=Paper=
{{Paper
|id=Vol-3742/paper9
|storemode=property
|title=Demodulation of the simulated periodically non-stationary random signal with Hilbert transform
|pdfUrl=https://ceur-ws.org/Vol-3742/paper9.pdf
|volume=Vol-3742
|authors=Ihor Javorskyj,Roman Yuzefovych,Oleh Lychak,Pavlo Semenov,Roman Slyepko
|dblpUrl=https://dblp.org/rec/conf/citi2/JavorskyjYLSS24
}}
==Demodulation of the simulated periodically non-stationary random signal with Hilbert transform==
https://ceur-ws.org/Vol-3742/paper9.pdf
Demodulation of the simulated periodically non-stationary
random signal with Hilbert transform
Ihor Javorskyj1,2,†, Roman Yuzefovych1,3,∗,†, Oleh Lychak1,†, Pavlo Semenov4,† and
Roman Sliepko1,†
1 Karpenko Physico-mechanical Institute of NAS of Ukraine, 79060, 5 Naukova Str., Lviv, Ukraine
2 Bydgoszcz University of Science and Technology, 85796, 7 Al. Prof. S. Kaliskiego, Bydgoszcz, Poland
3 Lviv Polytechnic National University, 79013, 12 Bandera Str., Lviv, Ukraine
4 Odessa National Maritime University, 65029, 34 Mechnikova Str., Odessa, Ukraine
Abstract
Demodulation procedure of the periodically non-stationary random signals (PNRSs), with
carrier harmonics, modulated by stationary, high-frequency random processes was explained.
Band-pass filtration and Hilbert transform were used for separation of spectral components of
PNRS and quadratures extraction from amplitude-phase modulated signal. Possibility to extract
and right estimate the quadratures of narrow-band high frequency modulation stochastic
processes was demonstrated. Presented technology will be useful for vibration-based
diagnostics of mechanisms.
Keywords
periodically non-stationary random process, amplitude-phase high-frequency modulations,
Hilbert transform, quadratures, stochastic simulation, vibrations1
1. Introduction
The effectiveness of investigation of the properties of physical systems and processes
based on experimental data on their behavior over time is largely determined by the
ability to identify and evaluate their main characteristic features [1–3]. These features are
obviously the repeatability and stochasticity of their behavior, and these features appear
in the properties of the processes not independently but in interaction. Many processes in
geophysics, meteorology, radiophysic signals and technical mechanics are characterized
by this behavior [4–6]. Periodically non-stationary random processes (PNRP) are
mathematical models describing the complex interaction of repeatability and stochasticity
CITI’2024: 2nd International Workshop on Computer Information Technologies in Industry 4.0, June 12–14, 2024,
Ternopil, Ukraine
∗ Corresponding author.
† These authors contributed equally.
ihor.yavorskyj@gmail.com (I. Javorskyj); roman.yuzefovych (R. Yuzefovych); oleh.lychak2003@yahoo.com
(O. Lychak); p.a.semenoff@gmail.com (P. Semenov); romasrt@gmail.com (R. Sliepko)
0000-0003-0243-6652 (I. Javorskyj); 0000-0001-5546-453X (R. Yuzefovych); 0000-0001-5559-1969
(O. Lychak); 0000-0003-4121-6011 (P. Semenov); 0000-0003-1418-128X (R. Sliepko)
© 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
[7–9]. The methods of analysis of such signals (processes) should obviously include in
their structure both periodic functions and purely stationary random processes [10–12],
both extreme cases and various models of the interaction of periodicity and stochasticity
[13–15]. Vibration signals acquired in machines and mechanisms during diagnostics are
one example of such processes [16–18].
Vibration signal can be modeled as the superposition of harmonics with multiple
frequencies which are stochastically amplitude- and phase-modulated [19–21]. Simulation
of PNRP signal as well as its demodulation procedure involving a Hilbert transform are
presented in this article.
