=Paper=
{{Paper
|id=Vol-3187/short7
|storemode=property
|title=Problematic Issues of Approximation and Interpolation in Signal Processing in Secure Information Systems (short paper)
|pdfUrl=https://ceur-ws.org/Vol-3187/short7.pdf
|volume=Vol-3187
|authors=Olena Nehodenko,Svitlana Shevchenko,Nataliia Trintina,Volodymyr Astapenia,Oleksandr Tereshchenko
|dblpUrl=https://dblp.org/rec/conf/cpits/NehodenkoSTAT21
}}
==Problematic Issues of Approximation and Interpolation in Signal Processing in Secure Information Systems (short paper)==
https://ceur-ws.org/Vol-3187/short7.pdf
Problematic Issues of Approximation and Interpolation
in Signal Processing in Secure Information Systems
Olena Nehodenko1, Svitlana Shevchenko2, Nataliia Trintina1, Volodymyr Astapenia2,
and Oleksandr Tereshchenko3
1
State University of Telecommunications, 7 Solomyanska str., 03110, Kyiv, Ukraine
2
Borys Grinchenko Kyiv University, 18/2 Bulvarno-Kudriavska str., 04053, Kyiv, Ukraine
3
National Technical University of Ukraine βIgor Sikorsky Kyiv Polytechnic Institute,β 37 Peremohy ave., 03056,
Kyiv, Ukraine
Abstract
One of the basic indicators of information systems security is the availability and integrity of
information. When transmitting a visual message, the signals are distorted due to the influence of
various random factors. It is established that an ideal channel must have an absolutely rectangular
frequency characteristic for the transmission of all components of the amplitude-frequency
spectrum of a continuous random process without distortion and loss. Such frequency response is
the result of a complex interaction of the parameters of the set of devices which are the part of the
information transmission channel and perform the above procedures for conversion and signal
processing. Some information and cybersecurity tasks require efficient interpolation, and in some
situations a combination of interpolation and approximation techniques. The paper proposes a
mathematical model of information signals based on fundamental trigonometric splines, which
allow to take into account the differential properties of information signals. It is shown that it is
feasible to use trigonometric splines as mathematical models of information signals, and it is more
feasible to apply fundamental approximation trigonometric splines for recovery of signals as
components of filters. The importance of this approach is explained by the fact that only
fundamental functions are subject to processing when applying linear methods. This fact allows
performing the necessary calculations for processing experimental data in two stages. In the first
stage, calculations are performed related to the processing of fundamental functions (these
calculations can be performed in advance). In the second stage, calculations are performed that
take into account the values of the reproduced functions. It is shown that the method of phantom
nodes should be used for interpolation of the useful signal in information networks, which allows
to increase the accuracy of information processing.
Keywords1
Information system, interpolation, approximation, fundamental trigonometric splines, signal
processing.
1. Introduction
In information and communication systems, there are many problems where it is necessary to provide a
high-quality approximation and interpolation of processes or objects with limited or inaccurate primary
data. Thus, in the vast majority of information systems used by humans, the input and output signal is a
CPITS-II-2021: Cybersecurity Providing in Information and Telecommunication Systems, October 26, 2021, Kyiv, Ukraine
EMAIL: negodenkoav@i.ua (O. Nehodenko); s.shevchenko@kubg.edu.ua (S. Shevchenko); trintina2015@gmail.com (N. Trintina);
v.astapenia@kubg.edu.ua (V. Astapenia); alexandr.tereschenko2014@gmail.com (O. Tereshchenko)
ORCID: 0000-0001-6645-1566 (O. Nehodenko); 0000-0002-9736-8623 (S. Shevchenko); 0000-0001-6827-4030 (N. Trintina); 0000-0003-0124-
216X (V. Astapenia); 0000-0003-0536-2708 (O. Tereshchenko)
Β©οΈ 2022 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
s
276
continuous function. Examples are voice messages, visual images, etc. (within the macroscopic model of
the organs of formation and perception). It is important to note that these are random processes in all cases
for the recipient. It is known that only such processes carry information [1, 2]. When communicating using
telephone, our voice message (elastic vibrations of air) is converted into a continuous electrical signal in
the microphone, which is a random process π΄(π‘). Further transformations in modern systems include
analog-to-digital conversion and a number of other procedures (error correction and cryptographic coding,
modulation of carrier harmonic motion, amplification, radiation, and inverse transformations). The user
(recipient) eventually needs the message in the "ideal" original form, but due to the influence of various
random factors, they receive it in a distorted, somewhat different from the original form A(t ) .
