=Paper= {{Paper |id=Vol-2577/paper6 |storemode=property |title=Protection of data transmission systems from the influence of intersymbol interference of signals |pdfUrl=https://ceur-ws.org/Vol-2577/paper6.pdf |volume=Vol-2577 |authors=Volodymyr Yartsev,Dmitry Gololobov |dblpUrl=https://dblp.org/rec/conf/its2/YartsevG19 }} ==Protection of data transmission systems from the influence of intersymbol interference of signals== https://ceur-ws.org/Vol-2577/paper6.pdf

Protection of data transmission systems from the
influence of intersymbol interference of signals

© Volodymyr Yartsev 1 and © Dmitry Gololobov 2
1 State University of Telecommunications, Kyiv, Ukraine
2 Taras Shevchenko National University of Kyiv, Ukraine

gololobov.dma@meta.ua, jvp57@ukr.net

Abstract. The problems of creating a regenerator of an optical system for trans-
mitting discrete messages with the aim of protecting against the influence of in-
tersymbol interference of signals using the methods of statistical theory for a
mixture of probability distributions are considered. A demodulator adaptation
model that calculates the weighted sum of 4 samples of the input signal and com-
pares the resulting sum with a threshold value has been created and studied. The
main operations for adaptation of the regenerator during demodulation of signals
distorted by intersymbol interference are assigned under certain transmission
conditions, which are formulated as the task of evaluating some parameters of
the probability distribution of the mixture. A block diagram of a regenerating
optical signal in which an algorithm for digital processing of a mixture is imple-
mented is proposed. The analysis is carried out and the requirements for the pa-
rameters of the analog-to-digital converter used in the regenerator are deter-
mined. The results of modeling digital processing of a discrete test message to
verify the process of adaptation of the device to the influence of intersymbol dis-
tortions of signals are presented. The block diagram of a digital processing device
is substantiated, in which high-speed nodes that perform real-time processing are
combined with a microprocessor, which implements an algorithm for adapting
the regenerator to specific message transmission conditions.

Keywords: mixture of probability distributions, fiber optic communication
lines, intersymbol interference of signals, demodulator adaptation, optical signal
regenerator.

1 Using statistical theory for mixtures of probability
distributions to eliminate intersymbol interference of signals

1.1 Basic concepts of mixtures
Protection of information systems for transmitting discrete signals from the effects of
intersymbol interference (ISI) remains one of the pressing communication problems.
ISI causes distortion of the signal corresponding to the i-th transmitted symbol, by sig-
nals corresponding to the i-1, i + 1, ... symbol. This makes it difficult for the demodu-

Copyright © 2019 for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0).
60

lator to make the right decision about the meaning of each transmitted symbol. To elim-
inate the influence of ISI, analog and digital signal correctors are used. Analog linear
distortion correctors that compensate for the imperfection of the frequency characteris-
tics of the communication channel are used to neutralize ISI. Operations with discrete
samples of the received signal are performed by a demodulator, which actually per-
forms the correction functions in the time domain.
The transmission of discrete information is often accompanied by ISI, which may be
caused by imperfection of the frequency characteristics of the channel or is artificial.
In any case, the presence of ISI makes signal demodulation difficult and forces the use
of rather complex processing algorithms. [1].
The principles of constructing correctors and demodulators to neutralize ISI do not
take into account the stochastic nature of the signal, which is caused by additional signal
distortions due to random noise and the pseudorandom nature of the sequence of trans-
mitted symbols. The concept of ISI neutralizing using correctors loses its meaning in
situations where interference is artificially created using signals with a partial response.
Solving the problems of ISI is possible while simultaneously performing the tasks of
correction and demodulation with minimization of errors. This is possible when using
a protection method based on the probabilistic interpretation of ISI phenomena and us-
ing the concepts and methods of statistical theory for mixtures [2].
Let a parametric family F = {f (x; A): A ∈A} of probability distributions of random
variables be given. The elements f (x; A) of this family are described by an identical
known analytical dependence from x (value of a random variable) and differ only by
values of some parameters A, which belong to a set A of admissible values. In the gen-
eral case, the probability distribution fP(x) of an infinite mixture from the family F is
defined by the expression:
f P ( x) =  f ( x; A)dP( A),
A (1)

where P(A) is a distribution function of the mixture. If the entire mass of the distribution
P(A) is concentrated in a finite number Q of points A1,...,AQ, which correspond to the
probabilities p1,...,pQ, then expression (1) will be transformed into the probability dis-
tribution of the final mixture Q
f P ( x; p1 ,..., pQ , A1 ,..., AQ ) =  p j f ( x; A j ).
j =1
(2)

