Some features of Hilbert transform and their use in energy informatics
Information-measuring technologies (IMT) are an important instrument for solving problems of energy informatics. They allow to form primary information based on the interaction of energy facilities with IMT sensors that form information signals. In many practical applications, the constructive model...
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| author | Babak , Vitalii Zaporozhets , Artur Kuts , Yurii Shcherbak , Leonid |
| author_facet | Babak , Vitalii Zaporozhets , Artur Kuts , Yurii Shcherbak , Leonid |
| author_institution_txt_mv | [
{
"author": "Vitalii Babak ",
"institution": null
},
{
"author": "Artur Zaporozhets ",
"institution": null
},
{
"author": "Yurii Kuts ",
"institution": null
},
{
"author": "Leonid Shcherbak ",
"institution": null
}
] |
| author_sort | Babak , Vitalii |
| baseUrl_str | https://systemre.org/index.php/journal/oai |
| collection | OJS |
| datestamp_date | 2026-07-18T12:57:42Z |
| description | Information-measuring technologies (IMT) are an important instrument for solving problems of energy informatics. They allow to form primary information based on the interaction of energy facilities with IMT sensors that form information signals. In many practical applications, the constructive model of information signals is the model of narrowband signals. The article summarizes the features of the discrete Hilbert transform and its application to obtain the primary characteristics of information signals – bypass and phase as functions of time. The main advantages of using the discrete Hilbert transform in signal processing for energy informatics are considered, including the consistency of obtaining frequency and time characteristics, high information content, the ability to analyze the dynamics of changes in signal characteristics, the possibility of obtaining samples of characteristics of information signals of significant volumes, etc. It is proposed to use a phase characteristic to select the time interval that limits the signal sample and sets it to a multiple of the signal period, and the sampling rate of information signals to reduce the errors in estimating their spectrum. The possibility of obtaining on their basis secondary deterministic (voltage level, voltage deviations from the nominal level, attenuation coefficient, signal period, signal phase shift, oscillation frequency, etc.) and statistical (sample characteristic, sample variance, sample median, sample circular variance, sample circular median, sample circular kurtosis, etc.) of signal information characteristics, which allows more complete to use their information resource. These characteristics can be used both for assessing power quality characteristics and for monitoring and diagnosing of energy facilities. |
| doi_str_mv | 10.15407/pge2022.01-02.090 |
| first_indexed | 2026-03-24T02:02:13Z |
| format | Article |
| fulltext |
90
V. BABAK, A. ZAPOROZHETS, Yu. KUTS, L. SCHERBAK
ISSN 1562-8965. The Problems of General Energy, 2022, issue 1-2(68-69)
ІНФОРМАЦІЙНО-ВИМІРЮВАЛЬНІ ТЕХНОЛОГІЇ,
МОНІТОРИНГ ТА ДІАГНОСТИКА В ЕНЕРГЕТИЦІ
ISSN 2522-4344 (Online), ISSN 1562-8965 (Print). The problems of general energy, 2022, 1-2(68-69): 90–96
doi: https://doi.org/10.15407/pge2022.01-02.090
UDC 519.6:621.3 Vitalii Babak1, Dr. Sci. (Engin.), Professor, https://orcid.org/0000-0002-9066-4307
Artur Zaporozhets1*, PhD (Engin.), Senior Researcher, https://orcid.org/0000-0002-0704-4116
Yurii Kuts2, Dr. Sci. (Engin.), Professor, https://orcid.org/0000-0002-8493-9474
Leonid Scherbak1, Dr. Sci. (Engin.), Professor, https://orcid.org/0000-0002-1536-4806
1Institute of General Energy of NAS of Ukraine, 172, Antonovycha Str., Kyiv, 03150, Ukraine
2National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, 37, Prosp. Peremohy,
Kyiv, 03056, Ukraine
* Corresponding author: a.o.zaporozhets@nas.gov.ua
SOME FEATURES OF HILBERT TRANSFORM AND THEIR USE IN ENERGY INFORMATICS
Abstract. Information-measuring technologies (IMT) are an important instrument for solv-
ing problems of energy informatics. They allow to form primary information based on the in-
teraction of energy facilities with IMT sensors that form information signals. In many practical
applications, the constructive model of information signals is the model of narrowband sig-
nals. The article summarizes the features of the discrete Hilbert transform and its application
to obtain the primary characteristics of information signals – bypass and phase as functions
of time. The main advantages of using the discrete Hilbert transform in signal processing for
energy informatics are considered, including the consistency of obtaining frequency and
time characteristics, high information content, the ability to analyze the dynamics of changes
in signal characteristics, the possibility of obtaining samples of characteristics of information
signals of significant volumes, etc. It is proposed to use a phase characteristic to select the
time interval that limits the signal sample and sets it to a multiple of the signal period, and
the sampling rate of information signals to reduce the errors in estimating their spectrum.
