NOISE IMMUNITY RESEARCH FOR NONLINEAR DYNAMICAL SYSTEMS IDENTIFICATION BASED ON VOLTERRA MODEL IN FREQUENCY DOMAIN

computig@computigolie.et www.computigolie.et ISSN 1727-6209 Iteratioal Joural of Computig NOISE IMMUNITY RESEARCH FOR NONLINEAR DYNAMICAL SYSTEMS IDENTIFICATION BASED ON VOLTERRA MODEL IN FREQUENCY DOMAIN Vitaliy Pavleko 1), Sergei Pavleko 1), Viktor Speraskyy 2) 1) ICS, Odessa Natioal Polytechic Uiversity, 65044, 1, Shevcheko av., Odessa, Ukraie, pavleko_vitalij@mail.ru, psv85@yadex.ru 2) BEITI, Odessa Natioal Polytechic Uiversity, 65044, 1, Shevcheko av., Odessa, Ukraie, speraskyy@gmail.com, http://study.cc.ua Abstract: The accuracy ad oise immuity of the iterpolatio method of oliear dyamical systems idetificatio based o the Volterra model i the frequecy domai is studied i this paper. The polyharmoic sigals are used for the testig the method. The algorithmic ad software toolkit i Matlab is developed for the idetificatio procedure. This toolkit is used to costruct the iformatioal models of test system ad commuicatio chael. The model is built as a first-, secod- ad third-order amplitude frequecy ad phase frequecy characteristics. The compariso of obtaied characteristics with previous works is give. Wavelet deoisig is studied ad applied to reduce measuremet oise. Copyright Research Istitute for Itelliget Computer Systems, 2013. All rights reserved. Keywords: idetificatio; oliear dyamic systems; Volterra models; multifrequecy characteristics; polyharmoic sigals; wavelet deoisig; commuicatio chaels. 34 1. INTRODUCTION It is ecessary to cosider techical coditios of the commuicatio chaels (CC) operatio for effective data trasfer. Chages i evirometal coditios cause reducig the trasmissio data rate: i the digital CC up to a full stop of the trasmissio, i aalog CC to the oise ad distortio of the trasmitted sigals. The ew methods ad supportig toolkit are developig to automate the measuremet of parameters ad takig ito accout the characteristics of the CC. This toolkit allows obtaiig the iformatioal ad mathematical model of such oliear dyamic object, as the CC [1], i.e. to fid the idetificatio problem solutio. Moder cotiuous CCs are oliear stochastic iertial systems. The model i the form of itegro power Volterra series used to idetify them [2 5]. The oliear ad dyamic properties of such system are completely characterized by a sequece of multidimesioal weightig fuctios Volterra kerels. Buildig a model of oliear dyamic system i the form of a Volterra series lies i the choice of the test actios form. Also it uses the developed algorithm that allows determiig the Volterra kerels ad their Fourier images for the measured resposes (multidimesioal amplitude frequecy characteristics (AFC) ad phase frequecy characteristics (PFC)) to simulate the CC i the time or frequecy domai, respectively [6, 7]. The additioal research of oise immuity to measuremet oise for oliear dyamical systems idetificatio method, based o the Volterra model i the frequecy domai is proposed. The developed idetificatio toolkit used to build iformatio model of the test oliear dyamic object i the form of the first, secod ad third order model [14] where updated. 2. INTERPOLATION METHOD OF NONLINEAR DYNAMICAL SYSTEMS IDENTIFICATION The presetatio of the iput output type oliear dyamical system preseted by Volterra series were give i previous work [6]. A iterpolatio method of idetificatio of the oliear dyamical system based o Volterra series is used [7 8, 14]. It is used fold differetiatio of a target sigal o parameter amplitude a of test actios to separate the resposes of the oliear dyamical system o partial compoets yˆ ( t) [8]. The ax(t) type test sigal is set to iput of the system, where x(t) ay fuctio; a 1 scale

