Estimation of the Noise Immunity Characteristics of
Telecommunication Network
MEHDIYEVA ALMAZ, BAKHSHALIYEVA SEVINJ
Department of Electronics and Automation,
Azerbaijan State Oil and Industry University,
Baku, Azadliq Avenue, 20,
AZERBAIJAN
Abstract. The harmful impacts of various interference sources in the communication channel of data
transmission systems and an increasing noise immunity in linear distortion conditions are important problems
in the development of information transmission systems. To improve reliability and the data receiving
accuracy, a mathematical model is developed, which allows to assessment of the survivability of
telecommunication systems to interference at incoherent receiving. In this case is assumed that system impact
the unintentional sources. The mathematical model considers the communication quality, spectral efficiency,
coding method, and modulation principle used during the multimedia services. Presently, the development of
the corporate multiservice communication networks is in conditions where the size of user and service traffic
packets is rapidly increase. To ensure transmission reliability is promising of an implementation of the coherent
modem. It will allow a decrease in the failures of such systems. To increase the transmission rate used coherent
modem with noise-resistant characteristics. However, the communication quality in this case is decreased, due
to more influence of the inter-symbol interference to the data quality receiving. The synthesis of optimal signals
based on rectangular pulses with different shapes by the coherent modem for the data transmission system aims
is solved by using of matching filter that has a pulse transient response. To obtain this the frequency limiter is
used at the output of the channel digital demodulator. One of the ways to properly solve the problem noted
above is by applying in the development of telecommunication multiservice networks the mathematical model
with correction ability. This one can be developed based on the optimal signal-receiving theory.
Key-Words: - quality of communication, final signals, telecommunication systems, transmission systems,
communication channel, mathematical model, spectral efficiency, communication network.
Received: July 26, 2022. Revised: September 18, 2023. Accepted: November 11, 2023. Published: December 31, 2023.
1 Introduction
The quality control of the transmission and
communication in a data transmission system (DTS)
with a coherent modem (CM) in complex conditions
are used by several means. These means provide
adaptive protection from interference in real-time
mode during the document and voice transmission.
This allows us to minimize the interference sources'
impact to CM. Due to this during the provision of
the multimedia services there is cause of a delay of
the useful and service traffic flow. The service
traffic is developed to increase the operation
efficiency of the multiservice telecommunication
networks (MSTN). For the assessment of the noise
immunity characteristics of the receiving of a DTS
with a CM, it is necessary to consider the bit error
probability
BER
P
and the data transfer rate (DTR)
b
V
of the CM in critical states. Further, we assume that
interference created unintentionally is organized,
and the different sources of interference are random,
[1], [2].
2 Problem Statement
Considering the components of the noise immunity
vector
),( nmöktR
of data receiving at time t, and the
transmission rate of the CM under the intentional
interference influence can be described by the
following expression:
ö
, (1)
Where
)(tNnn
function that considers the
intentional interference sources at time t;
),( 0
NESNR b
signal-to-noise ratio. It characterizes
the complex communication quality (CQ) indicators
with the CM considering the energy
b
E
of one bit
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signal and the spectral density of power of
interference
0
N
.
Dependence (1) presents the essence of the
approach, which considers the complex indicators of
the CQ in DTS. Based on dependence (1) a
mathematical model (MM) to assess the stability of
CM to interference during the data receiving is
proposed. During the use of this receiving model, is
assumed that any
nibi1,
parameter of the signal
),( i
btu
carries information and this is a process
with random characteristics. In the proposed
approach, to effectively separate this parameter
),( i
btS
from the time interval
],0[ c
T
is required:
,
c
Tt 0
(2)
An expression (2) can be used to analyse of the
optimal realization of binary signals at the time
interval, [3] and allows to decide on
)(
0tu
or
)(
1tu
.
The considered models are the main part of the
noise-immunity problem of data receiving and
binary signals processing in DTS with a CM. Let us
consider the process of building the MM to assess
the receiving noise immunity.