The same situation happens when transmitting a continuous visual message. This means that the channel
between the source and the recipient of the continuous message is not ideal and needs to be improved to
increase the availability and integrity of information as the basic indicators of the security of information
systems. The ideal channel should have an absolutely rectangular frequency response to transmit all
components of the amplitude-frequency spectrum of a continuous random process without distortion and
loss. Such frequency response is the result of a complex interaction of the parameters of the set of devices
which are the part of the information transmission channel and perform the above procedures for conversion
and signal processing.
This requires:
1. To know the essential components and the width of the amplitude-frequency spectrum of a continuous
random process that is formed by the source (the width of this spectrum is formally infinite) with sufficient
accuracy to efficiently recover the process.
2. to determine and minimize potential losses in the quality of recovery of the initial process after its
sampling under conditions of artificial limitation of the spectrum when determining the sampling interval
(sampling frequency) within the framework of Kotelnikovβs theorem.
3. To identify and minimize potential losses in the quality of recovery of the initial process due to noise
that occurs during quantization.
4. to ensure the formation of a rectangular frequency response of the channel in this band, taking into
account the influence of all its components.
5. To identify and adequately take into account the impact on the frequency response of additive and
multiplicative interferences which are inherent in the relevant channels (especially radio channels).
2. Some Examples which Demonstrate the Need of Identification of Processes
and Images in Information Systems
All of the above cases are part of the problems of identification of processes and images.
This category of problems includes the approximation of the autocorrelation function (ACF) RΠ°(Ο)
according to the the results of its calculations based on the set of samples of the random process Π(t) in
conditions when part of the samples is lost or distorted. Often this ACF is used to determine the amplitude-
frequency spectrum of a random process GΠ°(Ο) according to Khinchin's theorem [3].
ο₯
Ga (ο· ) ο½ ο² Ra (ο΄ ) exp(ο jο·ο΄ )dο΄
οο₯
where π = π‘2 β π‘1 ;; Ο = 2Οf is the angular frequency.
An example of the use of the ACF is also the prediction of a random process based on the parameters of
this function. Such prediction (extrapolation of processes) is necessary for the formation of artificial
intelligence algorithms in a broad sense and to solve relatively narrow problems of adaptation of individual
devices ranging from industrial robots, drones, adaptive surveillance systems ending with smart home
equipment.
277
When analyzing and processing visual information, the raster image can be considered as a random
field, for which there is also the concept of spatial correlation function [4], and taking into account that such
an image is a dynamic process in most cases, it is a spatial-temporal correlation function.
A number of tasks in the field of information and cyber security require effective interpolation, and in
some situations a combination of methods of interpolation and approximation [5].
Such tasks include the following:
1) from the standpoint of ensuring the availability and integrity of information:
- restoration of the voice message in case of the loss of a certain part of its samples under
the influence of interference or as a result of intentional distortion;
- restoration of the visual picture in case of the loss of a certain part of the fragments (pixels)
under the influence of interference or as a result of intentional distortion (for example, during
identification of a person, car numbers when monitoring the movement using surveillance cameras,
etc.);
- as an alternative to error correction coding;
- recovery of information from the damaged drive.
In such cases (in digital representation) we are dealing with a random sequence of rectangular (in the
first approximation) pulses, some of which are distorted or lost (when there are "windows" of damaged
characters).
2) from the standpoint of information security, there are inverse tasks:
- determination of the boundaries of artificial distortions, beyond which the secrecy in the
potential channel of information leakage can be considered reliable. That is, distortion of the
message (voice, visual image, etc.) in such a channel, so that the opposing party does not recover
the message. This may also be the case when you need to make the voice or appearance of an
important witness, or certain details of the interior of a room involved in IoT technology
unrecognizable;
- as a method of testing cryptographic procedures (ciphers) to assess their effectiveness;
- in cryptanalysis (decryption while the key is unknown).
In these situations, it is necessary to distinguish the true (which contains information) random sequence
from a mixture of it with another interfering random sequence. The frequency spectra of such sequences
have an envelope according to the shape of the spectrum of a rectangular pulse with chaotic filling. As an
example, Fig. 1 presents the experimentally obtained spectrum of a pseudo-random sequence [6].
f
1/ΟΡ
Figure 1: The spectrum of the pseudo-random sequence
Thus, the solution of these problems either in the time or in the frequency domain requires the
involvement of a powerful mathematical apparatus.
Directional antenna systems play an important role in information retrieval and transmission
technologies. They determine the angular coordinates of objects that reflect (radar, optoelectronic location,
sonar) or emit waves (radio reconnaissance, radio-technical reconnaissance, ultrasonic reconnaissance,
etc.), and in satellite, radio relay and tropospheric communication systems they ensure the availability and
integrity of information for the consumer, which is in a certain angular direction. In such systems, the
278
accuracy of the orientation of the maximum of the antenna directionality characteristic f (ΞΈ, Ο), i.e. the
angular spectrum that creates the antenna opening, is due to the amplitude-phase current distribution [7].
