Method of Inverse Operator for the Recover Input Signal
There have been considered the methods of construction of the inverse operator for the restoration of signals in conditions of weak nonlinear dynamic and nonlinear static distortions in the registration devices and transmission of continuous signals are considered. The simplest structure of the impl...
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Інститут кібернетики ім. В.М. Глушкова НАН України
2018
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| Цитувати: | Method of Inverse Operator for the Recover Input Signal / V.A. Ivanyuk // Математичне та комп'ютерне моделювання. Серія: Технічні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2018. — Вип. 17. — С. 63-70. — Бібліогр.: 3 назв. — англ. |
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irk-123456789-1621502020-01-04T01:25:22Z Method of Inverse Operator for the Recover Input Signal Ivanyuk, V.A. There have been considered the methods of construction of the inverse operator for the restoration of signals in conditions of weak nonlinear dynamic and nonlinear static distortions in the registration devices and transmission of continuous signals are considered. The simplest structure of the implementation block of the inverse operator is based on the adder, whose second inverse communication competitor includes the simulation model of the direct operator. A linear inertia-free unit with a set transmission factor is used as regularizer. Розглянуто методи побудови оберненого оператора для відновлення сигналів при умовах слабких нелінійних динамічних і нелінійних статичних спотворень у пристроях реєстрації та передачі неперервних сигналів. Найпростіша структура блока реалізації оберненого оператора будується на основі накопичуваного суматора, у другий контур оберненого зв’язку якого включена імітаційна модель прямого оператора. В якості регуляризатора використано лінійний безінерційний блок із визначеним коефіцієнтом передачі. 2018 Article Method of Inverse Operator for the Recover Input Signal / V.A. Ivanyuk // Математичне та комп'ютерне моделювання. Серія: Технічні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2018. — Вип. 17. — С. 63-70. — Бібліогр.: 3 назв. — англ. 2308-5916 http://dspace.nbuv.gov.ua/handle/123456789/162150 004.94 en Математичне та комп'ютерне моделювання. Серія: Технічні науки Інститут кібернетики ім. В.М. Глушкова НАН України |
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There have been considered the methods of construction of the inverse operator for the restoration of signals in conditions of weak nonlinear dynamic and nonlinear static distortions in the registration devices and transmission of continuous signals are considered. The simplest structure of the implementation block of the inverse operator is based on the adder, whose second inverse communication competitor includes the simulation model of the direct operator. A linear inertia-free unit with a set transmission factor is used as regularizer. |
| format |
Article |
| author |
Ivanyuk, V.A. |
| spellingShingle |
Ivanyuk, V.A. Method of Inverse Operator for the Recover Input Signal Математичне та комп'ютерне моделювання. Серія: Технічні науки |
| author_facet |
Ivanyuk, V.A. |
| author_sort |
Ivanyuk, V.A. |
| title |
Method of Inverse Operator for the Recover Input Signal |
| title_short |
Method of Inverse Operator for the Recover Input Signal |
| title_full |
Method of Inverse Operator for the Recover Input Signal |
| title_fullStr |
Method of Inverse Operator for the Recover Input Signal |
| title_full_unstemmed |
Method of Inverse Operator for the Recover Input Signal |
| title_sort |
method of inverse operator for the recover input signal |
| publisher |
Інститут кібернетики ім. В.М. Глушкова НАН України |
| publishDate |
2018 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/162150 |
| citation_txt |
Method of Inverse Operator for the Recover Input Signal / V.A. Ivanyuk // Математичне та комп'ютерне моделювання. Серія: Технічні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2018. — Вип. 17. — С. 63-70. — Бібліогр.: 3 назв. — англ. |
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Математичне та комп'ютерне моделювання. Серія: Технічні науки |
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AT ivanyukva methodofinverseoperatorfortherecoverinputsignal |
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2025-07-14T14:42:15Z |
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1837633776369270784 |
| fulltext |
Серія: Технічні науки. Випуск 17
63
UDC 004.94
V. A. Ivanyuk, Cand. of Techn. Sciences,
Kamianets-Podilskyi National Ivan Ohiienko University,
Kamianets-Podilskyi
METHOD OF INVERSE OPERATOR
FOR THE RECOVER INPUT SIGNAL
There have been considered the methods of construction of the in-
verse operator for the restoration of signals in conditions of weak non-
linear dynamic and nonlinear static distortions in the registration devic-
es and transmission of continuous signals are considered. The simplest
structure of the implementation block of the inverse operator is based
on the adder, whose second inverse communication competitor in-
cludes the simulation model of the direct operator. A linear inertia-free
unit with a set transmission factor is used as regularizer.