The possibility of obtaining on their basis secondary deterministic (voltage level, voltage
deviations from the nominal level, attenuation coefficient, signal period, signal phase shift,
oscillation frequency, etc.) and statistical (sample characteristic, sample variance, sample
median, sample circular variance, sample circular median, sample circular kurtosis, etc.) of
signal information characteristics, which allows more complete to use their information re-
source. These characteristics can be used both for assessing power quality characteristics
and for monitoring and diagnosing of energy facilities.
Keywords: energy informatics, information signals, signal processing, discrete Hilbert
transform, amplitude signal characteristics, phase signal characteristics.
1. Introduction
The current stage of development of energy sys-
tems is characterized by an increase in the amount
of electricity consumed, the integration of various
sources of electricity generators into a single sys-
tem, an increase in the number of energy compa-
nies and consumers of electricity, an increase in the
length of electricity networks, etc. At the same time,
such systems are constantly under the infl uence of
natural and anthropogenic factors that can destabi-
lize their functioning. Ensuring the stable and reli-
able operation of energy systems, the safety of en-
ergy facilities and systems (including environmen-
tal safety), the early detection of critical situations
that require a prompt response, the maintenance of
technological processes for managing them in nor-
mal and emergency modes and a guaranteed level of
quality of power parameters requires that the infor-
mation fl ows steady increase, which in turn requires
not only the development and improvement of the
processes of monitoring, diagnostics, and control in
the energy sector [1] but also a change in the par-
adigm of operation and development in the energy
sector as a whole.
To overcome new challenges in recent years, a
new scientifi c branch has emerged and is actively
developing – energy informatics [2], covering the
use of information-measuring and information-com-
munication technologies to solve the problems of
reducing specifi c energy consumption, increasing
energy effi ciency, integrating sources of decentral-
ized renewable energy into a single system, etc. In-© V. BABAK, A. ZAPOROZHETS, Yu. KUTS, L. SCHERBAK, 2022
91
Some features of Hilbert transform and their use in energy informatics
ISSN 1562-8965. The Problems of General Energy, 2022, issue 1-2(68-69)
creasing attention is being paid to various aspects
of this issue in scientifi c periodicals [3–5]. An im-
portant component of energy informatics is infor-
mation-measuring technologies (IMT), which allow
to generate and research of the information signals
to obtain objective information about the current
state of energy facilities [6]. An information signal
in the IMT of the power engineering industry is un-
derstood as both the electricity generated and trans-
mitted over the network, the parameters and quality
characteristics of which must be controlled, and the
signals generated by the IMT sensor systems that
characterize the current state of individual elements
and components of the electric power equipment.
The fi rst step in the study of information signals
is to obtain their parameters and characteristics. The
accuracy of their assessment largely determines the
reliability of the conclusions and the correctness
of technological solutions. Therefore, methods and
means of information signal processing are import-
ant components of IMT.
Modern methods for measuring signal parameters
in the time domain of its representation are focused
on using the capabilities of digital signal processing.
Usually, an analog variable frequency signal is con-
verted into digital, and its parameters – frequency,
period, phase shift, amplitude or eff ective voltage
value, etc., are obtained from its instantaneous val-
ues. Such a simplifi ed approach leads to the simplest
and most economical technical solutions, but it is
focused on the analysis of stationary signals with a
low level of noise and interference.
In many practical applications of IMT in energy,
information signals can be represented by a model of
narrowband signals. An effi cient method for charac-
terizing such signals in the time domain is the dis-
crete Hilbert transform. This digital signal processing
method is widely used in telecommunication tech-
nologies. However, in information and measuring
technologies, including those focused on the energy
industry, it has not yet found proper distribution, and
its capabilities have not yet been fully disclosed.
