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ESTIMATION THE NOISE IMMUNITY OF CONTINUOUS PHASE MODULATION SIGNALS WITH FULL RESPONSE BASED
SIMULATION MODEL
Andrey I. Azarov,
Military Academy of the Strategic Missile Forces named after Peter the Great, Balashikha, Russia, azarofff1985@yandex.ru
Evgenyi V. Budarin,
West Department of Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation, Russian Academy of Sciences, Kaliningrad, Russia, zhenyoker@yandex.ru
DOI: 10.36724/2072-8735-2021-15-1-52-56
Manuscript received 14 September 2020; Revised 28 October 2020; Accepted 22 December 2020
Keywords: noise immunity, continuous phase modulation signals with full response, Euclidean distances, modulation index, bit error probability
The problem of spectral load of communication channels is now becoming urgent, which determines the efforts of the world's leading tech communication companies to search for perspective signals. These signals must ensure compliance with international standards governing the use of frequency bands and the permissible level of out-of-band emissions and the required noise immunity of information transmission. The article deals with the noise immunity of continuous phase modulation signals research. A preliminary estimation is carried out by calculating the Euclidean distances between signals. The estimates of the noise immunity were obtained experimentally. The results of simulation are presented for fixed value of the signal to noise ratio for signals. Modulation indices and phase pulse were taken into account. The results of the study make it possible to emphasize the modulation indices for CPM FR signals, having more noise immunity then the known BPSK and MSK signals. Increasing the noise immunity becomes possible due to intersymbol phase communication. These signals are received according to the Viterbi algorithm. The results of the study show the values of the modulation indices to be of practical interest.
Information about authors:
Andrey I. Azarov, Cand. Sci. (Engineering) researcher Military Academy of the Strategic Missile Forces named after Peter the Great, Balashikha,
Evgenyi V. Budarin, engineer of the West Department of Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation, Russian Academy of Sciences, Kaliningrad, Russia
Для цитирования:
Азаров А.И., Бударин Е.В. Оценка помехоустойчивости частотно-модулированных сигналов с непрерывной фазой и полным откликом на основе имитационного моделирования // T-Comm: Телекоммуникации и транспорт. 2021. Том 15. №1. С. 52-56.
For citation:
Azarov A.I., Budarin E.V. (2021) Estimation the noise immunity of continuous phase modulation signals with full response based simulation model. T-Comm, vol. 15, no.1, pр. 52-56. (in Russian)
Introduction
In modem digital communication systems (DCS) noise immunity is determined by the BEP (bit error probability) value. In general the BEP value depends on the bit energy to the noise power spectral density at the input of the receiverEb/N0, the parameters of communication channel, the modulation method, the number of levels use in a particular modulation method, the signal reception method.
The most attractive to research are CPM (continuous phase modulation) signals. Due to the constancy of their envelope, a higher rate of decrease in power spectral density is provided compared to BPSK (binary phase shift modulation). Moreover, the use of CPM signals when choosing certain parameters and an effective algorithm allows achieving better noise immunity than using BPSK signals. The increase of the noise immunity in this case is explained by the fact that many different symbols differ not only in frequency, but also in the initial phase. Hence, adjacent symbols CPM signals have an intersymbol phase relationship (IPR), since the initial phase of any symbol depends on several previous information symbols [2,5,6].
This study focuses on CPM FR (full response) signals. In this case one symbol of the information sequence is transmitted during the symbol time. However, CPM PR (partial response) signals can be used to conserve frequency resources and improve power performance. In this case, the information transfer rate remains the same and the symbol time is increased by a factor L, where L is an ineteger [1,7,10].
This article is devoted to the estimation of the noise immunity of binary CPM signals and to the study of the influence of the parameters of these signals on the noise immunity at fixed values of Eb/N0.
Potential noise immunity analysis of the CPM signals
In general CPM signal can be written:
s(t) = l—cos (wt + ^(t: /„))
Al i.ç
0 <t<NT,
(1)
where E is a bit energy;
^(t: /„) - function is used to describe the phase trajectory over a given time interval [7,9,10,12]:
(t: /„) = 9n + 2nhq(t - nT) 0<t< NT, (2)
In - data symbols, each taking one of the values ±1; dn determines the depth of IPR depending on the modulation index h used in the DCS:
dn = nh 4 (3)
d\ 2.43
2
1.5 1
0.5 0
h - modulation index. This parameter determines the spectral properties and energy efficiency of CPM signals.
In modern DCS MSK (minimum shift keying) is widely used. It is a CPM signal with modulation index 0.5.
q(t) - phase pulse (PP) [2,5,6]. It defines the law of phase change and the level of out-of-band emissions:
9(0 = Jo a(t)dt 0 <t<T,
(4)
Formulas (5) determine the PP for a rectangular baseband pulse (\REC) and a pulse with non-linear phase law of the «raised cosine» class (1RC). This class is used to reduce the level of out-of-band emissions:
0, (t < 0),
Rl REc(t) —
2^.(0 <t<T)
0, (t < 0), ±(l-cosf),(0<t<T)
l(t>T) .
