Modification of the Gelfand-Levitan Method for 1-D Multylaered Structure Inverse Problem
For dielectric slab with step profile of dielectric constant the Gelfand-Levitan method is correct if peaks of time-domain reflected signal are close to δ-pulses. Combination of parametric spectral methods for obtaining time-domain signal from frequency domain data and Gelfand-Levitan method for tim...
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| Zitieren: | Modification of the Gelfand-Levitan Method for 1-D Multylaered Structure Inverse Problem / O.O. Drobakhin, M.V. Andreev, A. Novomlinov, V. Korotkaya, A.Z. Sazonov // Радиофизика и радиоастрономия. — 2002. — Т. 7, № 4. — С. 459-461. — Бібліогр.: 4 назв. — англ. |
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Drobakhin, O.O. Andreev, M.V. Novomlinov, A. Korotkaya, V. Sazonov, A.Z. 2017-07-02T16:53:37Z 2017-07-02T16:53:37Z 2002 Modification of the Gelfand-Levitan Method for 1-D Multylaered Structure Inverse Problem / O.O. Drobakhin, M.V. Andreev, A. Novomlinov, V. Korotkaya, A.Z. Sazonov // Радиофизика и радиоастрономия. — 2002. — Т. 7, № 4. — С. 459-461. — Бібліогр.: 4 назв. — англ. 1027-9636 https://nasplib.isofts.kiev.ua/handle/123456789/122360 For dielectric slab with step profile of dielectric constant the Gelfand-Levitan method is correct if peaks of time-domain reflected signal are close to δ-pulses. Combination of parametric spectral methods for obtaining time-domain signal from frequency domain data and Gelfand-Levitan method for time-domain signal processing can help to improve the solution of the problem. Results of numerical simulation are presented. Для диэлектрической плиты со ступенчатым профилем диэлектрической постоянной применим метод Гельфанда-Левитана, если пики отраженного сигнала близки к δ -импульсам. Комбинация параметрических спектральных методов для получения сигнала во временной области по данным из частотной области и метод Гельфанда-Левитана для обработки сигнала во временной области позволяют получить усовершенствованный алгоритм решения задачи. Приведены результаты численного моделирования. Для діелектричної плити зі східчастим профілем діелектричної сталої метод Гельфанда-Левітана застосовний, якщо піки відбитого сигналу близькі до δ - імпульсів. Комбінація параметричних спектральних методів для отримання сигналу в часовій області та метод Гельфанда-Левітана для обробки сигналу в часовій області дозволяють отримати удосконалений алгоритм розв’язання задачі. Наведено результати чисельного моделювання. en Радіоастрономічний інститут НАН України Радиофизика и радиоастрономия Modification of the Gelfand-Levitan Method for 1-D Multylaered Structure Inverse Problem Модификация метода Гельфанда-Левитана для обратной задачи одномерной многослойной структуры Модифікація методу Гельфанда-Левітана для зворотної задачі одновимірної багатошарової структури Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Modification of the Gelfand-Levitan Method for 1-D Multylaered Structure Inverse Problem |
| spellingShingle |
Modification of the Gelfand-Levitan Method for 1-D Multylaered Structure Inverse Problem Drobakhin, O.O. Andreev, M.V. Novomlinov, A. Korotkaya, V. Sazonov, A.Z. |
| title_short |
Modification of the Gelfand-Levitan Method for 1-D Multylaered Structure Inverse Problem |
| title_full |
Modification of the Gelfand-Levitan Method for 1-D Multylaered Structure Inverse Problem |
| title_fullStr |
Modification of the Gelfand-Levitan Method for 1-D Multylaered Structure Inverse Problem |
| title_full_unstemmed |
Modification of the Gelfand-Levitan Method for 1-D Multylaered Structure Inverse Problem |
| title_sort |
modification of the gelfand-levitan method for 1-d multylaered structure inverse problem |
| author |
Drobakhin, O.O. Andreev, M.V. Novomlinov, A. Korotkaya, V. Sazonov, A.Z. |
| author_facet |
Drobakhin, O.O. Andreev, M.V. Novomlinov, A. Korotkaya, V. Sazonov, A.Z. |
| publishDate |
2002 |
| language |
English |
| container_title |
Радиофизика и радиоастрономия |
| publisher |
Радіоастрономічний інститут НАН України |
| format |
Article |
| title_alt |
Модификация метода Гельфанда-Левитана для обратной задачи одномерной многослойной структуры Модифікація методу Гельфанда-Левітана для зворотної задачі одновимірної багатошарової структури |
| description |
For dielectric slab with step profile of dielectric constant the Gelfand-Levitan method is correct if peaks of time-domain reflected signal are close to δ-pulses. Combination of parametric spectral methods for obtaining time-domain signal from frequency domain data and Gelfand-Levitan method for time-domain signal processing can help to improve the solution of the problem. Results of numerical simulation are presented.
