The articlies devotedto the nfudeamntal proЫemsolutionon an efective digitalmatrix signal processindgevelopme nint the ftamework of researcwhorksupporteЬdу theRussianFederationMinistry of Scienc eand Нigher Education,relatetdo thenewmaterialfosr аnewmemorygeneratiounsingdirect recordinmgethodswith ultrashortlaseprulsetso solvperoЫemsprovidedЬу the end-to-enddigital technology"Neurotechnolodgany rAtificial Intelligence". Thepeapr substantiatetsheneedto develoаp newenergytheoryof multidimensionaldigitalrepresentationdan conversioonf realsignal sto creat efast algorithmswhichreducecomputationalcomplexity and improvetheaccuracoyf signalrecoverSyp.ectral algorithmfors simulationofmultidimensionadliscretdeeterministicdan randombandpasssignals are describeudsingtheexample oftwodimensionaldiscretebasisnfuctions dan Fourie rdan Hartley transformations, considerintgheenergycharacteristicsdan autocoпelationfunctions ofthesesignals.Communicationequationsaregivenforthe developmenotf two-dimensionalsimulationalgorithmsrfo signals with а non-axialftequency spectrum. Schemesof algorithmsfo rsimulating real-timebandpass signalsand software implementation of the algorithm rae presentedT.he developesdoftware ofr the simulatedsignalcharacteristicsresearch is describedT.hesecharacteristicsare theboundyar ftequenciesofthe function dan theshapeofthe signalpowerspecrtal density.
The article is devoted to the fundamental proЫem solution on an efective multidimen sional digital signal processing development in the framewor k of research wor k sup ported Ьу the Russian Federation Ministry of Science and Higher Education, related to the new materials for а new memory generation using direct recording methods with ultrashort laser pulses to solve proЫems provided Ьу the end-to-end digital technology "Neurotechnology and Artificial Intelligence".
Ьу its tauhors. Use permitted duner 4.0 lntemational (СС ВУ 4.0).
The relevance of the research topic described in this article is due to the need of new scientific methodology development for the synthesis of high-precision and high-per ofrmance algorithms for simulating deterministic and random signals of lgare dimen sions within the afrmework of classical and generalized coпelation theory in the spec rtal domain of harmonic bases, using the original algorithcrni relationship of models of pseudo-random and deterministic signals, which will allow to create common simula tion models оп а single mathematical and software basis, as well as will provide an ecfetive tool for statistical research of real-time systems for viarous purposes.
The first section of the article justifies the need to develop а new energy theory of digital matrix representation and conversion of real signals to create fast algorithms that reduce computational complexity and improve the accaurcy of signal recovery. А step by-step research plan is described, including experimental verification of theoretical results in comparison of simulation models of siagnls and systems. The nrniizrniation of conversion time and increasing staЬility and reliaЬility have been considered.
The second section of the article describes spectral algorithms rfo simulating multi dimensional discrete deterministic and random bdanpass signals оп the example of two dimensional discrete basis functions and Fourier and Hartley transformations, account ing the energy characteristics and autocoпelation functions of these signals. Commu nication equations are given that allow authors to develop two-dimensional simulation algorithms for signals with а non-axial frequency spectrum.
In the third section of the article, the algorithms rfo silmuating real-time bdanpass signals are given.
The turfoh section contains а description of the developed software (application), uselfu ofr researcher having an opporitunty to set and research the characteristics of the silmuated signal. 2
The Need to Develop а New Energy Theory of Real-Signal's Digital Matrix Representation and Conversion There are certain forms of data that are especially well studied and therefore have cer tain general methods of analysis, such as time series, working with large amounts of data [1]. In other cases, the input data is more complex in shape or size, today we сап get data from any source, starting from the genome [2] and ending with the media [3], in these cases more complex data get а more specific approach.
Big data means not only extended in computational space but is also in time. Time data characteristics may render traditional algorithms fundamentally useless. New data could соте continuously and it could Ье not only required to Ье stored [4], there should Ье а method of dividing the input data into operational events in real time with further intent of using these events rfo forecasting [5]. Another level of complication is а mul tidimensional data. Finance uses а multidimensional dynacrni analysis [6], the response timelines to threats in computer networks [7], articulated thinking visualization, all these areas need new approach of research.
One of the proЫems with multidimensional analysis stems from it being visually un intuitive. While it is possiЫe to imagine the nature behind time series or to locate the connection between reallife phenomenon and its two-, three- or even four-dimensional representation, higher numbers of dimensionsтау often leadto confusion. However, computational power availaЫetoday does not onlyallowus to nfially work with data as Ьigas it comes but alsoallows usto disengage from our own ЬiasesЬу placingmore work onto the machine. Indeed, such areas as machine learning and neural networks era not lirnited Ьу executing calculationspreassignedЬу the researcher but can to some extent choose their own mode of actions clairning levelsof eflxiЬility previouslyunat tainaЫe.
