CC BY
10
УДК 519.873
Nonlinear Methods of Statistical Analysis of Dynamics of the Tracking Systems in Radio Receivers
Vladimir A. Mironov*
Military educational scientific center air force «VVA» Bolshevikov, 54a, Voronezh, 394064
Russia
Dmitriy D. Dmitriev^ Valeriy N. Tyapkin* Aleksey Yu. Ershov§
Military Engineering Institute Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041
Russia
Received 20.03.2018, received in revised form 26.03.2018, accepted 10.07.2018 The article assesses the criteria ratios (required protection ratios) of the signal / noise that ensure the functioning of the radio receiving device in conditions of a noise immunity deficit with a probability of not less than 0.9. The assessment has been performed using Markov's chain (processes) on the basis nonlinear methods for analyzing the statistical dynamics of tracking systems and the theory of optimal filtering of data processes.
Keywords: Adaptive radio receivers, noise immunity, optimal algorithm, synthesis, tracking. DOI: 10.17516/1997-1397-2018-11-5-627-633.
Introduction
Modern ground-based command and measuring systems for managing the flight of spacecraft, besides their standard task of receiving command and program data and telemetry, are often required to measure navigational parameters (pseudodistance and pseudospeed) via a received useful signal for the purpose of ephemeris provision [1]. In order to solve this task it is necessary to construct a radio receiver capable of tracking the parameters of the received signal with a high degree of accuracy. Measurable parameters for navigational support include time delay and phase. Both parameters are tracked using delay tracking systems (DTS) and phased automatic frequency adaptors (PLL). The measurement accuracy of these parameters and the possibility of tracking them strongly depend on the noise immunity of the radio receiver.
* mirvam@live.ru
t dmitriev121074@mail.ru
^ tyapkin58@mail.ru
§ alexeyworking@mail.ru © Siberian Federal University. All rights reserved
1. Estimate of required protection ratio signal / noise
In this section we shall discuss the procedure for evaluating the immunity ratio of signal / noise which enables radio receivers to operate in conditions of a deficit of interference resistance with a probability of less than 0.9. The time and probability characteristics of the failure of the signal tracking process for a radio receiver it determined using nonlinear methods for analyzing the statistical dynamics of the tracking systems; these methods are based on the Markov chain [2,3,4]. The application of these methods allows us to determine the algorithm for calculating the average time and time dispersion to the tracking failure simultaneously for both PLL and DTS considering coupling. Simultaneous signal tracking failure in DTS and PLL is understood as the first exit of the trajectory of the phase p and delay Td beyond the limits of the discrimination characteristics. According to [4,5,6] joint statistical dynamics for DTS and PLL can be viewed in the coordinates
P=P - P, Td = T - t,
where p is the phase difference, Td is the difference of the time position of the base and input signals, p,T are their values.
The characteristics of tracking failure were determined using the following differential equations:
dW .
— = —aw — Aw dt
4 cos w
sin2w +--(- ) ni(t) +
ATo (l — M)
8
+ -
4 sin w
ATo i1 — ^)
m(t) +
ddTd = —ßT—at dt
A2To2( 1 — ^ )
■ ni (t) U2 (t)
— n3 (t) ,
F(Td) +
2F (Td)
(1)
+
2r„
-ns (t) +
ATo (l — cos w
4Te
ATo cos W A2T2 (l — cos:
n2 (t) +
- n2 (t) ns (t)
w
—nr (t),
where N0 is the power spectral density of interference; A is signal amplitude; Ap is the synchronization (locking) band of the PLL system; At is the locking band of the DTS; Te, T0 is the duration of the impulse of the pseudorandom sequence and, accordingly, the duration of the data symbol.
