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E. I. Algazin, A. P. Kovalevsky Novosibirsk State Technical University, Russia, Novosibirsk
VB.Malinkin
Siberian State University ofTelecommunications and Computer Science, Russia, Novosibirsk
SIGNALS TRANSMISSION BY INVARIANT METHOD WITH FURTHER NON-LINEAR PROCESSING UNDER WEAK CORRELATION
The invariant system of information processing based on square-law characterized non-linear processing has been synthesized. In calculating of the parameters of such system it is assumed that the readings of the sub-carrier are interfered with additive noise and weakly correlated with each other. The quantitative estimation of the operation of such system is compared with the quantitative indications of the classical system with amplitude modulation and with the characteristics of the invariant system on the basis of extended synchronous detection.
Keywords: noise immunity; invariant; probability of pairwise transition; signal/noise relation.
The analysis of the qualitative parameters of the invariant system with non-linear processing of the readings of the sub-carrierunderweak correlativity of the noise readings is carried out. The analytical expression of calculation of the probability density of the invariant estimation is found. All this allows to use the given structure for qualitative transmission of information.
In papers [1-5] the invariant systems of transmission of information having different probabilities of pairwise transition were investigated.
It should be pointed out that the mentioned above invariant systems have appreciably better characteristics in comparison with the classical systems of amplitude modulation under complex influence of noise.
The advantage in noise immunity of the invariant systems is explainedby thefactthatthe modulatingparameteris included in the relation of energies of informative and training signals.
However, it should be pointed out that the search for construction of such invariant systems is not stopped. The given paper is devoted to the further investigation of characteristics of the invariant system using square-law characterized detection, that is non-linear processing of signals.
Statement of Purpose. There is a channel of communication, limitedby the frequencies^ andf. Temporal dynamics of the channels with variable parameters can be conventionally divided into the intervals of stationarity. Consider the reception of the informative and training signals withinthe extractedintervals of stationarity. Withinthe abovementioned intervals the influence of multiplicative noise is describedby the constancy of the coefficientof transmission k(t) on a certain frequency. The algorithm of reception is determined by the carrying frequency, given as an average frequency of the channel, whose amplitude is modulated by the sub-carrier.
Each transmitting block will contain an informative part and a sequence of training signals Sr. The quantity of the elements of the information sequence related to the quantity of the elements of the training sequence is equal to:
2 1
Ninf :Ntr = 3:3.
Due to the changing of parameters of the communication channel to informative and training signals, the additive noise is interfering.
The Solution of the Stated Problem. Onthe receiving side the training signals are averaged and used for demodulation of the informative part of the block and for reduction of additive noise influence of the communication channel.
In figure 1 the structure of the receiving part of the invariant relative amplitude modulation is represented. Such kind of structure contains a synchronous detector (multiplier, PLL and LPF) and a special calculator.
Due to the equal influence of multiplicative noise onboth parts of each transmitted block, the algorithm of demodulation of the reception signals with the chosen way of signals processing will consist in calculation of the invariant estimation.
Since non-linear square-law characterized algorithm is used in calculation of the estimation of the invariant, the following relationship is true:
((k• INVl• S(i )+&(i ))2
INV*2 =tL!Ln-------------------------------S2 2)
L (((k $Str •S ( j #+"( mj #)2
L m=1 j=1
In the numerator of the expression (1) there is a sum of N squares of instantaneous readings of the signal of information impulse. The information signal is formedby the sub-carrier of the kind
S ( i ) = A sin ( 2nfs - At • i ), where A is amplitude; f is frequency of oscillations of the sub-carrier; At is digitization interval and is equal to the following expression:
C(i ) = k• INV• S(i )+&(i ), where k • INVl • S ( i ) is instantaneous reading of the signal of the information part of the block, coming from the channel; x(i) - additive noise readings, distributed according to the normal law; k is the coefficient of transmission of the communication channel on the interval of stationarity.
In the denominator of the expression (1) there is a sum of N squares of instantaneous readings of the signal of training impulse, formedby the sub-carrier
G(m ) = k• Str • S(i ) + " (m, j ), where "(m, j) is noise inm-realizationof the training signal, distributed according to the normal law; L is the quantity of accumulation G(m).