The inverse operator, based on the output signal of an object and
its mathematical model, is able to restore the input signal of an object.
Depending on the method of construction of the inverse feed-
back, there are considered various methods of the inverse operator
construction.
It is necessary to specify that the amplitudes of input and output
signals should be agreed. Failure to meet this condition leads to the in-
verse operator accuracy loss. The regularization parameter should dif-
fer from the unit, and the accurate functioning of the structure may go
beyond the stability limits. Physical dimension of variables differ, but
the ranges of their changes in numbers should coincide.
When using a regularizer with a value above than one, the compu-
tational process may vary however there is a small amount of stability
thanks to digitalization being used as a regularization parameter.
The algorithms for the implementation of the inverse operator may
be used while improving the efficiency of the operation of energy-satu-
rated equipment, improving the resolution of surveillance systems, imp-
roving the information exchange rate in communication lines, impro-
ving the information capacity of the information registration means, etc.
The effectiveness of the offered approaches is studied on mod-
el problems implemented in the Matlab/Simulink system. There
have been made the computational experiments that demonstrated
the efficiency of the method of inverse operators for the signals
restoration in the presence of weak nonlinear dynamic and nonline-
ar static distortions in the registration devices and transmission of
continuous signals in real-time mode.
Key words: restoration of signals, nonlinear dynamic distor-
tions, stability.
Introduction. At present the application of computers in various tech-
nical devices and control systems has become the habitual phenomenon. The
special interest is represented with cases where the computing way of im-
© V. A. Ivanyuk, 2018
Математичне та комп’ютерне моделювання
64
provement of their physical and economical properties has no alternative. Ap-
plication of analogue or digital filters for correction of dynamic characteristics
of the element in system allows to lower the technical requirements to this
element. It frequently allows not only to reduce the cost of the whole system,
but also to improve its basic technical parameters essentially. It is possible to
note as examples the inertial measuring converters or executive elements,
communication line with large attenuation etc [1–3].
There is the perfect computer equipment for solving such problems.
However, the known computer algorithms do not always provide the ef-
fective using of this equipment. This problem is not only the question how
to solve a computing problem, but mainly how to formulate it.
The majority of computing problems arising in this case may be di-
vided into two classes: the direct problems and the inverse ones. In tech-
nical systems the solution of an inverse problem usually names as recover-
ing of signal x = By. The initial data are the output signal y and mathemat-
ical model of investigated object A. The block which solves this problem
is named as the block of realisation of the inverse operator B = 1 / A, and
the block, which solves a direct problem of simulation of object y = Ax, is
named as the block of realisation of direct operator A.
One of the effective methods of solving the signal recovering problem is
the inverse operator’s method. The essential feature of this method is the ex-
plicit using of the direct operator for obtaining the inverse one. As a rule, block
A is an element of one of feedback circuits in the structure of block B [2, 3].
The concept of direct and inverse operators is especially often used in
the theory of ill posed problems solving. The solution of these problems is
characterised by instability or high sensitivity to errors of the initial data.
These problems are also called as inverse because their sense consists in
recovering the input signal of the inertial measuring device from its regis-
tered output signal. On the contrary, the direct problem of simulation of
such measuring device (or realisation of the direct operator) is correct.
This problem is characterised by insensibility toinitial data errors, that is
the rather rough measurements are allowed.
