CC BY
6
Circuits and Systems for Receiving, Transmitting and Signal Processing
DOI: 10.18721/JCSTCS.14304 УДК 621.37
SEMI-NATURAL MODELING FOR GNSS INTEGRITY MONITORING ALGORITHM
A.P. Rachitskaya
Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russian Federation
The paper considers the suboptimal version of GNSS integrity monitoring algorithm involving multichannel signal processing. This algorithm was examined in terms of probability-based characteristics obtained during semi-natural modeling. Such modeling assumes that multichannel snapshots are getting from real channels of multichannel GNSS receiver including antenna array when all subsequent procedures are implemented in Matlab later. Probability-based characteristics obtained in such way consequently checked with similar characteristics obtained by Matlab simulation ideal model, which ignored probable effects of signal transmission and reception in real environment. It was shown the level of similarity between characteristics of both types, and also clarified the conditions when the characteristics are close to each other, and the conditions when the difference between them is significant. The main reason of such difference was found out empirically.
Keywords: generalized maximum likelihood ratio test, probability-based characteristics, confidence interval, non-identity of channels, multichannel receiver.
Citation: Rachitskaya A.P. Semi-natural modeling for GNSS integrity monitoring algorithm. Computing, Telecommunications and Control, 2021, Vol. 14, No. 3, Pp. 43-55. DOI: 10.187-21/JCST-CS.14304
This is an open access article under the CC BY-NC 4.0 license (https://creativecommons.org/ licenses/by-nc/4.0/).
ПОЛУНАТУРНОЕ МОДЕЛИРОВАНИЕ АЛГОРИТМА КОНТРОЛЯ ЦЕЛОСТНОСТИ НАВИГАЦИОННОГО ПОЛЯ ГНСС
А.П. Рачицкая
Санкт-Петербургский политехнический университет Петра Великого,
Санкт-Петербург, Российская Федерация
Рассмотрен алгоритм контроля целостности навигационного поля (КЦНП), синтезированный в соответствии с обобщенным критерием отношения правдоподобия для прямой многоканальной обработки сигналов с элементов антенной решетки (АР). Произведена оценка эффективности алгоритма с помощью полунатурного моделирования, с использованием записи реальных навигационных сигналов с элементов АР и их обработкой в среде МаШЬ в соответствии с рассматриваемым алгоритмом КЦНП. Проведено сравнение результатов измерения вероятностных характеристик алгоритма, полученных на основе такого моделирования, с аналогичными результатами идеализированного моделирования в среде МаШЬ, когда формирование сигналов производится искусственно без учета возможных факторов (на пути распространения сигналов или при их приёме), возникающих при работе алгоритма в реальных условиях. Выявлена степень соответствия результатов обоих типов моделирования. Определены условия, когда получаемые результаты оказываются близки и когда расхождения значительны. Экспериментальным путем установлена одна из возможных причин найденных различий между характеристиками.
Ключевые слова: обобщенный критерий максимального правдоподобия, вероятностные характеристики, доверительный интервал, неидентичность каналов, многоканальный приёмник.
Ссылка при цитировании: Rachitskaya A.P. Semi-natural modeling for GNSS integrity monitoring algorithm // Computing, Telecommunications and Control. 2021. \Ы. 14. No. 3. Pp. 43—55. DOI: 10.18721/JCSTCS.14304
Статья открытого доступа, распространяемая по лицензии CC BY-NC 4.0 (https://creative-commons.org/licenses/by-nc/4.0/).
Introduction
Various faults occurring in the process of receiving and processing signals of the global navigation satellite system (GNSS) [2—5] can lead to significant errors in the navigation object (NO) devices. If the errors exceed a certain permissible level (for example, the level of normal errors caused by the thermal noise of the NO receiver), it is interpreted as a breach of the integrity of the navigation field (NF) [1, 6—9]. To detect such situations, specialists develop methods of GNSS integrity monitoring [9—16]. Each of the methods is usually optimized for specific types of faults. In addition, the effectiveness of the developed methods of GNSS integrity monitoring aimed at identifying critical failure caused by the false navigation signal sources (FNSS) in a number of cases either proves to be unacceptably low or inapplicable for all NO types [13, 15, 17—19]. Failures caused by spoofing are especially hazardous as the NO may receive signals identical to the signals of navigation satellites (NS) at the input. However, they have different values of time-frequency parameters, which results in bias errors in the results of the NO devices operation [11, 13, 20—22].
There are various approaches that consider the features of such faults [13, 14, 20, 23, 24, 26], but in the majority of cases, the task of GNSS integrity monitoring can be re reduced to decision-making on whether or not there is any spoofing involved. To improve the effectiveness of such a check, it is possible to use statistical synthesis of a decision-making algorithm based on the a priori known data on the position of GNSS satellites at the orbit. Moreover, the best effectiveness is achieved by "direct" analysis of the processes observed in the antenna array (AA) elements [27, 28].
