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Visnyk N'l'UU KP1 Seriia Radiolekhnika tiadioaparat.obuduuannia, "2023, Iss. 92, pp. 23—27
UDC 621.396
Error Probability of a Multipath Communication Channel With Inaccurate Estimation of the Impulse Characteristic of Such Channel
Pochernyaev V. N.1, Syvkova N. M.1, Mahomedova M. S.2
1 National Academy of the Security Service of Ukraine, Kyiv, Ukraine 2Kyiv Professional College of Communications, Kyiv, Ukraine
E-mail: nalsivonaiMgmail. com
The possibility of error in the case of imprecise estimation of the impulse characteristic of a multipath channel is investigated in the article. The study was carried out for a multipath communication channel of discrete channel models, which corresponds to the mapping of a continuous two-pat.li channel onto a discrete channel with an impulse characteristic. Numerical results of calculations are obtained, which can be used to calculate the error probability in the cases indicated in the article, which differ in the ratio of the amplitudes of the interfering beams. Formulas for calculating probability integrals are presented in the article. The influence of the accuracy of estimating the components of the impulse characteristic vector 011 the error probability in a two-beam channel with constant parameters is studied. The results of a study of the influence of the communication channel model 011 the error probability for different models of the communication channel for 8PSK modulation are also presented. With the "deterioration" of the type of the channel impulse characteristic (an increase in the number of channel amplitude-frequency characteristic dips in the signal band and an increase in tlioir depth), the decrease in the error probability characteristic due to begins at lower estimation error values. The results of studying the error probability of 8PSK and 64QAM signals in a single-beam channel with Rayleigli fading are presented. It is determined that, the influence of errors becomes more noticeable with an increase in the signal-t.o-noise ratio in the channel and with an increase in the number of dips in the amplitude-frequency characteristic of the channel in the signal band and an increase in their depth.
Keywords: error probability: impulse characteristic: normalized standard deviation: communication channel model: multipath communication channel: one-path channel with Rayleigli fading: t.wo-pat.h channel with constant, parameters
DOI: 10.20535/RADAP. 2023.92.23-27
Introduction
The components of the sampling vector of the impulse characteristic (IC) of the multipath channel are independent random variables, and the errors of their estimates are considered to be mutually independent of each other and of the present values. This assumption is valid if the optimal filter matched with the distorted multipath channel is used at the receiver input, with subsequent decoration of original samples. Such a filter is very difficult to implement, and instead of it. in practice, a filter is used that is consistent not with the received signal, the distorted channel, but with the transmitted one. Usually a raised cosine filter or some kind of it is used [1,2].
1 Literature analysis
In the article [1]. two options for combating intersymbol interference in multipath communication channels were studied: the nse of an equalizer and time orthogonal multiplexing. The nse of equalizers and orthogonal time multiplexing is compared, and the possibility of using equalizers to create controlled intersymbol interference as a way to deal with such interference that occurs in the troposcatter communication channel is shown.
The article [2] considers the system of control, monitoring and diagnostics of a combined radio engineering system from the standpoint of the theory of complex systems. Specific examples of promising combined radio engineering systems operating via multipath communication channels with the nse of equalizers built according to the minimum standard deviation (MSD) criterion are given.
The expediency of using combined radio engineering systems has been repeatedly noted at the annual international conferences MILCOM [5 7].
An analysis of bibliographic sources showed that the closest in terms of the method of solving such problems are studies conducted in [4.8 12].
In all considered cases, the readings of the IC vector of a discrete communication channel, formed by mapping a continuous nmltipath channel onto a discrete channel, cannot be considered independent values. Analytical calculation of the dependence of characteristics on the accuracy of estimating the channel parameters in this case, which is of practical interest, is extremely difficult duo to the lack of analytical expressions.
The purpose of the article is to determine Perror in the case of inaccurate estimation of the IC of a nmltipath channel.
2 The problematic part
When considering nmltipath channels with Rician fading for arbitrary signals with incoherent reception, the error probability is determined through integrals of the form [3]. Arbitrary signals mean orthogonal, opposite, simplex, and others. The problem is that there are no general functions for Perror for arbitrary signals. Therefore, one should consider integrals of functions of the form:
F[±1 — erf « + ft)]m,
where n = 0,1, 2,..., m = 1, 2, 3,..., a = 0 and ft - continuous parameters, £ - independent variable, erf (u) = Jq exp(-t2)dt - integral of probability (a, ft and t can be both real and complex independently of each other). The differences [±1 — erf (at + ft)] the complements of er f K + ft) to
±1
the sign choice rule for the cases Re(a£) ^ as
t approaches infinity of the given trajectory on the complex planes.