The most important advantage of the inverse operator’s method is the
technology of construction of stable computing process which realises un-
stable (complex) inverse operator on the base of explicit application of sta-
ble (simple) direct operator. Moreover, the inverse operators method allows
to organise two independent loops for adaptation the block B to the process-
es of ageing of the object A model and changing the errors quality [3].
1. Examples of application in technical systems. In control systems
the problem of maintaining the stability is solved rather good by known
classical methods with the selection of regulator parameters at realisation
of a principle of a deviation control. However, the necessity of use of the
inverse operator arises at realisation of more simple principle of specifying
influence control. It is necessary not for increasing the stability of a con-
Серія: Технічні науки. Випуск 17
65
trol system, but contrary for decreasing its roughness. Introduction of the
inverse operator allows to solve automatically one more problem which is
paid not enough attention in the theory of automatic control. It is a prob-
lem of dimensions. And not only in sense of discrepancy in dimensions of
physical values on the input and output of controlled object, but also in the
sense of discrepancy in number of inputs and outputs of this object.
The usual combined control system from proportional and specifying
action can be represented in two variants: Fig. 1 and Fig. 2. Here X is input,
and Y is output signal of controlled object B, RB is computing model of the
inverse operator of this object, I is inverter, S is adder, R is regulator.
BS RB I S R
x y x
x
y
Fig. 1. Control by input
RBS B I S R
y x y
y
y
Fig. 2. Control by output
In the case when block RB is instantaneous element with unit gain
(RB = 1) the both structures Fig. 1 and Fig. 2 coincide completely.
In case when R = 0 the specifying action control is realised. In case
of switching-off of the first adder input, the second input of which is con-
nected to output of regulator, the proportional action control is realised.
The structures Fig. 1 and Fig. 2 differ only by a place of connection of
block RB. Block RB carried out some of standard functions from the block
R, for example, differentiation or integration. A case when R = 1 is quite
possible, when block RB carried out all functions according to robustness
and stability balance of control.
The organisation from output is especially convenient at control of mul-
ticonnected objects. So, for example, usual carburettor engine has at least two
outputs: tachometer (crankshaft rotation speed) and flowmeter (fuel consump-
tion). Inputs are the structure of a combustible mix (ratio petrol/air) and the
ignition timing. The classical control circuit consists in control from the first
input (at RB = 1) with organisation of a feedback of the first output on the sec-
ond input. However, it is possible to organise the control from the first, or on
the second output. In the first case the dynamics of automobile control is im-
proved, and in the second case the fuel consumption decreases.
Математичне та комп’ютерне моделювання
66
The algorithms of realisation of the inverse operator are important
not only for increasing the efficiency of power-intensive equipment opera-
tion. Represent some other examples of their use. These are: increasing the
resolution of observation systems; increasing the speed of information
interchange in communication lines (compression of spectrum at digital
transmitting of continuous signals, multitone coding); increasing the ca-
pacity of telephone exchange by reduction of signal recognition time in
tone dialling; increasing the information capacity of information recording
devices (magnetic and thermoplastic record); for images recognition in
technical diagnostics by transients; for synthesis of band-pass filters [3].
The separate attention can be paid to full-scale `semivirtual reality’-type
simulators, where simplified (base) object plus computer plays the role of com-
plex object. On Fig. 3 the block diagram of such simulator is presented, where
U are control signals, which the person generates acting on controls, block RB
realises the inverse operator the base object mathematical model B, M is com-
puting model of simulated object, Z are output signals of object or model.
MU RB B
u z u
Fig. 3. Simulation model
The inverse operator’s method also useful in construction of numeri-
cal algorithms for solving the various applied problems.
2. Simple numerical experiment. As simple example we can consider
the calculation of square root as inverse operation to square. This numerical
experiment demonstrates an opportunity of application of the inverse opera-
tor’s method not only to solving the problems for linear dynamic systems or
systems with weak nonlinearity usual in applications mentioned above, but
also for essentially nonlinear problems. Certainly, the known classical algo-
rithms for of a square root calculation are better in many senses, except one:
they cannot be applied to calculation of other inverse functions. There is one
more parameter, on which there may be superiority in comparison with tra-
ditional algorithms. It is balance of accuracy and speed.