Direct suboptimal algorithm is of special practical interest. It is obtained as a result of statistical synthesis taking into account several simplifications, in particular, an assumption of significant difference between the NO coordinates estimates and its actual position in case of FNSS impact, as well as some other conditions (see below) [28]. The feasibility of using this algorithm is due to its low computational complexity in comparison with other similar algorithms (the number of multiplications omitted can reach up to one order or more) combined with its high level of effectiveness, which is proved with modeling in the Matlab environment [27, 28]. At the same time, this modeling was conducted in ideal conditions disregarding possible factors emerging in real conditions (features of satellite and FNSS signals propagation, nonidentity of NO device channels, etc.). Therefore, the conclusions drawn on the basis of such an ideal (hereafter referred to as "conventional") modelling cannot be considered fully objective. We can make the results obtained before more accurate by means of semi-natural modeling, when we use the records of actual navigation signals from AA elements and process them in the Matlab environment according to the GNSS integrity monitoring algorithm under study.
We evaluated the effectiveness of the considered GNSS integrity monitoring algorithms [27] based on the analysis of such characteristics as:
1) the probability of a false decision that there is interference, while it is actually absent (PpA probability of false alarm);
2) the probability of a false decision that there is no interference, while it is actually present (PMD probability of missed detection).
Both probabilities are calculated, if there are GNSS signals present. In addition, according to the Neyman — Pearson criterion [25], the considered algorithms were optimized by means of minimizing the P probability at a fixed value of the PpA probability, which determines the value of decision-making threshold A0.
Direct suboptimal algorithm of GNSS integrity monitoring
Let us consider a direct suboptimal algorithm of GNSS integrity monitoring obtained on the basis of a comparison between the likelihood ratio and the threshold for the processes snapshots in the AA elements.
The likelihood functions for column-vector x (t ) = [ x1 (t), x2 (t),..., xM (t )J of xm (t) processes snapshots on each mth of M antenna elements is formed separately if one of the two possible hypothesis HQ and H is valid in case of observing a constellation from L satellites and the presence of additive white Gaussian noise (AWGN) in the AA elements. HQ corresponds to the situation when the FNSS influence is absent, while HJ takes place when there is FNSS influence present, moreover, we consider a single FNSS emitting all L radio-navigation signals from one point [11]. The parameters of the satellite radio-navigation signals (initial phases, amplitudes), as well as the NO parameters (its coordinates PNO, velocity, attitude angles a^ a2, a3) are considered unknown. We assume the FNSS parameters (coordinates PS of the FNSS itself), false coordinates P' and velocity vector v' of the NO generated by the FNSS to be unknown as well. We exclude the indicated unknown ("interfering") parameters in accordance with the generalized likelihood ratio test [25].
The test involves maximization (either analytical directly during the algorithm synthesis or numerical during the consequent algorithm running) with respect to the values of these parameters. The "initial" direct optimal algorithm obtained in this manner requires considerable computations as the majority of the maximization procedures with respect to unknown parameters can be solved numerically only in the process of running the considered algorithm. We can significantly simplify it by a conversion into a suboptimal algorithm: to reduce the computing costs, we use a justified assumption on a significant deviation of the NO coordinates from the actual position if the H hypothesis is valid [29]. We can achieve additional simplification by means of substituting the numerical maximization with respect to unknown values of P' coordinates and v' velocity vector with a numerical maximization directly with respect to time-fre-
quency parameters tJ, Aw'd of the FNSS radio-navigation signals. Here, TJ = tJ(1), tJ ,..., tJ(L) is a vector the elements of which comprise the propagation times of /th FNSS signal to the AA antenna element
AQ'^ , AQ
'(2)
q
'( L)
is a vector of Doppler shifts of the
chosen as a reference one, while Aw 'D = FNSS signals frequencies.
The "direct suboptimal algorithm" obtained this way presupposes a numerical maximization with respect to angular directions and (azimuth and incline respectively) in the FNSS [28]:
ME, N0
mAax 3 H,
Ks
Hi
>
<
H
A.