The purpose of the article is to analyze the fea-
tures of using the discrete Hilbert transform to ob-
tain the characteristics of information signals gener-
ated in power systems during their operation.
2. Model of information signals and their in-
formation resource
In a broad sense, a signal is understood as a pro-
cess that characterizes a change in time and space of
the physical state of an object and is used to obtain
information about this object, as well as to display,
register, transmit, receive and process messages.
Signals can have diff erent physical nature. In energy
informatics, both electrical signals in the power grid
and signals in systems for measuring, controlling,
and diagnosing power system equipment are consid-
ered as informational, which are formed by sensors
in the form of electrical signals – time-varying elec-
trical voltage or current, and which are most conve-
nient for processing, storage, and transmission.
A necessary step in the measurement of informa-
tion signals is the presentation of their model in an
analytical form [7, 8]. To solve a signifi cant range of
problems in the analysis of energy informatics sig-
nals, a narrowband signal model can be used. Its char-
acteristic feature is that the energy spectrum of such
signals is concentrated in a small frequency band f
in the vicinity of a certain center frequency 0f f .
The analytical model of such signals has the form
c, , cos , , ,u t p U t p t p t T (1)
where , , ,U t p t p ‒ respectively bypass (am-
plitude or just amplitude characteristic) and phase
of the signal (phase-time or simply phase character-
istic); , ct T ‒ respectively, the current time and the
time of signal observation; p ‒ vector of informa-
tive signal parameters. As informative parameters,
there can be quantities and characteristics of techno-
logical processes in the energy sector that are diff er-
ent in physical nature, for example, the frequency of
the generated electrical signal and its initial phase,
the temperature of the coolant in the reactor cooling
system and its speed, the rotational speed of turbine
rotors, diagnostic parameters of power equipment,
level of mechanical vibrations, etc. If necessary (for
example, during diagnosing energy systems and
networks branched in space), the arguments of the
model (1) can be the spatial coordinates of receiving
an information signal in the accepted coordinate sys-
tem. The ,u t p signal is considered as a realization
of t random process belonging to the class of pro-
cesses with a fi nite power, i.e. 2
c,t t TM
(M ‒ mathematical expectation operator).
Electric currents and voltages in power transmis-
sion networks are generally represented as periodic
polyharmonic signals with a multiple frequency ratio
1
1
cos 2 , , ,
G
g g g c g
g
u t U f t t T f f N
(2)
where N – set of natural numbers, , ,g g gU f ‒
amplitude, frequency, and initial phase of the g-th
harmonic respectively. As primary parameters,
they are of the greatest interest as objects of mea-
surement for determining the quality of electricity.
Based on these signal parameters, some power qual-
ity characteristics regulated by the standard [9] can
be determined, including the frequency of the power
supply voltage, frequency deviation from the nomi-
nal value, harmonic voltage, voltage dip, etc. Since
1, 1gU U j condition is satisfi ed for electrical
92
V. BABAK, A. ZAPOROZHETS, Yu. KUTS, L. SCHERBAK
ISSN 1562-8965. The Problems of General Energy, 2022, issue 1-2(68-69)
network signals, we can assume that the energy
spectrum of signals (2) is concentrated in the vicin-
ity of the frequency of the fi rst harmonic 1f , there-
fore, such signals can be considered as narrow-band
signals with a certain level of approximation.
3 Results
3.1 Determination of the primary characteris-
tics of information signals
One of the eff ective methods for analyzing the
signals of type (1) is the integral Hilbert transform
[10,11]. It makes it possible allows to unambigu-
ously determine the amplitude ,U t p and the phase
,t p characteristic of the signal (1). This possi-
bility arises because the Hilbert transform allows
determining the quadrature signal c, ,u t p t T , all
frequency components of which are shifted by 2
relative to the corresponding components of the in-
formation signal c, ,u t p t T . This allows to con-
sider and determine the characteristics of ,U t p
and ,t p for the so-called analytical complex
signal of the form
c, , , , , 1.z t p u t p iu t p t T i (3)
The procedure for determining the analytical sig-
nal is much faster and easier in the discrete version
of the frequency domain using the discrete Fourier
transform (DFT) [12].
