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IMPROVING THE ACCURACY OF DETERMINING THE SINGULARITY BY THE DETAIL WAVELET-COEFFICIENTS OF THE REFLECTOGRAM DECOMPOSITION
Irina V. Manonina,
MTUCI, Moscow, Russia, ivm@mtuci.ru
Vladimir V. Shestakov,
MTUCI, Moscow, Russia, shvvov@mtuci.ru
DOI 10.24411/2072-8735 -2018-10011
Keywords: Lipschitz exponent, singularity, wavelet-coefficients, reflectogram, accuracy.
When measuring communication lines with "classical" TDR and OTDR, the results obtained in the form of reflectograms contain certain errors. These errors depend on communication lines discontinuity, type of measured device, length of the measured line, parameters of the pilot pulse (wavelength, duration and pulse shape), dispersion of optical fiber, and methods for acquisition of the reflectogram on the screen and its further processing. An overview of modern methods of measuring communication lines has shown that known methodology and devices based on it make it possible to detect discontinuities and damage to communication lines with errors, therefore, it is necessary to improve the methods of measuring communication lines using modern methods of processing data from OTDRs that can significantly reduce the error of localization of damage.
The method for determining the singularity of reflectograms based on the analysis of the detail coefficients of the wavelet decomposition of the reflectogram to improve measurement accuracy is proposed in the article, and it allows to significantly increase the localization of damages and discontinuities. The method is based on the mathematical processing of the signal with the determination of the points of discontinuity or changes in the regularity of the signal (finding the exact localization of the singularity) pointing to the places of damage, discontinuities or specialized connection (splitters) by the wavelet coefficients. The presence of a singularity is characterized by the Lipschitz exponent at a given point. Two types of typical damages (discontinuities, connection) were simulated as the analyzed signal, which were processed using different wavelets (Haar wavelet, Daubechies wavelet D4-DI8, symmlets) and at the following comparison of the results for this type of discontinuities, the most accurate values for the location of damage were obtained using Daubechies wavelets D6-DI0 - having 3, 4 and 5 vanishing moment. To accurately define singularity, a procedure written in the MATLAB environment is used. In conclusion, an example of the application of the developed method for the real reflectogram is given for conducting a comparative analysis of the results obtained from the reflectometer and by the wavelet processing. The proposed method made it possible to improve the accuracy of instrumental methods for localizing damage and discontinuities (singularity) of communication lines by a factor of 1.5-2.
Information about authors:
Irina V. Manonina, Moscow Technical University of Communications and Informatics (MTUCI), associate professor of the MS and MI department, candidate of technical sciences, Moscow, Russia
Vladimir V. Shestakov, Moscow Technical University of Communications and Informatics (MTUCI), associate professor of the MS and MI department, candidate of technical sciences, Moscow, Russia
Для цитирования:
Манонина И.В., Шестаков В.В. Повышение точности определения нерегулярности по детализирующим вейвлет-коэффициентам разложения рефлектограммы // T-Comm: Телекоммуникации и транспорт. 2018. Том 12. №1. С. 53-58.
For citation:
Manonina I.V., Shestakov V.V. (2018). Improving the accuracy of determining the singularity by the detail wavelet-coefficients of the reflectogram decomposition. T-Comm, vol. 12, no.1, рр. 53-58.