factor for th order partial compoet allocatio yˆ ( t) from the measured respose of oliear dyamical system y[ax(t)]. I such case it is ecessary to fid th private derivative of the respose o amplitude a at a=0 t t ˆ y ( t )... w (,..., ) x ( t ) d 1 l l 0 times 0 l1 1 y a x( t)! a a0. Partial compoets of resposes yˆ ( t) ca be calculated by usig the test actios ad procedure (1). Diagoal ad subdiagoal sectios of Volterra kerel are defied o basis of calculated resposes. Formulas for umerical differetiatio usig cetral differeces for equidistat kots yr y[ ar x( t)] y[ rhx( t)], r r1, r1 1,..., r2 with step of computatioal mesh o amplitude h= a were received [8]. Certai limitatios should be imposed o choice of frequecy polyharmoic test sigals while determiig multidimesioal AFC ad PFC [6]. That s why the values of AFC ad PFC i this limited poits of multidimesioal frequecy space ca be calculated usig iterpolatio oly. I practical realizatio of oliear dyamical systems idetificatio it is eeded to miimize quatity of such udefied poits at the rage of multidimesioal frequecy characteristics determiatio. This doe to provide a miimum of restrictios o choice of frequecy of the test sigal. It is show that existed limitatio ca be weakeed. New limitatios o choice of frequecy are reducig quatity of udefied poits. It is defied: to obtai Volterra kerels for oliear dyamical system i frequecy domai the limitatios o choice of frequecies of test polyharmoic sigals have to be restricted. This restrictios provide iequality of combiatio frequecies i the test sigal harmoics. It is ecessary to cosider the imposed costraits o choice of the test polyharmoic sigal frequecies durig determiatio of multidimesioal trasfer fuctios of oliear systems. It provides a iequality of combiatio frequecies i output sigal harmoics: ω 1 0, ω 2 0 ad ω 1 ω 2 for the secod order idetificatio procedure, ad ω 1 0, ω 2 0, ω 3 0, ω 1 ω 2, ω 1 ω 3, ω 2 ω 3, 2ω 1 ω 2 +ω 3, 2ω 2 ω 1 +ω 3, 2ω 3 ω 1 +ω 2, 2ω 1 ω 2 ω 3, 2ω 2 ω 1 ω 3, 2ω 3 ω 1 ω 2, 2ω 1 ω 2 +ω 3, 2ω 2 ω 1 +ω 3 и 2ω 3 ω 1 +ω 2 for the third order idetificatio procedure. (1) 3. THE TECHNIQUE OF THE TEST OBJECT IDENTIFICATION Described method was fully tested o a oliear test object (Fig. 1) described by Riccati equatio: dy( t) 2 y( t) y ( t) u( t)., (2) dt Fig. 1 Matlab-Simulik model of the test object. Aalytical expressios of AFC ad PFC for the first, secod ad third order model were received ad preseted i [8]. The mai purpose was to idetify the multifrequecy performaces characterizig oliear ad dyamical properties of oliear test object. Volterra model i the form of the secod order polyomial is used. Thus, test object properties are characterized by trasfer fuctios of W 1 (jω 1 ), W 2 (jω 1,jω 2 ), W 3 (jω 1,jω 2,jω 3 ) by Fourier images of weight fuctios w 1 (t), w 2 (t 1, t 2 ), w 3 (t 1, t 2, t 3 ). Structure chart of idetificatio procedure determiatio of the 1 st, 2 d or 3 rd order AFC of CC is preseted o Fig. 2. The weighted sum is formed from received sigals resposes of each group (Fig. 2). As a result the partial compoets of CC resposes y 1 (t), y 2 (t) ad y 3 (t) are got. For each partial compoet of respose the Fourier trasform (the FFT is used) is calculated, ad from received spectrum oly a iformative harmoics (which amplitudes represet values of required characteristics of the first, secod ad third orders AFC) are take. Fig. 2 The structure chart of idetificatio procedure usig the k- order Volterra model i frequecy domai, umber of experimets N=4, k order of the estimated Volterra kerel. The first order AFC W 1 (jω 1 ) ad PFC argw 1 (jω 1 ), where ω 1 =ω are received by extractig the harmoics with frequecy f from the spectrum of the CC partial respose y 1 (t) to the test sigal x(t)=a/2(cosωt). 35