In this case, a sample of values of the sam-
pled information signal , , 1,u j p j J is an-
alyzed, and the corresponding analytical dis-
crete signal is represented by a sequence of val-
ues , , , ,z j p u j p iu j p 1,j J , where
pj t T is the signal sample number in the sam-
ple, c pJ T T , pT is the signal sampling period,
p 1 gT f .
Using the ,z j p signal, the estimates of the
desired discrete amplitude characteristics and the
phase characteristic limited by the 0, 2 interval
are determined by the known formulas [10], re-
spectively, as
2 2ˆ , , , , 0, 1 ,U j p u j p u j p j J (4)
ˆ , arctg , ,j p u j p u j p
0.5 2 sign , 1 sign , ,u j p u j p
0, 1 .j J
(5)
In formulas (4), (5), and below, the symbol “^”
denotes estimates of the corresponding character-
istics obtained from the results of processing mea-
surement data.
Function (5) has a sawtooth shape and periodi-
cally changes within [0, 2π), i.e. represents the so-
called non-unwrapped phase of the signal. The dis-
crete instantaneous unwrapped phase is obtained
from (2) as
ˆ ˆ ˆ, , 2π , , 0, 1 ,j p j p q j p j J (6)
where ˆ ,q j p ‒ the step function that increas-
es by one each time the phase changes from 2π to
0. Equation (6) allows us to estimate the reversed
phase of the signal as a function of time.
If it is necessary to evaluate phase shifts between
two signals of the same frequency (for example, if
phase angles between successive linear voltages in
three-phase networks are determined), the estimates
of the unwrapped phase 1 2
ˆ ˆ,j j of two coher-
ent signals 1 2,u j u j are determined according to
formulas (5), (6), and the phase shift between them
for all j points as the diff erence
2,1 2 1
ˆ ˆˆ , 0, 1 .j j j j J (7)
Using the unfolded phase (6), it can be got the
instantaneous frequency of the signal using the fol-
lowing formula
ˆ ˆ, 1, mod 2ˆ , ,
2π p
j p j p
f j p
T
1, .j J (8)
Thus, the use of DHT allows to simultaneously
obtain samples of signifi cant amounts of instanta-
neous values of the amplitude, phase, phase shifts,
and frequency of information signals for their sub-
sequent use.
3.2 Determination of secondary characteris-
tics of information signals by their amplitude and
phase characteristics
Solving the urgent important problems of energy
informatics requires the most complete use of the
signal information resource. This requires not only
an assessment of the primary parameters and char-
acteristics of information signals but also the iden-
tifi cation and selection of secondary characteristics,
which may be more informative for some cases of
monitoring and diagnostics. The general concept of
using information signals in energy informatics is
shown in Fig. 1.
Variants of secondary characteristics that can be
obtained from the ˆ ,U j p and ˆ ,t p functions are
presented below. Table 1 shows the secondary de-
terministic characteristics of information signals.
These characteristics change during the operation of
energy systems as a whole and their components as
a result of the action of various destabilizing factors
– fl uctuations in load power, the presence of distur-
bances from operating equipment, the infl uence of
meteorological factors, changes in technological
93
Some features of Hilbert transform and their use in energy informatics
ISSN 1562-8965. The Problems of General Energy, 2022, issue 1-2(68-69)
operating modes of equipment, changes in the me-
chanical or electrophysical parameters of parts and
materials of energy equipment, etc.
In emergencies, these characteristics may go be-
yond the permissible values.
Since information signals in the process of their
formation are infl uenced by various random factors,
and their transmission is accompanied by the action
of noise, therefore, they can be considered as real-
izations of random processes, then not only deter-
ministic but also random characteristics should be
included to the secondary characteristics. Table 2
shows a list of the most promising secondary statis-
tical characteristics of information signals for use.
It is assumed that statistical properties are obtained
from samples of corresponding features of a certain
size in a stationary mode.
Secondary statistical characteristics for voltage
are determined by well-known algorithms for cal-
culating sample characteristics of random variables.
The defi nitions of circular characteristics for sam-
pling the diff erence in phase shifts of signals are giv-
en in [13]. It do not diff er from the sampling circular
characteristics of random angles on the plane [14,
15, 16], and the analysis of their content is beyond
the scope of the article.
In general, the obtained results expand the possi-
bilities of practical use of the discrete Hilbert trans-
form in the implementation of information-mea-
suring signal processing technologies in the energy
sector.
