrow><mo>~</mo></mover></mrow><mrow><mi>i</mi></mrow></msub></math> (Fig. 8).

57	Подпись к рисунку/медиа
58	Fig. 8. Spectrogram of GDP oscillations	<strong>Fig. 8.</strong> Spectrogram of <em>GDP</em> oscillations <strong>Fig. 8.</strong> Spectrogram of <em>GDP</em> oscillations

59	Spectral analysis of oscillations ${\tilde{g}}_{i}$ for the period 1871–2007 (Korotaev, Tsirel, 2010) determined that the duration of the Kondratiev cycle is 52–53 years. In that instance, the Kuznets swing was interpreted there as the third harmonic of the Kondratiev cycle with a relative natural frequency θ₀ ≈ 3/(52×8) ≈ 0.0072. From the above spectrogram the estimate ${\tilde{θ}}_{K}$ ≈ 0.01.	Spectral analysis of oscillations <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mover accent="true"><mrow><mi>g</mi></mrow><mo>~</mo></mover></mrow><mrow><mi>i</mi></mrow></msub></math> for the period 1871–2007 (Korotaev, Tsirel, 2010) determined that the duration of the Kondratiev cycle is 52–53 years. In that instance, the Kuznets swing was interpreted there as the third harmonic of the Kondratiev cycle with a relative natural frequency θ<sub>0</sub> ≈ 3/(52×8) ≈ 0.0072. From the above spectrogram the estimate <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mover accent="true"><mrow><mi mathvariant="normal">θ</mi></mrow><mo>~</mo></mover></mrow><mrow><mi>K</mi></mrow></msub></math> <sub> </sub>≈ 0.01. Spectral analysis of oscillations <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mover accent="true"><mrow><mi>g</mi></mrow><mo>~</mo></mover></mrow><mrow><mi>i</mi></mrow></msub></math> for the period 1871–2007 (Korotaev, Tsirel, 2010) determined that the duration of the Kondratiev cycle is 52–53 years. In that instance, the Kuznets swing was interpreted there as the third harmonic of the Kondratiev cycle with a relative natural frequency θ<sub>0</sub> ≈ 3/(52×8) ≈ 0.0072. From the above spectrogram the estimate <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mover accent="true"><mrow><mi mathvariant="normal">θ</mi></mrow><mo>~</mo></mover></mrow><mrow><mi>K</mi></mrow></msub></math> <sub> </sub>≈ 0.01.

60	The low pass (LP) filtering of the Kuznets swing may be performed by the FIR filter ${\tilde{g}}_{K i} = \sum_{l = 1}^{128} c_{l} {\tilde{g}}_{i + l - 128} (i = 128, \dots, 175)$ with the cut off frequency ${\tilde{θ}}_{0}$ = 0.018 (Fig. 9).	The low pass (<em>LP</em>) filtering of the Kuznets swing may be performed by the <em>F</em><em>IR</em> filter <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mover accent="true"><mrow><mi>g</mi></mrow><mo>~</mo></mover></mrow><mrow><mi>K</mi><mi>i</mi></mrow></msub><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>l</mi><mo>=</mo><mn>1</mn></mrow><mrow><mn>128</mn></mrow></msubsup><msub><mrow><mi>c</mi></mrow><mrow><mi>l</mi></mrow></msub><msub><mrow><mover accent="true"><mrow><mi>g</mi></mrow><mo>~</mo></mover></mrow><mrow><mi>i</mi><mo>+</mo><mi>l</mi><mo>-</mo><mn>128</mn></mrow></msub><mi mathvariant="normal"> </mi><mfenced separators="\|"><mrow><mi>i</mi><mo>=</mo><mn>128</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>175</mn></mrow></mfenced></math> with the cut off frequency <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mover accent="true"><mrow><mi>θ</mi></mrow><mo>~</mo></mover></mrow><mrow><mn>0</mn></mrow></msub></math> <em> </em>= 0.018 (Fig. 9). The low pass (<em>LP</em>) filtering of the Kuznets swing may be performed by the <em>F</em><em>IR</em> filter <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mover accent="true"><mrow><mi>g</mi></mrow><mo>~</mo></mover></mrow><mrow><mi>K</mi><mi>i</mi></mrow></msub><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>l</mi><mo>=</mo><mn>1</mn></mrow><mrow><mn>128</mn></mrow></msubsup><msub><mrow><mi>c</mi></mrow><mrow><mi>l</mi></mrow></msub><msub><mrow><mover accent="true"><mrow><mi>g</mi></mrow><mo>~</mo></mover></mrow><mrow><mi>i</mi><mo>+</mo><mi>l</mi><mo>-</mo><mn>128</mn></mrow></msub><mi mathvariant="normal"> </mi><mfenced separators="\|"><mrow><mi>i</mi><mo>=</mo><mn>128</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>175</mn></mrow></mfenced></math> with the cut off frequency <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mover accent="true"><mrow><mi>θ</mi></mrow><mo>~</mo></mover></mrow><mrow><mn>0</mn></mrow></msub></math> <em> </em>= 0.018 (Fig. 9).

