Frequency analysis of the stochastic filtering using transfer functions. Part I: Sinusoidal input
Volterra-series-type analyses to communication systems with stochastic resonance driven by sine waves result to. The n -hold Fourier transform of the n -th Volterra kernel plays an important role in the analysis. Methods of computing transfer functions from the system equation are described and ob...
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Kharchenko, O. I. Lonin, Yu.F. Ponomarev, A.G. 2019-02-14T18:47:09Z 2019-02-14T18:47:09Z 2018 Frequency analysis of the stochastic filtering using transfer functions. Part I: Sinusoidal input / O. I. Kharchenko, Yu.F. Lonin, A.G. Ponomarev // Вопросы атомной науки и техники. — 2018. — № 4. — С. 249-251. — Бібліогр.: 9 назв. — англ. 1562-6016 PACS: 05.45 https://nasplib.isofts.kiev.ua/handle/123456789/147447 Volterra-series-type analyses to communication systems with stochastic resonance driven by sine waves result to. The n -hold Fourier transform of the n -th Volterra kernel plays an important role in the analysis. Methods of computing transfer functions from the system equation are described and obtaining results are considered. It is shown, if the transfer functions are known, then the output signal can be obtained by substitution the transfer functions in general formulas derived from Volterra series representation. The obtained results showed that the amplitude of the sinusoid should not be greater than one to obtain a minor third harmonic. Besides, with an increase in the frequency of the sinusoid, the power of the output signal decreases sharply. Numerical calculations of the output signal driven by sinusoidal input were made to improve the accuracy and reliability of the obtained results. Comparative analysis showed the coincidence of the results of the calculation by different methods. Наведено аналіз на основі рядів Вольтера стосовно телекомунікаційних систем, що володіють ефектом стохастичного резонансу, які збуджуються синусоїдальним сигналом. Важливу роль при такому аналізі грає багатовимірне перетворення Фур'є. Описано способи обчислення передаточних функцій, виходячи з рівняння системи, і розглянуто результати розрахунків. Порівняльний аналіз чисельних розрахунків і розрахунків на основі рядів Вольтера показав збіг результатів розрахунку. Приведен анализ на основе рядов Вольтера применительно к телекоммуникационным системам, обладающим эффектом стохастического резонанса, возбуждаемым синусоидальным сигналом. Важную роль при таком анализе играет многомерное преобразование Фурье. Описаны способы вычисления передаточных функций, исходя из уравнения системы, и рассмотрены результаты расчетов. Сравнительный анализ численных расчетов и расчетов на основе рядов Вольтера показал совпадение результатов расчета. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Нелинейные процессы Frequency analysis of the stochastic filtering using transfer functions. Part I: Sinusoidal input Частотний аналіз стохастичної фільтрації за допомогою передаточних функцій. Частина i: Синусоїдальний вхідний сигнал Частотный анализ стохастической фильтрации с помощью передаточных функций. Часть I: синусоидальный входной сигнал Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Frequency analysis of the stochastic filtering using transfer functions. Part I: Sinusoidal input |
| spellingShingle |
Frequency analysis of the stochastic filtering using transfer functions. Part I: Sinusoidal input Kharchenko, O. I. Lonin, Yu.F. Ponomarev, A.G. Нелинейные процессы |
| title_short |
Frequency analysis of the stochastic filtering using transfer functions. Part I: Sinusoidal input |
| title_full |
Frequency analysis of the stochastic filtering using transfer functions. Part I: Sinusoidal input |
| title_fullStr |
Frequency analysis of the stochastic filtering using transfer functions. Part I: Sinusoidal input |
| title_full_unstemmed |
Frequency analysis of the stochastic filtering using transfer functions. Part I: Sinusoidal input |
| title_sort |
frequency analysis of the stochastic filtering using transfer functions. part i: sinusoidal input |
| author |
Kharchenko, O. I. Lonin, Yu.F. Ponomarev, A.G. |
| author_facet |
Kharchenko, O. I. Lonin, Yu.F. Ponomarev, A.G. |
| topic |
Нелинейные процессы |
| topic_facet |
Нелинейные процессы |
| publishDate |
2018 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Частотний аналіз стохастичної фільтрації за допомогою передаточних функцій. Частина i: Синусоїдальний вхідний сигнал Частотный анализ стохастической фильтрации с помощью передаточных функций. Часть I: синусоидальный входной сигнал |
| description |
Volterra-series-type analyses to communication systems with stochastic resonance driven by sine waves result
to. The n -hold Fourier transform of the n -th Volterra kernel plays an important role in the analysis. Methods of
computing transfer functions from the system equation are described and obtaining results are considered. It is
shown, if the transfer functions are known, then the output signal can be obtained by substitution the transfer functions in general formulas derived from Volterra series representation. The obtained results showed that the amplitude of the sinusoid should not be greater than one to obtain a minor third harmonic. Besides, with an increase in the
frequency of the sinusoid, the power of the output signal decreases sharply. Numerical calculations of the output
signal driven by sinusoidal input were made to improve the accuracy and reliability of the obtained results. Comparative analysis showed the coincidence of the results of the calculation by different methods.