3 Solution of the Problem
The MM to assess the receiving noise immunity
proposed to solve this problem includes the methods
of discrete signals receiving, correction codes had
large effectiveness, and amplitude modulation (AM)
with quadrature law in a DTS using a CM. So as the
noise immunity of receiving depends on random
characteristics of the used modem that is approved
by system analysis in, [4], [5], so quantitative
assessment of it can be the operation failure
probability of DTS. This probability is a monotonic
function of the SNR at the input of the receiver and
can be estimated based on the average erroneous
receiving probability of a discrete signal:
, (3)
To increase the receiving noise immunity of the
CM in a DTS, it is necessary to consider the main
indicators which characterize the operating quality
of the MSCN, [6], [7], [8]. As a criterion for the
quality indicator of the studied communication
network, is accepted its bandwidth, considering at
the channel level the parameters of the system noise
immunity under the intentional interference
influence. The maximum bandwidth of the CM in a
DTS is defined by the following expression:
,(4)
where - the energy to transmission of one bit of
the discrete signal and defined as follow:
]/[ mogREE kcb
,
)/( nkRk
, (5)
k
R
the rate of Reed-Solomon (RS) code; ;
- the bandwidth of the signal.
The expressions (4) and (5) is obvious
following. The main parameter which determines
the maximum bandwidth and CQ in DTS is a
complex indicator of the SNR, which is determined
as:
, (6)
The last expression characterizes the CQ of the
receiving by coherent modem and shows of the
signal power to noise level ratio in DTS. Expression
(6) is also an energy indicator of noise immunity of
receiving and describes the use of the CM in terms
of power. Considering the limit value qtr of SNR in
DTS, the preventing conditions of the degradation
of the CQ can be defined by the following
expression:
кр, (7)
where qtr - is the limit value, the critical of the SNR
value. It provides a preset quality of a DTS using a
CM:
)]},([{min nmbкр
E
tr kESNRq b
, (8)
where
kr
a modem's resource coefficient, which
considers the energy losses during the received
signal processing in a real DTS relatively on ideal
receiving (
1
kp
).
The mathematical model is developed to assess
of the noise immunity of receiving by using a CM in
a DTS is used to solve this problem, [9], [10]. A
mathematical interpretation of this can be expressed
by the following function:
ö , (9)
Under the following restrictions
),(),( .bb
bb
makbbmak VECVEC
,
bb
BERBER PP
,
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),(),( nmb
bb
nmbkESNRkESNR
, (10)
The expressions (9) and (10) are the basis of the
proposed approach, [11], [12], [13], [14]. The
mathematical model developed based on it considers
the indicators of CQ, energy effectiveness, coding,
and modulation methods to assess the noise
immunity of receiving. As well as expressions (9)
and (10) are a simple analytical description of the
function of noise immunity when assessing the
operation quality of the DTS with a CM.
Considering the formulation of the problem given
above, let us study and assess the bit error
probability during the discrete signals receiving. By
the Neumann-Pearson (NP) criterion and the model
to assess the noise immunity receiving, as the noise
immunity characteristic of the given modem is
accepted the discrete signal receiving probability at
the output of the demodulator (DM), and the total
power of an intentional interference which coming
to the input.
As research result in, [4], to assess the discrete
signal probability at the output of the DM proposed
the following expression:
t
i
i
te
iN
te
iN
Nqe PPCP
1
)1(1
, (11)
where - the data distortion probability of the
traffic packets ransmitted them by the DTS;
-
binomial coefficient; t - the correcting capacity of
the RS code, expressed considering the code
sequence length N and code rate, as follow:
)1)((5,0)1(5,0 kk RrkRNt
(12)
- the RS code rate, that determined as follows:
1)/( NkRk
.
The study aim is to use polynomial, where
)12(
min
td
is the minimum of code distance, the
number of errors in packet.
The total interference power at the input is
determined based on the bandwidth
c
F
of the
signal, interference
n
F
, and the average external
interference power expressed by spectral density, as
following way, [6]:
nncnog FNFp )(
, (13)
where
)(F
cn
matching coefficient of the
interference and signal in frequency that defined as
follow:
ccncn FFFF /)]()[()(
, (14)
Considering the (7), (8), and expression noted below
]/[]/[ mncmnbРPNE
, (15)
a bit error probability for the CM can be determined
as, [6]:
5,0
2
1
nm
mn
b
BER k
N
E
N
k
erfcP
, at М=2 (16)
where
)(xerfc
an additional error function integral,
that is defined as, [3]:
хdttxerfc )exp()/2()( 2
k and N - the noise immunity parameters of the RS
code (k - number of information bits include in the
packet). Expression (16) is used to determine as
main noise immunity indicator when applying
M_RS modulation and RS code.