Modelingdan simulatio ngrant us the aЬilityto study any realtime processes virtually saving costs for the physicalexperiments. The usage of higher dimensionscontributes to the accuracy deliveredЬу the new algorithms.The usage of energy spectra places the scientific research rfo these algorithmsinthe well-researchedarea of spectraltheory [7-10].
Spectraltheory provides new approaches of research based on matrix mathematical apparatus that renders the new algorithmsprepeard for tfurher automatizationЬу the means of tooldevelopedin such areas as machine learning, artificial intelligencea,nd Ьig data processing.
The tauhors consider the basics of the theory of specatrl simulation ofmulti-dimen sionalsignalsin harmonic bases continuingtheir research [11- 15]. Properties of two dimensionalharmonic discrete basisfunctions and transformations necessary for the developmentof specatrl algorithmsfor simulatingtwo-dimensionalsignalsare consid ered in this articleas wellas its experimentalsoftware realization. 3 3.1
First Steps to the New Theory: Spectrum Simulation
Algorithms for High Dimensional Discrete Determine and
Random Bandpass Signals
Two-Dimensional Discrete Basis Functions for Fourier's and Hartley's
Transformation
The two-dimensiona ltrigonometric nfuctions are the basis of the two-dimensionalhar
monic ones:
(
Usingfunctions (2.1.1) three two dimensionalcomplete basic systems can Ье created.
The rfist basis system is been created Ьу altematingthe evenfunctions:
COS [2n е;:1 + ki:2)], where the number ofeven functions in it is greater than the number The coersponding spectra willhave altemating even (marked Е) dan coefficients as well,presentedas follows: ofodd functions. odd (marked О) к 1 к 2 к 1 2 к 1 ехр 2п ( ;: + ;: )] = cos [2n ( ;: + к;: )] + jsin [2n ( ;: +
2 + к;: )]; k1, i1 Е [О, N1); k2, i2 Е [О, Nz). are even and odd elementsofthe basisfunctions' power; x(i1, i2 ) - two-dimensional determine signal, defшedat the systemofNl xN2 points, aswellas k1 Е [О, : 1); k1 Е [о, :2).
А discretetwo-dimensionalFourier series(inverseDFT)in а itrgonometric basishas the lfolowing form: x(i1 , i2) = Хн(О,О) + r,: 1: I№k22: {хн(k1, k2)cos [2n (к; :.1 + к; :. 2)]} + +Хн (: 1, :2) cos[n (i1, i2)]. x(i1, i2 ) = Хн(О,О).
The colupe of DFTs (
к;:z)], and is а two-dimensionalHartley functions' modicfiation.
This system is orthonormal
and obeys а pair of discrete two-dimensionaltransformations:
(
orthonormal and multiplicative. The couple of discrete transformations
for this system has following form: the given Fourier transformations
are oriented to the determined signals. However, they will also apply to random signals y(i1, i2) with spectra Y(k1, k2). 3.2
Energy Characteristics of Two-Dimensional Signals and their Relation to Fourier Coefficients The energy properties of deterministic
two-dimensionalsignals, as well as their one dimensional counterparts, are characterized
using SPDF (specatrl power density func tion) 1 S(w1, w2) = li➔mо [-1тт-2 IXF (w1, w2) 1 2 ], 1Т zT ➔o hwere
X(w1, w2) is а continuous spectrum mensional values of frequency: denfied on an infinite interval of two-di with hwere
At the discrete points of the ftequency range
(
XFE(k1, k2), Хю(k1, k2) determine the even and odd components of the coпesponding complex spectra.