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Introduction
Singularities and irregular structures often cany the most important information in signals. In images, the discontinuities of the intensity provide the locations of the object contours, which are particularly meaningful for recognition purposes. For many other types of signals, from electrocardiograms to radar signals, the interesting information is given by transient phenomena such as peaks. In physics, it is also important to study irregular structures to infer properties about the underlined physical phenomena. The Fourier transform was the main mathematical tool for analyzing singularities. But the Fourier transform is global and provides a description of the overall regularity of signals, and it is not well adapted for finding the location and the spatial distribution of singularities. The wavelet transform decomposes signals into elementary building blocks that are well localized both in space and frequency, thus characterizing the local regularity of signals. In mathematics, the local regularity of a function is often measured with Lipschitz exponents. The local maxima of the wavelet transform modulus provide enough information for analyzing these singularities [1,2],
Mathematical definition of signal singularity
The wavelet transform can be used to search singularities or irregular structures of a function, that is, to search for a singularity in the signal structure. The singularity of the signal on the graph, as well as its determination by the detail coefficients of the wavelet decomposition, is shown in Figure I. A function/(.v) is pointwise Lipschitz/> 0 at if there exist A > 0 and a polynomial p of degree n such that
XQ
|/(*)-Pv wjs/ijjt-*^. If fix) satisfies this relation for all
jT(j e [ai, o:], with a constant A that is independent of xo, then the function /(.y) is uniformly Lipschitz x over [ou ûj. Wavelet transform Wvf{a, b) of the function /iv) is
1
W.
,f(aM=~r f fWV
x-b
The decay of the wavelet transform amplitude across scales is related to the uniform and pointwise Lipschitz regularity of the signal. Measuring this asymptotic decay is equivalent to zooming into signal structures with a scale that goes to zero.
It is assumed that the wavelet i// has M vanishing moments
OS
jx?tf/(x)dx = 0) A = 0, A/-1 and M times continuously dif-
ferentiahle (M > x) with derivatives that have a fast decay (that
. . r
is, for all 0 < k< M there is a constant C,„: <--—■)■
l+\x\"
The connection between the global and local regularity of the function fix) and the rate of decrease of its wavelet transform as a —* 0 is as follows: if the timet ion fix) is uniformly Lipschitz regular with exponent / < M on the interval [a\, a?], then there
t
exists a constant A > 0 such that: |^/(<2}1£>)|¡g Aa 2 ■ If the
function fix) is Lipschitz regular with exponent/< M at the point then there exists a constant A > 0 such that:
i r
x+n
\Wvf(a,b)\<LAa ^
1 +
b-x„
Thus, these expressions evaluate the regularity of the signal function, that is, the maximum value of % for which fix) is Lipschitz regular with exponent X- In order to estimate / at a certain point, for those k, that correspond to the position of this point, the binary logarithms of the empirical wavelet coefficients and x are estimated by the least-squares method.
Investigated signal
Detail coefficients at the first level of the wavelet decomposition
jr = 2.8
-10 12 3 4
Figure 1. Determination of a signal singularity
Near the signal singularity, there will be a sharp increase in the amplitude of the wavelet coefficients. And regions of uniform regularity will give coefficients that decrease to zero with decreasing scale a. The rate of this decrease depends not only on the regularity of the signal, but also on the number of vanishing moments of the wavelet used. If the function has a singularity at the point .io, then the Lipschitz exponent at a given point will characterize the singularity [I-6J. In [1-3], examples are given of determining the singularity of a function using the Daubechies wavelet.
Determination of the singularity by the detail wavelet
coefficients of the reflectogram decomposition
Accuracy of localization of damage and discontinuities to communication lines by reflectogram depend on the type of measured device, length of the measured line, parameters of the pilot pulse (wavelength, duration and pulse shape), dispersion of optical fiber, and for used TDR and OTDR can be in the range from 0,4 to 80 meters [7, 8]. On the rellectogram the accuracy of the measurement can be represented as the distance between two neighboring points.
Figure 2 displays the part of the reflectogram showing the area with the beginning of the damage, OTDR in the automatic operation mode determines location of the damage (the point at which the back scattering level changes) at a point with a value of 194, while, as can be seen from the figure, the change in the backscattering (lifting) level occurs at a point with a value of 189. Thus, the difference between the value determined by the OTDR and the actual value is 5 units. With a different resolution of OTDR, this difference can be in the range of 2 to 400 meters and, accordingly, the measurements are made with such errors, which greatly complicates the operation of maintenance services.
To reduce the measurement error, it is suggested to use the following scheme for processing the rellectogram. To localize the exact location of damage and discontinuities by the reflectogram, it is necessary to determine the location of the singularity: the reflectogram must be decomposed into approximating and detailed wavelet coefficients by means of a multilevel one-
7TT
points arc processed. This leads to false leaps and a slightly lower value of the level of detail coefficients than with the use of Haar wavelet.