(5)
In according to the geometric interpretation of signal theory the potential noise immunity is determined by the minimum d2min between the ends of two signal vectors (two phase trajectories). For the binary case d2min is defined:
dfnin = minij ± J™7 1 - cos (cp ((t: /;) - ^(t: /,))) dt, (6)
where N - the number of symbol intervals involved in deciding on the value of the first symbol of the transmitted information sequence. The values obtained by the formula (6) are normalized to the 2E value [2,6,7]
Of particular interest is the upper bound d2B of the Euclidean distances. The upper bound determines the maximum for a given modulation index energy difference between the two signals before the first merging (d2min=0) of the phase trajectories [2,6,7]. It is used to determine the potential noise immunity of CPM signals and is a critical value. The upper bound value for 1REC for corresponding h is defined by (7):
J2fu\ _ 0/-1
dl(h) = 2(1
2 nh
(7)
When calculating the minimum d2min, all values exceeding the upper bound d2B are not taken into account in the future. The algorithm for calculating the dependence of the minimum Euclidean distances on the modulation index is considered in detail in [2,4].
Figure 1 is a good illustration of d2min versus h for a given N (bit decision intervals) and phase pulse 1REC, 1RC.
d1 (N=1) 1REC
d2 (N=2)
d3 (N=3) J
d4 (N=5)
de (N=16)
• MSK
• BPSK
• CPM (A-0.715)
• CPM
0.2
0.5 0.715
1
1.2
Fig. 1. Euclidean distances versus modulation index for binary CPM FR
Analysis of d2min allows selecting the values of h, the use of which makes it possible to increase the noise immunity of CPM signals compared to BPSK and MSK. It also makes it possible to emphasize the values of the modulation indices h, the use of which is impractical. However, BEP estimates are needed to decide whether a specific modulation index in DCS is appropriate. BEP estimates determine the DCS immunity at fixed values of Eb/N0.
Experimental estimation of CPM signals noise immunity. Study of the influence of modulation index and phase pulse shape on the BEP value.
Determination of the dependence of BEP on Eb/N0 for CPM signals is closely connected with finding the dimension of the signal space, determining the spatial packing of signals vectors in it. Then it's necessary to divide the signal space into solution regions and to enumerate all variants of errors. It's extremely difficult to obtain an analytical solution to this problem. For this reason, the decision was made to abandon the exact calculation of the BEP. This difficulty is overcome in practice by calculating the upper limit of this probability.
In [2,6] it is proposed to estimate the probability of error using the combined upper limit:
' \ 2 x t,N'PIS,N
BEP <
iMfl-^-
Euclidean
(8)
information
J2N0
D(Pu,N',Pp:N) - Euclidean distance between sequences P1 and P2 of length N with a different first symbol; ®(x) - tabulated Crump function:
= = i + (9) However, the combined upper limit turns out to be too overestimated and can undeservedly exclude CPM signals with a sufficiency high but difficult to calculate noise immunity out of consideration. Thus, the simplified method for evaluating the noise immunity isn't suitable for analyzing the noise immunity of CPM signals.
Therefore, a more accurate determination of the dependence of BEP on Eb/N0 for CPM signals is possible through simulation only. For this, the authors developed a simulation model of a radio channel with CPM using the Matlab 2016b trial package [3]. Decoding of the received sequences is carried out according to the Viterbi algorithm, which allows taking into account the IPR.
The number of symbols of the transmitted sequence is chosen based on the authenticity requirements (confidence level) of obtaining estimates of the noise immunity of DCS with CPM. According to the theory of mathematical statistics, the value V of the number of transmitted symbols is a function of the error probability per bit and confidence level.
BER (bit error rate) is taken as an estimate of the BEP. BER is the rate m/V of the occurrence of an event consisting in the erroneous reception of m symbols from the total number of V received symbols [8, 11].
The following to the random reasons, the BER estimate may generally differ from BEP. Values are introduced: s - the accuracy of the estimate h a - the authenticity of estimate:
\BEP-BER\<t h a = P{BEP-BER\< e} (10)
In this case, the number V of transmitted symbols required to obtain an estimate with accuracy e and authenticity a is given by: v=t2BBRa-BBV (ii)
ta Student's coefficient. With a confidence level of 0.95 and a sufficiently large value of the transmitted symbols V the Student's coefficient will be 0.95.
Table 1 shows the data that make it possible to determine the minimum number of symbols V f or various values of the BEP.