Modeling of the parameters of the modems in
the DTS performed in MATLAB environment.
Simulating of transmitting and receiving of signals
used the Communications Toolbox, [7]. To
determination of the dependency, the bit error
probability and signal-noise ratio used the BERTool
graphical package. The BERTool environment is
allowed to calculate the values of
and modem's noise immunity for a given frequency
range. The graphical curve for these dependents
)],([ nmbBER kESNRFP
was built, which used to calculate of the theoretical
value of the RS-code parameters and bit rate. The
graphical curve is built by the BERTool on
analytical expressions (14-16). The following values
were obtained based on calculations:
RS code modulation performed according to the
RS algorithm: at
75,0
k
R
;
24,4,32 kdN
.
The bit error probability improved at
175,0
k
R
.
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An analysis of results of performed calculations
shows the following: an increase of
),( nmbkESNR
leads to a decreasing in the bit error probability.
This matches with the requirements on the CQ and
the noise immunity level of DTS tracts at the given
rates for bit and code. Thus, the proposed
mathematical model allows to assessment of noise
immunity of the CM receiver when impacts of the
unintentional interference. This MM considers the
energy indicators of the modems, manipulation type
at modulation, bit rate, and coding rate.
The results of research and analysis presented in
the article can be used in a mathematical model for
assessment of the noise immunity (NI) at incoherent
receiving in DTS. These research results increase
the reliability and accuracy of data receiving. To
provide the CQ of DTS under the influence of
unintentional interference, and to ensure noise
immunity of DTS with the required quality of
service (QoS) at the channel and physical levels, it
is necessary to develop and apply the new receiving
models. Regarding this, the problems of assessment
of the noise immunity parameters during the design
of DTS are urgent. Investigation results on the
optimal receiving of discrete signals and
determination of an error probability in coherent
receiving were presented in, [4], [5]. The aim of this
work is to develop a MM to assess of the noise
immunity during the incoherent receiving in DTS
that operate under the impacts of unintentional
interference.
Let now us analyze the operating quality of the
DTS, under the influence of unintentional
interference sources.
At the channel and physical level in
telecommunication system (TCS) Modularly Phase
Shift Keying (M-PSK) modulation and the
polynomial RS code
),,( dkNGF
are used. The
parameters of the RS code are [8]. The RS
codes that used in the given system are directly
cyclic codes, with an error correction set, which can
be defined as presented in, [2]. As the main
indicator to assess the quality of TCS during the
impacts of interference, the average value of an
error probability is accepted. Thus, the need of
development a MM to improve the QC by applying
effective coding and modulation algorithms, as well
as for assessment of the noise immunity, is the
actuality of the problem. So, if the noise immunity
of receiving depends on the random parameters of
the receiver, then it’s the quantitative measure can
be the malfunction probability of the system. This
probability is assessed by the average probability,
which shows of an erroneous receiving and depends
on SNR at the input of the receiver, also is a
monotonic function as:
},)],(,[(,{][ bbEbkse VMVkESNRRFPE
, (17)
where
k
R
is the Reed-Solomon coding rate,
1
k
R
.
An energy transmitting coefficient
)( bE Vk
is
determined considering the bit signal energy and the
signal-to-noise ratio as follows:
1)/()( inbbE EEVk
, (18)
where
in
E
the bit signal energy is at the input.
Considering the above and expression (18), the
mathematical statement of the given problem on the
development of MM, the objective function to
assess noise immunity of the TCS can be recorded
as follow:
)]}([min{)( 2
cse
h
nhPEArgWЕK
c
, (19)
at the following restrictions
, ,
С
, (20)
where - maximum throughput for the given
TCS during the multimedia services;
BER
P
bit
errors probability;
)( cSE F
spectral efficiency of
TCS at the incoherent receiving. which shows the
using efficiency of the bandwidth
of the binary
signal;
..bbBER
P
,
)(
.. cbbСE F
the admissible
values respectively.