À
H
f \
; 1 V /
i
\ /
v
» g W * » K tu 1 J » 4 * ■ ( ■
a)
1
\ ___.
\ / t
w i
/
1« v Y
A.....1.
!\ ;
! \|
i .......1..........
_ --- _L 1 'H \ /v> _
----- N ^ rt \ / [ V7^
! V 1 \ /
.........!.....
i
e)
Figure 6. Detail coefficients of the reflectogram wavelet decomposition:
(a) Haar; (b) Daubechies wavelets with 3 vanishing moments; (c) Daubechies wavelets with 9 vanishing moments; (d) Sy mm lets with 2 vanishing moments; (e) Sy mm lets with 6 vanishing moments
For the damage shown in Figure 5, the most accurate values were obtained using Daubechies wavelets D6, D8 and D10. This is due to the number of readings from the beginning of the rise of the backseattering level to the point with a sharp leap. In this case, using decomposition of the re Hectogram with the Haar wavelet, it is also possible to determine the irregularity near point 805. Figure 7 shows the detail coefficients for the decomposition of the reflectogram with the use of wavelets: Haar, Daubechies D8 and 1)14, symmlcts of 2, 4 and 7 order. The solid line denotes the first coefficient, the dotted line indicates the second detail coefficient. As can be seen from Figure 7{d)-7{f) the use of the symmlct for determining the irregularity of the reflectogram with a smooth increase in the backscattering level is inexpedient.
A
J >/\ _
1 \ /1 I \ /X ---\
. :...... • , i \ \ y fV
la « • 1 u* « Ut IU 1 b) 1 114 114 13»
: ! A !
V \ \
....... J../. v /ty.....
• i— s — -Jy ••V I r~
■ Î \ \j
' i
W « » t< M * e » i > a «î* * i ■
1 \
¿r
i
N
\ /
\ V j^J
*........... •••
* a m U* ÉH I:M 111 1 d) 14 ■ i 111 IS
! A
_ / W ^ \ Y
/
\ V"" ( !
\ f
i V
Figure 7. Detail coefficients of the reflectogram wavelet decomposition:
(a) Haar; (b) Daubechies wavelet with vanishing moments; (e) Daubechies wavelet with 7 vanishing moments; (d) Symmlcts with
2 vanishing moments; (e) Sy mm lets with 4 vanishing moments; (f) Sy m inlets with 7 vanishing moments
To accurately determine the irregularity of the reflectogram, it is necessary to find the numerical values of the detail coefficient by the above procedure.
Over the entire reflectogram, the detail coefficients of the first decomposition level have values of the order of tO"3 - 10°. A sharp change in the coefficients is determined at point 809, which corresponds to a sharp leap on the reflectogram (Figure 5) and for decomposition of the reflectogram by the Daubechies wavelet D8 has a value -1,2272. Consequently, the previous value corresponds to the irregularity of the reflectogram and has a value of-0,2564 at point 806. Thus, deviations near the point 806, larger than 10"1, should he considered as sought, i.e. the
irregularity of the refleetogram is defined in the range of values ¡805,37-806,53}. The absolute error is in the range {1,37-2,53}.
With a different resolution of OTDR, for example, from 0,4 to 80 meters, the error in determining the irregularities of the refleetogram by the proposed method will be in the interval from {0,547-1,012} m at a resolution of 0,4 m to {27,4-50,6} m at a resolution of 20 m. In this case, the OTDR detects a damage at a point 809, because the limit of the admissible absolute error in the measurement of the attenuation in this case is 0,95. That is, when localizing the damage the OTDR determines this damage only when the hackscattering level is changed by more than 0,95 dB, In this case, the localization error has values from 2 m with a resolution of 0,4 m, up to 400 m at a resolution of 80 m.
Accordingly, the localization of the damage by the wavelet analysis makes it possible to reduce the error from 2 to 4 times in the considered case, depending on the OTDR resolution.
