CC BY
30
work of the adaptive filterbecause of truncation of its pulse reaction to size N; - is the capacity of noise of a
communication channel.
At an intake input (a subtracter exit) in classical algorithm the level of noise of a communication channel doubles. It is caused by the fact that undercompensation noise /2 A _ ,
J A AF undercomp
and communication channel noise /2 are not correlated.
cc
The gainvalue in relative sizes will be equal to
_2
*A = 10lg—f-, (18)
where /2 - defines either the self-noise level or noise level of undercompensationinthe invariantecho-jack; ct2ao - defines eitherthe self-noise level or noise level of undercompensation in Widrow algorithm.
Fore = 0.9; N = 100; n = 1,000; m =12thesizeof *A will
7 777 own
be equalto21.79db.
Similarly,forc = 0.9; N = 100; c = 0.05;/* =40db;the
j > > > i > m.cc ’
size *A , will be equal to5.3db.
undercomp
The received gain canbe explainedby several reasons: Firstly, the invariant echo-jack is controlled by a transmission signal. The classical echo-jack is controlledby a signal of an error from a subtracter exit.
Secondly, the invariant echo-jack uses the readings of hindrance taken directly from a communication channel. The classical echo-jack calculates the echo-signal estimation artificially.
Thirdly, the work of the invariant echo-jack does not depend on correlation communications of signals of two directions.
The structure of the invariant echo-jack of the second orderis synthesized. The overall performance of the invariant echo-jack is proved. The invariant echo-jack of the second order can find application in systems of telecommunications and objects control.
Bibliography
1. Levin, D. N. Aninvariantecho-jack withaprotective time interval / D. N. Levin, V. B. Malinkin, S. S. Abrams // Telecommunication. 2008. № 2. C. 48-49.
2. Malinkin, V. B. An invariant echo-jack without a protective time interval / V. B. Malinkin, D. N. Levin, S. S. Abrams // Scientific bulletin of Novosibirsk State TechnicalUniversity, 2007. № 2. P. 25-29.
3. Malinkin, V. B. Increase of noise stability of modified Kalmanfilters in relative compensatory methods: the Thesis foradoctor’s degree/V. B. Malinkin. Omsk, 2003.
4. Goldenberg, L. M. Digital processing of signals / L. M. Goldenberg,B. D. Matyushkin, M. N. Pole. M.: Radio andSvyaz. 1990.
5. Mueller, K. A new digital echo canceler for two-wire full duplex data transmission / K. Mueller // IEEE Trans. on comm. 1976. Vol. № 24. № 9. P. 956-962
© Malinkin V.B., KulyasovE. V.,MalinkinE. V., PavlovI. I., 2009
E. I. Algazin, E. G. Kasatkina, A. P. Kovalevsky Novosibirsk State Technical University, Russia, Novosibirsk
VB.Malinkin
Siberian State University ofTelecommunications and Computer Science, Russia, Novosibirsk
THE NOISE IMMUNITY OF THE INVARIANT SYSTEM OF INFORMATION TRANSMISSION BASED ON COHERENT RECEPTION UNDER WEAK CORRELATION COMMUNICATIONS
The invariant system ofinformation processing based on obtaining of the rectangular envelope by using a synchronous detector is considered. The indexes of the noise immunity of such system are calculated. It is supposed that the closest readings of the rectangular envelope are interfered with the additive noise whose readings are weakly correlated with each other. The quantitative estimation of the operation of such system is compared with the quantitative indexes of the known invariant system under non-correlativity of the noise readings.
Keywords: noise immunity, invariant, invariant relative amplitude modulation, probability of pairwise transition, signal/noise relation, coefficient of correlation.
The method of analysis of the qualitative parameters of the invariant system using synchronous detector under the weak correlativity of the noise readings is developed. The analytical expression of calculation of the probability density of invariants transition is worked out on the basis of the expression of invariant estimation.
The results obtained under non-correlativity of the noise readings are presented. All this helps to use the offered structure for qualitative transmission of information.
The invariant systems of information transmission can be based on different methods of information processing. The aim is to reduce the influence of the multiplicative noise using the algorithm of the particular information parameter to the training one [1].
The authors considered the four ways of signals processing with the help of invariant relative amplitude modulation and noise readings of different correlativity.