Для диэлектрической плиты со ступенчатым профилем диэлектрической постоянной применим метод Гельфанда-Левитана, если пики отраженного сигнала близки к δ -импульсам. Комбинация параметрических спектральных методов для получения сигнала во временной области по данным из частотной области и метод Гельфанда-Левитана для обработки сигнала во временной области позволяют получить усовершенствованный алгоритм решения задачи. Приведены результаты численного моделирования.
Для діелектричної плити зі східчастим профілем діелектричної сталої метод Гельфанда-Левітана застосовний, якщо піки відбитого сигналу близькі до δ - імпульсів. Комбінація параметричних спектральних методів для отримання сигналу в часовій області та метод Гельфанда-Левітана для обробки сигналу в часовій області дозволяють отримати удосконалений алгоритм розв’язання задачі. Наведено результати чисельного моделювання.
|
| issn |
1027-9636 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/122360 |
| citation_txt |
Modification of the Gelfand-Levitan Method for 1-D Multylaered Structure Inverse Problem / O.O. Drobakhin, M.V. Andreev, A. Novomlinov, V. Korotkaya, A.Z. Sazonov // Радиофизика и радиоастрономия. — 2002. — Т. 7, № 4. — С. 459-461. — Бібліогр.: 4 назв. — англ. |
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Radio Physics and Radio Astronomy, 2002, v. 7, No. 4, pp. 459-461
MODIFICATION OF THE GELFAND-LEVITAN
METHOD FOR 1-D MULTYLAERED STRUCTURE
INVERSE PROBLEM
O. Drobakhin, M. Andreev, A. Novomlinov, V. Korotkaya, A. Sazonov
Radiophysics Dept., Dniepropetrovsk State University,
13 Nauchny str., 49050, Dniepropetrovsk, Ukraine
For dielectric slab with step profile of dielectric constant the Gelfand-Levitan method is correct if peaks of
time-domain reflected signal are close to δ-pulses. Combination of parametric spectral methods for obtaining
time-domain signal from frequency domain data and Gelfand-Levitan method for time-domain signal processing
can help to improve the solution of the problem. Results of numerical simulation are presented.
1. Introduction
1-D inverse problem for multylayered dielectric
structures is a fundamental problem of physics. The
profile of dielectric constant as function of distance
has to be reconstructed. A well-known approach to
the solution of the problem is the Gelfand-Levitan
method [1]. The main advantage of this approach is
capability of discrete (step) profile recovering with
taking into account multiple reflections.
2. Basic Relations
The Gelfand-Levitan method can be applied to dis-
crete multylayered dielectric structures with step pro-
file of dielectric permittivity. It can be used success-
fully if reflectogramma of structure is obtained even
for situation with multiple reflections in every layer
but only for ultrashort time pulses.
Let us consider a multylayered dielectric structure
which consists of n layers with 1 2, nε ε ε… and thick-
nesses 1 2, nd d d… ; 0ε is dielectric constant of air.
The essence of the Gelfand-Levitan method
consists in serial exception of the upper most layer
influence in relation to direction of the falling wave.
For this purpose the linear equation system is made
reflected wave data iR with unknown function
( ),2K k j k− for system consisting of the k top
layers.
0
0 1
0 1 1
0 01 0 0
0 1 0 0
0 0 1
( , 2 )
( , 4 )
( , )
k
R
R R
R R R
K k k
K k k
K k k
−
+
− −
……
… …
i
… …
0
1
1
.
k
R
R
R −
− − = −
(1)
After solving this system of equations, we will find
value of ( ),K k k and then after substitution of this
value to the formula
( ) ( )[ ]
2
2
1
0
1 1 1 , 1
k
k i
i
r r K k k
−
−
=
= − + −
∏ , (2)
we will find the reflectivity factor of the ( )1k −
layer.