Наведено аналіз на основі рядів Вольтера стосовно телекомунікаційних систем, що володіють ефектом стохастичного резонансу, які збуджуються синусоїдальним сигналом. Важливу роль при такому аналізі грає багатовимірне перетворення Фур'є. Описано способи обчислення передаточних функцій, виходячи з рівняння системи, і розглянуто результати
розрахунків. Порівняльний аналіз чисельних розрахунків і розрахунків на основі рядів Вольтера показав збіг результатів
розрахунку.
Приведен анализ на основе рядов Вольтера применительно к телекоммуникационным системам, обладающим эффектом стохастического резонанса, возбуждаемым синусоидальным сигналом. Важную роль при таком анализе играет
многомерное преобразование Фурье. Описаны способы вычисления передаточных функций, исходя из уравнения системы, и рассмотрены результаты расчетов. Сравнительный анализ численных расчетов и расчетов на основе рядов Вольтера показал совпадение результатов расчета.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/147447 |
| citation_txt |
Frequency analysis of the stochastic filtering using transfer functions. Part I: Sinusoidal input / O. I. Kharchenko, Yu.F. Lonin, A.G. Ponomarev // Вопросы атомной науки и техники. — 2018. — № 4. — С. 249-251. — Бібліогр.: 9 назв. — англ. |
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| fulltext |
ISSN 1562-6016. ВАНТ. 2018. №4(116) 249
FREQUENCY ANALYSIS OF THE STOCHASTIC FILTERING USING
TRANSFER FUNCTIONS. PART I: SINUSOIDAL INPUT
O. I. Kharchenko, Yu.F. Lonin, A.G. Ponomarev
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
E-mail: dthnbycrbq@gmail.com; lonin@kipt.kharkov.ua
Volterra-series-type analyses to communication systems with stochastic resonance driven by sine waves result
to. The n -hold Fourier transform of the n -th Volterra kernel plays an important role in the analysis. Methods of
computing transfer functions from the system equation are described and obtaining results are considered. It is
shown, if the transfer functions are known, then the output signal can be obtained by substitution the transfer func-
tions in general formulas derived from Volterra series representation. The obtained results showed that the ampli-
tude of the sinusoid should not be greater than one to obtain a minor third harmonic. Besides, with an increase in the
frequency of the sinusoid, the power of the output signal decreases sharply. Numerical calculations of the output
signal driven by sinusoidal input were made to improve the accuracy and reliability of the obtained results. Compar-
ative analysis showed the coincidence of the results of the calculation by different methods.
PACS: 05.45
INTRODUCTION IN VOLTERRA SERIES
ANALYSIS
In communication systems often it is necessary to
deal with the devices executing non-linear conversions.
Volterra series are usually used for calculation of such
devices. Wiener introduced Volterra series into nonline-
ar circuit analysis [1].
The object of this paper is to present results of ap-
plying Volterra-series-type analyses to systems driven
by sine waves.
Volterra series describe the output of a nonlinear
system in degrees of input ( )x t . A substantial number
of the communication system can be represented as
Volterra series. The series for typical system can be
writing as [2]
( ) ( )1 1
1 1
1( ) ... , ...,
!
n
n n n r
n r
y t du du g u u x t u
n
∞ ∞ ∞
−∞ −∞
= =
= −∑ ∏∫ ∫ ,
(1)
where ( )y t is the output, ( )x t − the input and the ker-
nels ( )1, ...,n ng u u describe the system. The first-order
kernel 1 1( )g u is simply the familiar impulse response
of linear network. The higher order kernels of higher
order impulse responses and serve to characterize the
various orders of nonlinearity.