This expression statistically describes a mixed sample XP=X1∪...∪XQ with volume
N = N1 +... + NQ, in which elements of partial samples X1,...,XQ are mixed in an arbitrary
sequence. Each partial sample Xj consists of Nj random variables that obey the distri-
bution f(x;Aj) and mix in the ratio pj=Nj/N [3].
In the general case, the statistical theory for a mixture of probability distributions
allows us to solve the following problems by processing a specific XP sample:
1) to determine the number Q of partial samples from which the mixture was
formed;
61

2) to find estimates of the probabilities p1,...,pQ and the parameters A1,...,AQ of the
partial distributions;
3) to decide on the membership of each element xi (i = 1,...,N) of the XP mixed
sample (that is, what partial sample of X1,...,XQ does it belong to).
The statistical theory for mixtures of distributions determines a set of conditions, the
implementation of which provides the possibility of solving problems 1 and 2 (the task
3 reduces to a typical problem of statistical solutions). The main of them are the condi-
tions of recognition, consisting in the requirement of linear independence of the ele-
ments of the family F [1]. Many families of probability distributions satisfy these con-
ditions, in particular, normal and uniform distributions [2,4].
It is important to note two circumstances. Firstly, the property of identifiability is
caused not by the stochastic nature of the function f (x; A), but by the peculiarity of the
dependence on the arguments x and A. Secondly, both occurrences – intersymbol inter-
ference and mixing samples of random variables – are the result of linear transfor-
mations of partial actions. These circumstances justify the formal identification of phys-
ical signals with probability distributions of random variables.

1.2 An example of setting the signal demodulation problem
Consider a communication channel with amplitude modulation, through which a se-
quence of symbols "1", "0" is transmitted at a speed of 1/T bit/sec (where T is the dura-
tion of the bit interval). Let a impulse g(t,θ) of a Gaussian form be formed for"1", where
the function g(t,θ) of the Gaussian form is formed, where the function g(t,θ) is identical
to the analytical description of the normal probability distribution with parameters m
=θT, σ = 0.425T. Moreover, the impulse width at the level of 0.5 is T. The values of
the function g(t,θ) are truncated to zero at t =m±T. Such a signal occupies 2-bit intervals
and reproduces well the effect of dispersion distortion in fiber-optic communication
systems. Let G(t) be the signal that will be formed from the transmitted sequence "1"
and "0" as a result of superposition of the corresponding impulses g(t,θ). Additional
distortions are caused by random noise N(t) with zero expectation and standard devia-
tion σn.
Suppose a demodulator provides signal samples
S (t ) = G (t ) + N (t )
(3)

at discrete time instants tm with a periodicity of dt= 0.5T, and this receiver is synchro-
nized with the signal transmitter, and samples are produced at the boundary and in the
middle of bit intervals. Denote the samples as sm= S(tm).
The elementary method of demodulation is the comparison of values sm with a thresh-
old value sd (CWT method). That is, the solution ri about the presence of " 1 "or" 0 "
in the bit position with the number i, should be: ri = 1, if s2i+2 >sd, otherwise ri = 0.
However, this does not take into account that useful information is contained not in one
but in all samples on stretch of an interval 2T of the existence of the signal g(t,θ), as a
result of which the reliability of demodulation is significantly reduced.
62

The best results can be obtained if to accept maximum likelihood solutions (MLS
method). For a solution on the position of bit number i it is necessary to analyze M = 4
samples
s2i , s2i +1 , s2i + 2 , s2i + 3 ,
(4)

which fall into a " a sliding window " of 2T duration, and also to take into account
influence of the following i+1-st and previous i-1-st of a symbol. Thus, the values of
the samples in the "window" of the analysis depend on the combination "1" and "0" at
3 positions. For all possible 23 = 8 combinations are necessary to generate the etalons
of a signal eh,0,...,eh,3 (h = 0,...,7) as the values of the process G(t2i), ..., G(t2i+3) in the
expression (3). For normal distributions of noise N(t) the solution accepts such aspect
that ri is central bit of 3-digit binary number j
M −1
j = arg min
h = 0 ,..., 7
 ( s 2 i + m − eh , m ) 2
m=0 (5)