61	Подпись к рисунку/медиа
62	Fig. 9. The AF and PhF characteristics of the LP filter	<strong>Fig. </strong><strong>9</strong><strong>.</strong> The <em>AF</em> and <em>P</em><em>h</em><em>F</em> characteristics of the <em>LP</em> filter <strong>Fig. </strong><strong>9</strong><strong>.</strong> The <em>AF</em> and <em>P</em><em>h</em><em>F</em> characteristics of the <em>LP</em> filter

63	It follows from the figure above that the amplitude of oscillations $S (t) = 2 π {\tilde{f}}_{0} t ({\tilde{f}}_{0} = 0.125 {\tilde{θ}}_{0})$ does not change in the bandpass of the filter. However, the phase of the filtered process S_f(t) modified as shown in Fig.10.	It follows from the figure above that the amplitude of oscillations <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>S</mi><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mn>2</mn><mi>π</mi><msub><mrow><mover accent="true"><mrow><mi>f</mi></mrow><mo>~</mo></mover></mrow><mrow><mn>0</mn></mrow></msub><mi>t</mi><mo>(</mo><msub><mrow><mover accent="true"><mrow><mi>f</mi></mrow><mo>~</mo></mover></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mi mathvariant="normal"> </mi><mn>0.125</mn><msub><mrow><mover accent="true"><mrow><mi>θ</mi></mrow><mo>~</mo></mover></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math> does not change in the bandpass of the filter. However, the phase of the filtered process <em>S</em><sub><em>f</em></sub>(<em>t</em>) modified as shown in Fig.10. It follows from the figure above that the amplitude of oscillations <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mi>S</mi><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced><mo>=</mo><mn>2</mn><mi>π</mi><msub><mrow><mover accent="true"><mrow><mi>f</mi></mrow><mo>~</mo></mover></mrow><mrow><mn>0</mn></mrow></msub><mi>t</mi><mo>(</mo><msub><mrow><mover accent="true"><mrow><mi>f</mi></mrow><mo>~</mo></mover></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mi mathvariant="normal"> </mi><mn>0.125</mn><msub><mrow><mover accent="true"><mrow><mi>θ</mi></mrow><mo>~</mo></mover></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math> does not change in the bandpass of the filter. However, the phase of the filtered process <em>S</em><sub><em>f</em></sub>(<em>t</em>) modified as shown in Fig.10.