A4To (l - ^2) A4 To cos2 p (l -Ap =-' } kn; At =-^-w k22,
2N22
2 N22T e
k • To k • To
n1 (t) = np (t) g (t — t) sin $ dt, n2 (t) = nd (t) g (t — t) cos $ dt,
(k-1) • To
(k-i) • To
k • To
ns (t) = I nd (t) dg (t,— t) cos $ dt
dr
(k-1) • To
- noise, which is a normal random process; g(t) is the pseudorandom sequence of impulses;
F(tp) is a function which characterizes the discriminate feature of the DTS; it has the following
•p view
k • To k • To / „ i \ / „ 1 \
F( . Te r dg (t - f) dt T e r ff[t-f + 2ie - ff[t-f-
F( Td)= ¥o J g (t - T)-df-= T0 J g (t - T)-^-dt
(k-1) To (k-1) To
After a piecewise linear approximation, the function F(Td) can be viewed as such
| Td + 2t e| 3 1
1--provided --Te < Td < --Te,
Td 2 2
ITd - 2Tel ITd + 1 Tel . 1 1
provided - -Te < tp < —— Te,
F (td) =
e Te 2 2
I Tp - 1tI ^ ., , 1 3
--1 provided --Te < Tp < --Te,
t 2 2
33 0 provided - 2Te > Tp > 2Te.
For equation (1) we have considered the product of the accepted oscillation £i(i) and the base signal can be shown as:
Z1 (t) g (t - T) cos (wot + jj =2A^ 1 - ~Tcos ^ + nd (t) g (t - T) cos (wot + jj ,
Z1 (t) g (t - T) sin (wot + j) =2A ^ 1 - sin ^ + nd (t) g (t - T) sin (wot + jj ,
Z1 (t) dg T) cos (lot + j) = 1 Ag (t - t) dg ^ T) cos y + nd (t) dg ^ T) cos (wot + j) .
Besides, during time 0 the variable y and Td it should be noted that the constant time for PLL and DTS in operating electronics is typically greater than the duration of the information symbol.
Note that F(Td) is wider than the dynamical characteristics of the DTS DTS Fdyn, which is described by the expression
Fdyn = 1 - — F (td), (2)
e
the limits of which correspond to point Td = ±Te.
The dispersion of random the processes n1(t), n2(t) end n3(t) is determined using the following formulas:
1 = ^ No T0, c2n2 = 4 NoTo, v2n3 = 2 NoTo , kn1n2 = kn1n3 = kn2n3 =0, (3)
where a^ ,a'22 ,a'2nz are respective to the dispersion of the random processes n1(t), n2(t), n3(t); kn1n2, kn1n3, kn2n3 are the functions of their mutual correlation.
The differential equations (1) fully describe the statistical dynamics of the tracking systems in a radio receiver for noise-like signals possessing inverse modulation with phase fluctuation and signal delay in an additive noise environment. For time intervals ti lg At, where At >> tc, tc = To is the random processes correlation time n1(t),n2(t),n3(t), equation (1) describes a bidimen-sional Markov chain (y, Td), the deviation coefficient and diffusion of which are determined by
the following formulas
M [Xj (t + At) - Xj (t)] I (XjXj) At '
b(XiXj )
M [Xj (t + At) - Xj (t)] [Xj (t + At) - Xj (t)] | (XjXj) At '
(4)
where Ai = w, Ai = rd, i = 1, 2, j = 1, 2. Using a technique for calculating the deviation coefficient and the diffusion of the Markov chain [7], and considering (3), we will get for equation (1):
i(w) = -aw - Awsin2W' a(Td) = -ßTd + Atf(Td)'
b(Te' Te)
b(W' w) = b(W' Td) No
2A2 tS
cos2w(i 1 -8A2wS
0 - $
F2(Td) + 2 1 - ^ +
2S
To cos2w
+ - NT'
+ 2 '
(1 - s)
1+
Toa - Td) J
+2N■»
(5)
4 sin w Aw AtF (t d)S - cos w
where S = — is the ratio of the spectral noise density to the square of the signal amplitude.
A2
The coefficients in equation (5) enable us to determine the probability and time characteristics for signal tracking failure in both DTS and PLL simultaneously; this is performed on the basis of the numerical solution for Pontryagin's second equation [3,4,5], which relative to k-th moment
of tk time of the first advance of the phase trajectory and delay of the work zone limits signal
n
\p\ = 2, \Td\ = Te from the starting point (p0,Td0) G {\p\ < n/2, \Td\ < Te} will become
4A2wS
(Tö)
1+
To (l ) .