The analogue prototype of our digital filter is the known circuit with
connection of the direct operator in a feedback of operational amplifier
Fig. 4, where B is the object, described by direct operator, M is mathemat-
ical (electronic) model of this object, OA is operational amplifier, OR are
operational resistors, I is inverter. The dotted line leads round blocks,
which form block RB which carries out the inverse function relative to
object. In other words, according to the object signal and its mathematical
model this block restores the input signal of object. But this structural dia-
gram cannot be converted into discreet form explicitly. That is why the
similar analogue structures received the name ‘nonalgorithmic’ [2, 3].
Серія: Технічні науки. Випуск 17
67
B OR OA I
x y
OR M
x
– x– y
Fig. 4. Structural diagram of the inverse operator
On Fig. 5 the other variant of this computing structure transformed to al-
gorithmic form is represented. Here S is adder, I is inverter, R is regularizator
(linear instantaneous element with gain r). In fact, the second adder is accumu-
lator (there is the feedback on its second input). This circuit is realized in digi-
tal computing elements, but already it cannot be realized in analogue ones.
B S R S
x y
I M
x
– y x
Fig. 5. Structural diagram of the inverse operator (short loop)
In other cases, e.g. for dynamic objects (differential or integral equa-
tions) the block RB has another structure (Fig. 6) and allows both types of
realization (analogue and digital or nonalgorithmic and algorithmic). Here
the regularizator not only limits the magnitude of residual between the
output signals of object and its model, but also limits the amplification in a
loop of positive feedback by restored signal. The modification of these
structures with several regularizators also represent the practical interest.
B
S R
S
x y
I M
x
– y
r(x–y)
xy
Fig. 6. Structural scheme of the inverse operator (long loop)
Математичне та комп’ютерне моделювання
68
The structural diagrams Figs. 5 and 6 can be easily converted into
corresponding recurrent formula
( ) ( 1) ( ) ( ( 1))x i x i r y i M x i , (1)
( ) ( ) ( ) ( ( 1))x i = y i r x i M x i , (2)
where y is output signal and x is input signal of computing process, M
realizes the mathematical model of direct operator, and r is regulariz-
ing parameter. In the case of square root calculation M(x) = x2, and we
have two additions and two multiplications at the iteration step. Be-
cause of the double recursion by calculated output signal the term bire-
cursive digital filter can be used to characterize such algorithms real-
ized the inverse operator’s method in application to signal recovering
problems. In mathematical literature (1) corresponds to Friedman itera-
tive regularization, and (2) corresponds to Lavrentyev algorithm re-
spectively.
Fig. 7. The initial signal sinusoidental type
The graphs on Figs. 7 and 8 represent the results of restoring the
squared signal with one iteration at every time step. The initial time inter-
val was divided into 100 discretes. On Fig. 7 the initial signal x(t) was
rectangular meander with addition of normal white noise having the ampli-
tude 0.1 of initial signal. Fig. 8 corresponds to the unnoised sinusoide. In
both cases magnitude of initial signal was 4. Dotted lines correspond to
standard function sqrt(x).
Серія: Технічні науки. Випуск 17
69
Fig. 8. The initial signal rectangular meander with addition normal white noise
From graphs Figs. 7 and 8 we can see that parameter r has the influ-
ence on the stability of computing process. At variations of magnitude and
type of signal or characteristics of noise it is possible to choose such r,
which provides the best balance of stability and accuracy of computing pro-
cess. The instability is especially appearing in places of fast amplitude varia-
tions or sign of initial signal derivative. At change of digitalisation step the
balance of accuracy and stability of calculations on BRDF algorithm will
change, the traditional algorithm has not such property. With increasing the
quantity of digitisation units, the accuracy, naturally, grows, and for provid-
ing the stability it is necessary to change the regularizing parameter r.