The secod order AFC W 2 (jω 1,jω 2 ) ad PFC argw 2 (jω 1,jω 2 ), where ω 1 =ω ad ω 2 =ω 1 +Ω 1, were received by extractig the harmoics with summary frequecy ω 1 +ω 2 from the spectrum of the CC partial respose y 2 (t) to the test sigal x(t)=(a/2)(cosω 1 t+cosω 2 t). The third order AFC W 3 (jω 1,jω 2,jω 3 ) ad PFC argw 3 (jω 1,jω 2,jω 3 ), where ω 1 =ω, ω 2 =ω 1 +Ω 1, ω 3 =ω 2 +Ω 2 were received by extractig the harmoics with summary frequecy ω 1 +ω 2 +ω 3 from the spectrum of the CC partial respose y 2 (t) to the test sigal x(t)=(a/2)(cosω 1 t+cosω 2 t+cosω 3 t). The results (first, secod ad third order AFC ad PFC) which had bee received after procedure of idetificatio are represeted i Fig. 3 5 (umber of experimets for the model N=4). The surfaces show o Fig. 6 9 are built from subdiagoal cross sectios which were received separately. Ω 1 was used as growig parameter of idetificatio with differet value for each cross sectio i secod order characteristics. Fixed value of Ω 2 ad growig value of Ω 1 were used as parameters of idetificatio to obtai differet value for each cross sectio i third order characteristics. Fig. 6 Surface of the test object AFC built of the secod order subdiagoal cross sectios received for N=4, Ω 1 =0,01 rad/s. Fig. 3 First order AFC ad PFC of the test object: aalytically calculated values (1), sectio estimatio values umber of experimets for the model N=4 (2). Fig. 4 Secod order AFC ad PFC of the test object: aalytically calculated values (1), subdiagoal cross sectio values with umber of experimets for the model N=4 (2), Ω 1 =0,01 rad/s. Fig. 7 Surface of the test object PFC built of the secod order subdiagoal cross sectios received for N=4, Ω 1 =0,01 rad/s. Fig. 5 Third order AFC ad PFC of the test object: aalytically calculated values (1), subdiagoal crosssectio values with umber of experimets for the model N=6 (2), Ω 1 =0,01 rad/s, Ω 2 =0,1 rad/s. 36 Fig. 8 Surface of the test object AFC built of the third order subdiagoal cross sectios received for N=6, Ω 1 =0,01 rad/s, Ω 2 =0,1 rad/s.

measuremets i model. The sum of these two sigals for the liear test model is show i Fig. 11. Fig. 10 The Simulik model of the test object with oise geerator ad osillosopes. Fig. 9 Surface of the test object PFC built of the third order subdiagoal cross-sectios received for N=6, Ω 1 =0,01 rad/s, Ω 2 =0,1 rad/s The secod order surfaces for AFC ad PFC had bee received after procedure of the test object idetificatio ad are show i Fig. 6 7 (umber of experimets for the model N=4). The third order surfaces for AFC ad PFC had bee received after procedure of the test object idetificatio ad are preseted i Fig. 8 9 (umber of experimets for the model N=6). Numerical values of idetificatio accuracy usig iterpolatio method for the test object are represeted i Table 1. Table 1. Numerical values of idetificatio accuracy usig iterpolatio method. Kerel order, k 1 2 3 Experimets quatity, N AFC relative error, % PFC relative error, % 2 2.1359 2.5420 4 0.3468 2.0618 6 0.2957 1.9311 2 30.2842 76.8221 4 2.0452 3.7603 6 89.2099 5.9438 4 10.9810 1.6280 6 10.7642 1.5522 4. THE STUDY OF NOISE IMMUNITY OF THE IDENTIFICATION METHOD Experimetal research of the oise immuity of the idetificatio method were made. The mai purpose was the studyig of the oise impact (oise meas the iexactess of the measuremets) to the characteristics of the test object model usig iterpolatio method i frequecy domai. The first step was the measuremet of the level of useful sigal after test object (Out2 i Fig. 10). The amplitude of this sigal was defied as the 100% of the sigal power. After that procedure the Radom Noise sigal where added to the test object output sigal. This where made to simulate iexactess of the Fig. 11 The oised sigal of the test object, the level of oise is 50% of source sigal. The simulatios with the test model were made. Differet oise levels were defied for differet order of the model. The automatic wavelet deoisig were used to reduce the oise impact o fial characteristics of the test object. The Daubechie wavelet of the 2 ad 3 level were chose ad used for the AFC ad PFC deoisig respectively [9, 13]. The first order (liear) model was tested with the level of oise 50% ad 10% ad showed excellet level of oise immuity. The secod order (oliear) model was tested with the level of oise 10% ad 1% ad showed good level of oise immuity. The oised (Fig. 12) ad de-oised (filtered) (Fig. 13) characteristics (AFC ad PFC) with level of oise 10% are preseted. The third order (oliear) model was tested with the level of oise 10% ad 1% ad showed good level of oise immuity. Fig. 12 Noised characteristics (AFC top, PFC bottom) of the 2d order for the test object model with oise level 10%. 37