64	Подпись к рисунку/медиа
65	Fig. 10. Effect of filtering a harmonic process	<strong>Fig. 10.</strong> Effect of filtering a harmonic process <strong>Fig. 10.</strong> Effect of filtering a harmonic process

66	Due to quarterly (∆T ≈ 0.25) estimates G_i, the value $A_{h} ({\tilde{θ}}_{0}) = 0.25 s i n (0.01 π) / 0.01 π \approx 0.25 .$ Thus, the recovered trajectory of the Kuznets swing is ${\tilde{ξ}}_{K} (t - 0.125) \approx 4 {\tilde{g}}_{K} (t),$ , and in the case of a discrete representation of data is reduced to the form ${\tilde{ξ}}_{K (i - 1)} \approx 4 {\tilde{g}}_{K i}$ (Fig. 11).	Due to quarterly (∆<em>T</em> ≈ 0.25) estimates <em>G</em><sub><em>i</em></sub>, the value <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>A</mi></mrow><mrow><mi>h</mi></mrow></msub><mo>(</mo><msub><mrow><mover accent="true"><mrow><mi>θ</mi></mrow><mo>~</mo></mover></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo><mo>=</mo><mn>0.25</mn><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfenced separators="\|"><mrow><mn>0.01</mn><mi>π</mi></mrow></mfenced><mo>/</mo><mn>0.01</mn><mi>π</mi><mi mathvariant="normal"> </mi><mo>≈</mo><mn>0.25</mn><mo>.</mo></math> Thus, the recovered trajectory of the Kuznets swing is <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mover accent="true"><mrow><mi>ξ</mi></mrow><mo>~</mo></mover></mrow><mrow><mi>K</mi></mrow></msub><mfenced separators="\|"><mrow><mi>t</mi><mo>–</mo><mn>0.125</mn></mrow></mfenced><mo>≈</mo><mn>4</mn><msub><mrow><mover accent="true"><mrow><mi>g</mi></mrow><mo>~</mo></mover></mrow><mrow><mi>K</mi></mrow></msub><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo></math> , and in the case of a discrete representation of data is reduced to the form <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mover accent="true"><mrow><mi>ξ</mi></mrow><mo>~</mo></mover></mrow><mrow><mi>K</mi><mo>(</mo><mi>i</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow></msub><mo>≈</mo><mn>4</mn><msub><mrow><mover accent="true"><mrow><mi>g</mi></mrow><mo>~</mo></mover></mrow><mrow><mi>K</mi><mi>i</mi></mrow></msub></math> (Fig. 11). Due to quarterly (∆<em>T</em> ≈ 0.25) estimates <em>G</em><sub><em>i</em></sub>, the value <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mi>A</mi></mrow><mrow><mi>h</mi></mrow></msub><mo>(</mo><msub><mrow><mover accent="true"><mrow><mi>θ</mi></mrow><mo>~</mo></mover></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo><mo>=</mo><mn>0.25</mn><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfenced separators="\|"><mrow><mn>0.01</mn><mi>π</mi></mrow></mfenced><mo>/</mo><mn>0.01</mn><mi>π</mi><mi mathvariant="normal"> </mi><mo>≈</mo><mn>0.25</mn><mo>.</mo></math> Thus, the recovered trajectory of the Kuznets swing is <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mover accent="true"><mrow><mi>ξ</mi></mrow><mo>~</mo></mover></mrow><mrow><mi>K</mi></mrow></msub><mfenced separators="\|"><mrow><mi>t</mi><mo>–</mo><mn>0.125</mn></mrow></mfenced><mo>≈</mo><mn>4</mn><msub><mrow><mover accent="true"><mrow><mi>g</mi></mrow><mo>~</mo></mover></mrow><mrow><mi>K</mi></mrow></msub><mfenced separators="\|"><mrow><mi>t</mi></mrow></mfenced><mo>,</mo></math> , and in the case of a discrete representation of data is reduced to the form <math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><msub><mrow><mover accent="true"><mrow><mi>ξ</mi></mrow><mo>~</mo></mover></mrow><mrow><mi>K</mi><mo>(</mo><mi>i</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow></msub><mo>≈</mo><mn>4</mn><msub><mrow><mover accent="true"><mrow><mi>g</mi></mrow><mo>~</mo></mover></mrow><mrow><mi>K</mi><mi>i</mi></mrow></msub></math> (Fig. 11).

