Signal imitation iswidely usedtoday sinceit helpsto bring the experimentto the virtualdomain thus eliminatingrisksof damaging realequipment. At the sametime all signals usedin the physical worldare limited Ьу the fшite band of frequencies renderinbgandpasssignalstudiesespeciallyimportant. Тhе method for imitating bandpasssignals in complexbasisis vfaoraЫe in the case of а bdanpass signalasit usesresourceesffectivelyand provide seth desiredaccuracy. Тhе authorhas implemented the method in the form of the РС application generatingsignalsaccordingto eth characteristicssetЬу theuserТ.hesecharacteristicsare:bordersdenfiing the signal's rfequency band, the time period,the numberof stepsfordiscretizationt,he specatrl densityform. Тhе РС application usesthe characteristics to generatethe signaland its experimentaalutocorelation. Тhе paplication calculatestheoreti cand algorimthic autocorelations in orderto evaluatethe quality of the imitation Ьу computing the eorr function. Тhе application visualizesall theresultinginformation via thesimple interface. Тhе application wasusedto generattewo-dimensionalsignals to highlight the presenltimitationsand to sketchthe directionforthe future. Тhе paplication is laterto Ье adapted completely to imitating multidimensionalsignals. Тhisworkisfinancially supportedЬу theRussianFederationMinistryofScienceand Higher Education intheframework of theResearcPhrojectitled"Component'sdigital transformation methods'fundeamntal resecarh for micro-dan nanosystems("Project#0705-2020-0041).
Bauman
The word "signal" today is oknwn to everyone and is used regularly but often we don't
even suspect how often this word could Ье used but wasn't. Temporal changes of some
physical value сап Ье represented as time series or as signals. The term "signal pro
cessing" is applicaЫe to any processes that change in time [
Increasing volumes of data breed growing amotuns of information all of which have to Ье contained
in the form of а sign al (or а time series), and as you have to represent
more and more linked processes the dimensions of sign
als being used grow [
The complex basis has shown itself to Ье uselfu ofr imitating one-dimensionalband
pass sign als [
Section 2 shows the results gained Ьу using the program in the case of one-dimen sional sign als. Section 3 embarks upon settling ewhther the method of signal imitation in colmpex
basis described in section 2 can Ье used to generate two-dimensional sign als and what changes have to Ье made to increase quality of such generating. The etrfu si
ng als separately and do not the design ed program for imitating are described in the conclusion.
One-Dimensional Signallmitation in Complex Basis
Complex Basis Imitation Algorithm Banadpss si
ng al's spectrum
spectral density of the banadpss
is constrained within two border efrequncies
[
with discrete So
S(ro) borders NLdan NR. "L" stands for "Left" and "R'' stands for "Right". ХFE and ХFO are even and odd Fourier coeficients.
Formula of the random complex spectrum is as follows:
The formula derived for calculating the resulting signal is presented below:
The spectrum and the signal are connected through the Fourier rtansform. When imi
tating random signal, coefficients µk and Ykrandomly take on values of "1" or "-1".
When imitating determined signal all of themjust remain set to "1". The values ofYF
on the borders depending on whether the N is odd or even are to Ье considered sepa
rately which is dropped here in favor of the general method. These formulas to Ье used
in the program were derived Ьу Professor Syusev V. V. [
Applying the Method Three the signals and also three experimental autocorrelations generated Ьу the pro gram are presented on the figure 2. random signals (on the left) and three resulting autocoпelations (on the right)
based on the same spectral density The resulting experimental autocorrelation is calculated as follows:
1 =-
N - m
I- y(i)y(i + m), N-1-m i=O m Е [О,М).
The program calculates diferent autocoпelations (gfiure 3). The rfist one (figure 3, а) is an а priori theoretical autocoпelation derived directly ofrm the spectral density. The second one (figure 3, Ь) is an algorithmic autocoпelation that uses Fourier coeficients. The third one (figure 3, с) substitutes Fourier coefficients with their complex basis ver sions, this is the resulting experiment autocoпelation that is compared to the other two in order to estimate the quality ofimitation.