4 Analysis of the characteristic features of the
DPG for use in energy informatics
Even though considerable attention has been
paid to the problem of information signal processing
in general, the issues of using DHT as a component
of energy informatics have not yet been adequately
covered. Let’s look at the main advantages of using
DHT for signal processing in energy informatics.
1. Consistency in obtaining frequency and time
characteristics. Since it is convenient to determine
Table 1. Secondary deterministic characteristics of information signals
Primary characteristics of the information signal
Amplitude characteristic Phase characteristic
Se
co
nd
ar
y
de
te
rm
in
is
tic
ch
ar
ac
te
ris
tic
s • Voltage level
• Voltage deviation from the nominal value
• Symmetry of line voltages
• Long and short voltage interruptions
• Amplitude modulation signal function
• Signal amplitude-shift keying function
• Attenuation coeffi cient (decrement)
• Phase shift of signals
• Phase angles between series line voltages
• Oscillation frequency (fi rst harmonic)
• Frequency deviation from the nominal value
• Signal phase modulation function
• Signal phase-shift keying function
• Signal period
Fig. 1. The general structure of information signal processing in energy informatics
Table 2. Secondary statistical characteristics of information signals
Primary characteristics of the information signal
Amplitude characteristic Diff erence in phase characteristic
Se
co
nd
ar
y
st
at
is
tic
al
ch
ar
ac
te
ris
tic
s
• Sample characteristic
• Sample mean
• Sample dispersion
• Sample standard deviation
• Sample median
• Sample moments of higher orders
• Sample kurtosis
• Sample asymmetry factor
• Empirical distribution (histogram)
• Sample characteristic function
• Sample sine-moments and cosine-moments
• Sample circular average of the phase shift
• Sample length of the resulting vector
• Sample circular dispersion
• Sample circular standard deviation
• Sample circular median
• Sample circular kurtosis
• Sample circular asymmetry coeffi cient
• Empirical distribution of phase shifts (pie chart)
• Diff erence of trends of phase characteristics
94
V. BABAK, A. ZAPOROZHETS, Yu. KUTS, L. SCHERBAK
ISSN 1562-8965. The Problems of General Energy, 2022, issue 1-2(68-69)
the DHT through the discrete Fourier transform
(DFT), the integration of information signal pro-
cessing processes in the frequency and time domains
practically does not require additional hardware and
software costs (Fig. 2).
Fig. 2 shows the structure where the ana-
log-to-digital converter (ADC) provides a digital
copy ,u j p of the analog information signal
,u t p . The DFT block provides the calculation
of the signal spectrum. Transformation of the
signal spectrum (TSS) makes it possible to ob-
tain the spectrum of the discrete analytical signal
,z j p , the samples of which are calculated in
the inverse DFT (IDFT) block. The discrete di-
rect and inverse Fourier transform is performed
according to [10]:
21
0
, , , 0, 1 ;
iJ kj
J
j
U k p u j p e k J
(9)
21
т
0
1, , , 0, 1 .
iJ kj
J
k
z k p U k p e j J
J
(10)
where ,U k p ‒ transformed signal spectrum
,u j p .
Calculation of estimates ˆ ,U j p and ˆ ,j p
occurs in the block for determining the discrete
characteristics of the signal (DDCS).
The phase characteristic ,j p can be used to
select the time interval that limits the signal sam-
pling and makes it a multiple of the signal period,
and the ADC sampling rate implemented by the ad-
aptation block AB. This is necessary to eliminate
the eff ect of splitting the spectral components of the
,u j p signal and provide the required frequency
resolution. The idea of determining the current value
of the signal period by the analog function ˆ ,t p is
illustrated in Fig. 3.
To determine the current period of the signals, the
operation of sliding scanning of the values of the sig-
nal phase characteristic is performed with a rectangu-
lar window with an aperture of 2π (or a multiple of
2π), which displays the sections of the phase charac-
teristic selected by the window on the time axis, de-
termine the corresponding time intervals and evaluate
the current values of the period of the studies signal.
12
2 1
ˆ ˆ, ,1ˆ , .