Comparing them with each other, а record of a discrete spectral density is been created in the form of
However, one equation (17) is not suficient to determine even dan odd components of the spectrum. Therefore, we introduce а phase two-dimensional density for modeling 1Т
zT ф(k1 zп-, k2 zn), Ъу setting it in the rfom
Then get Solving the equations (18) and (19), we get new equations for the relationship between the Fourier coefficients and the spectral density power function (SDPF): (18) (20)
These coupling equations allowto develop two-dimensional simulation algorithms ofr signalswith а non-axial frequency spectr.um
Given non-axial signalsare а specialcase of bandpass signals,they are described in additional materialsenclosed with the paper (zip file ), so tfurher we consider only two dimensional bandpass signalswhose SPDF and ACF are easilyassociated with Fourier coeficients, replacingin equations (20) and (21) the definition intervals[О, N_l) and [О, N_2 ) with [N_lL, N_lR] and [N_2L,N_2R], respectively.Here in а following sub section 2.3 there are obtained equations for two-dimensional signals' autocoпelation ufnctions. 3.3
The Two-Dimensional Signals' Autocorrelation Functions Suggested spectral algorithms of imitation may Ье illustratedwith the helpof а signal with а two-dimensional spectral density having the shape of а parallelepiped. Such spectral density is visualized on the figure 1, this spectral density is technicallythe spectral density of а bandpass two-dimensional white noise with the intensity of So. ofr the even N1 and
N2: ofr the odd N1 and N2: denfied as follows: where
W1 , _/ ------ -----. -J.(__________ ------- ,.· -----------------------У
Fig 1. Functional scheme ofthe two-dimensional specatrl density's visualization. The equation rfo random two-dimension signal's calculation with rfom shown at the У(.l1, lz . ) -- у; F L, k1--N1L L, k2--N2L
N1R N2R +;у F (N1R + N1L - k1, N2R + N2L - -k2) ехр [-].2 П:
{у; F (k1, k2) ехр + (N2н+;:-k2)i2)] }.
[·2 ] П: - N ( k1i1 + -kN2i2)] + 1
2 (( N1л+N1L-k1)i1 +
N 1 Even and odd Fourier coecfiients and the right boundary Fourier coecfiients are Т1Т2
Sо-_ Z(N1R-N1L)(N2R-N2L)
li. N1R,NzR = О, the coeficients = 1 with all k1, k2 apar t fr om k1 = NlR, k2 = N2R, wh ere are defined diferently:
4 4.1
Experimental Research: SoftwareRealization ofthe Random Bandpass Signal Simulation Algorithm with Complex Basis
Accuracy can Ье measured Ьу comparing with theoretical values for а specific signal, and speed can depend on the mathematical form of the algorithm - rmufolas are com puted faster than complex algorithms using matrices. Mathematical equations provide low memory size requirements as well. А possiЫe disadvantage of the described algo rithms is the need rfo impressive mathematical training before programming.
The output data of the software is the theoretical, algorithmic, and experimental au tocorrelation functions, as well as the errors, as the diference between aoutcoerlation ufnctions, and the simulated signal itself.
The developed application allows us to simulate signals based on specified char acter istics with the aЬility to simulate one- dan two-dimensional signals. The following case proves the efficiency of the algorithm for one-dimensional signals' simulation. Two dimensional signal simulation output will Ье availaЫe in tfuure articles. 4.2
The Fig. 2 shows the functional scheme descriЬing the basic path of work with the algorithm. The input contains borders limiting the band of the signal being generated, the time period, the number of discretization steps. These characteristics, also known as input characteristics, allow us to display the spectral density diagram.
hТeoretical autocorrelation
coefcfiients -+ "-'
Fig 2. Functional scheme of the algorithm.
The spectral density is нsed to compute the theoretical and algorithmic aoutcorrela tions used to evaluate the quality ofthe imitation, it is also used to imitate the signal according to the set input characteristics and to form its experimental autocorelation. 4.3
The Algorithm The set ofsteps undertaken to generate the signal, its energetic characteristic and to estimate the quality ofthe imitation process is shown on the Fig. 3. Every step ofthe algorithm implements mathematical formulas described earlier. Yes (__ s__art_)
Formin s ectral densit Calculating фe ?retical auto
corre1atюn Calculating algorithmic auto
correlation and errors Acquiring the imitated signal Calculating experimental auto correlation
Finish Fig 3. Flowchart ofthe random signal imitation algorithm. 4.4
Software Implementation The software implemeanttion has been done using Lazarus IDE, supporting Free Pascal programing language. The chosen IDE provides all the necessary mathematical func tions and instruments allowing to build desired interfaces. The workflow in the program mirrors shown at the figure 3. The user sets the input characteristics which are used to picture the spectral density, later the autocorrelation, the signal and the error hapgr era calculated and visualized.
Left Ь о rder E:J Right bordeг Т: N: n:
:EJ 5qua,e form v
I 0,8
+'1 ----;--1--- ---- +1--+1 --+ -1-+--1 +
!-----1-----1-- --i----+---- -----i-----i----j-----j-----i0,2 :1----- 1----- f1-- --1 -----1:----- -----1 -----1 ---- 1----- -----1 о ' о 1 1 10 --------1----15 20 25 1 30 1 35 1 40 1 1 50 Тheoretical autocorrelation 10 20 30 40 50 &О liг-:r"- --:1----: - 1 170 180 190 20 Computational complexityof the method used to calculatetheoreticalis О(М). Com putational complexities for calculating algorithmicand experimental are both O(M*(N2-Nl)).