1*, I d2tk
+ 1 N M+
+
A2tS
cos2 wo [1 y
dd2tk , 4 sin w o Aw At F (t po )s d2tk
F2 (t d0 ) + 2[1 -
(1 )
+
2S
To cos2 wo
dT d0
+
+ 4 Nt\ x
dtk
(1 -^y cos w o
dw odT d0
- (awo +sin2w o^--
dw
- [ßTdo - AtF (tdo )]
dt
dT
-kt
do
k-1'
(6)
where t0 = 1,k = l, 2,..., tk =0 provided \p\ = n/2, \tp0\ = Te.
According to (6), the calculation of the probability and time characteristics of the signal tracking failure incorporates the following expressions that characterize the equivalent PLL and DTS bands
fnP = Ap + a/2; fnT = ^ + (7)
2t e 2
where fnp and fnT are the equivalent PLL and DTS bands.
2
à
à
2
2
If we substitute in (5) the values Ay, At, kn,k22 and considering the fact these expressions have been determined for the lesser values y and tp, we will get
, 1 0 L V222NT if
fnT = 2ßd1 - 2 ; fn¥ =2ay1 -
2^/ 2ft2 ' 2a2 ' (8)
A4T0 _ A4T0 '
1/211 = -Nf5 k222 = - Nf2'
In order to bring the discussed optimal PLL and DTS models in line with existing tracking systems, it is desirable during calculating the probability and time characteristics of tracking failures to fix their equivalent bands at a level, which provides maximum interference immunity of the radio receiver at given instabilities of the base generator.
An admissible error due to the instability of the base generator in equal to 0, In, where n s the width of the discriminate characteristics of the tracking system. From this condition, for the given instability of the base generator Afg we shall obtain a minimal admissible value of the equivalent band
fn = Af, (9)
where fg is the operating frequency of the base generator.
Thus, when calculating the probability and time characteristics of tracking failure (6), which are used to determine the effectiveness of the impact of noise on the receiver, the following procedure is used to acquire initial data:
1) the instabilities of the base generators, used in the device, are understood;
2) the equivalent bands PLL and DTS (fgyfgT) are determined using the equation (9);
3) in equation (6), the following is accepted
Ay = fny - a)(l - T1) ; At = (2t efnT - /3t e) f 1 - cos2y).
N0
From equation (7) with the ratio —< 1, which corresponds to the operation of a radio receiver
A2
during natural noise, we find the values Nv and NT. To simplify the calculation, we can accept that a = fny, ft = fnT.
Solving elliptical differential equations (6), having substituted the numerical values of the parameters, which characterize the satellite signals, allows us to obtain the quantitative values of the average time and time dispersion to the moment of tracking failure in radio receivers of command and measuring systems in noise environments, having an intensity of N0.
Since it has not been possible to solve this equation analytically [3, 8], the solution is sought for using numerical methods and the famous tridiagonal matrix algorithm [3]. In order to enable a high quality analysis of the influence of inverse modulation and cross coupling between PLL and DTS on the effectiveness of noise influence on the receiving device, the solution of the equation (6) is performed for the following variants:
- for PLL supposing that DTS has an ideal performance, i.e.
dtk „ d2tk d?t ~d<_
for DTS supposing that PLL has an ideal performance, i.e.
Tdo = 0, -y^- = 0, -fr- = °>, j =°;
d-rdo dTd0 dyodTd0
dtk d?tk d?tk W 0 = 0, -T^ =0, k "
d<fo ' d<f2 ' d<f0drdo
- for joint performance of PLL and DTS.
When solving the equation, the values of the parameters of the radio signals are taken as: fgv = 1 500 MHz; fgT = 10 MHz; Te= 0,1 us, where fgv is the frequency of the DTS base generator; fgT is the frequency of the PLL base generator. The frequency instability of the base generator is accepted as equal to 10-9. The solution of the equation (6) was performed for two values of T0 : 10-3 s and 2 • 10-2 s. The initial tracking errors for frequency and phase were supposed as equal to errors caused by the instability of base generators, i.e. = 0, In, To = 0, 01 us.