(1)
where E0 = ^C^t) I dt; N0/2 is the spectral density of the average AWGN power at the input of
(T )V 1=J J
the receiver channels (the channels are assumed to be identical); CO') (t) is a pseudorandom sequence modulating signal of the /th satellite; HS = [e;5Tl, e
;'ST2 „MM , ..., c
is the directing vector column for a pos-
1
(i )
лк0 > Jk0 > ^k0
sible direction ns in the FNSS, where S^ = (ks (ps, %))T pm; k^Vs, % )
x[0 = sin ps cos ;yk0 = cosps cos;= sin ; c is the speed of light in vacuum, l = 1 ... L, ©0 is the carrier frequency of the radio signal emitted by the Ith satellite; Pm = [ xm, ym, Xm ] are the coordi nates of the AA elements in the local coordinates system of the NO [2, 3];
v;=
V) V,(1 ) V'
v 1 ' r 2 ' M
(i )'
(2)
where V''1 ) = j* Fxm (t)C0(l) (t — Tj(l) ) e jAw° le ;Affl° *dt and integration are performed in the analysis
T )
interval the Ta value of which is defined by the duration of the pseudorandom sequence C0) (t); Fxm (t) is a complex envelope of the xm (t) snapshot; Aq'J) = ш0') — o0 is a deviation of the lth satellite radio signal frequency from the carrier receiver frequency ю0.
Semi-natural modeling
In the course of the semi-natural modeling, a GPS simulator of L1 range (1575.42 MHz) emits L signals of FNSS. An experimental simulation device of GNSS integrity monitoring includes a multichannel signal recorder, 6-element AA receiver (layout presented in Fig. 1), and a standard navigation receiver (Ublox-M8T) for the monitoring of radio navigation signals present. The multichannel signal recorder makes snapshots of the processes from the AA elements via a multichannel radio frequency (RF) receive path and digitizes the respective xm (t) processes using an analog-to-digital converter. The reference values of the Fxm (t) complex envelope of the xm (t) processes digitized at the sampling rate of 2.046 MHz are stored as .dat-files.
In the course of the semi-natural modeling, the GPS simulator emitted the FNSS signals with the power level significantly exceeding the radio navigation signals of the satellite (the interference/signal ratio was у ~ 6 dB). The signals were received and recorded in real conditions in the presence of tress, urban buildings, moving objects, etc. Fig. 2а shows the satellite group observed in the process of the experiment.
Fig. 1. Configuration of the 6-element AA used in the study
Fig. 2. View of the observed satellite constellation (a), visual representation of the detected radio navigation signals in the absence of FNSS signals (b) and in their presence (c) in the indicator of the Ublox-M8T receiver
Simultaneously with recording the signals from the AA elements, we were monitoring the readings of the standard navigation receiver connected to one of the AA elements. Fig. 2c shows an example of the results of processing legitimate GNSS satellite group signals in the interface of the standard navigation receiver Ublox-M8T in case there is no GNSS integrity failure. Each column depicted in Fig. 2b corresponds to the observed radio navigation signal the number of which (satellite number) is located in the bottom; the height of a column is proportional to the value of the C0 /N0 ratio of C0 carrier power of the radio navigation signal under consideration to the doubled spectral power density of the AWGN. Fig. 2b shows an example of indicating the results of processing the radio navigation signals by the same navigation receiver under the influence of FNSS. There is an obvious absence of any identifying attributes of the fact that in the second case the coordinates were measured according to the FNSS signals with significant mistakes. At the same time, in the presence of the FNSS the measured values of the NO coordinates considerably differ from the coordinates measured in case the FNSS influence is absent.
Thus, Table 1 shows an example of the measurements for the coordinates in both cases under study obtained in one of the experiments. The measured coordinates clearly differ by approximately 1600 m.
Table 1
Results of processing navigation signals by the Ublox-M8T receiver in the absence and in the presence of FNSS signals
Presence of FNSS signals Latitude, deg Longitude, deg Altitude, m Standard deviation of measurements caused by AWGN, m Absolute deviation from the actual NO position caused by FNSS, m
— 59.9937998 30.3795120 56.5 3...7 —
+ 59.9874327 30.3536052 2.7 4...8 1611.5
Results of the semi-natural modeling
We used Matlab algorithm model (1) for the semi-natural modeling, while the recordings of the real signals from the AA elements were incoming to the input of the model for analysis. The obtained charac-
10'
10
10"
0.4
0.45
0.5
0.55
0.6
A.
0.65
0.7
0.75
Fig. 3. Probability of false alarm at M = 3, L = 3, 4 and C0 / N0 = 45...50 dB-Hz
10'
10"J
io-2
10
10
0.4
0.45
0.5
0.55
A,
0.6
0.65
Fig. 4. Probability of false alarm at L = 3, M = 4, 5 and C0 /N0 = 45...50 dB-Hz
teristics (PFA and PMD) of algorithm (1) were compared with the similar characteristics resulting from the "conventional" modeling in the identical conditions, when C0/N0 = 40...50 dB-Hz, y = 6 dB. Fig. 3 presents dependencies of the PpA probability on the decision-making threshold A0 forM = 3, L =3 (satellites no. 5, 7 and 8 in Fig. 2).