Analysis of the results of the refleetogram processing
This method of determining the irregularities in the refleetogram was applied to the actual refleetogram obtained as a result of measuring the PON line with the splitter, and shown in Figure 8, to determine the exact location of the splitter. As a base wavelet for localizing the irregularities the Daubechies wavelet D4 was used, which demonstrated the best results in the course of experiments. In addition, the parameters defined in [10] were used to remove noise from the refleetogram.
The noise variance, determined from the refleetogram, is cr = 0,076\ The value of the location of the splitter, determined by the OTDR, corresponds to the reading 8971,2. The resolution of the OTDR is 0,16 m. The real place of the splitter corresponds to the reading 8969,6,
1 ;
-33 -36 -39
............J.......
-48 \ .................
-17 ! !
Figure 8. The studied refleetogram
After removing the noise from the refleetogram using a coiflet of the first order, and symmlets of 2 and 5 order, three cleared re Hectograms were obtained from which the exact localization value of the splitter was determined using the Daubechies wavelet D4. For the first refleetogram obtained after the removal of noise using a coiflet of the first order, the damage is deter-
mined in the range 8970,56-8971,20. For the second refleetogram, the damage is determined in the range 8970,40-8971,20. For the third refleetogram, the damage is determined in the range 8970,72-8971,20.
Thus, the difference between the value determined on the OTDR and the values determined according to the proposed method is, respectively: up to 0,1024 m, 0,1280 m, 0,0768 m for each wavelet used. In this case, the error in determining the damage relative to the actual location is: for a OTDR - 0,256 m, for a coiflet of the first order - from 0,153 m, for a symmlet of the second order - from 0,128 m, for a symmlet of the fifth order -0,179 m.
Thus, the proposed method makes it possible to improve the accuracy of instrumental methods of localizing the damage and irregularities (singularities) of communication lines by a factor of 1,5 - 2. That can significantly reduce the measurement errors for OTDRs with different resolutions.
References
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3. Boggess A,, Narcowicli F,J. (2001), A first course in wavelets with Fourier analysis. Upper Saddle River, Prentice Hall.
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5. Tu C.L., Hwang W.L. (2005), «Analysis of singularities from modulus maxima of complex wavelets», IEEE Trans. Inf, Theoty, vol. 51, no. 3. pp. 1049-1062.
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7. «OTDR FOD 7xxx. OTDR MTSe 5000. TDR REYS-205. Technical description and manual», Opticheskiy reflektometr FOD7xxx. Op-ticheskiy reflektometr MTSe serii 5000. Reflektometr eifrovoy REYS-205. Tekhnicheskoe opisanie i instrukciya po ekspluatacii (in Russian)
8. Listvin A.V., Listvin V.N. (2005), Reflektometrija opticheskih volokon, [Reflectomeiry of optical fibers], WELCOME, Moscow (in Russian)
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ПОВЫШЕНИЕ ТОЧНОСТИ ОПРЕДЕЛЕНИЯ НЕРЕГУЛЯРНОСТИ ПО ДЕТАЛИЗИРУЮЩИМ ВЕЙВЛЕТ-КОЭФФИЦИЕНТАМ РАЗЛОЖЕНИЯ
РЕФЛЕКТОГРАММЫ
Манонина Ирина Владимировна, МТУСИ, Москва, Россия, ivm@mtuci.ru Шестаков Владимир Владимирович, МТУСИ, Москва, Россия, shvvov@mtuci.ru
Aннотация
При измерениях линий связи "классическими" импульсными и оптическими рефлектометрами результаты, получаемые в виде ре-флектограмм, содержат определенные погрешности. Данные ошибки зависят от неоднородности линии связи, от типа измеряемого прибора, длины измеряемой линии, параметров зондирующего импульса (длина волны, длительность и форма импульса), дисперсии оптического волокна и также от методов получения рефлектограммы на экране и дальнейшей её обработки. Обзор современных методов измерения линий связи показал, что известные методики и основанные на них приборы позволяют выявлять неоднородности и повреждения линий связи с погрешностями, поэтому необходимо совершенствовать методы измерений линий связи с использованием современных методик обработки данных рефлектометров, позволяющих существенно повысить точность локализации повреждений.