3 Main part
The accuracy of estimates of the IC communication channel will be characterized by the valne of the normalized MSD of the estimate e
Ontchannel is carried based on the model of a discrete channel corresponding to the mapping of a continuous two-path channel onto a discrete channel with an IC of the form:
hn = i
K
a16n + a2ejv ^^ sin(n — k — t/T)
(1)
where n = 1,...; «^d a2 - real ray amplitudes; f - random phase shift between beams; t/T relative delay in units of symbol length T between beams; K ^ 1. The value of K characterizes the accuracy of mapping a continuous channel to a discrete. The value a is a normalization constant chosen from the condition:
E |h"|2 = 1,
n=1
where L - number of rays.
We will specifically consider two options that differ in the ratio of the amplitudes of the interfering beams: la) a1 = a2 = 1/\f2 and 1 b) a1 = 2a2 = 2/\[3. These options are among the most difficult to evaluate, for example, option la duo to the equality of the amplitudes of the interfering beams, leading to channel frequency response dips to zero.
Let's move on to the analytical part of the work.
We present the formula for indefinite integrals of the above functions for arbitrary n > 0 and its separate forms for ft = 0 for even n = 2k and odd n = 2k + 1 (m = 0, 1, 2,...):
k
1
/cn+1 f
T[±1 —erf «+ft№ = ±— J Ferf «+ft) d^ =
tn+1 n! [g ftn+1-2k
[±1—erf (a£+ft)]
n +1 (—a)n+^^o 4kk!(n+1 —2k)\
t2k[±1 — erf «M
£2k + 1 2k + 1
[±1 —erf«)] —
E
+ O
e3
k!exp(—a2£2)
(2k j!a2k+1-2i
+ o
(N 3 ) ,
r f2k+2
J t2k+1[±1 —erf(a£)]dz = ±+¿[±1 —er/«)] +
(k + 1)!
erf (at) exp(—a2t2)
(2a)2k+2
E
k=0
(j + 1)!t2j+1
(2j + 2)!a2k+1-2i
+ o
(N3 ) .
(2)
(3)
x
x
X
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In formula ( ), [n/2] is the integer part of the number n/2. These expressions are obtained using formulas for integrating the probability integral itself and its derivatives with a power function. The general formula (2) is written here (for convenience) with the addition of an integral constant equal to (n, (3)/(-a)n+1. The simpler formulas ( ) and (4) are given, taking into account the greatest practical interest in the corresponding cases.
Let us pass to integrals of the form £n [±1- erf (a£+ p)]mi for mi > 1.
These expressions are easily obtained using integrals of derivatives of statistics functions with square and cnbe of the probability integral. As a result, we arrive at cumbersome formulas, which, taking into account the physical conditions of the problem, can be represented as:
Г[±1-ег/К + P)]Ч = [±1 -erf « + x
e+1
n + 1
[n/2]
E
¡3'
>n+1-2k
(-a)n+1 ^ 4fck!(n+ 1-2fc)!
+ О
(n3) , (5)
Г C2rn+1 ( ™ ¿2k
jem[±1-erf юм=2— [±1 -erf к)]2 - (2w+1)A ^м-^ет-erf ю]х£
V2
V2 .( 2 ^ (2k)! e Xp(-2a2a ^ (2k)! ^
2m+1 J (V2a^^8k(k!)2 ^ (k!)2^
k=0 v ' v fc=1 v ' 1 = 1
(I - 1)£
21-1
(fc!)2 8k-l(2l - 1)!a2m+2-
j+O (N2) ,
/Г ¿2m+
em+1 [±1-er f(aO]2d£ = [±1 -erf (a£)]2 5 (2m +^^exp[-(a2e)][±1-erfK)] x £
(2 m + 1)!
(m + 1)!
2 m + 2 (m + 1)!(2 a)2m+2_
- Щп-2к '
4m-fc (2 к + 1)!a2m+1-2fc
fc=Q
+ o(Nt) .
(6)
(7)
The results obtained in the form of formulas (l)-(7) The error probability curves are shown in Fig. 1 can be used to calculate Perror in the indicated cases, for an inaccurate estimate of the component of the Numerical calculation results are shown in Figures 1, 2 vector IC. and 3.
a) 8 PSK
> Eb/No=10dB ,Eb/No=16dB >Eb/No=
b) 64 QAM
Fig. 1. Influence of the accuracy of estimating the components of the vector IC on Perror in a two-beam channel
with constant parameters (model lb)
П!
a
Curves Perror for different communication channel models for 8PSK modulation are shown in Fig. 2. It can be seen that with the "deterioration" of the type of channel IC (an increase in the number of channel frequency response dips in the signal band and an increase in their depth), the decrease in the characteristic Perror due to estimation errors begins at lower error values.