2. Simulation model of PNRP signal
For simulation we select the following model [22, 23]:
nh c nh cos0nh s nh sin 0nh , (1)
where 0 is base cyclic frequency, 0 is a central frequency of high-frequency modulation
stochastic process.
The quadratures of modulating process have a following form:
c nh pc nh cos 0nh ps nh sin 0nh , (2)
s nh qc nh cos 0nh qs nh sin 0nh , (3)
and Epc ,s nh Eqc ,s nh 0 , R pc ,s jh Epc ,s nh pc ,s n j h , R pc jh R ps jh ,
Rqc ,s jh Eqc ,s nh qc ,s n j h , Rqc jh Rqs jh , R pc jh Rqc jh , c ,s
R pq jh 0 ,
cs
R pq jh 0 , R pcs jh Rqcs jh 0 . Assume that quadratures in (2) and (3) have different
autocovariance functions, and are non-correlated. The covariance function of (1) is equal
to:
b nh , jh B 0 jh C 2 jh cos 20nh S 2 jh sin 20nh ,
where
1
B 0 jh R pc jh Rqc jh cos 0 jh cos 0 jh , (4)
2
1
C 2 jh R pc jh Rqc jh cos 0 jh cos 0 jh , (5)
2
1
S 2 jh Rqc jh R pc jh cos 0 jh sin 0 jh . (6)
2
Now, let's assume that
R pc jh Ape , Rqc jh Aq e
p j h p j h
,
For the simulation of PNRP series the following parameter values were used Ap 10 ,
Aq 4 , p q 0.01 , 0 102 , 0 2 103 , series length K 5104 . A fragment of the
simulated series is shown in Fig. 1. The estimators of the covariance components were
calculated using the following equations:
1 K 1
Bˆ0 jh nh n j h ,
K n 0
C 2 jh
ˆ cos 20nh
2 K 1
nh n j h ,
Sˆ jh K n 0 sin 20nh
2
are presented in Fig. 2. As we can see, the forms of the estimators behavior and their
values have insignificant difference from their theoretical values substituted in (4)–(6).
a) b)
Figure 1: Simulated PNRP series (a) and its local segment (b)
a) b)
c)
Figure 2: Estimators of the covariance components: (a) Bˆ0 u ; (b) Cˆ2 u ; (c) Sˆ2 u .
Now let us obtain of the Hilbert transform H for simulated PNRP series [24, 25]
nh H nh . Calculations of the estimators of the covariance components based on
the Hilbert transform for series
nh c nh cos0nh s nh sin0nh ,
confirm that the differences between the values of the covariance components for the
signal and its Hilbert transform are negligible (Fig. 3).
a) d)
b) e)
c) f)
Figure 3: Estimators of the covariance components for the Hilbert transform of
simulated series (signal) (a) Bˆ0 u ; (b) Cˆ2 u ; (c) Sˆ2 u and differences
Bˆ u Bˆ u (d), Cˆ u Cˆ u (e), Sˆ u Sˆ u (f)
0 0 2 2 2 2
The zeroth spectral component for series was estimated using the equation
h L ˆ
fˆ0 B0 nh k nh cos nh ,
2 L
u
where k nh is the Hamming window, L m , and um is the cut-off point of the
h
correlogram.. The graph of fˆ shown in Fig. 4 explains two clear peaks.
0
Figure 4: Estimator of the power spectral density zeroth spectral component for signal
3. Hilbert transform - based demodulation method
To separate two spectral components of PNRP signal [16, 25] we use band-pass
filtering with the rectangular transfer functions
1, for 9 102 Ηz , 103 Ηz
1 ( ) 2 ,
0, for other frequencies ,
and
1, for 103 Ηz , 11 103 Ηz
2 ( ) 2 .
0, for other frequencies ,
Two obtained filtered signals can be represented in the form
nh c nh cos 0 0 nh s nh sin 0 0 nh , (7)
nh c nh cos 0 0 nh s nh sin 0 0 nh ,
(8)
where nh , nh are upper and lower frequency bands respectively, and
1 1
c t pc t q s t , s t qc t ps t ,
2 2
1 1
c t pc t q s t , s t ps t qc t .