The coefficient 1 !n inserted A. Bedrosian and D.
Rice [2] it because it simplifies many of equations.
The n -fold Fourier transform have the form [2, 3]
( )
( ) ( )
1
1 1 1 1
,...,
... ,..., exp ... .
n n
n n n n n
G f f
du du g u u j f u f u
∞ ∞
−∞ −∞
=
= − + ∫ ∫
(2)
0G is identically zero because our Volterra series
starts with 1n = . ( )1 1G f is the transfer function of
linear network. For linear systems, the possible output
frequencies are the same as the frequencies in the input.
For non-linear systems, however, the relationship be-
tween the input and output frequencies is more compli-
cated [4, 5].
Thus the transform of the n th-order Volterra kernel
is seen to be analogous to an n th-order Volterra trans-
fer function. In many cases Gn can be obtained without
first computing gn.
The complete formulas are infinite series. Fortunate-
ly, in the study of communication system it is often pos-
sible to neglect terms of the Volterra series of order
higher then the second or third. They are usually used
because of fast increase in complexity [2, 3]. The n -
fold Fourier transform considerably simplifies the solu-
tion of a large number of problems.
To calculate the transfer functions, we use the har-
monic input method [2]. This method relies on the fact
that a harmonic input must result in a harmonic output
when (1) holds. System specified by the nonlinear dif-
ferential equation considered [2, 3]
2
( / ) ( )l
l
l
F d dt y a y x t
∞
=
+ =∑ , (3)
with the condition that system causally ( ( )y t vanish
identically when ( )x t does). It is assumed that one and
only one such solution exists (it is proved in [3]) and the
system is stable. ( / )F d dt is a polynomial in /d dt ,
and the coefficients in ( / )F d dt and the coefficients la
are independent of , ,t x and y .
The Volterra transfer functions for (3) can be written
as [2]
( )
1
2
1
1
( , ..., )
( , ..., )
( ... )
n
l
l n n
l
n n
n
a G f f
G f f
F j jω ω
== −
+ +
∑
. (4)
The last equation is recurrence relation because ( )l
nG
is given by
( )
1 1 1 1
( ; , ) 2
2 1 1 1 2
( ,..., ) ! ( ,..., )
( ,..., )... ( ,..., )
n
l
n n l
l n l
l n
G f f l a G f f
G f f G f f
n n
n
n n n n n µ
=
+ +
= ×
×
∑ ∑
,
for the n -fold Fourier transform of the n -th kernel in
the Volterra series for [ ( )]ly t , l being a positive inte-
ger, and 1 l n≤ ≤ . ( )
1( , ..., )l
n nG f f is zero for l n> and
( )
1( , ..., )n
n nG f f is equal to 1 1 1 2 1! ( ) ( )... ( )nn G f G f G f .
mailto:dthnbycrbq@gmail.com
mailto:lonin@kipt.kharkov.ua
ISSN 1562-6016. ВАНТ. 2018. №4(116) 250
SINUSOIDAL INPUT FOR STOCHASTIC
RESONANCE
Consider a general bistable dynamic system which
can be described by the following stochastic differential
equation [6, 7]
3 ( )dy ay by x t
dt
= − + , (5)
where a and b are positive, usually given in terms of
system parameters,
( ) ( ) ( )x t s t n t= + , 0( ) sin(2 )s t A f tπ ϕ= + is the driving
signal, ( )n t is the input noise,
[ ( ) ( )] 2 ( )E n t n t Dt δ t+ = , D is the noise intensity.
The input signal consists of the driving signal ( )s t and
the additive noise ( )n t [6, 8]. This equation describes a
stochastic resonance effect (SR) [6 - 8]. Volterra trans-
fer function for ( )y t are given in table 1 for the general
case (the equation 3) and for SR equation.
Table 1
Volterra transfer function
Volterra transfer function for eq. 3 [2] Volterra transfer function for eq. 4
1G 11/ ( )F jω
1
1
a jω− +
2G 2 1 1 1 2 1 22 ( ) ( ) / ( )a G f G f F j jω ω− + 0
3G
'
2 1 1 2 2 3 3 1 1 1 2 1 3
3
1 2 3
2 ( ) ( , ) 6 ( ) ( ) ( )
( )
a G f G f f a G f G f G f
F j j jω ω ω
+
−
+ +
∑
1 2 3 1 2 3
6
( )( )( )( )
b
a j a j a j a j j jω ω ω ω ω ω
−
− + − + − + − + + +
When noise is absent, input signal can be writing as 0( ) ( ) sin(2 )x t s t A f tπ ϕ= = + . We will solve SR equation
for this case. The leading terms output signal are given in Table 2.