а)
=...... ... ..... , - .c... = 0, i --- ----r----:----1---)- --1----1---- --- ----r----r-- -1----j---·i----1----r ----i·---1----r---. ,o ±iJ +-:- т-· ·-··:-· -,- :- . -: -: ;. : : :>i+·:·:····
=············ ····· ····=[J j:···· 1:····i(··t:"···1:····1 О 10 20 30 40 50 бО 70 80 90 10 110 120 130 140 150 160 170 180 190 20 с) The епоr function and the mean епоr ear computed Ьу nfiding the diference between the two autocoпelation being compared. An example ofthe епоr function calculated is presented on the figure 4. Due to the symmetry ofthe digital spectrum the right halfof the епоr function plot with the peak on the very right could Ье ignored. When generating determined signals comparison is done between the resulting auto coпelation and the theoretical autocoпelation that is derived а priori. The random sig nals are qualified on the dirfeence between the experimental uatocoпelation and the algorithmic autocoпelation. 3 3.1
Two-Dimensional Signal lmitation in Complex Basis
The Specifics of Two-Dimensional Signal Processing
The structure of multidimensional signals presents the certain level of difficulty when
it comes to both representing and processing [
Ki
Ai - 1, i = 1, 2, ... , п. Inverse transform: where О а1, ... , й-п А с1,...,п) - 1.
However, before advancing into two-dimensional domain it was decided to study the specifics of the "quasi-two-dimensional" signals that are obtained Ьу stacking to gether random broadband one-dimensional signals generated earlier. 3.2
Applying One-Dimensional Algorithm to Two-Dimensional Signals
Despite the need of readjusting the method for two-dimensionalsignals this method can
already Ье used. То do so we just have to transform the two-dimensionalspectral den
sity into an raray of one-dimensionalbroadband ones stacked together. Resulting two
dimensional specatrl density is presented on figure 6.
Then the one-dimensionalsignals comprising the two-dimensionalone can Ье gener
ated separately and stacking them together side Ьу side provides us with а two-dimen
sional signal (gufire 7). This signal inherits the quality of either being determinate or
random Ьу the virtue of its coeficients.
Signals generated while being two-dimensionalare unidirectional as clear ofrm the fig
ure 7 - the most obvious trends are visiЫe on the main horizontal axis so the so called
waterfall plot appears. Waterfall plots are encountered in medicine [
This paper is а part of а new development for high-dimensional signal simulation that is presented in the conference Ьу the paper where the author was involved too. The method of imitation developed earlier for one-dimensional imitation was used to imitate two-dimensional signals. Further research and adaptation of this method is to Ье per ofrmed in due cosure.
The method of imitation in complex basis reduces algorithmization to the execution of pre-derived mathematical equations, which reduces the computational complexity and resource intensity of the algorithm, and the use of linear data structures positively afects the scalaЬility of the developed solution.
The software solution was implemented in the Lazarus DIE which allowed to meet all the accuracy criteria and to create the interface. Free Pascal language used in Lazarus IDE is very clear as it was designed Ьу mathematicians to Ье understood Ьу their col leagues. This language is also widely used in education field in Russia so the program developed could Ье studied Ьу the future students during their digital signal processing cosure.
Since the in-box work with two-dimensional signals is not supported yet and to Ье added later the results in section 3 were obtained Ьу putting the one-dimensional signals comprising the two-dimensional one through the software and later stacking the results back together for the visualization through MS Excel 2010.
The first test of the one-dimensional algorithm being expanded to imitate two-di mensional signals highlighted the direction rfo tufeur development: the algorithm should Ье adopted to allow rfo signals with diferent numbers of dimensions, the visu alization facilities should Ье expanded. The method as it is can Ье used for modeling the unidirectional two-dimensional data in the form of а waterfall plot.
This work was supervised Ьу professors Syuzev V. V. and Smimova Е. V. of the Bau man Moscow State Technical University. The project was made possiЫe with the fi nancial support of the Russian Federation Ministry of Science and Higher Education in the framework of the Research Project titled "Component's digital transformation methods' fundamental research for micro- and nanosystems" (Project #0705-20200041). Special gratitude goes to the organizers ofDDAМDI conference for providing а medium suitaЫe rfo exchanging ideas and results and advancing the quality of scien tific work.