2
t p t p
T t p
t t
(11)
2. High information content through the abili-
ty to analyze all information related to the integral
infl uence of the parameter vector p on the charac-
teristics of the information signal, in contrast to the
case of analyzing individual harmonic components
obtained using the DFT, when this infl uence is dis-
tributed among diff erent frequency components of
the spectrum.
3. The ability to analyze the dynamics of chang-
es in the discrete amplitude, phase, and frequency
characteristics of signals over the time interval of
their observation cT , which provides new opportu-
nities for improving the methodology for processing
information signals.
4. The ability to obtain a sample of the character-
istics of information signals of signifi cant volumes,
which creates the prerequisites for a more correct
application of statistical methods for processing
characteristics.
Fig. 2. Obtaining primary information characteristics of signals in the frequency and time domains
Fig. 3. Graphical representation of the process
of determining the period of a signal by its
phase characteristic
95
Some features of Hilbert transform and their use in energy informatics
ISSN 1562-8965. The Problems of General Energy, 2022, issue 1-2(68-69)
5. The ability to synchronously calculate the am-
plitude and phase characteristics of the signal for
sharing, extracting new information features, and
searching for their functional or correlation relation-
ships with the desired parameters of the processes
and objects under study.
6. Since the DHT has the property of linearity,
and the modulus of the Hilbert transform complex
coeffi cient is equal to 1, the formation of the Hilbert
image of the signals ( , )u t p occurs without distort-
ing the voltage degree (ADC quantization step) with
which the ( , )u t p values are measured.
7. The value of the phase characteristic in radians
is a value with a dimension of 1, determined by the
ratio of two quantities of the same kind – ( , )u t p
and ( , )u t p without any use of a separate measure
for measuring the signals phase shift (5).
In general, the use of DHT as the basis of the sig-
nal processing methodology in energy informatics
creates favorable conditions for minimizing the an-
alog part of signal processing systems by expanding
digital processing and complicating measurement
information processing algorithms, which increases
the fl exibility of control, diagnostics, and monitor-
ing systems and the possibility of their improvement
through modernization software.
5. Conclusions
One of the topical areas in the development of in-
formatics is energy informatics, aimed at the use and
management of energy, increasing energy effi ciency,
integrating sources of decentralized renewable en-
ergy into a single energy system, reducing specif-
ic energy consumption, and reducing the negative
impact of energy systems on the environment. The
main results of this research are:
1. It is shown that information and measurement
technologies are an important segment of energy in-
formatics, which allow obtaining objective informa-
tion both about the current state of energy systems
and both about the current state of energy systems
and networks and about the electricity quality.
2. For a large number of information signals in
the energy sector, there is a model of narrowband
signals.
3. The characteristic features of the discrete Hil-
bert transform are considered, which make it possi-
ble to determine the primary characteristics of nar-
row-band signals, their amplitude, and phase char-
acteristics.
4. For the fi rst time it is proposed to use the
phase characteristic to select the time interval that
limits the signal sampling and sets it to a multiple
of the signal period, and the sampling rate of infor-
mation signals to reduce the errors in estimating
their spectrum.
5. The DHT, unlike other well-known signal
processing methods, makes it possible to obtain not
only the bypass and phase of information signals,
but also to determine new secondary deterministic
and statistical characteristics due to the obtained
signifi cant data arrays. These include, fi rst of all,
such sample characteristics as trigonometric mo-
ments, circular mean, length of the resulting vector,
circular dispersion, circular median, obtained from
the phase of the signal, etc. These characteristics can
be recommended for use in energy informatics to-
gether with generally accepted characteristics in the
formation of databases describing the state of power
equipment, the quality of electricity and the dynam-
ics of their change.
In general, the obtained results expand the
possibilities of practical use of the discrete Hil-
bert transform in the implementation of informa-
tion-measuring signal processing technologies in
the energy sector.