Table 2
The leading terms output signal
The output signal for eq. 3 [2] The output signal for eq. 4
2
2 ( , ) ...
4 p p
P G f f
− +
0
3
1 3( ) ( , , ) ...
2 16
jpt
p p p p
P Pe G f G f f f
+ − +
+
3
1 3( ) ( , , ) ...
2 16
jpt
p p p p
P Pe G f G f f f−
+ − + − − +
2 2 2
0
02 2 2 2 2
0 0
3 ( )
cos
( ) 4( )
A b aA a t
a a
ω
ω
ω ω
− +
− +
+ +
+
2
0
0 02 2 2 2 2
0 0
3
sin
( ) 2( )
A bA t
a a
ω
ω ω
ω ω
+ +
+ +
2
2
2 ( , ) ...
8
j pt
p p
Pe G f fπ
+
+
2
2
2 ( , ) ...
8
j pt
p p
Pe G f fπ−
+ − − +
0
3
3
3 ( , , ) ...
48
j pt
p p p
Pe G f f fπ
+ +
+
3
3
3 ( , , ) ... ...
48
j pt
p p p
Pe G f f fπ−
+ − − − + +
4 2 2 43
0 0 0
2 2 3 2 2 2 2
0 0 0 0 0
( 12 3 ) cos 3
4( ) ( 9 ) 2 (5 3 ) sin 3
a a tA b
a a a a t
ω ω ω
ω ω ω ω ω
− − + +
+ + + −
We will define the output signal of non-linear sys-
tem by the Runge-Kutta method and by Volterra series.
Results of calculations are given in Figure for 1,A =
0.5f = Hz.
Results from the Figure show, that in the numerical
calculation by Runge-Kutta method takes place transient
that lasts about two periods of a signal. Further the re-
sults of calculation of an output signal received by both
methods match.
We will determine the power of an output signal.
Powers of the first and third harmonica are specified in
Table 3.
Analyzed results of the output signal using
different methods: Volterra series (solid line);
Runge-Kutta method (dotted line)
ISSN 1562-6016. ВАНТ. 2018. №4(116) 251
Table 3
The power of the output signal
The power of the output signal for eq. 3 [2] The power of the output signal for eq. 4
0
Sω
23
1 0 3 0 0 0( ) ( , , )
2 16
A AG f G f f f+ −
2 22 2 4 6 2
0
2 2 2 2 2 3 2 2 4 2 2 2
0 0 0 0
3 9
4( ) 8( ) 64( ) 4( )
Aa A A ba A b
a a a a
ω
ω ω ω ω
− + +
+ + + +
03S ω
23
3 0 0 0( , , )
48
P G f f f
6 2
2 2 3 2 2
0 064( ) ( 9 )
A b
a aω ω+ +
Results of Table 3 show that, with an increase in the
frequency of the sinusoid, the power of the output signal
decreases drastically.
We can calculate the powers relation of the first and
third harmonicas as:
2 2 2 2 2 2 2 2 2 2 2 2 2 2
0 0 0 0 0 0 0
30 0 4 2 2 2 2 4 2
0
16( )( 9 ) 24 ( 9 ) 9( 9 ) 16 ( )( 9 )
/
( )
a a a a a a a a
S S
A b A b a A b
ω ω
ω ω ω ω ω ω ω
ω
+ + + + + +
= − + +
+
.
The obtained results showed that the amplitude of
the sinusoid A should not be greater than one to obtain
a minor third harmonic.
CONCLUSIONS
The Volterra series is a powerful tool that can be
used to describe a wide class of non-linear systems.
Results of applying Volterra series analysis to sys-
tems with SR effect driven by harmonic input showed
what output signal contains the 1st and 3rd harmonicas,
that matches the known results [6]. In addition, with an
increase in the frequency of the sinusoid, the power of
the output signal decreases sharply and with an increase
in the frequency of the sinusoid, the power of the output
signal decreases sharply.