In the paper [1] the invariant relative amplitude modulation (IRAM) under ideal conditions is considered.
In the paper [2] the invariant non-coherent system of information processing is considered.
In the paper [3] the qualitative characteristics of the invariant relative amplitude modulation with the noise in generator are considered.
In the paper [4] the qualitative characteristics of the invariant relative amplitude modulation of the near readings of the information and training signals are obtained. The given paper completes the investigation of the invariant relative amplitude modulation behavior in case of weak correlativity of the readings of information and training signals.
We have a communication channel limited by the frequencies f and fh. The condition of the communication channel is determined by the interval of stationarity inside which the influence of multiplicative noise is described by the channel transmission k(t) on a certain frequency.
Multiplicative noise equally corrupts informational and training parts of the block of transmitted signals. However the ratio of the energy of the information signal to the energy of the training signal is constant on the interval of stationarity.
Besides, each transmitting block is influenced by the additive noise.
It is supposed that the nearest readings of the additive noise are weakly correlated with each other.
It is necessary to calculate the probability of the pairwise transition of invariants in the system under consideration. For this purpose the analytical expression of the density of probability of invariant estimation is to be found.
In figure 1 the structure of the receiving part of the invariant relative amplitude modulation is presented. Such structure contains synchronous detector (multiplier, AFCc, LPF) and special calculator.
Fig. 1. Block diagram of invariant system of information transmission: AFCc is an automatic - frequency control circuit, LPF is a low - pass filter, SC is a special calculator
On the transmitting side, the modulating parameter is put into the ratio of the energy of the information signal to the energy of the training signal.
Owing to the fact that multiplicative noise equally influences both parts of each transmitted block, the algorithm of demodulation of reception signals taking into consideration the chosen method of signals processing will consist in the expression of invariant estimation:
INV, =■
( (k • INVl + £,(i))
i =1___________________________
1 ( ((k • S,r +ii( m j))
• Str.
(1)
In the numerator (1)theN sum of instantaneous readings of the signal of the information impulse is presented. The information signal is formed by the rectangular envelope with the amplitude k • INV +\(i), where & (i) is additive noise readings distributed according to the normal law [5].
In the denominator (1) the sum N of instantaneous readings of the signal of the training impulse formed by the rectangular envelope is represented: k$Str+ "(m, j), where "(m, j) is the noise the m-realization of the training signal, distributed according to the normal law [5].
Without loss of generality we suppose that Ste =1.If Str F1, then all outcome parameters, namely INV and /& (root-mean-square deviation of the noise &(i), "(m, j)) canbe scaledby the quality Str. In this case the formula (1) will be changed due to the introduced restrictions as follows:
INV, =
( (k • INV; +S (i))
i=1______________________ .
1 ( ( (k • Str +11(^ j))
A
B
(2)
j =1 m =1
In the formula (2) k-INVl is the instantaneous reading of the signal of the information part of the impulse, coming from the channel; &(i) - i - is instantaneous value of the noise in the information signal; k is coefficient of transmission of the channel; "(m, j) - j - is instantaneous value of the noise in m-realization of the training signal.
Let us suppose that the occasional values &(i) and "(m, j) are equally distributed according to the normal law with the zero mathematic expectation and the dispersion /&2. Besides, it is supposed that in each block only the next occasional values are dependent. Then
CORR (&(i), &(i -1)) = CORR ("(m j), "(m j -1)) = R, where R is coefficient of correlation.
All the other occasional values entering each receiving block will be independent. For the realization of this model it is necessary that [6]
|R|< 1/V2.
Let us use the known way of estimation of the probability of the pairwise transition described by the formula of the full probability.
Zip @
Pr = P H wt (z)dz + P H wj (z)dz,
(3)
Zp
j=1 m=1
where Pr is the probability of transition of INVl into INV. and vice versa; P1 is the probability of appearing of INVr P. is the probability of appearing ofINV, where INV} is sent. The second integral the probability of appearing of INV} when INV, is sent; Zthi is threshold value, necessary for calculating Ptr; whenP1 and P. are known it is calculated with the help of the best bias estimation by minimization of Pr on Zthr. When
P, andP areunknownwe choose P, = P =0.5.