The disadvantage of this method of demodulation is a fairly large amount of compu-
tation.
Now we consider a solution to the demodulation problem based on the statistical
theory for mixtures (method STM). As is known, the number of pulses in the “window”
of analysis (Q = 3), their shape and position, it is enough to solve only problem 2, where
is necessary to estimate the probabilities p1, p2, p3 . In this case, the reference sequence
(4) plays the role of the empirical probability distribution of a sample of random varia-
bles from the general population, which obeys the discontinuous distribution of the
mixture probabilities Q
f m ( p1 ,..., pQ ) =  p j f m , j ,
j =1
(6)

where pj is the parameter of the presence of the symbol "1" or "0" at position j (respec-
tively pj≠ 0 or pj =0), and fm,j are the values of the partial discontinuous distribution
equal to the values of the function g(m×0.5, j-1).
It is necessary to find estimates p1(i),...,pQ(i) of the parameters p1,...,pQ by handling the
empirical distribution (4). For the least squares estimation [3] they correspond to the
minimum of the criterion M −1 M −1 Q
D ( p1 ,..., pQ ) =  ( s 2i + m − f m ( p1 ,..., pQ )) 2 =  ( s 2i + m −  p j f m, j ) 2 .
m =0 m=0 j =1
(7)

If we equate the partial derivatives of the criterion (7) on p1,...,pQ to zero, we obtain a
system of simple equations Q

 p j c j ,h = bh(i ) ,
j =1
(8)
63

where
M −1 M −1
c j , h =  f m , j f m , h , bh(i ) =  s2i + m f m , h
m =0 m =0 (9)

(i) (i)
The solution of the system of equations (8) gives all estimates p1 ,...,pQ however,
(i)
it is advisable to use only one of them p2 which can be calculated by the formula
Q
p2( i ) =  ci2, j b (j i ) ,
j =1
(10)

c j ,h
where ci - are elements of the matrix inverse to the matrix . These matrixes can
be calculated in advance. The solution rule for this method is to check the inequalities:
if p2(i)>pd , then ri = 1, otherwise ri = 0. It is easy to see, that the volume of calculations
of magnitudes bh(i) , p2(i) is much smaller in comparison with the volume of calculations
for the MLS method.

1.3 Statistical Modeling Results
For the considered demodulation methods, a statistical experiment is conducted, the
results of which are presented in Fig. 1. The abscissa axis corresponds to the ratio SN
of the useful signal power G(t) to the noise power N(t) (in decibels). The ordinate axis
corresponds to the probability of demodulation errors for the CWT, MLS and STM
methods. In the experiment, it was taken into account that the moments of the reference
signals are shifted by 0.2T relative to the above positions due to phase lag of the syn-
chronization circuit. The threshold values are sd = pd = 0.5. As expected, the CWT
method is the least efficient, and the MLS method is the most efficient and can be con-
sidered as the optimal demodulation method. With an increase in the signal-to-noise
ratio, the gain in efficiency increases, and at SN = 18 dB the MLS method reduces
demodulation errors by 3 orders of magnitude compared to the CWT method. The ef-
fectiveness of the STM method is slightly inferior to the MLS method, but also signif-
icantly superior to the CWT method.

Fig.1. Statistical experiment results
64

On Fig.2 the diagrams of time of simulation are represented. The lower part of the
column of each diagram corresponds to the time spent on signal modeling, the upper
part to the time spent on the demodulation operation. The diagrams show that the STM
method requires about 8 times less computation, which is an important argument for
the benefit of its choice.

Fig.2. Diagrams of time of simulation

The results of statistical modeling showed that:
- the proposed STM method, based on the use of the statistical theory of mixtures,
provides almost the same demodulation efficiency as the MLS, but it requires approx-
imately an order of magnitude less computation;
- the STM method can be extended to more complex cases, for example, when it is
necessary to estimate the position of signals in the “window” of analysis (when solving
problem 2, it is necessary to evaluate not only the probabilities p1,..., pQ, but also pa-
rameters A1,...,AQ).