To get the results of the value,NT and Nv, were determined for d = 10-4. The further decrease of d while finding the values of Nv and NT did not significantly affect the results if the calculation; this allows us obtain guarantied assessment during investigations in conditions of greater ambiguity for possible real values of Nv and NT.
During calculations, we have determined that the value of the average time to tracking failure corresponds to the mean square value of time deviation to tracking failure.
Conclusion
Based on the conducted analysis, we can conclude that during the evaluation of the noise immunity of a radio receiver operating in the tracking mode in noise environments, it is enough to study the tracking failure in the PLL at real initial shifts of signal frequency and phase and the ideal operating of the DTS. Let us determine the threshold value of the noise to signal ratio th or a noise interference which matches the signal spectrum. Considering that actual interfering signal can be replaced by a fictive flat noise with a correlative function,
Kth (t2 - ti) = Nos (t2 - ti),
2Pn
where N0 = , Pn is the noise level at output from the receiver.
^Jn
In result we get Kth = (Pn/Ps)th = 4Sn/Afn, where Pn/Ps is the ratio between the noise level to the signal at output.
Considering that in real receivers more than half of the signal power is lost during processing [9], we shall get
Kth = 2Sn/Afn. (10)
The results of the calculations using the formula (10) and considering the calculated above
N0
threshold values of the ratio , are given in the Tab. 1.
A2±o
Table 1. Threshold values of the noise to signal ratio for noise
/n^i Hz 6 10 25 40 50 70
Kth, dB 37 36 30 27 26 25
The realizable values of the minimal width of the equivalent band of the PLL system for radio receiving devices of command and measuring systems are estimated at 20 Hz. Consequently, the threshold ratio of noise to signal is respectively 42 dB and 32 dB.
This study has been funded by the Ministry of Education and Science of the Russian Federation (Agreement 14.577.21.0220 dated 03.10.2016; unique project identifier: RFMEFI57716X0220) .
References
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[2] Y.Ephraim, N.Merhav , Hidden Markov processes, IEEE Transactions on Information Theory, 48(2002), no. 6, 1518-1569.
[3] R.Rudnicki, M.Tyran-Kaminska, Markov processes, SpringerBriefs in Applied Sciences and Technology, Issue 9783319612935, 2017, 33-62.
[4] V.I.Tikhonov, M.A.Mironov, Markov Processes, Sov. Radio, Moscow, 1977 (in Russian).
[5] W.C.Lindsey Synchronization Systems in Communication and Control, Englewood Cliffs NJ, Prentice-Hall, 1972.
[6] J.J.M.Wang, R.D.Cideciyan, W.C.Lindsey, Phase and frequency transfer with automatic doppler and delay compensation, GLOBECOM'86, IEEE Global Telecommunications Conference: Communications Broadening Technology Horizons, Conference Record., Houston, TX, USA, 1986.
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Нелинейные методы статистического анализа динамики следящих систем радиоприемных устройств
Владимир А. Миронов
Военный учебно-научный центр ВВС Старых Большевиков, 54а, Воронеж, 394064
Россия
Дмитрий Д. Дмитриев Валерий Н. Тяпкин Алексей Ю. Ершов
Военно-инженерный институт Сибирский федеральный университет Свободный, 79, Красноярск, 660041
Россия
В статье проведена оценка критериальных отношений (требуемых защитных отношений) сигнал/помеха, обеспечивающих функционирование радиоприемного устройства в условиях дефицита помехозащищенности с вероятностью не менее 0,9. Оценка проведена с использованием теории марковских процессов на основе нелинейных методов анализа статистической динамики следящих систем и теории оптимальной фильтрации информационных процессов.
Ключевые слова: адаптивные радиоприемные устройства, помехоустойчивость, слежение, оптимальный алгоритм, синтез.