A comparison of the dependencies of the PpA probability on the decision-making threshold obtained in the process of the semi-natural modeling with the results of the "conventional" modeling revealed a certain divergence (less than a half an order of magnitude). Similar conclusions can be also drawn while using the signals of 4 satellites (L = 4 in Fig. 3), as well as with a larger number of elements (M = 4, 5 in Fig. 4).
At the same time, when calculating the PMD probability, we registered strong influence of the AA design on the degree of proximity of the characteristics obtained by two types of modeling under study. Thus, we found little difference in the P values obtained in the course of the "conventional" and
' MD
semi-natural modeling at the small number of the AA elements (M = 3 in Fig. 5); at the larger numbers (M = 4 in Fig. 6), this difference was growing. Therefore, at M = 5 (Fig. 7), the difference in the PMD characteristics obtained by means of two different ways of modeling exceeded several orders.
10
10"J
10
10
0.97
A„
Fig. 5. Probability of missed detection at M = 3, L = 4, Cj N0 = 45...50 dBHz and y = 6 dB
Semi-natural modeling
L=3.M =4
0.95 0.955 0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995
Fig. 6. Probability of missed detection at M = 4, L = 3, Cj N0 = 45...50 dBHz and y = 6 dB
10'
10
Semi-natural modeling ,
L-3. M=5 ,[' X'
x
r
/
"Conventional" modeling /. - 3. M=5
_j_i_i_
0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98
Fig. 7. Probability of missed detection at M = 5, L = 3, Cj N0 = 45...50 dBHz and y = 6 dB
We can assume that the cause of the discovered divergence between the probability characteristics obtained by both the above mentioned types of modeling is a possible nonidentity of the channels of the receive path used for recording of the signals from the respective AA elements. Testing this assumption is of interest.
Influence of nonidentity of the receive path channels on the efficiency of the GNSS integrity monitoring system
As additional researched results [30] showed, the semi-natural modeling employed a multichannel receiver with a significant difference in the gain characteristics between the channels, so the phase difference Amm between the channels reached 2n rad and more (Table 2).
Table 2
Change in the unknown A^ phase tune-outs between the AA channels
Channel 1 Channel 2 Channel 3 Channel 4 Channel 5 Channel 6
Am , rad t m' 0 1.699130 -1.650571 0.985480 -1.128182 -0.255712
By compensating the phase tune-outs (according to Table 2) during the semi-natural modeling, we can improve the characteristics of the GNSS integrity monitoring algorithm and make them approach the respective characteristics obtained by means of the "conventional" modeling (Fig. 8—10). The improvement of the probability characteristics after compensation is especially significant at M > 3, so at M = 5 it reaches 3 or more orders. Thus, we can confirm the assumption that the main reason for the divergence in the results of the semi-natural modeling from the "conventional" modeling lies in the nonidentity of the receiver channels.
10"
ioJ
10
0.94
0.95
0.96
0.97
A„
0.9S
0.99
Fig. 8. Probability of missed detection at M = 3, L = 3, Cj N0 = 45...50 dBHz and y = 6 dB
10"
10"J
0.95 0.955 0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995
Fig. 9. Probability of missed detection at M = 4, L = 3, Cj N0 = 45...50 dBHz and y = 6 dB
10'
10
0.8 0.82 0.84 0.86 0.88 0.9
0.92
0.94 0.96
0.98
Fig. 10. Probability of missed detection at M = 5, L = 3, C0 /N0 = 45...50 dBHz and y = 6 dB
Conclusions
In the process of the conducted research, we showed that the "conventional" modeling in ideal conditions (rectilinear propagation of satellite signals, absence of the multipath propagation effect, identity of the receive path channels, etc.) allows us to evaluate the considered GNSS integrity monitoring algorithm for the simplest antenna arrays (2—3 elements) quite accurately in real receiving conditions even in case of a considerable nonidentity of the receive path channels. On the other hand, when more complex antenna arrays (4 or more elements) are involved, the results of the "conventional" modeling adequately reflect the efficiency of the GNSS integrity monitoring algorithm built only on the basis of radio receive path with a compensation of phase tune-outs between the channels.
The considered method of the semi-natural modeling obviously allows evaluating the efficiency of the GNSS integrity monitoring algorithm with various geometric properties of the GNSS groups and in different conditions of receiving signals from satellites and false sources.
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Статья поступила в редакцию 17.05.2021.
THE AUTHOR / СВЕДЕНИЯ ОБ АВТОРЕ
Rachitskaya Antonina P. Рачицкая Антонина Павловна
E-mail: antonina [email protected]
© Санкт-Петербургский политехнический университет Петра Великого, 2021