N ο1 ο© 2ο° οΉ
j οͺοͺi ο« ο² cos ο§ i οΊ
f (ο± , οͺ ) ο½ ο₯ Ai e ο« ο¬ i ο»
i ο½0
where: π΄π is the normalized current amplitude in the antenna array element; ΟΡ is the phase of the current
in the element; Ξ» is the wavelength; ΟΡ β the distance from the central to the i-th element of the array; Ξ³Ρ β
the angle between the direction to the point of space with the angular coordinates ΞΈ, Ο and the direction ΟΡ.
Under the influence of random factors, the amplitude-phase distribution in the opening may be distorted
and the question arises of determining of the directional characteristic. These issues are considered using
classical methods in the paper [8]. The results of measuring the angular distribution of the field strength,
which created by the antenna in space, can be obtained with errors; here we can talk about identification of
distortions on an aperture or detection of non-working vibrators [9,10].
At the present stage, these problems have not been fully resolved. One of the mathematical methods that
can help to solve such problems is the use of spline approximation and spline interpolation [11,12].
3. Spline approximation and spline interpolation as a method of signal
processing in secure information systems
Based on the analysis of the scientific literature [13β17], it was determined that there are properties of
signals, without which the very statement of many tasks of signal processing does not make sense. Such
properties of information signals are their smoothness properties, which characterize the behavior of the
signal in some neighborhood of an arbitrary point belonging to the signal interval. These properties contain
information about the existence of a certain number of continuous derivatives of the studied signal, as well
as information about some analytical properties of these derivatives. On the basis of this theory the
mathematical model of information signals was developed and explained using fundamental trigonometric
splines which allow to take into account differential properties of information signals.
It is established [18] that the signal represented by the Fourier series can only be periodic. Signals of
arbitrary shape can be represented by a Fourier series only approximately, since this provides for the
periodic repetition of the signal interval outside its setting. At the junctions of the periods there may be
breaks and fractures of the signal, as well as processing errors caused by the Gibbs phenomenon, to
minimize which certain methods are used.
The paper proposes a method for attenuating the Gibbs phenomenon based on trigonometric Fourier
series. This method allows to periodically extend the signal of arbitrary nature and at the same time get rid
of breaks and fractures of the signal at the junctions of periods.
An improved discrete version of this method will be the basis for reproducing a useful signal in
information transmission systems.
Consider the method of periodic continuation of non-periodic functions for the case when trigonometric
splines are used as an approximating function.
Consider the function
f (t ) on the interval ο0,2ο° ο . Set N interpolation nodes, N ο½ 2n ο« 1 ,
i ο1
h ο½ 2ο°
where
n ο½ 1,2,..., the uniform grid step is N , where i ο½ 1,2,..., N .
The value of the function in the interpolation nodes is calculated. A sequence of values of the function
N
{f(h(k-1))}k=1 = {fk }Nk=1 is obtained. Next, a trigonometric interpolation spline is constructed based on
these nodes, which has the form:
279
ο ο
n
a0
S tr ( f , ο N , t ) ο½ ο« ο₯ ο‘ k (r , N ) a kοͺ ο ck ο¨r , N , t ο© ο« bkοͺ οks ο¨r , N , t ο© ,
2 k ο½1 (1)
where
cos kt ο₯ ο© cosο¨mN ο« k ο©t cosο¨mN ο k ο©t οΉ
ο ck ο¨r , N , t ο© ο½ ο« ο₯οͺ ο«
k r ο«1 m ο½1 ο« ο¨mN ο« k ο©r ο«1 ο¨mN ο k ο©r ο«1 οΊο» ,
sin kt ο₯ ο© sin ο¨mN ο« k ο©t sin ο¨mN ο k ο©t οΉ
οks ο¨r , N , t ο© ο½ ο« ο₯οͺ ο
k r ο«1 m ο½1 ο« ο¨mN ο« k ο©r ο«1 ο¨mN ο k ο©r ο«1 οΊο» ,
ο₯ ο© οΉ
οο‘ k ο¨r , N ο©οο1 ο½ 1r ο«1 ο« ο₯ οͺ 1
ο«
1
οΊ
ο« ο¨mN ο« k ο© ο¨mN ο k ο©
r ο«1 r ο«1
k m ο½1 ο»,
2 N
a 0 ο½ ο₯ f (t i ) a k ο½ ο₯ f (t i ) cos kt i
οͺ 2 N
N i ο½1 ,
N i ο½1 ,
2 N
b ο½ ο₯ f (t i ) sin kti
οͺ
k ο½ 1,2,..., n .
k
N i ο½1 ,
S
The interpolation trigonometric spline tr interpolates the function π(π‘) at π + 1 points, given on the
segment 2π. Since the value of this trigonometric spline is definite at the point 2π and, due to the
S
periodicity, it follows that ππ‘ (2π) = ππ‘ (0). Therefore, we will consider the spline interpolation tr only on
the interpolation segment 2π β β.