Thus, computationalcomplexityreaches onlyO(M*(N2-Nl)), which is closerto lin ear time complexityrather than to О(М2) - such resultmay Ье deemed sufficient. 5 ocL
Comparison Between Theoretical and Experimental Results In the followingexamplecharacteristics of the signalunder generation are: coR = 6;1t = З1t;
Т = 2 s; N = 51; n = 4. Fig. 4 displaysthe spectraldensity printed according to the characteristics and generated theoreticaland algorithmicautocoпelations. 10 10 1 45 □ д1] О 1 Shape: . :--1:
О : shown on the figure on the Fig. 6.
Fig 4. Signal's specatrl density and theoretical and algorithmic autocoпelations.
In the case of the random signalone set of characteristics can describe diefrent sig nals. Three signalswith characteristics coR = бп; coL
= Зп; Т = 2 s; N = 51; n = 4 are 5. Three diferent
generated experimentalcoпelations are shown 1 1 1 1 1 1 1 1 1 1 -' 1 1 1 1 1 1 1 1 1 1 , 1 1 1 1 1 1 t 1 ': " : --О 1 1 1 1 1 1 1 1 1 1 О
Graph visualization of errors between autocorrelation functions is shown at the Fig. 7. О 1 1 1 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 160 160 160 1 1 1 1 1 1 1 1 1 1 20 1 1
1 The tаЫе 1 contains mean errors calculated with diferent iunpt characteristics and allows to estimate the quality of the algorithm for dirfeent
spectral densities. ТаЫе 1 - Errors' testingwith diferent characteristics
Characteristics OIL = 2,51t; OIL = 2,51t; Т = 3,2 s; Т = 3,2 s; OIL = 3,0n; Т = 2,О s; ЮL = 3,Оп; Т = 2,О s;
Square spectral density method
N = 16; N = 57; N = 51; N = 50; n= n= n= n= n= ROI = 101t; ROI = 101t; ЮR= 6п; ЮR= 6п; 2 3 4 4 2
Right
rtiangle spectral density method ROI = 101t;
OIL = 2, 51t; Т = 3,2 s;
N = 32;
The acquired mean errors meet the requirements set at the beginning of the project. Factor analysis shown that separate input characteristics do not afect the mean eorrs directly which was expected ftom the random signal. 1 Fig 7. Error graph.
Meanerors 0,03335 0,03052 0,04902 0,01060 0,03335
Conclusion This paper starts new specatr theory development for high-dimensional signal simula tion. First steps using two-dimensional simulation proofed new direction of research. The paper shows that the spectral theory provides new approaches of research based on matrix mathematical paparatus.
The new algorithms could Ье prepared for further au tomatization Ъу the means of tool developed in such areas as machine learning, ratificial intelligence, and Ьig data processing.
The method of two-dimensional simulation of signals in а complex basis reduces algorithmizing to the execution of pre-derived mathematical formulas, which reduces the computational complexity and resource intensity of the algorithm, and the use of linear data structures positively acfet the scalaЬility of the developed solution. The use ofа complex basis that more accurately describes the nature ofongoing processes pro vides higher silmuation accuracy.
The software solution implemented in the Lazarus environment in the Free Pascal language meets the requirements and allows generating deterministic and random sig nals, as well as evaluating the quality ofsimulation Ьу displaying the епоr graph nad/or displaying the average епоr number.
The simulation method in the Hartley basis and the software solution based on it, as in the case ofthe complex basis, make the algorithm less resource intensive. The soft ware solution is implemented in Microsoft Visual Studio using the С# language. For both bases, both deteпninistic and random signals сап Ье silmuated (at the user's dis cretion), and the shape ofthe SPDF signal сап Ье selected: rectangular and rectangular irtangul.ar The signals obtained meet the expectations for the spectruщ and ewhn com peard with theoretical and algorithmic ACF, they show епоr levels ofless than 0.05. In ufture studies, it is planned to expand the choice ofsignal forms, as well as to test new methods on а wider range of tasks. It is also planned to create а library of obtained algorithms comЬined in а single solution that provides silmuation in diferent bases. It is worth noting that formulas era easier to implement in low-level programming lan guages, which means that the methods listed in section 4 сап Ье used in embedded and nricroprocessor technology.
Acknowledgments. This work is been financially supported Ьу the Russian Federation Ministry of Science and Higher Education in the framework of the Research Project titled "Compone'nts digital transformation methods' fundamental research rfo nricro and nanosystems" (Project #0705-2020-0041).