Without loss of generality it is supposed that St =1.If Sr ^ 1 then all the initial parameters, namely INVl and ! (root-mean-square deviation of the noise "(i), #(m, j)) can be scaled by the quantity Str.
In accordance with the restrictions introduced, formula (1) willbe as follows:
)(k• INVl• S(i )+"(i ))2 A
INV*2 & —^---------------------------------------------&-, (2)
t))(k•S(j%+#(m,j%)2 B
L m&1 j&1
where variables are described above.
Let us assume that the occasional quantities "(i) and #(m, j) are equally distributed according to the normal law with the zero mathematic expectation and dispersion !"2. Besides, it is supposed that in each block only the next occasional quantities are dependent. Then corr ("(i %,"(i-1 %)&
& corr (#(m, j ), #(m, j -1 ))& R,
where R is the coefficient of correlation.
All the other occasional quantities entering each receiving blockwillbe independent. To realize this model, itis necessary to have
IrI < 1/V2.
Let us use the known approach of estimation of pairwise transition probability, described by the formula of average probability [6]
Z 2
Z thr ,
P & P - Wt ( z ) dz + Pt - W ( z ) dz , (3)
0 Zthr
where Pt is the probability of transition ofINVt2 to INV.2 and vice versa; Pj is the probability of appearing INV^; P. is the probability of appearing INV,2.
The first integral is probability of appearing INV.2 when INVj2 is sent.
The second integral is probability of appearing INVt2
when INV2 is sent.
-----i
z2thr is the threshold value, necessary to calculate Ptr;
with the known P. and P.
1 i.
It is calculated by the best bias estimation using minimization P on z2 .
tr thr
HavingunknownPj andP.we choose Pj =P. = 0,5. From analysis (3) it is evident that to calculate Pte it is necessary to know the analytical expression W1(z) and W.(z) of the density of probability of invariant estimation.
At non-linear processing and calculation of quantities of the invariants the shift appears. This shift is stipulated by the fact that in the formula (3) the quantities W1(z) and W.(z)
are calculated for the squares of invariants. The threshold value zthi in the expression (3) is also squared. The shifted squares of the invariants in the formula (3) are marked as INV^hINV2.
On the basis of the expression (2) mathematic expectation and dispersion of instantaneous values ^4 and B are calculated. Mathematic expectation of the numerator is equal to [7]:
= )(k2 INV,2 S(i)2 +ct2 ).
Mathematic expectation of the denominator is [7]:
mB &) (k2S (i)2 +!2 ) .
i&1
Dispersion of the numerator is equal to [7]:
N
Da & 4k2 INV2!2) S2 (i)+2N!4 +
i&1
N-1
+8) k2 INV2 S (i) S (i +1)ct 2 R+4( N-1) R
i&1
Dispersion of the denominator is equal to [7]:
. N /
4k2!2) S2 (i) +
(4)
(5)
2 ! 2.
(6)
+8k2 !2 R) S ( i ) S ( i + 1 )-
+2 N !4 + 4 ( N-1 ) R2 !4
(7)
The calculation of the quotient of two accidental values is made on the basis of the formula givenbelow [7]:
1
W ( z)= -
-,
( zx-mA )2 ( x-mB )•
25!a!b
c\dx,
(8)
where ! and aB are calculated using expressions (6) and (7), mA and mB are calculated using expression (4) and (5).
It should be noted that in the formula (3) INVj is used calculating W1(z), and INV. is used calculating W.(z). The value of the pairwise probability Ptr was calculated using the method of numerical integration. The number of accumulations with averaging is equal to 40.
The data obtained are restricted by the first six pairs of comparedinvariants, whenlNVj = 1, INV. = 2; 3;4;5;6;7.
The probability of the pairwise transitionwas calculated by the values h-relations of signal/noise withthe help of the formula defined by the relation of the signal intensity to the noise intensity
,2 k2 INV2 a h &--------2—.