1 i 1 i
From the analysis (3) we can see that for calculation of Ptr it is necessary to know the analytical expression W1(z) and W(z) of the probability density of the estimation of the invariant. Onthe basis of (2) letus calculate the mathematic expectation and dispersions of the instantaneous values A and B.
Mathematic expectation of the numerator is equal to [7] mA = N-k-INVr (4)
But mathematic expectation of the denominator is equal to [7]
mB = N-k. (5)
The dispersion of the numerator is [7]
gA = D(((i)) = N - o2 + 2(N - 1)cov(^(i),£(i +1)) =
i= 1
= N + 2(N -1)R = a^ (N + 2(N - 1)R). (6)
The dispersion of the denominator is equal to [7]
1
= (N + 2(N -1)R).
(7)
The quotient of the two occasional values is calculated by the formula [7]
W ( z ) = H
1
_ (zx_ mA )
2aA
_ ( x-mB )
2<A
2mS B
x \dx,
(8)
where / and /B are calculated by the expressions (6) and (7); mAw.mB - by the expression (4,5).
It should be pointed out that in the process of calculation inthe formula (3) W.(z) -INV, is usedwhere i = 2,3,4,5,6,7. The value of the probability of the pairwise transition Pt was calculated using the method of numerical integration. The number of accumulations with averaging is 40 [1].
The received data are limited by the first 6 pairs of the compared invariants, whenlNVj = 1, INV. = 2,3,4,5,6,7.
The probability of the pairwise transition is calculated at the value h - signal/noise relations whichwas calculatedby the formula expressed by the relation of the power of signal to the power of noise [5]
, 2 k2inv,2 h = -
N«*2
The transmission of these signals is carried out on the basis of classical algorithms of information processing and has no high noise immunity.
The curve 3 in figure 2 and figure 3 corresponds to the errorprobability Per, which is the analogue of the probability of the pairwise transition Pand is calculated by the known formulas (5) and only after processing of these signals in accordance with the algorithm of the quotient by the expression (1),we obtainthe invariant estimationwhichis in reality a number but not a signal. As we can see from figures 2 and 3, the probability of the pairwise transition in the invariant systemis definedby the quantities (10-1...10-33). At the same values the signal/noise probability of the inaccurate reception of the single symbol in classical systems is within the limits (10-1...10-7).
The given analysis of the invariant system shows that the invariant system of information transmission under the weak correlation of the readings of the additive noise has a high noise immunity. The error probability of the classical algorithm with the amplitude modulation is at least twice as large as the probability of the pairwise transition in invariant system.
Therefore, the given system should be used in telecommunication systems, telecontrol systems and other systems, placing exacting demands uponthe noise immunity.
The threshold values Zto were calculated by minimization Pt inthe formula (3). The results of the calculationfordifferent values of the coefficient of the channel transmission k, the coefficient of correlation R and the quantity INV, = 1, INV. = 2,3,4,5,6,7 are placed in tables 1,2.
If in the formulas (6) and (7) R = 0 (the readings of the noise are non-correlated) the general expression of probability density of the invariant estimation obtained in the paper turns into the relation of the calculation of the analogous parameter received under the operation of the non-correlated readings of the white noise [1].
However, the received expression of the density of the probability in the given paper is redetermining and most fully reflects the real situation.
The peculiarity of any invariant system based on the principle of the invariant relative amplitude modulation is the fact that the amplitude modulated signals formed by INV, and S are transmitted along the channel.
Fig. 2. The noise immunity of IRAM at k = 1 and INV, = 1; INV. = 2; 3; 4; 5; 6; 7.
l 7 . 7 7 7 7 7
Curve 1 is the probability of the pairwise transition into IRAM under non-correlativity of the noise readings (theoretical limit); Curve 2 is the probability of the pairwise transition of IRAM at R = 0,7; Curve 3 is the error probability of the classical AM
Table 1
The value Z at the given K, R
K = 1 R = 0.7
Zthr 1 1.627 | 2.057 | 2.488 3.062 | 3.516 | 4.162
The value Zthr at the given K, R Table 2
K = 0.7 R = 0.7
Zthr 1 1.820 | 2.226 | 2.632 3.038 | 3.646 | 4.078
Fig. 3. The noise immunity of IRAM at k = 0,7 and INV, = 1; INV. = 2; 3; 4; 5; 6; 7.