Figure shows the curves Perror of 8PSK and G4QAM signals in a single-path channel with Rayloigh fading. As can be seen from Fig. 3. there is a certain zone of estimation errors, within which their influence on the characteristics of immunity to interference is insignificant, and when leaving it. a sharp increase in the value of Perror is observed. Moreover, the errors in estimating e 1 in the larger direction are more critical.
p
1 error
LOOE+OO
1.00E-01
1.00E-02 l,OOE-<)3
1.00E-04 1.00E-05
-30
-20
-L0 0 10
/ M
1 /
2
1 la model
2 =— lb model
Fig. 2. The impact of the communication channel model on Pe,
Fig. 3. Influence of the estimation accuracy of the vector component IC on Perror in a single-beam channel with
Rayloigh fading
Conclusion
Dnal-nse communication systems, which include troposcatter systems, operate over a nmltipath communication channel. Such a troposcatter communication channel is characterized by signal fading and intersymbol interference. Note that the impact of errors becomes more noticeable when: a) increasing the
signal-to-noise ratio in the channel: b) ''deterioration" of the type of channel IC (an increase in the number of dips in the frequency response of the channel in the signal band and an increase in their depth).
Modems in combinations of troposcatter-radiorolay communication systems use various typos of PSK and QAM. The requirements for the troposcatter components of combined systems in terms of error
Ймов1ршсть помилки багатопроменевого каналу зв'язку при неточному оцшюванш ¡миульсно! характеристики.
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probability are very high, at the level Perror 10-5 degree. Therefore, the graphs in Fig. 1,2,3 have practical applications. Since the requirements for troposcatter communication systems in terms of error probability increase, theoretically obtaining the results (formulas 1-7) will allow determining the error probability for more complex modulation types than those indicated in the article.
References
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[12] Zhang W.. Zhang Z.lia .1., Qi L. (2016). STC-CFDM systems with Walsh-Hadamard transform. IEEE International Conference on Electronic Information and Communication Technology (1CE1CT), pp. 162-165. DOl: 10.1109/1CE1CT.2016.7879674.
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Bastos L.. Wietgrefe H. (2012). Tactical troposcatter applications in challenging climate zones. Military communications conference (M1LCOM), p. 1-6. DOl:10.1109/MlLCOM.2012.6415601.
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Ймов1ршсть помилки багатопроменевого каналу зв'язку при неточному оцшюванш 1мпульсно1 характеристики такого каналу
Почернясв В. М., Сивкова Н. М., Магомедова М. С.
У статт! досл!джуеться ймотмршеть помилки у ви-падку неточного оцшюваппя !мпульспо1 характеристики багатопроменевого каналу. Досл1джеппя проведено для багатопроменевого каналу зв'язку па баз! модел! дискретного каналу, що в!дпов!дае в1дображепшо безпе-рервпого двопромепевого каналу па дискретпий капал з !мпульспою характеристикою. Отримаш результати роз-рахушав. як! можпа використовувати для розрахупку ймов1рпост! помилки у зазпачепих в робот! випадках. що в1др!зпяються ствв1дпошеш1ям амшнтуд штерферу-ючих промешв. В робот! паведеш формули для розра-хупк!в штеграл!в ймов!рпост! та проведено досл1джеппя впливу точпост! оцшкп компопепт!в вектора 1мпульспо1 характеристики па ймов!ршсть помилки у двопромепе-вому капал! з постшпими параметрами. Також наведено результати досл!джеипя вплпву модел! каналу зв'язку па ймов!ршсть помилки при р!зпих моделях каналу зв'язку для модуляци 8РвК та величин! в!дпошеппя сигпал/шум 8 дБ. При «попршепп!» виду !мпульспо1 характеристики каналу (зб!лынепш шлькост! провал!в ампл!тудпо-частотпо1 харакетрнстпкн каналу у смуз! сигналу та зростапш 1х глибппн) зпижеппя характеристики ймов!риост1 помилки почипаеться при мепших зпачеппях оц!пки помилок. Надаш результати досл!-джеппя ймов1рпост! помилки сигпал1в 8РвК та 64С^АМ в одпопромепевому капал! з релеевськими завмирап-пями. Визпачепо. що вплив помилок стае пом!тшшпм при збглынепш в!дпоше1Ц1я сигпал/шум у капал! та при збглынепп! шлькост! провал!в ампл!тудно-частотпо1 характеристики каналу у смуз! сигналу та зростапш 1х глибгши.
Клюноог слова: ймов!ршсть помилки: !мпульспа характеристика: штеграл: пормовапе середпьоквадрати-чпе в!дхилеппя: модель каналу зв'язку: багатопромепе-вий капал зв'язку: одпопромепевий капал з релеевськими завмираппями: двопромепевий капал з постшпими параметрами