В статье для повышения точности измерений предложено использовать метод определения нерегулярности (сингулярности) ре-флектограмм, основанный на анализе детализирующих коэффициентов вейвлет-разложения рефлектограммы, и позволяющий существенно повысить локализацию повреждений и неоднородностей. Метод основан на математической обработке сигнала с определением по вейвлет-коэффициентам точек разрыва или изменения регулярности сигнала (нахождения точной локализации нерегулярностей), определяющие места повреждений, неоднородностей или специализированных соединений (сплиттеров). При этом наличие сингулярности характеризуется показателем Липшица в данной точке. В качестве исследуемого сигнала смоделированы два вида типовых повреждений (неоднородностей, соединений), которые подвергались обработке с использованием различных вейвлетов (вейвлет Хаара, вейвлеты Добеши 2-го - 9-го порядков, симлет 2-го - S-го порядков), и при последующем сравнении результатов для данного вида неоднородности наиболее точные значения местоположения повреждений получены с использованием вейвлетов Добеши 3-го, 4-го и Б-го порядков. Для точного определения нерегулярностей применяется процедура, написанная в среде MATLAB. В заключении приведен пример применения разработанного метода к реальной ре-флектограмме с целью проведения сравнительного анализа результатов, полученных от рефлектометра и с помощью вейвлет-обработки. Показано, что предложенный метод позволил повысить точность инструментальных методов локализации повреждений и неоднородностей (нерегулярностей) линий связи в I,S-2 раза.
Ключевые слова: показатель Липшица, нерегулярность (сингулярность), вейвлет-коэффициенты, рефлектограмма, точность. Литература
1. Mallat S. A wavelet tour of signal processing. - Second Edition. Academic Press. I999. SSI с.
2. Mallat S., Liang H.W. Singularity detection and processing with wavelets // IEEE transactions on information theory. I992. Vol. 3S, no. 2, рр. 6I7-643.
3. Boggess A., Narcowich F.J. A first course in wavelets with Fourier analysis. Upper Saddle River: Prentice Hall, 200I. Р. 2S3.
4. Venkatakrishnan P., Sangeetha S., Muthukumaran M. Performance analysis of life time efficiency of Machines using Wavelet Transform Modulus Maxima // International Journal of Scientific & Engineering Research. 20I2. Vol. 3, Iss. 6.
5. Tu C.L., Hwang W.L. Analysis of singularities from modulus maxima of complex wavelets // IEEE Trans. Inf: Theory. 200S. Vol. SI, no. 3, рр. I049-I062.
6. Кудрявцев А.А., Палионная С.И., Титова А.И., Шестаков О.В. Оценивание показателя локальной регулярности функции сигнала по коэффициентам вейвлет-разложения // T-Comm: Телекоммуникации и транспорт. 20I6. №I0. С. 43-46.
7. Оптический рефлектометр FOD 7xxx. Оптический рефлектометр MTSe серии S000. Рефлектометр цифровой РЕЙС-205. Техническое описание и инструкция по эксплуатации.
S. Листвин А.В., Листвин В.Н. Рефлектометрия оптических волокон. М.: ВЭЛКОМ, 200S. 20S с.
9. Смоленцев Н.К. Основы теории вейвлетов. Вейвлеты в MATLAB. М.: ДМК Пресс, 200S. 304 с.
10. Манонина И.В. Определение оптимальных параметров для вейвлет-обработки рефлектограмм // Наукоемкие технологии в космических исследованиях Земли. 20I6. Том S, № S. С. 2S-3S.
Информация об авторах:
Манонина Ирина Владимировна, Московский технический университет связи и информатики, доцент каф. МСиИИ, к.т.н., Москва, Россия Шестаков Владимир Владимирович, Московский технический университет связи и информатики, доцент каф. МСиИИ, к.т.н., Москва, Россия
T-Comm ^м 12. #1-2018