2 Technical solution for eliminating intersymbol interference of
signals in a communication channel

2.1 Model of a device for reducing the influence of inter-symbol interference
in an optical information transmission channel
The traditional means of combating intersymbol interference of signals are the use of
various analog linear distortion correctors that compensate for the imperfection of the
amplitude-frequency characteristics of the communication channel. Digital corrections
are also used, which directly operate on the signal as a function of time [5].
The general block diagram of an optical signal regenerator with digital processing
that uses a conversion algorithm based on a mixture theory is shown in Fig. 3.
The regenerator circuit consists of such elements. The regenerator circuit consists of
such elements:
AID – analog input devices that receive Si(t) optical signals, detect them and filter
them. Received optical signals Si(t) is converted using a fiber optic converter (OE
converter), at the output of which an electric signal S(t) is generated, which contains
information about the sequence of transmitted symbols “0” and “1”, as well as a syn-
chronization signal C(t).
65

Fig. 3. Block diagram of an optical signal regenerator

CG is a clock generator that generates the clock sequences t1k (one pulse per bit in-
terval) and tm (nb pulses per bit interval) from the signal C(t).
ADC is an analog-to-digital converter that encodes samples sm = S(tm) in the form of
an nr-bit binary code.
DPD is a digital processing device that implements adaptation and demodulation op-
erations; in the demodulation mode, it gives a decision rk = 1 or rk = 0 for each k-th bit
interval;
OPG is an optical pulse generator that, in accordance with the solution rk, generates
the reconstructed optical pulses So(t) at the output of the regenerator.
Regardless of the technical implementation of the CPU, it is important to determine
the time discreteness of the counts of the signal S(t), that is, the number of readings nb
during the bit interval and the discreteness of the readings of the amplitude sm, that is,
the number of nr encoding bits. The first parameter must satisfy the inequality nb= 2.
However, the increase in nb necessitates an increase in the speed of ADC and DSP,
which complicates the regenerator. Moreover, the increase in nb is justified only if the
independence of the random distortion of the signal S(t) by the noise N(t) is maintained
for each sample. If the WUA performs a consistent filtering of the signal S(t), then the
random components of the counts are found to be correlated more significantly, the
greater the nb. Given these features, it is advisable to take nb = 2 [5].
Increasing the number of nr encoding bits complicates ADCs and DTCs if the demod-
ulation algorithm has a technical implementation. Due to low-bit coding, rather rough
"images" of the signal are formed during the adaptation phase and this reduces the re-
liability of demodulation. If, as the number of digits increases, the encoding discrete-
ness decreases to values smaller than the root mean square noise σn, then a further
increase in nr becomes useless. Thus, determining the minimum number of encoding
bits that provides acceptable demodulation accuracy is one of the important tasks of the
technical implementation of the proposed processing algorithm [6,7].
The signal amplitude sampling in the ADC can be described by the following func-
tion:
 Nr − 1 Nr − 1
66  , if x >
Nr Nr
 [ x ⋅ Nr ] Nr − 1
kvant( x) =  , if 0 ≤ x ≤
 Nr Nr
 0, if x < 0


where [n] is the integer part of n; Nr = 2nr is the number of quantization levels. The
operational value of argument x is in the range 0≤ x ≤ 1.
When processing digital sample values, the required number of bits must exceed nr
by several units. If the CPU is based on microprocessors, then this requirement is met,
since modern signal processors have at least 64 bits. To test the effect of the number of
coding bits nr, statistical modeling was performed.
The discretization of the readout values is taken into account in the form of replace-
ment of sm by quant (sm) in the calculation of the array of accumulated readings sm for
adaptation of the regenerator, which is performed according to the formula:
1 Nw −1
smm =  s6 j + m , m = 0, ..., 5.
Nw j = 0
(11)

Quantum (sm) is also used to calculate the probability estimate p2(k) that the symbol
"1" is in the kth position when demodulating the signals:
M
p2 ( k ) =  c12 ,m ⋅ sk ⋅nb + m −1.
m =1 (12)

All other calculations were performed with the maximum accuracy possible in
MathCAD14 [8].
To adapt the regenerator, a sequence of repeating triples of symbols "010" (TPS1)
was used, which consisted of Nw = 257 triples of symbols "010". In each demodulation
simulation session, a sampling of signals over a 256-bit interval was generated. For the
demodulation error statistics set, the simulation sessions were repeated 400 or 4000
times. Thus, the total number of characters that were modeled in each session group
reached 100,000 or 1,000,000. The simulation results are shown in Table. 1.