67	Подпись к рисунку/медиа
68	Fig. 11. Estimates of the Kuznets swing trajectory	<strong>Fig. </strong><strong>1</strong><strong>1</strong><strong>.</strong> Estimates of the Kuznets swing trajectory <strong>Fig. </strong><strong>1</strong><strong>1</strong><strong>.</strong> Estimates of the Kuznets swing trajectory

69	4. CONCLUSION	4. CONCLUSION 4. CONCLUSION

70	A method for recovering the trajectory of the economic cycle, represented by the GDP function, was developed. It is based on the interpretation of cycles in the form of random oscillations in income with a certain natural frequency, which is also referred to as a narrowband random process. The method consists of bandpass filtering of the GDP function and subsequently parrying the influence of the estimator in the passband of the filter. The peculiarities of narrowband processes made it possible to create a simplified procedure for recovering the cycle trajectory. In the example of the Kuznets swing, the feasibility of this approach is demonstrated.	A method for recovering the trajectory of the economic cycle, represented by the GDP function, was developed. It is based on the interpretation of cycles in the form of random oscillations in income with a certain natural frequency, which is also referred to as a narrowband random process. The method consists of bandpass filtering of the GDP function and subsequently parrying the influence of the estimator in the passband of the filter. The peculiarities of narrowband processes made it possible to create a simplified procedure for recovering the cycle trajectory. In the example of the Kuznets swing, the feasibility of this approach is demonstrated. A method for recovering the trajectory of the economic cycle, represented by the GDP function, was developed. It is based on the interpretation of cycles in the form of random oscillations in income with a certain natural frequency, which is also referred to as a narrowband random process. The method consists of bandpass filtering of the GDP function and subsequently parrying the influence of the estimator in the passband of the filter. The peculiarities of narrowband processes made it possible to create a simplified procedure for recovering the cycle trajectory. In the example of the Kuznets swing, the feasibility of this approach is demonstrated.

71	The developed method can be used in the problems where the solution requires knowledge of the trajectory of the cycle under consideration. An example of such a problem is predicting the trajectory of a cycle (Karmalita, 2022).	The developed method can be used in the problems where the solution requires knowledge of the trajectory of the cycle under consideration. An example of such a problem is predicting the trajectory of a cycle (Karmalita, 2022). The developed method can be used in the problems where the solution requires knowledge of the trajectory of the cycle under consideration. An example of such a problem is predicting the trajectory of a cycle (Karmalita, 2022).

Библиография

1. Павлейно М.А., Ромаданов В.М. (2007). Спектральные преобразования в MATLAB. Учеб-но-методическое пособие. Санкт-Петербург: Санкт-Петербургский государственный университет. 160 c.

2. Bolotin V.V. (1984). Random vibrations of elastic systems. Heidelberg: Springer. 468 p.

3. Brandt S. (2014). Data analysis: Statistical and computational methods for scientists and engi-neers. 4th ed. Cham, Switzerland: Springer. 523 p.

4. Karmalita V. (2020). Stochastic Dynamics of Economic Cycles. Berlin: De Gruyter. 106 p.

5. Karmalita V.A. (2022). Predicting the trajectory of economic cycles // Экономика и математи-ческие методы. Т. 58. № 2. С. 140–144.

6. Korotaev A.V., Tsirel S.V. (2010). Spectral analysis of world GDP dynamics: Kondratieff waves, Kuznets swings, juglar and kitchin cycles. In: Global economic development, and the 2008–2009 economic crisis. Structure and Dynamics, 4 (1), 3–57.

7. Schlichtharle D. (2011). Digital filters: Basics and design. 2nd ed. Berlin: Springer–Verlag. 527 p.

8. Cho S. (2018). Fourier transform and its applications using Microsoft EXCEL®. San Rafael: Mor-gan & Claypool. 123 p.

9. Tikhonov A.N., Arsenin V.Y. (1997). Solution of ill-posed problems. Washington: Winston & Sons. 258 p.

Библиография

Комментарии

Войти через