Fig. 13 Deoised characteristics (AFC top, PFC bottom) of the 2d order for the test object model with oise level 10% The umerical values of stadard deviatio (SD) of the idetificatio accuracy before ad after wavelet deoisig procedure are show i Table 2. Table 2. Stadard deviatio with oise impact. k N Noise level = 10% Noise level = 1% Improvemet SD for SD for SD for SD for for for AFC PFC AFC PFC AFC, PFC, (without / with deoisig) times times 2 0.000097 0.09031 /0.000063 /0.07541 1,540 1,198 1 4 0.000271 0.07804 /0.000181 /0.06433 1,497 1,213 6 0.000312 0.12913 /0.000223 /0.09889 1,399 1,306 2 0.000920 0.52063 /0.000670 /0.51465 1,373 1,012 2 4 0.001972 0.28004 /0.001663 /0.06877 1,186 4,072 6 0.004165 0.39260 /0.003908 /0.19237 1,066 2,041 0.000288 0.89857 4 1,003 1,467 /0.000288 /0.61251 3 0.000461 0.84868 6 1,310 1,431 /0.000352 /0.59319 The diagrams, showig the improvemet of stadard deviatio for idetificatio accuracy usig the automatic wavelet deoisig of the received characteristics (AFC ad PFC) are show i Fig. 14 ad Fig. 15 respectively. Fig. 14 Stadard deviatio chagig for AFC usig automatic Wavelet-deoisig. Fig. 15 Stadard deviatio chagig for PFC usig automatic Wavelet-deoisig. 5. HARDWARE-SOFTWARE TOOLKIT AND TECHNIQUE OF RADIOFREQUENCY CC IDENTIFICATION Experimetal research of the Ultra High Frequecy rage CC were doe. The mai purpose was the idetificatio of multifrequecy performaces that characterize oliear ad dyamical properties of the CC. Volterra model i the form of the secod order polyomial is used. Thus physical CC properties are characterized by trasfer fuctios of W 1 (j2πf) ad W 2 (j2πf 1,j2πf 2 ) by the Fourier images of weightig fuctios w 1 (t) ad w 2 (t 1, t 2 ). Implemetatio of idetificatio method o the IBM PC computer basis has bee carried out usig the developed software i Matlab software. The software allows automatig the process of the test sigals formig with the give parameters (amplitudes ad frequecies). Also this software allows trasmittig ad receivig sigals through a output ad iput sectio of PC soudcard, to produce segmetatio of a file with the resposes to the fragmets, correspodig to the CC resposes beig researched o test polyharmoic effects with differet amplitudes. I experimetal research two idetical marie trasceivers S.P.RADIO A/S SAILOR RT2048 VHF (the rage of operatioal frequecies is 154,4 163,75 MHz) at 16 th operatioal chael ad IBM PC with Creative SBLive! soudcards were used. These trasceivers are ow used at most ships for commuicatio with coast port statios. Sequetially AFC of the first, secod ad third orders were defied. The method of idetificatio with umber of experimets N=4 was applied. Geeral scheme of a hardware software complex of the CC idetificatio, based o the data of iput output type experimet was studied i [6]. The CC received resposes y[ ai x( t)] to the test sigals a i x(t), compose a group of the sigals, 38

which amout is equal to the used umber of experimets N (N=4), show i Fig. 16. I each followig group the sigals frequecy icreases by magitude of chose step. А cross correlatio was used to defie the begiig of each received respose. Iformatio about the form of the test sigals give i [7] were used. The wavelet oise suppressio was used to smooth the output data of the experimet [9]. The results received after digital data processig of the data of experimets (wavelet Coiflet de oisig) for the first, secod ad third order AFC are preseted i Fig. 17 20. Fig. 17 AFC of the first order after wavelet Coiflet secod level deoisigю Fig. 16 The group of sigals received from CC with amplitudes: -1 (1); 1 (2); -0,644 (3); 0,644 (4); N=4. I