Table 1. Results simulation regenerator test operation
σν S/ N nr = 2 nr = 3 nr = 4 nr = 6 nr = 10 nr= 15

0.2 14,00 0.0043 0.0030 0.0028 0.0030 0.0030 0.0035

0.18 14,90 0.0019 0.0011 0.0011 0.0011 0.0011 0.0015

0.16 15,90 0.00062 0.00034 0.00024 0.00024 0.00025 0.00041

0.141 17,00 0.00015 0.00008 0.00002 0.00004 0.00004 0.00008

0.12 18,40 0.00003 0.000005 0.000001 0.000001 0.000001 0.000001
67

The first column presents the noise intensity in the form of a value of σn. In the second
column, the signal-to-noise ratio in dB is calculated as 20lg (1 / σn), given that the
maximum amplitude of the useful signal C(t) is approximately 1. Other columns pro-
vide estimates of the probability of demodulation errors for different numbers of nr
encoding bits in the ADC.
As we can see, with a small number of digits nr = 2, the errors are, as expected, max-
imal. However, at nr = 4, they reach a minimum. Further increase in the number of digits
of the coding does not actually improve the accuracy of demodulation, but on the con-
trary, worsens it. For example, for the highest possible accuracy of calculations in
MathCAD14, which corresponds to 15 decimal places (nr = 15), the results are even
worse than for nr = 4.
This can be explained by the fact that quantization makes the "image" of the signal
coarser. On the other hand, quantization negates the effect of noise in the range of ΔS
values between quantization levels. If S << n, no leveling occurs. At the same time, it
is obvious that, as noise levels decrease, it is advisable to reduce ΔS, that is, to increase
nr. Given that the simulation noise level exceeds the actual noise level in the regenera-
tors (since it is impossible to obtain statistically reliable estimates of error probabilities
at low noise), 5 digits of coding in the ADC can be considered sufficient.
Digital processing operations associated with adaptation of the regenerator and de-
modulation of the signal can be performed using modern signal processors.
Such a microprocessor embodiment of digital processing of the received signal is
technically the simplest and cheapest. However, in this case, only sequential execution
of operations is possible. This limitation is related to the actual DPD speed. However,
the most critical in terms of performance are two groups of operations directly related
to the processing of sm counters. These are operations of accumulation of sums (11) at
the stage of adaptation and operations of calculating the probabilistic estimate p2(k)
(12).
At the demodulation stage, high speed is required in the decision-making operation
rk, where it is necessary to compare the estimate obtained with the decision threshold
ppor in terms of:
1 if p 2 ( k ) > p por ,
rk = 
0 — else.
(13)

Due to the simplicity of these operations, it is advisable to create specialized nodes
for their execution, which provide minimal processing time, in particular due to the
organization of parallel calculations. Such a principle and the order of processing can
be recommended. Firstly, it is necessary to form a sufficiently long test signal sequence
(TPS1) to adapt the regenerator. Moreover, only the initial fragment with a length of
3⋅Nw characters is actually used to obtain information about the properties of the signals.
On this fragment, it is necessary to form partial sums (for each m = 0, ..., 5) in expres-
sion (11). Corresponding operations must be performed in real time by the accumula-
tion unit (АU). After the initial fragment, the sequence of TPS1 should continue for a
time exceeding the time required by the microprocessor calculator (MPC) to perform
all subsequent operations, including the calculation of the coefficients c12.1 ..., c12.4,
68

which are used in expression (12). During this time, which may be called the "adapta-
tion period" (PA), the input signals of the regenerator are ignored. To determine the
duration of the PA interval need to carry out simulations of MPC. After the end of the
PA interval, regenerator enters the operating mode of signal processing in accordance
with expressions (11), (12). This processing is performed by a separate demodulation
node (DN).
The block diagram of a possible option for constructing such a hardware-software
DPD is shown in Fig. 2. According to the results of simulation, we take the number of
bits of the ADC coding nr = 5, and the length of the initial fragment of TPS1 is 768
characters (Nw = 256).