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ОСОБЛИВОСТІ ПЕРЕТВОРЕННЯ ГІЛЬБЕРТА
ТА ЇХ ВИКОРИСТАННЯ В ЕНЕРГЕТИЧНІЙ ІНФОРМАТИЦІ
Віталій Бабак1, д.т.н., професор, https://orcid.org/0000-0002-9066-4307
Артур Запорожець1*, к.т.н, ст.досл., https://orcid.org/0000-0002-0704-4116
Юрій Куц2, д.т.н., професор, https://orcid.org/0000-0002-8493-9474
Леонід Щербак1, д.т.н., професор, https://orcid.org/0000-0002-1536-4806
1Інститут загальної енергетики НАН України, вул. Антоновича, 172, м. Київ,
03150, Україна
2Національний технічний університет України «Київський політехнічний інститут
імені Ігоря Сікорського», проспект Перемоги, 37, м. Київ, 03056, Україна
* Автор-кореспондент: a.o.zaporozhets@nas.gov.ua
Анотація. Інформаційно-вимірювальні технологій (ІВТ) є важливим ресурсом
для розв’язання задач енергетичної інформатики. Вони дають змогу формувати
первинну інформацію на основі взаємодії об’єктів енергетики з сенсорами IВT,
які формують інформаційні сигнали. У багатьох практичних застосуваннях кон-
структивною моделлю інформаційних сигналів є модель вузькосмугових сигналів.
В статті узагальнено особливості дискретного перетворення Гільберта та його
застосування для отримання первинних характеристик інформаційних сигналів –
обвідної та фази, як функцій часу. Розглянуто основні переваги використання
дискретного перетворення Гільберта в обробленні сигналів для енергетичної ін-
форматики, серед яких: узгодженість отримання частотних та часових харак-
теристик, висока інформативність, можливість аналізу динаміки зміни харак-
теристик сигналів, можливість отримання вибірок характеристик інформацій-
них сигналів значних обсягів та ін. Запропоновано використання фазової харак-
теристики для вибору часового інтервалу, який обмежує вибірку сигналу і задає
його кратним періоду сигналу, та частоти дискретизації інформаційних сигналів
з метою зменшення похибок оцінювання їх спектру. Показана можливість отри-
мання на їх основі вторинних детермінованих (рівень напруги, відхилення напруги
від номінального рівня, коефіцієнт загасання, період сигналу, фазовий зсув сигна-
лу, частота коливань та ін.) та статистичних (вибіркова характеристика, ви-
біркова дисперсія, вибіркова медіана, вибіркова кругова дисперсія, вибіркова кру-
гова медіана, вибірковий круговий ексцес та ін.) інформаційних характеристик
сигналів, що дає змогу більш повно використовувати їх інформаційний ресурс. Ці
характеристики можуть бути використані як для оцінювання показників якості
електроенергії, так і для контролю та діагностування об’єктів енергетики.
Ключові слова: енергетична інформатика, інформаційні сигнали, обробка сигна-
лів, дискретне перетворення Гільберта, амплітудна характеристика сигналів, фазова
характеристики сигналів.
Resived to the Editorial Board: 22.03.2022
|
| id | systemreorg-article-558 |
| institution | System Research in Energy |
| keywords_txt_mv | keywords |
| language | Ukrainian |
| last_indexed | 2026-07-19T01:17:54Z |
| publishDate | 2022 |
| publisher | General Energy Institute of the National Academy of Sciences of Ukraine |
| record_format | ojs |
| resource_txt_mv | systemreorg/96/6001446e8cf6204ca7c744d038153196.pdf |
| spelling | systemreorg-article-5582026-07-18T12:57:42Z Some features of Hilbert transform and their use in energy informatics Особливості перетворення гільберта та їх використання в енергетичній інформатиці Babak , Vitalii Zaporozhets , Artur Kuts , Yurii Shcherbak , Leonid energy informatics, information signals, signal processing, discrete Hilbert transform, amplitude signal characteristics, phase signal characteristics енергетична інформатика, інформаційні сигнали, обробка сигналів, дискретне перетворення Гільберта, амплітудна характеристика сигналів, фазова характеристики сигналів Information-measuring technologies (IMT) are an important instrument for solving problems of energy informatics. They allow to form primary information based on the interaction of energy facilities with IMT sensors that form information signals. In many practical applications, the constructive model of information signals is the model of narrowband signals. The article summarizes the features of the discrete Hilbert transform and its application to obtain the primary characteristics of information signals – bypass and phase as functions of time. The main advantages of using the discrete Hilbert transform in signal processing for energy informatics are considered, including the consistency of obtaining frequency and time characteristics, high information content, the ability to analyze the dynamics of changes in signal characteristics, the possibility of obtaining samples of characteristics of information signals of significant volumes, etc. It is proposed to use a phase characteristic to