The restoration of dielectric permeability is de-
scribed by the formula:
( ) ( )( )21 1 11 1i i i ir rε ε − − += − + . (3)
The Gelfand-Levitan method has a disadvantage
for situation of peaks with non-similar to δ -peak
Fig. 1. Time-domain reflection coefficient for two-
layered structure calculated using discrete Fourier
transform ( 1 0.499f = GHz, 50N = , 1 1d = ,
2 2d = cm)
O. Drobakhin, M. Andreev, A. Novomlinov, V. Korotkaya, A. Sazonov
460 Radio Physics and Radio Astronomy, 2002, v. 7, No. 4
form. The accuracy of the profile reconstruction is
being degraded. For finite frequency band signals
there are ripples in reconstructed profile of dielectric
constant. This situation is rather usual for microwave
measurements.
Parametric spectral methods such as the Prony
method, the generalized pencil of matrix method [2]
and the maximum of likelihood method [3] can be
applied instead of discrete Fourier transform for ob-
taining time-domain signal. There is a simple ap-
proach [4] for data interpretation but the approach is
correct in case of negligibly low level of reverbera-
tions. Applications of parametric spectral analysis
methods allows one to avoid disadvantages of tradi-
tional application of the Gelfand-Levitan method to
time-domain signal obtained by discrete Fourier
transform.
3. Numerical Simulation
For numerical simulation of the Gelfand-Levitan
method using different methods of parametric spec-
tral analysis the time-domain reflection coefficient
has been synthesized on the base of frequency-
domain data calculated under suggestion of planar-
wave approximation. The parameters of two-layered
structure were 1 9ε = and 2 5ε = . The thicknesses
were 1 1d = cm and 2 2d = cm. The values of step
in frequency domain were chosen equal to
1 0.499f∆ = GHz, 2 0.15f∆ = GHz. The number
of frequencies N was 50. Thus the frequency band
were 24.45 and 7.35 GHz correspondingly.
For data presented in fig. 2, 3, 5, 6 one sample
of x-axis corresponds to geometrical sample equal to
1
2 iN fπ ∆
with additional multiplier 1
iε . The
value iε was chosen equal to value in this point.
Time discrete for data in Fig. 1 were chosen in
the manner provided time-domain peak form closely
similar to form of δ -pulses.
Fig. 2 confirms Gelfand-Levitan method capa-
bility of correct reconstruction of dielectric constant
profile for case of sin /x x peaks which are closely
similar to δ -pulses. Non-sufficient ripples in the
reconstructed profile are occurred due to side-lobes
of functions of sin /x x . Results of dielectric con-
stant profile reconstruction by the Gelfand-Levitan
method for time-domain signals obtained by general-
ized pencil of matrix method is presented in Fig. 3.
The Prony method and the maximum of likeli-
hood method give similar. It is clear that application
of the three methods of parametric spectral analysis
allows one to reconstruct profile more correctly. The
parasitic ripples have been vanished. For some parts
of the profile the first approach provides more accu-
rate results but this situation occurred only for spe-
cific situation of δ -form of pulses. In practical situa-
tion δ -form of pulses can be occurred seldom.
Results for case 2 0.15f∆ = GHz are presented
in Fig. 4-6. After discrete Fourier transform obtained
peaks in time domain have form non-similar to δ -
pulses. Recovering the profile by direct application
of the Gelfand-Levitan method implies the appear-
ance of ripples with large amplitudes comparable
with values of the dielectric constant. These oscilla-
Fig. 2. Profile of dielectric constant recovered by
Gelfand-Levitan method for signal of Fig. 1
Fig. 3. Profile of dielectric constant recovered by
Gelfand-Levitan method for data received by gener-
alized pencil of matrix method
Fig. 4. Time-domain reflection coefficient for two-
layered structure calculated using discrete Fourier
transform ( 1 0.15f = GHz, 50N = , 1 1d = ,
2 2d = cm)
Modification of the Gelfand-Levitan Method for 1-D Multylaered Structure Inverse Problem
Radio Physics and Radio Astronomy, 2002, v. 7, No. 4 461
tions managed to be avoided by use of the spectral
parametrical analysis methods.