Demodulation node
Accumulation node

SRg
Sum
IRg RgP ARg 1...6
1
С124 Х S2K+3

sm0,…,sm5 С123 Х S2K+2

tm SRg

CD MPC
t1k
С122 Х S2K+1
AN

С121 Х S2k

Sum2

rk
SPP

Fig. 2. Block diagram of a digital processing device

The accumulation node (AN) consists of the following elements:
- IRg - input 5 - bit register, which receives sm signals from the ADC;
- ARg1 ..., ARg6 - 13-bit registers of accumulation of private sums (11). The
number of digits is defined as 5 + log2 (Nw) = 5 + 8 = 13;
- RgP - 13-bit intermediate register, with the help of which communication with
ARg1 ..., ARg6 and MPC is carried out;
- SUM1-13-bit adder adds the next signal from the ADC to the corresponding
partial sum.
The received amounts after completion of accumulation are transferred using RgP to
microprocessor calculator. The numbers with which this node operates are treated as
non-negative integers (with a point fixed after the least significant bit). The result of
the calculations of the operations performed by the MPC are the coefficients c12.1 ...,
69

c12.4, written to the corresponding registers C121, ..., C124 of the demodulation node (as
16-bit numbers with a fixed or floating point).
The demodulation node (DN) consists of the following elements:
- С121, ..., С124 - registers with weight coefficients, which are determined at the stage
of adaptation of the regenerator;
- SRg - shift register from 4 cells to 5 bits, where sm samples are received in the
operating mode (each clock pulse tm shifts information by one cell);
- MUX is a 16-bit multiplier that calculates each sum of (12), where sm is treated as
fractional numbers with a point fixed before the highest digit;
- SUM2 is a 16-bit adder that accumulates the sum (12);
- SPP is a comparison scheme with the threshold ppor = 0.5, which gives the solution
rk in accordance with (13).
The solution is the value of the discharge, which is located to the right of the point
separating the integer and fractional part of the number at the output of the adder. The
operations of multiplication and summation in the demodulation node are performed
once during the bit interval, that is, by the clock cycle t1k.
The control device (CD) synchronizes the interaction of all elements of the circuit
using clock pulses tm, t1k.
According to the logical sequence of signal processing in the DPU, the microproces-
sor calculator starts working after the accumulation node completes its work. However,
it must be taken into account that the MPC and AN start to function simultaneously
after turning on the power or the receipt of a special signal to start operation. While AN
carries out the accumulation of TPN1, it is advisable to provide for testing MPC. Since
the ratio of accumulation time and testing time is unknown, we will proceed from the
following interaction principle:
- AN, after completing its work, transfers to the MPC the accumulated amounts sm1,
..., sm6 and the readiness flag, which are recorded in the MPC memory in the interrupt
mode;
- After completion of testing, the MPC checks the sign of readiness and, if there is
one, proceeds to the processing of information, and in the absence, the signs of readi-
ness are expected.
After completing all the processing associated with the adaptation of the regenerator,
the MPC issues the obtained weight coefficients c121 ..., c124 to the demodulation unit
and gives the adaptation termination flag to the control device, by which the latter opens
the arrival of the sm signal samples to the shift register in the ID and after 2 bit intervals
(when the register is full) allows the operation of a comparison scheme with a threshold
that produces a solution rk.

Conclusions

The basic operations of protecting the regenerator of a signals transmission system
that have distortion due to intersymbol interference can be considered from the point of
view of the statistical theory of mixtures as a task of estimating some parameters of the
probability distribution of the mixture.
70

Due to a significant decrease in the effective bandwidth of the communication line,
as the length of the link increases, a significant pulse expansion occurs and its shape
becomes bell-shaped, which causes significant inter-symbol interference of the signals.
It is proposed to adapt the digital corrector to the features of a particular communica-
tion line using a special test sequence of characters that is generated at the beginning of
the communication session and allows you to create a "image" of the distorted signal
in the communication line.
To adapt the regenerator, need to accumulate 6 averaged real-time counts that corre-
spond to three bit intervals (2 counts per bit interval). Then need center and highlight 4
of the 6 averaged counts. Selected readings form an "image" of a real signal in the form
of a response of the transmission line to a single character "1", which, as a result of
inter-character distortions, takes about 2 bit intervals. Finally - to calculate the values
of the weighting coefficients for the obtained "image" of the real signal.

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