Table 1
The number V of symbols needed to estimate the BEP
Pb 8=0.05* Pb 8=0.1* Pb 8=0.15* Pb 8=0.2* Pb
10-1 1.38*104 3.46*103 1.54*103 8.65*102
10-2 1.52*105 3.8*104 1.69*104 9.51 *103
10"3 1.54*106 3.84*105 1.71 * 105 9.59*104
10-4 1.54*10' 3.84*106 1.71 * 106 9.6*105
10"5 1.54*108 3.84*10' 1.71*10' 9.6*106
The estimates of the BEP (BER) obtained in the course of simulation modeling for CPM FR signals with different modulation indices h at Eb/N0=8A dB are shown at figure 2.
The reliability of the results is ensured by the coincidence of the values of Eb/N0 of the known radio signals obtained during the simulation with their a priori known values. So, for BPSK and MSK signals having the same BEP value at a fixed value of Eb/N0, to ensure BEP = 10-4 the value Eb/N0 = 8.4 [7,10].
Fig. 2. Dependence of the BER of CPM FR signals on the modulation index h at a fixed value of Eb/N0=8.4 dB
The adequacy of the results obtained is confirmed by the fact that results of simulation and theoretical data correspond to each other. For a 1REC phase pulse, the maximum d2min, and, consequently, the maximum noise immunity is achieved for the modulation index h=0.715 at N=3 and reaches the value d2min=2.43. This result is the same as those obtained in the studies [2, 5]. Figure 2 shows that for h=0.715 there is a minimum value of the BER. In the range of h values from 0.5 to 0.715 an increase in the values of Euclidean distance is observed (see Figure 1). Figure 2 illustrates that for the specified range of values of h there is a decrease in the values of the BER. A further increase of h leads to a decrease in the values d2min, and an increase in the BER to the value of modulation index h=1. For this h the maximum value of the BER is observed in the range of modulation indices from 0.5 to 1. This fact is explained by the fact that when h=1 there is no IPR in the CPM signals [2,6,7].
Similarly, for the 1RC phase pulse in the modulation index range from 0.5 to 0.6, it can be seen that the corresponding Euclidean distances d2B increase, but starting from 0.6 and up to 1, a rather sharp decrease in the values of the Euclidean distances is observed. In the specified range, there are significant differences between the 1RC from the 1REC, which demonstrates an increase in d2e up to the value h=0.715. Further fall of d2B to h=1 is clearly less pronounced than 1RC phase pulse. When comparing the above dependence, it can be seen that both CPM FR signals have the same minimum Euclidean distances at the modulation index h=0.5 (MSK). With an increase in the value of h, the largest Euclidean distance has a signal with 1REC phase pulse.
Table 2 shows the values of the upper bound of the Euclidean distances d2B for CPM FR signals of which use is of practical interest, the values of Eb/N0 at which the BEP = 10-4 is provided, and error reduction factor k in comparison with the used BPSK, MSK.
Table2
Comparing CPM FR to BPSK, MSK signals
As can be seen from table 2 the use of CPM FR signals with the indicated modulation indices h can significantly increase the noise immunity of DCS in comparison with the used BPSK and MSK signals.
Estimation of the noise immunity CPM FR signals
It is proposed to pre-select signals for DCS using the d2min between signals, the final selection of signals using the BEP.
Estimation includes 5 points:
1. De nition of input data: choice of the value of Eb/N0 for which is necessary to obtain an estimate of the noise immunity, selection of phase pulse shape, bit rate, selection modulation index h range, the values of confidence level and authenticity during simulation.
2. D ermination of the potential noise immunity of CPM FR signals by calculating the Euclidean distances according to (6) within the specified range of the modulation index h.
3. hoice of the values of the modulation indices h with greater Euclidean distances than those of the used BPSK and MSK signals-.
4. mulation modeling for given input data and modulation index values defined in item 3.
5. nalysis of the results obtained. Checking the correspondence of theoretical data to the research results. Taking into account the complexity of the hardware and software implementation of signal shaping devices, the issue of the expediency of using one or another modulation index h of the phase pulse shape in the designed DCS is being resolved.
Conclusion
For a preliminary estimation of the noise immunuty of CPM FR signals using the geometric interpretation of the signal theory, an analysis of the potential noise immunity was carried out. Analytical dependences on the modulation indices h are obtained for 2 types of radio signals of the studied class. Analytically the presence of a number of indices by modulation has been established, the use of which in the designed DCS makes it possible to achieve more nise immunity than when using the known BPSK and MSK signals.
An estimate of BEP for a given class of signals was obtained using a simulation model. It was found that the results of simulation and theoretical data are consistent with each other. Estimates of the BEP are obtained for a CPM FR signal with different modulation indices h at a fixed value Eb/N0=8.4 dB. Better noise immunity in comparison with non-linear laws of phase change 1RC has 1REC phase pulse.