It is clear that the trigonometric interpolation spline is due to the fact that π(0) β π(2π), in the
neighborhood of the points 0 and 2π has the same defects as the Fourier series in the neighborhood of the
breakpoints. Therefore, it is advisable to use the method of improving convergence, which is called the
method of phantom nodes [13].
This method is as follows. An even number of phantom nodes is added to the sequence of interpolation
nodes; the values in these nodes will be chosen taking into account the estimates of the derivatives, which
we estimate using the divided differences in the neighborhoods of the points 0 and 2ο° ο h. . That is, we
construct the function π(π‘), π‘π(2π β πΌ, 2π) on the interval (2π β πΌ, 2π) based on the conditions
π(2π β πΌ) = π(2π); π(2π) = π(0);
πβ²(2πβπΌ) = π β²(2π) ; πβ²(2π) = π β²(0) ;
β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦
π(πβ1) (2π β πΌ) = π πβ1 (2π); ππβ1 (2π) = π (πβ1) (0)
and find the values of the derivatives of the function at the corresponding points.
Addition of 2π (π = 1,2, β¦ ) phantom nodes increases the number of interpolation nodes on the segment
πβ1
[0,2π], and reduces the step β of the interpolation grid, which now becomes equal to βπ = 2π . Since
π+2π
the number of values of the interpolated function does not change, the decrease in the step of the interpolated
grid leads to a decrease in the interpolation segment, which becomes equal to πβπ .
A linear function π(π‘) is constructed on the segment [2π β πβπ , 2π], that satisfies the necessary
conditions
π , π‘ = πβπ ;
π(π‘) = { π
π1 , π‘ = 2π.
280
Calculating the values of this function in the phantom nodes π‘π+1 , π‘π+2 , β¦ . , π‘π+2πβ1 , we can find the
required values in these nodes.
An even number of phantom nodes is chosen in order the total number of interpolation nodes to be odd,
because trigonometric interpolation of an odd number of points is more convenient. It is convenient to add
a small number of phantom nodes, i.e. to set π = 1,2.
Phantom nodes can be added on both sides of the interpolation sequence. However, we have to construct
two functions π1 (π‘) on the left and π2 (π‘) on the right. However, due to the periodicity of interpolation
trigonometric splines, it is more convenient to do it on the one side.
When constructing the function π(π‘), we can require that its derivatives of a certain order also take
certain values at the points πβπ and 2π. Divided differences of the interpolated function can be used to
find these values.
Also, in many cases, when constructing the function π(π‘), the exact values of the derivatives of the
interpolated function π(π‘) can be used. Figure 2 shows the algorithm for information processing.
Figure 2: The scheme of the information processing algorithm
This algorithm is repeated on each of the specified segments for processing discrete information.
A generalized method for information processing during signal excretion on spline filters is proposed,
which greatly simplifies the processing algorithm.
4. Conclusions
In the study of various errors of linear links in the theory of information systems for various purposes,
such as linear amplifiers, filters and up to the frequency response of the information transmission channel
in general, it is feasible to use periodic models of information signals. This feasibility is explained by the
fact that the trigonometric functions used in the construction of periodic models are Eigen functions of
281
linear operators, i.e. do not change within a constant when exposed to linear operators. Thus, it is proved
that it is feasible to use trigonometric splines as mathematical models of information signals, and it is more
feasible to use fundamental approximation trigonometric splines to recover signals as components of filters.
The importance of this approach is explained by the fact that when applying linear methods, only
fundamental functions are subject to processing. This fact allows performing the necessary calculations for
processing experimental data in two stages. In the first stage, calculations are performed related to the
processing of fundamental functions (these calculations can be performed in advance). In the second stage,
calculations are performed that take into account the values of the reproduced functions.
For the processing of information signals in practical calculations, an important place is occupied by
differentiation operations to find derivative functions at the stage of interpolation, using only the values of
these functions at individual points.
The improved method of signal processing in the information system on the basis of fundamental
trigonometric splines allows to periodically extend the signal of arbitrary nature and at the same time to get
rid of breaks and fractures of the signal at the junctions of periods. This method uses fundamental
trigonometric splines, which allow real-time calculations. The method of phantom nodes should be used to
interpolate the useful signal in information networks, which improves the accuracy of information
processing.
Further improvement of the considered technique and application of spline approximation and spline
interpolation in relation to the problems of signal processing listed above in the protected information
systems is expected in the future.
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