described experimet with use of soud card the maximum allowed amplitude was A=0,25V (defied experimetally). The rage of frequecies was defied by the soud card pass bad (20 20000 Hz), ad frequecies of the test sigals has bee chose from this rage, takig ito accout restrictios specified above. Such parameters were chose for the experimet: start frequecy f s =125 Hz; fial frequecy f e =3125 Hz; a frequecy chage step f=125 Hz; to defie AFC of the secod order determiatio, a offset o frequecy F 1 =f 2 -f 1 was icreasigly growig from 201 to 3401 Hz with step 100 Hz. The weighed sum is formed from received sigals resposes of each group (Fig. 2). As a result we get partial compoet s of respose of the CC y 1 (t) ad y 2 (t). For each partial compoet of respose a Fourier trasform (the FFT is used) is calculated, ad from received spectra oly a iformative harmoics (which amplitudes represets values of required characteristics of the first ad secod orders AFC) are take. The first order AFC W 1 (j2πf) is received by extractig the harmoics with frequecy f from the spectrum of the partial respose of the CC y 1 (t) to the test sigal x(t)=a/2(cos2πft). The secod order AFC W 2 (j2πf,j2π(f+f 1 )), where f 1 =f ad f 2 =f+f 1, was received by extractig the harmoics with summary frequecy f 1 +f 2 from the spectrum of the partial respose of the CC y 2 (t) to the test sigal x(t)=(a/2)(cos2πf 1 t+cos2πf 2 t). Fig. 18 Subdiagoal cross sectios of AFCs of the secod order after wavelet Coiflet secod level deoisig at differet frequecies: 201 (1), 401 (2), 601 (3), 801 (4), 1001 (5), 1401 (6) Hz. The surfaces show i Fig. 19 20 were built from subdiagoal cross sectios that have bee received separately. A growig parameter of idetificatio f with differet value for each sectio was used. Fig. 19 Surface built of AFC cross sectios of the secod order after wavelet Coiflet 3rd level deoisig. 39

Fig. 20 Surface built of AFC cross sectios of the third order after wavelet Coiflet 3rd level deoisig, where f 3 = f 1 +100 Hz. 6. CONCLUSION The method based o Volterra model usig polyharmoic test sigals for idetificatio oliear dyamical systems is aalyzed. The method based o compositio of liear resposes combiatio o test sigals with differet amplitudes were used to differetiate the resposes of object for partial compoets. New values of test sigals amplitudes were defied ad model were validated usig the test object. Excellet accuracy level for received model is achieved as i liear model so i oliear oes. Give values are greatly raisig the accuracy of idetificatio i compare to amplitudes ad coefficiets studied i [10, 11]. The idetificatio accuracy of oliear part for the test object has grow for 5-20% while the stadard deviatio i best cases is o more tha 10% that meas excellet adequacy of used method. The oise immuity is very high for the liear model, high eough for the secod order oliear model ad has moderate oise immuity for the third order model. The wavelet deoisig is very effective ad gives the possibility to improve the quality of idetificatio of the oisy measuremets up to 1,54 ad 4,07 times for the AFC ad PFC respectively. Iterpolatio method of idetificatio usig the hardware methodology used i [11, 12] is applied for costructig of iformatioal Volterra model as a APC of the first ad secod order for UHF bad radio chael. Received results had cofirmed sigificat oliearity of characteristics of the tested objects that leads to distortios of sigals i differet type radio devices. 