N ct2
Fig. 1. Structural circuit of invariant system of information transmission: PLL is a phase-lock loop; LPF is a low-pass filter
Threshold values z2te were calculated by minimization Ptr informula (3). For k = 1, R = 0,7 and INVt = 1;INV. = 2;3;4; 5; 6; 7 the calculations are z2 = 1,521;2,047;2,513;3,406;4,117; 4,595. For k = 0,7, R = 0,7 and INVt = 1;INV. = 2;3;4;5;6;7the calculationsare z2 = 1,341; 1,689;2,117;2',617;2,970;3,401.
thr
The peculiarity of any invariant system based on the principle of the invariant relative amplitude modulation is amplitude modulating signals formed by INV7 and transmitted through the channel.
The transmission of these signals is provided on the basis of classical algorithms of information processing and has low noise immunity. Only after processing of these signals in accordance with the algorithm of the quotient of the expression (2) we obtain the estimation of the invariant which is really a number, not a signal.
Curve 2 in figures 2 and 3 corresponds to the probability of error Per in classical systems, being an analogue of the probability of pairwise transition, and is calculated by the knownformulas [6].
As we can see from figures 2 and 3 the probability of pairwise transition in invariant system is determined by the quantity (10-1...10-18). At the same values signal/noise probability of inaccurate reception of a single symbol in the classical systems is inthe limits (10-1...10-5).
1 2 3 4 5 h
Fig. 2. Noise immunity of invariant system under the absence of multiplicative noise and INV1 = 1 ; INVi = 2;3;4;5;6; 7 Curve 1 is the probability of pairwise transition under weak correlation of readings of the noise and non-linear processing of signal readings Curve 2 is the probability of error of classical AM Curve 3 is the probability of pairwise transition under non-correlativity of noise readings and using extended synchronous detector
The carried out analysis shows that the invariant system of information transmission under additive noise with non-
correlated readings has high noise immunity. The probability of error of the classical algorithm with amplitude modulation is at least by two orders greater than the probability of pairwise transition in invariant system.
1 2 3 4 h
Fig. 3. The noise immunity of the invariant system under multiplicative noise and k = 0,7; INV1 = 1; INVi = 2; 3; 4; 5; 6; 7 Curve 1 is the probability of pairwise transition under weak correlativity of the readings of noise and non-linear processing of signal readings Curve 2 is the probability of error
of the classical AM Curve 3 is the probability of pairwise transition under non-correlativity of the readings of the noise and application of the extended synchronous detector
We should like to emphasize that the system with the square-law characterized non-linear processing is much simpler in realization in comparison with the invariant systems, developed by the authors earlier [1-5]. Simplification presupposes that in the developed above algorithm the extended synchronous detection is not required. Therefore, the system can be used in telecommunication systems, telecontrol systems and other systems requiring high level of immunity to noise.
Bibliography
1. Algazin, E. I. Estimation of Noise Immunity of the Invariant System of Information Processing under Noncoherent Reception / E. I. Algazin, A. P. Kovalevsky, V. B. Malinkin // Vestnik of Siberian State Aerospace University named after academician M. F. Reshetnev : Sci. Papers. Iss. 2 (19). Krasnoyarsk, 2008. P. 38-41.
2. Algazin, E. I. Comparative Analysis of the Ways of Increasing Noise Immunity of the Invariant System of Information Processing / E. I. Algazin, A. P. Kovalevsky, V. B. Malinkin // Materials of the IX Intern. Conf. “Actual Problems ofElectronic Instrument-Making” (APEIM-2008), Novosibirsk, 24-26 Sept. Novosibirsk, 2008. P. 17-19.
3. Algazin, E. I. The Noise Immunity of the Invariant Relative Amplitude Modulation / E. I. Algazin, A. P. Kovalevsky, V B. Malinkin//Materials of the IX Intern. Conf. “Actual Problems ofElectronic Instrument-Making”
(APEIM-2008), Novosibirsk, 24-26 Sept. Novosibirsk, 2008. P. 20-23.