/5 j 5 5 5 5 5
Curve 1is the probability of the pairwise transition into IRAM under non-correlativity of the noise readings (theoretical limit; Curve 2 is the probability of the pairwise transition of IRAM at R = 0,7; Curve 3 is the error probability of the classical AM
Bibliography
1. The Invariant Method of Analog ofTelecommunication System of Information Transmission : monograph /
V. B. Malinkin, E. I. Algazin, A. N. Levin, N. V. Popantonopulo. Krasnoyarsk, 2006.
2. Algazin, E. I. The Estimation of the Noise Immunity of the Invariant System oflnformationProcessing underNon-Coherent Reception / E. I. Algazin, A. P. Kovalevsky,
V B. Malinkin. Vestnik SibSAU, Krasnoyarsk, 2008. Iss. 2(19). P. 38-41.
3. Algazin, E. I. The Noise Immunity of the InvariantRelative Amplitude Modulation / E. I. Algazin, A. P. Kovalevsky,
V B. Malinkin//MaterialsofIXIntem Conf. “ActualProblems of Electronic Instrument Engineering” (APEIE-2008). Novosibirsk, 2008. Vol. 4. P. 20-24.
4. Algazin, E. I. The Noise Immunity of the Invariant System of Information Transmission under the Weak Correlation Communications /E.I. Algazin, A. P. Kovalevsky, V. B. Malinkin // Vestnik of SibSAU. Krasnoyarsk,2008, P. 29-32.
5. Teplov, N. L. The Noise Immunity of the Systems of TransmissionofDiscrete Information/N.LTeplov. M., 1964.
6. Borovkov, A. A. The Theory of Probabilities / A. A. Borovkov. M., 1999.
7. Levin, B. P. Theoretical Foundations of Statistic Radio Engineering/B. P. Levin. M., 1989.
© A/gazin, E. I., Kasatkina E. G., Kova/evskyA. P., Ma/inkin V. B., 2009
A. E. Novikov SiberianFederal University, Russia, Krasnoyarsk
E. A. Novikov
Institute of Computational Modelling, RussianAcademy of Sciences, Siberian Branch, Russia, Krasnoyarsk
THE NUMERICAL MODELING OF A CESIUM CYCLE IN THE UPPER ATMOSPHERE BYAN L-STABLE METHOD OF SECOND-ORDERACCURACY*
An algorithm of right-hand side and Jacobian formation of differential equations of chemical kinetics is described. Numerical simulation of the cesium cycle in the upper atmosphere is conducted by means of the L-stable method of the second order of accuracy with the control accuracy. The results of the computation are presented.
Keywords: chemical kinetics, cesium cycle, stiff problem, L-stable method, accuracy control.
The modeling of chemical reactions kinetics’ is applied in the studies of various chemical processes. The subject of this study is the time dependence on concentration of reagents being a solution for the Cauchy problem and for systems of ordinary differential equations. Difficulties in solving such problems are related to stiffness and a large scale. In modern methods forthe solving stiff problems, an inversion of the Jacobi matrixoftheequations’ systemisused. Inthecaseoftheoriginal problem’s large scale, the decomposition of the given matrix essentially defines a total of computational work. To improve calculation efficiency, in a number of algorithms the freezing of the Jacobi matrix, i.e.the application of the same matrix at several
iteration steps, is used [1-2]. This approach is used in advantage to algorithms based on the multistep methods, in particular the formulaforbackwarddifferentiation [3]. The situationis worse in algorithms for integrations based on known iteration-free methods among which are the Rozenbrok type methods and theirmodifications [1].Here isanalgorithmforthe construction of the right-hand side and the Jacobi matrix of the differential chemical kinetics’ equations. Results of numerical modeling of an ionization-deionization cesium cycle in the upper atmosphere withan L-stable method of second-order accuracy, in which the numerical freezing as well as analytical Jacobi matrix is allowed, aregivenhere.
* The work was supported by the Russian Foundation of Fundamental Researches (grant №08-01-00621) and by the Presidential grantNSh-3431.2008.9.