select the time interval that limits the signal sample and sets it to a multiple of the signal period, and the sampling rate of information signals to reduce the errors in estimating their spectrum. The possibility of obtaining on their basis secondary deterministic (voltage level, voltage deviations from the nominal level, attenuation coefficient, signal period, signal phase shift, oscillation frequency, etc.) and statistical (sample characteristic, sample variance, sample median, sample circular variance, sample circular median, sample circular kurtosis, etc.) of signal information characteristics, which allows more complete to use their information resource. These characteristics can be used both for assessing power quality characteristics and for monitoring and diagnosing of energy facilities. Інформаційно-вимірювальні технологій (ІВТ) є важливим ресурсом для розв’язання задач енергетичної інформатики. Вони дають змогу формувати первинну інформацію на основі взаємодії об’єктів енергетики з сенсорами IВT, які формують інформаційні сигнали. В багатьох практичних застосуваннях конструктивною моделлю інформаційних сигналів є модель вузькосмугових сигналів. В статті узагальнено особливості дискретного перетворення Гільберта та його застосування для отримання первинних характеристик інформаційних сигналів – обвідної та фази, як функцій часу. Розглянуто основні переваги використання дискретного перетворення Гільберта в обробленні сигналів для енергетичної інформатики, серед яких: узгодженість отримання частотних та часових характеристик, висока інформативність, можливість аналізу динаміки зміни характеристик сигналів, можливість отримання вибірок характеристик інформаційних сигналів значних обсягів та ін. Запропоновано використання фазової характеристики для вибору часового інтервалу, який обмежує вибірку сигналу і задає його кратним періоду сигналу, та частоти дискретизації інформаційних сигналів з метою зменшення похибок оцінювання їх спектру. Показана можливість отримання на їх основі вторинних детермінованих (рівень напруги, відхилення напруги від номінального рівня, коефіцієнт загасання, період сигналу, фазовий зсув сигналу, частота коливань та ін.) та статистичних (вибіркова характеристика, вибіркова дисперсія, вибіркова медіана, вибіркова кругова дисперсія, вибіркова кругова медіана, вибірковий круговий ексцес та ін.) інформаційних характеристик сигналів, що дає змогу більш повно використовувати їх інформаційний ресурс. Ці характеристики можуть бути використані як для оцінювання показників якості електроенергії, так і для контролю та діагностування об’єктів енергетики. General Energy Institute of the National Academy of Sciences of Ukraine 2022-05-28 Article Article application/pdf https://systemre.org/index.php/journal/article/view/558 10.15407/pge2022.01-02.090 System Research in Energy; No. 1-2 (68-69 (2022): The Problems of General Energy; 90-96 Системні дослідження в енергетиці; № 1-2 (68-69 (2022): Проблеми загальної енергетики; 90-96 2786-7102 2786-7633 uk https://systemre.org/index.php/journal/article/view/558/490 Copyright (c) 2022 Vitalii Babak , Artur Zaporozhets , Yurii Kuts , Leonid Shcherbak https://creativecommons.org/publicdomain/zero/1.0 |
| spellingShingle | energy informatics information signals signal processing discrete Hilbert transform amplitude signal characteristics phase signal characteristics Babak , Vitalii Zaporozhets , Artur Kuts , Yurii Shcherbak , Leonid Some features of Hilbert transform and their use in energy informatics |
| title | Some features of Hilbert transform and their use in energy informatics |
| title_alt | Особливості перетворення гільберта та їх використання в енергетичній інформатиці |
| title_full | Some features of Hilbert transform and their use in energy informatics |
| title_fullStr | Some features of Hilbert transform and their use in energy informatics |
| title_full_unstemmed | Some features of Hilbert transform and their use in energy informatics |
| title_short | Some features of Hilbert transform and their use in energy informatics |
| title_sort | some features of hilbert transform and their use in energy informatics |
| topic | energy informatics information signals signal processing discrete Hilbert transform amplitude signal characteristics phase signal characteristics |
| topic_facet | energy informatics information signals signal processing discrete Hilbert transform amplitude signal characteristics phase signal characteristics енергетична інформатика інформаційні сигнали обробка сигналів дискретне перетворення Гільберта амплітудна характеристика сигналів фазова характеристики сигналів |
| url | https://systemre.org/index.php/journal/article/view/558 |
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