7. REFERENCES [1] G. B. Giaakis, E. Serpedi, A bibliography o oliear system idetificatio ad its applicatios i sigal processig, commuicatios ad biomedical egieerig, Sigal Processig EURASIP, Elsevier Sciece B.V., (81) 3 (2001), pp. 533 580. [2] D. T. Westwick, Methods for the Idetificatio of Multiple Iput Noliear Systems, Departmets of Electrical Egieerig ad Biomedical Egieerig, McGill Uiversity, Motreal, Quebec, Caada, 1995, pp. 192 232. [3] F. J. Doyle, R. K. Pearso, B. A. Oguaike, Idetificatio ad Cotrol Usig Volterra Models, Published Spriger Techology & Idustrial Arts, 2001, pp. 58 72. [4] S. Boyd, Y. S. Jag, L. O. Chua, Measurig Volterra kerels, IEEE Trasactios o Circuits ad Systems, (CAS-30) 8 (1983), pp. 571 578. [5] K. V. Peddaarappagari, M. Bradt-Pearce, Volterra series approach for optimizig fiber optic commuicatios system desigs, J. Lightwave Tech., (16) 11 (1998), pp. 2046 2055. [6] V. D. Pavleko, V. I. Lomovoy, V. O. Speraskyy, Modellig of radiofrequecy commuicatio chaels usig Volterra model, Proceedigs of the 6th IEEE Iteratioal Coferece o Itelliget Data Acquisitio ad Advaced Computig Systems: Techology ad Applicatios, Vol. 2, 2011, pp. 574-579. [7] V. D. Pavleko, Idetificatio of oliear dyamic systems i the form of the Volterra kerels o the basis of the data of pulse respose measuremets, Electroic Modelig, (32) 3 (2010), pp. 3 18. (i Russia). [8] M. Schetze, The Volterra ad Wieer Theories of Noliear Systems, Wiley & Sos, New York., 1980, pp. 321 360. [9] J. G. Goswami, A. K. Cha, Fudametals of Wavelets: Theory, Algorithms, ad Applicatios, Publishig Joh Wiley & Sos, Ic., 1999, pp. 125 137. [10] L. V. Dailov, P. N. Mathaov, E. S. Philipov, The theory of oliear electrical circuits, Published Eergoatomizdat, Leigrad, 1990, pp. 136 148 (i Russia). [11] V. D. Pavleko, V. O. Speraskyy, Commuicatio chael idetificatio i frequecy domai based o the Volterra model, Proceedigs of the Iteratioal Coferece o Computers, Digital Commuicatios ad Computig (ICDCC'11), Barceloa, Spai, September 15-17, 2011, Published by WSEAS Press, 2011, pp. 218 222. [12] V. D. Pavleko, S. V. Pavleko, V. O. Speraskyy, Iterpolatio method of oliear dyamical systems idetificatio based o Volterra model i frequecy domai, Proceedigs of the 7 th IEEE Iteratioal 40

Coferece o Itelliget Data Acquisitio ad Advaced Computig Systems: Techology ad Applicatios (IDAACS 2013), Berli, Germay, 15 17 September 2013, pp. 173 178. [13] M. Misiti, Y. Misiti, G. Oppeheim, J.-M. Poggi, Wavelets toolbox users guide, The MathWorks. Wavelet Toolbox, for use with MATLAB, 2000. Vitaliy Pavleko, D.Sc., seior research associate. Educatio: 1970 Studet, Odessa Polytechic Uiversity, 1973 Post graduate Studet, Computers Departmet, Odessa Polytechic Uiversity. Curret positio: professor of Computer Systems Istitute. Scietific iterests: Modelig ad Simulatio for Idustrial Applicatios;No-parametric Idetificatio of Noliear Systems; Theory of the Volterra series; Mathematical methods, models ad techologies for complex systems' research; Problems of data ad kowledge miig; Statistic Classificatio; Neural ets for simulatio; Diagostic, Fault detectio, Forecastig; Computers Iformatio Techologies; Computer ad telecommuicatio systems; Parallel computig algorithms ad methods of parallel programmig. Sergey Pavleko, PhD studet. Educatio: 2007 Studet, Odessa Natioal Polytechic Uiversity, 2010 Post graduate Studet, Computer Systems Departmet, Odessa Natioal Polytechic Uiversity. Curret positio: Juior researcher of Eergy Maagemet Istititute. Scietific iterests: Modelig ad Simulatio for Idustrial Applicatios; No-parametric Idetificatio of Noliear Systems; Theory of the Volterra series; Mathematical methods, models ad techologies for complex systems' research; Problems of data ad kowledge miig. Viktor Speraskyy, PhD studet. Educatio: 2003 Studet, Odessa State Academy of Refrigeratio, 2007-2012 Post graduate Studet, Computer Systems Departmet, Odessa Natioal Polytechic Uiversity. Curret positio: seior lecturer of Ecoomic cyberetics ad iformatioal techologies departmet i the Busiess, Ecoomics ad Iformatioal Techologies Istitute. Scietific iterests: Modelig ad Simulatio for Idustrial Applicatios; No-parametric Idetificatio of Noliear Systems; Theory of the Volterra series; Mathematical methods, models ad techologies for complex systems' research; Computer Iformatio Techologies; Wired ad wireless commuicatios; Programmig algorithms; Operatio systems. 41