Thus, the results of the study make it possible to emphasize the modulation indices h for CPM FR signals, having more noise immunity then the known BPSK and MSK signals. In addition, the results become possible to estimate the BEP for CPM FR signals at fixed values of Eb/N0 and a selected phase pulse shape with a given accuracy.
References
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Modulation Phase The upper Eb/N0, Error reduction
index, h pulse, q(t) bound, (dB) factor, k
0.66 1REC 2.40 7.56 4.11
0.715 1REC 2.43 7.51 4.85
0.55 1RC 2.09 8.09 1.25
0.59 1RC 2.11 8.05 1.55
ОЦЕНКА ПОМЕХОУСТОЙЧИВОСТИ ЧАСТОТНО-МОДУЛИРОВАННЫХ СИГНАЛОВ С НЕПРЕРЫВНОЙ ФАЗОЙ И ПОЛНЫМ ОТКЛИКОМ НА ОСНОВЕ ИМИТАЦИОННОГО МОДЕЛИРОВАНИЯ
Азаров Андрей Игоревич, Военная академия РВСН имени Петра Великого, г. Балашиха, Россия, azarofff1985@yandex.ru Бударин Евгений Валерьевич, НИИ Калининградского филиала Института земного магнетизма, ионосферы и распространения радиоволн имени Н.В. Пушкова, г. Калининград, Россия, zhenyoker231@yandex.ru
Аннотация
Статья посвящена исследованию помехоустойчивости частотно-модулированных сигналов с непрерывной фазой и полным откликом. Предварительная оценка помехоустойчивости проведена путем расчета минимальных евклидовых расстояний между сигналами. Оценки помехоустойчивости исследуемых сигналов получены экспериментально. Представлены результаты имитационного моделирования при фиксированном значении отношения энергии бита к спектральной плотности мощности шума для исследуемых сигналов с заданными индексами модуляции и формой фазового импульса. Результаты исследования позволяют выделить сигналы, обладающие помехоустойчивостью лучшей, чем используемые в настоящее время сигналы с BPSK и MSK, а также получить для них оценки вероятности ошибки на бит с заданной точностью. Повышение помехоустойчивости становится возможным за счет межсимвольной фазовой связи, возникающей вследствие отсутствия разрывов фазы несущего колебания на границах символьных интервалов, и ее учета при декодировании принимаемой последовательности с использованием алгоритма Витерби.
Ключевые слова: помехоустойчивость, частотно-модулированные сигналы с непрерывной фазой и полным откликом, евклидово расстояние, индекс модуляции, вероятность ошибки на бит.
Литература
1. Aulin T., Sundberg W. Continuous phase modulation - Part II: Partial response signaling. IEEE. 1981, vol. com - 29, №3, p.210-225.
2. Aulin T., Sundberg W. Continuous phase modulation - Part I: Full response signaling. IEEE. 1981 , vol. com - 29, №3, p.196-209.
3. Азаров А.И. Программа имитационного моделирования радиоканала с непрерывной частотной модуляцией. Свидетельство о государственной регистрации программ для ЭВМ №2019617763 / Дата государственной регистрации в Реестре программ 20 июня 2019 г.
4. Азаров А.И., Аношкин А.В. Потенциальная помехоустойчивость частотно-модулированных сигналов с непрерывной фазой // Известия ТулГУ. № 7, 2018. C. 420-427.
5. Detwiler T. Continuous phase modulation for high speed fiber-optic links. - A thesis presented to the academy faculty. Georgia Institute of Technology. 2011. 177 p.
6. Емельянов П.Б., Парамонов А.А. Дискретные сигналы с непрерывной фазой. М.: Зарубежная радиотехника. № 12. 1990. С. 17-34.
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9. Semenov V.Y. Demodulation method for continuous phase modulation signals based on least-squares method // Radioelectronics and Communication Systems. No. 4. 2018. P. 153-156.
10. Скляр Бер. Цифровая связь. Теоретические основы и практическое применение, 2-е изд. М.: И. Д. Вильямс, 2016. 1104 с.
11. Вентцель Е.С. Теория вероятностей: учеб. для вузов. 6-е изд. М.: Высш. шк., 1999. 576 с.
12. You Zhou, Rongke Liu, Ruifeng Duan, Bofeng Jiang. A low-complexity noncoherent Maximum Likelihood Sequence Detection scheme for continuous phase modulation in aeronauticl telemetry // International Journal of Aerospace Engineering. №1. 2015. P. 1-16.
Информация об авторах:
Азаров Андрей Игоревич, научный сотрудник, к.т.н., Военная академия РВСН имени Петра Великого, г. Балашиха, Россия
Бударин Евгений Валерьевич, инженер, НИИ Калининградского филиала Института земного магнетизма, ионосферы и распространения радиоволн имени Н.В. Пушкова, г. Калининград, Россия