4. Algazin, E. I. The Invariant System of Information Processing under Non-Coherent Reception and its Quantitative Characteristics/E. I. Algazin, A. P. Kovalevsky, V. B. Malinkin // Materials of the IXth Intern.Conf. “Actual Problems of Electronic Instrument-Making” (APEIM-2008). Novosibirsk, 24-26 Sept. Novosibirsk, 2008. P. 13-16.
5. The Invariant Method of Analysis of Telecommunication Systems of Information Transmitting / V. B. Malinkin, E. I. Algazin, D. N. Levin, V. N. Popantonopulo //Monograph. Krasnoyarsk, 2006.140 p.
6. Teplov, N. L. The Noise Immunity of the Systems of Discrete InformationTransmission/N. L. Teplov. M.: Svyaz, 1964. - 359p.
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© Algazin E. I., KovalevskyA. P., Malinkin V. B., 2009
E. N. Garin
SiberianFederal University, Russia, Krasnoyarsk IMPLEMENTATION ERROR OF RELATIVE MEASUREMENTS
The RNE time scale parameters impact the NSC signal-tracking system operation and the forming ofradio navigation signal parameter evaluations had been studied.
Keywords: error, measurement, frequency, reference generator, navigation spacecraft, phase.
Pseudorange measurement errors are affected by the instability of an automatic voltage control reference-frequency generator. The purpose of the analysis is to measure the impact parameters on the automatic voltage control provisional scale for low critical range of signal tracking system operation; and the formation of radionavigation signal parameter evaluations.
An automatic voltage control provisional scale is formed onthebase of reference generatorfrequency. The reference generators used, are not ideal and the reference generator frequency is unstable. These disadvantages affect the automatic voltage control provisional scale. There are two kinds of reference-frequency generator instabilities: the shortterm instability and the long-term instability.
During fulfillment of the task, concerning relative coordinates’ position measurement, the short-term instability of frequency canbe of some interest. Short-term instability of frequency means that relatively quick changes of reference generator signal frequency that took place for example, during an interval of one second. Long-term instability of frequency does not impact tracking system operation; therefore it is possible to not consider it.
The frequency variation Afk that takes place during the time of a constant interval Atk is a random quantity within the Gaussian law and zero-centered. It is suggested that a value of frequency instability 8 at a correct time interval, determined in the reference generator ratings (for example,
1 • 10-11 per 1 second), is the limit (3") value of a frequency chancevariation Afk forthistimeinterval.
If a value of the reference generator instability during any time interval At is equal to 8, it means that by means of the completionmomentineveryfollowingtimeinterval Atk acurrent frequency value of the reference generator Afk can change, relatively to the frequency value f (k -1), affectingthe random value atthe beginning of interval Atk. The limitvalue of a random quantity is calculated by using the following formula:
A/max =8- fk . (1)
Considering the fact that frequency deviation of a reference generator is much lower than the nominal value of the reference generatorfrequency, we canreplace informula (1)the frequency value f (k -1) by the nominal frequency value of a reference generator fH .
Then:
Afmax =8- /h. (2)
It is recommended to study two variants of the frequency variation model during the time interval resulting from its instability (linearand steplike).
Underthe linearfrequencyvariation, afrequencyvariation of the reference generator takes place during the whole time interval Atk linearly,beginningfromthevaluenull;bytheend of the interval it reaches the value Afk, which accidentally occurred in the given interval [1]. This variant is probably the closest to the actual processes of a reference generator. A frequency variation in the first model will occur if a frequency derivative change discontinuously at the beginning of interval Afk and remains unchanged during all of its extent. In this case afrequency derivative value canbe the following:
f'=f k At '
(3)
According to the linear model a phase variation of a reference generator signals A&k in interval Atk the value will be the following:
. At
A&k = 2"fk — = 2"
. AfkAt
(4)
2 2
A phase variation limit value of a reference generator signal A&max is:
Af At S- At
A s*. O J max o —J H
A&max = 2"------------------ = 2"-
(5)
2 2 In the case of a steplike frequency variation; a frequency variation Afk taking place discontinuously at the beginning