Conclusion. Algorithm can be applied to solving the systems of line-
ar and nonlinear equations, including differential and integral ones. The
most positive moment is an achievement of high speed, which is very im-
portant for real time signal processing.
References:
1. Mathematical Methods for Signal and Image Analysis and Representation /
eds. L. Florack, R. Duits, G. Jongbloed, M. C. van Lieshout, L. Davies. —
New York : Springer London Dordrecht Heidelberg, 2012. — 316 p.
2. Ivanyuk V. A. The method of inverse operators for the restoration of signals at
the input of linear dynamic systems, given by transfer functions / V. A. Ivanyuk,
V. V. Ponedilok, O. A. Diachuk // Mathematical and computer simulation. Se-
ries: Engineering sciences: collection of research papers / V. M. Hlushkov Insti-
tute of Cybernetics of the National Academy of Sciences of Ukraine, Kamianets-
Математичне та комп’ютерне моделювання
70
Podilskyi National Ivan Ohiienko University / [editorship: Yu. H. Kryvonos
(ed.), et.al.] — Kamianets-Podilskyi : Kamianets-Podilskyi National Ivan
Ohiienko University, 2017. — Vol. 15. — P. 62–67. (Ukr.)
3. Verlan A. F. Methods and devices of experimental dependencies interpretation
at research and control of power processes / A. F. Verlan, B. B. Abdusatarov,
A. A. Ignatchenko, N. A. Maksimovich. — Kiev : Naukova dumka, 1993. —
208 p. (Rus.).
МЕТОД ОБЕРНЕНОГО ОПЕРАТОРА
ДЛЯ ВІДНОВЛЕННЯ ВХІДНОГО СИГНАЛУ
Розглянуто методи побудови оберненого оператора для віднов-
лення сигналів при умовах слабких нелінійних динамічних і неліній-
них статичних спотворень у пристроях реєстрації та передачі непере-
рвних сигналів. Найпростіша структура блока реалізації оберненого
оператора будується на основі накопичуваного суматора, у другий
контур оберненого зв’язку якого включена імітаційна модель прямого
оператора. В якості регуляризатора використано лінійний безінерцій-
ний блок із визначеним коефіцієнтом передачі.
Обернений оператор по вихідному сигналу об’єкта і його матема-
тичної моделі відновлює вхідний сигнал об’єкта.
В залежності від підходу до побудови оберненого зв’язку розгля-
даються різні методи побудови оберненого оператора.
Важливо відмітити, що амплітуди вхідного і вихідного сигналів по-
винні бути узгодженні. Якщо ця умова не виконується точність оберне-
ного оператора знизиться. Параметр регуляризації необхідно буде виби-
рати відмінним від одиниці, а точне функціонування структури може бу-
ти за межами стійкості. Фізичні розмірності величин можуть бути різни-
ми, але діапазони їх змін в числах повинні співпадати.
При використанні регуляризатора, значення якого більше одиниці,
обчислювальний процес може розходитись, хоча залишається невели-
кий запас стійкості за рахунок кроку дискретизації, який використо-
вується як регуляризуючий параметр.
Алгоритми реалізації оберненого оператора можуть бути викорис-
тані при підвищенні ефективності експлуатації енергонасиченого об-
ладнання, підвищення роздільної здатності систем спостереження,
підвищення швидкості обміну інформації в лініях зв’язку, підвищення
інформаційної ємності засобів реєстрації інформації тощо.
Ефективність запропонованих підходів досліджено на модельних
задачах, які реалізовано в системі Matlab/Simulink. Проведені обчис-
лювальні експерименти показали ефективність методу обернених
операторів для відновлення сигналів при наявності слабких неліній-
них динамічних і нелінійних статичних спотворень у пристроях ре-
єстрації та передачі неперервних сигналів в умовах реального часу.
Ключові слова: відновлення сигналів, нелінійні динамічні спотво-
рення, стійкість.
Отримано: 28.05.2018
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