The last expression (20) also (21) is the basis of
the new approach. Based on them a MM for
assessment of the NI of a TCS was developed. This
MM considers the CQ, coding methods, modulation
scheme, and spectral efficiency, [14]. At the same
time, obtained expressions allow us to analytically
represent the NI function to assess the performance
of the TCS enough simple. To implement the new
approach used in constructing the MM, a scheme for
the operation of an optical receiver with incoherent
receiving is proposed. To perform the task by the
method of random-phase incoherent receiving, it is
necessary to study the paths of the transmission and
receiving systems of discrete data. Then, by
comparison with the threshold estimation DQ, a
decision is made in favor of assumptions H0 and H1.
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The limiting level is determined by the similarity
criterion as follows:
)(
)(
)()(
0
1
0
1
0,1 up
up
SП
H
H
uП
, (21)
For signals that have the same probability
1
0П
, then the expression for the similarity
coefficient will take the following form:
)/2()/2( 000010 NSnINSnI
0
01
0
1
N
EE
H
H
, (22)
Expressions (21) and (22) determine the
structure of the optimal receiver of a binary signal
with an unknown initial phase in case of incoherent
receiving. At the output of ZS and MS, the binary
signal is essentially the energy of a discrete signal:
{П (0,1)}
c
TEdttuE 0
2
1,01,0 )(
(23)
Let's compare
)( 0
ПS
with the threshold signal at the
considered moment
c
T
. Usually, in these cases, the
value of the threshold signal is taken close to
)()2/( 0
ПSE
. Let us analyze the average values of
the probability of erroneous receiving of discrete
signals obtained as a result of MM research. Based
on similarity criteria (22) and (23), a formula is
proposed to estimate the code packet receiving
probability on the matched filter output based on the
noise immunity estimation model:
k
t
i
i
te
iN
te
iN
Nkqe PPCtNP
1
)1(1),(
, (24)
where
te
P
the distortion probability of the packet
data during transmission by the system;
-
binomial coefficient from N to N-i;
k
t
the
correction ability of the RS coding, which,
considering the code sequence length N. The code
rate is determined as (12). RS code rate and defined
as
1)/( NkRk
.
In this case, the polynomial
)7,64,127(),,( GFdkNGF
was used.
The expression (12) allows us to calculate the
probability of packet loss due to various data
distortions due to the lack of the recipient's ability to
correct the code:
),()/()( .. kqeteтрkkie tNРPNkttP
, (25)
On using the M-RSK modulation scheme, the
aggregate nominal power of the unintentional
interference at the receiver input is taken as a noise
immunity characteristic. The total interference
power
cmn
Р
at the DM input is determined by the
signal bandwidth
c
F
and interference bandwidth
n
F
, the spectral density of the average external
interference power
cп
N
, as the following expression:
шncncncmn PFNFР )(
, (26)
where
)(F
cn
the matching coefficient of the
interference and signal at a given frequency range,
is defined as:
ccncn FFFF /)]()[()(
, (27)
Considering the expressions (26) and (27), the SNR
takes a complex form:
]),(,,[)/( cncncmncсmncNFPBEPPSNR
, (28)
where
c
B
is the base of the received complex
signal and, considering the entire bit arrival time
b
T
,
is determined as follows:
cbcbc FVFTB )/1(
, (29)
Considering the obtained expressions (24)-(29) and
]/[]/[ cmnccmnb РPNE
,
for the M-RSK modulation at M = 2, a bit error
probability is determined by the following
expression:
)(
2
1
2
2
1
bE
cmn
in
k
c
BER Vk
N
E
Rerfc
h
erfcP
, (30)
where
)(1)( xerfxerfc
a residual error
function is determined as:
xdxerfc
)exp()(2)( 25,0
.
Calculations and simulating of a noise-
immunity of communication system during the
receiving incoherent receiver are performed also
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using MATLAB Toolbox. The optimal transmitting
and receiving algorithm is shown in Figure 1:
4 Conclusion
The decrease in the dynamic viscosity of oil
using the developed reagent in the temperature
range characteristic of oil transportation
processes at a shear rate corresponding to
starting loads was 55%.
The effectiveness of the ANA-10 reagent
as an inhibitor of ASF formation in comparison
with industrially produced analogs has been
experimentally proven.
The technical and economic feasibility of
using the developed reagent of complex action
when collecting borehole fluid and transporting
oil by pipeline transport is justified by a
reduction in operating costs when using it by
10...15%, considering the commercial cost of
chemicalization of the transportation process of
20 rub/m3.
Fig. 1: Optimal transmitting and receiving algorithm
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
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Creative Commons Attribution License 4.0
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