Results of profile reconstruction by the Gelfand-
Levitan method for time-domain data obtained from
the same data of Fig. 4 using parametric spectral
methods are presented in Fig. 6. Restored dielectric
permeability structure due to use of this methods
becomes more smooth and oscillations disappear.
Worse result was obtained for case of the
Prony’s method but even for this situation the ripples
were disappeared. The best results were obtained for
maximum likelihood method and generalized pencil
of matrix method. The approaches provide rather
high accuracy of estimation.
4. Conclusions
The results of numerical experiments allow to make
conclusions that the Gelfand-Levitan method is the
most simple in realization if time characteristic of the
wave reflected from the layered structure are ob-
tained by discrete Fourier transform from data ob-
tained experimentally or numerically in frequency
domain. This way is more economic concerning
computing expenses but it works adequately when
the width of frequency band is chosen in the manner
that peaks have δ -pulse form and non-overlapped. In
practice it is difficult enough for realizing if layers
have different electric thicknesses. The spectral pa-
rametrical analysis methods used allow one to solve
the problem.
References
1. K. Aki, P.G. Richards. Quantitative seismology. The-
ory and methods. W.H. Freeman and Company, 1980.
2. Y. Hua, T.K. Sarkar. Generalized Pencil-of-Function
Method for Extracting Poles of an EM System from
Its Transient Response. IEEE Trans. Antennas and
Propag, AP-37, No. 2, pp. 229-233, Feb. 1989.
3. H. Vanhamme. High resolution frequency-domain
reflectometry. IEEE Trans. Instrumentation and
Measurement, 39, No. 2, pp. 369-375, Apr.1990.
4. M.V. Andreev, V.F. Borulko, O.O. Drobakhin. One-
dimensional Inverse Problem Solution for Multilay-
ered Dielectric Structures Using Least-Square Spectral
Estimation Method. Proc. of the 1995 URSI Int.
Symp. on Electromagnetic Theory, St.Petersburg,
Russia, May 23-26, 1995, pp. 148-151.
МОДИФИКАЦИЯ МЕТОДА
ГЕЛЬФАНДА-ЛЕВИТАНА ДЛЯ
ОБРАТНОЙ ЗАДАЧИ ОДНОМЕРНОЙ
МНОГОСЛОЙНОЙ СТРУКТУРЫ
О. Дробахин, М. Авдеев, А. Новомлинов,
В. Короткая, А. Сазонов
Для диэлектрической плиты со ступенчатым про-
филем диэлектрической постоянной применим метод
Гельфанда-Левитана, если пики отраженного сигнала
близки к δ -импульсам. Комбинация параметрических
спектральных методов для получения сигнала во вре-
менной области по данным из частотной области и
метод Гельфанда-Левитана для обработки сигнала во
временной области позволяют получить усовершенст-
вованный алгоритм решения задачи. Приведены ре-
зультаты численного моделирования.
МОДИФІКАЦІЯ МЕТОДУ ГЕЛЬФАНДА-
ЛЕВІТАНА ДЛЯ ЗВОРОТНОЇ ЗАДАЧІ
ОДНОВИМІРНОЇ БАГАТОШАРОВОЇ
СТРУКТУРИ
О. Дробахін, М. Авдєєв, О. Новомлінов,
В. Короткая, А. Сазонов
Для діелектричної плити зі східчастим профілем
діелектричної сталої метод Гельфанда-Левітана засто-
совний, якщо піки відбитого сигналу близькі до δ -
імпульсів. Комбінація параметричних спектральних
методів для отримання сигналу в часовій області та
метод Гельфанда-Левітана для обробки сигналу в часо-
вій області дозволяють отримати удосконалений алго-
ритм розв’язання задачі. Наведено результати чисель-
ного моделювання.
Fig. 5. Profile of dielectric constant recovered by
the Gelfand-Levitan method for signal of Fig. 4
Fig. 6. Profile of dielectric constant recovered by
Gelfand-Levitan method for data received by gener-
alized pencil of matrix method
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