Results of calculations showed that numerical calcu-
lation and calculation by Volterra series match. At the
same time numerical calculation is followed by transient
which lasts about two periods of oscillations. Transient
take place in radioengineering devices [9].
In the present paper the method of transfer functions
is developed. The received gear Volterra transfer func-
tion for the systems with SR effect will allow to receive
further expressions for an output signal in case of input
white Gaussian noise and an input additive mix of a
harmonic signal and white Gaussian noise.
REFERENCES
1. N. Wiener. Nonlinear Problems in Random Theory.
Cambridge, Mass. Technology Press; and New York
Wiley, 1958.
2. E Bedrosian, S. Rice. The Output Properties of
Volterra Systems (Nonlinear Systems with Memory)
Driven by Harmonic and Gaussian Inputs // IEEE.
1971, v. 59, № 12, p. 58-82.
3. K.A. Pupkov, V.I. Kapalin, A.S. Yushchenko.
Funktsional'nyye ryady v teorii nelineynykh sistem.
M.: “Nauka” 1978, 448 p. (in Russian).
4. N.V. Zernov, V.G. Karpov. Teoriya radiotekhnich-
eskikh tsepey. Moskva-Leningrad: “Energiya”, 1965,
816 p. (in Russian).
5. Yu.I. Voloshchuk. Pidruchnik dla studentiv vich.
navch. zakladiv. Kharkiv: TOV “Kompania CMIT”,
2005, v. 3, 228 p. (in Russian).
6. V.S. Anishchenko, A.B. Neiman, F. Moss, L.
Schimansky-Geier. Stochastic resonance: noise-
enhanced order // Uspekhi Fizicheskikh Nauk, Rus-
sian Academy of Sciences. 1999, 42(1)7-36, p. 7-34
(in Russian).
7. O. Kharchenko. Simulation of the Stochastic Reso-
nance Effect in a Nonlinear Device // Global Jour-
nal of Researches in Engineering -F Volume 15 Is-
sue 7 Version 1.0 September 2015, p. 19-23.
8. Xiaofei Zhang, Niaoqing Hu, Zhe Cheng. Stochastic
resonance in multi-scale bistable array // Physics
Letter. 2013, A 377, p. 981-984.
9. B. Sklar. Digital Communication. Fundamentals and
Applications, Second Edition, Prentice Hall PTR,
2003, 1099 p.
Article received 04.06.2018
ЧАСТОТНЫЙ АНАЛИЗ СТОХАСТИЧЕСКОЙ ФИЛЬТРАЦИИ С ПОМОЩЬЮ ПЕРЕДАТОЧНЫХ ФУНКЦИЙ.
ЧАСТЬ I: СИНУСОИДАЛЬНЫЙ ВХОДНОЙ СИГНАЛ
О.И. Харченко, Ю.Ф. Лонин, А.Г. Пономарев
Приведен анализ на основе рядов Вольтера применительно к телекоммуникационным системам, обладающим эф-
фектом стохастического резонанса, возбуждаемым синусоидальным сигналом. Важную роль при таком анализе играет
многомерное преобразование Фурье. Описаны способы вычисления передаточных функций, исходя из уравнения систе-
мы, и рассмотрены результаты расчетов. Сравнительный анализ численных расчетов и расчетов на основе рядов Воль-
тера показал совпадение результатов расчета.
ЧАСТОТНИЙ АНАЛІЗ СТОХАСТИЧНОЇ ФІЛЬТРАЦІЇ ЗА ДОПОМОГОЮ ПЕРЕДАТОЧНИХ ФУНКЦІЙ.
ЧАСТИНА I: СИНУСОЇДАЛЬНИЙ ВХІДНИЙ СИГНАЛ
О.І. Харченко, Ю.Ф. Лонін, А.Г. Пономарьов
Наведено аналіз на основі рядів Вольтера стосовно телекомунікаційних систем, що володіють ефектом стохастично-
го резонансу, які збуджуються синусоїдальним сигналом. Важливу роль при такому аналізі грає багатовимірне перетво-
рення Фур'є. Описано способи обчислення передаточних функцій, виходячи з рівняння системи, і розглянуто результати
розрахунків. Порівняльний аналіз чисельних розрахунків і розрахунків на основі рядів Вольтера показав збіг результатів
розрахунку.
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