2 2
The estimators of the autocovariance functions for the series in (7) and (8) are slowly
decaying harmonic oscillations (Fig. 5), with frequencies of 950 Hz and 1050 Hz. Obtained
estimator values well coincide with the theoretical ones, calculated by the equations:
1
R (u ) R pc (u ) Rqc (u ) cos 0 0 u ,
4
1
R (u ) R pc (u ) Rqc (u ) cos 0 0 u .
4
The values of the second component estimators for the time series in (7) and (8) for the
arbitrary lag are smaller than 1 102 . Thus, the separated spectral components of (7) and
(8) can be considered as realizations of stationary random processes. Note that the sum of
their autocovariance functions is equal to the zero covariance components in (4).
a) b)
Figure 5: Estimators of the covariance functions of the separated components
(a) Rˆ u ; (b) Rˆ u
The cross-covariance functions of (7) and (8), are calculated as following:
1
R t ,u R pc u Rqc u cos 20t 0 0 u ,
4
1
R t ,u R pc u Rqc u cos 20t 0 0 u ,
4
It is clear that R t ,u R t u , u . Using following statistics:
Cˆ2 jh 2 K 1
cos 20nh
nh n j h
,
Sˆ2 jh K n 0 sin 20nh
Cˆ2 jh 2 K 1
cos 20nh
nh n j h
,
Sˆ2 jh K n 0 sin 20nh
we can estimate of the second cosine and sine harmonics [22] for the cross-covariance
functions of (7) and (8). Summing these quantities, we obtain the estimators of the
covariance components in (5) and (6), respectively, as shown in Fig. 2:
Cˆ2 jh Cˆ2 jh Cˆ2 jh ,
Sˆ2 jh Sˆ2 jh Sˆ2( ) jh .
A Hilbert transform of the processes in (7) and (8) gives:
nh c nh sin(0 0 )nh s nh cos(0 0 )nh ,
nh c nh sin 0 0 nh s nh cos 0 0 nh ,
and then we can extract of their quadratures as following:
c nh nh cos 0 0 nh nh sin 0 0 nh ,
s nh nh sin 0 0 nh nh cos 0 0 nh ,
c nh nh cos 0 0 nh nh sin 0 0 nh ,
s nh nh sin 0 0 nh nh cos 0 0 nh .
The segments of the quadrature time series are shown in Fig. 6. The estimators of the
covariance functions of the quadratures for t are presented in Fig. 7. These are
slowly decaying functions. Small differences between their values and the theoretical ones,
which were determined by the relations:
1 1
R c u R pc u Rqs u , R s u R ps u Rqc u ,
4 4
1 1
Rc u R pc u Rqs u , Rs u R ps u Rqc u ,
4 4
can be explained by statistical errors of calculations.
a) b)
Figure 6: Cosine (a) and sine (b) quadratures for the component t
a) b)
Figure 7: Autocovariance functions of the (a) cosine and (b) sine quadratures for t
The results of calculations of the cross-covariance functions shown on Figs. 8 and 9
1 K 1 1 K 1
Rˆ cs jh c nh s n j h , Rˆcs jh c nh s n j h ,
K n 0 K n 0
K 1 K 1
nh s n j h , Rˆcs jh K c nh s n j h
1 1
Rˆ
cs
jh
K n c 0 n 0
confirm that the respective quadratures are non-correlated.
a) b)
Figure 8: Cross-covariance functions of the quadratures for each component:
(a) Rˆcs u ; (b) Rˆ cs u
a) b)
Figure 9: Cross-covariance functions of the quadratures for different components:
(a) Rˆ csu ; (b) Rˆ scu
4. Conclusion
Demodulation of the simulated PNRP with Hilbert transform-based procedures
approved that it is possible to extract quadratures of the modulating processes and right
estimate their covariance properties with statistically satisfied accuracy. Such processing
technology can be useful for demodulation of the complex vibration signals for diagnostics
of mechanisms.
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