COMPUTATION OF DYADIC CONVOLUTION ON GPU FOR EFFICIENT MODELING OF DYADIC LTI SYSTEMS UDC :

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1 FACTA UNIVERSITATIS Series: Automatic Cotrol ad Robotics Vol. 10, N o 1, 2011, pp COMPUTATION OF DYADIC CONVOLUTION ON GPU FOR EFFICIENT MODELING OF DYADIC LTI SYSTEMS UDC : GPU Duša B. Gajić, Radomir S. Staković Departmet of Computer Sciece, Faculty of Electroic Egieerig, Uiversity of Niš, Aleksadra Medvedeva 14, Niš, Serbia dule.gajic@gmail.com, radomir.stakovic@gmail.com Abstract. Covolutio systems are a importat class of systems ofte used i may practical applicatios i differet areas of sciece ad egieerig. I the case of systems with biary ecoded iput ad output sigals, fiite dyadic groups are usually assumed as the uderlyig algebraic structure. Modelig ad hadlig of such systems requires frequet computatio of the dyadic covolutio. This paper discusses a method for efficiet computatio of the dyadic covolutio o Graphics Processig Uits (GPUs) with the related algorithm implemeted i Ope Computig Laguage (OpeCL). We cosider several importat issues cocerig the efficiet mappig of the algorithm to the GPU architecture. Performace of the proposed implemetatio is compared with the referet C/C++ implemetatio processed o the Cetral Processig Uit (CPU). Experimetal results cofirm that sigificat speedups are achieved by the applicatio of the proposed GPU calculatio method. Key words: Liear traslatio ivariat (LTI) systems, Dyadic covolutio, Fast Walsh trasform (FWT), GPU parallel programmig, OpeCL 1. INTRODUCTION Liearity, causality, ad traslatio ivariace are features ofte assumed for mathematical models of may systems useful i practice. These features directly lead to covolutio systems, sice ay causal liear traslatio ivariat (LTI) system ca be represeted by a covolutio system. At the same time, ay covolutio system is liear, causal, ad traslatio ivariat. Depedig o the class of sigals to be processed, systems ca be time-ivariat, shift-ivariat, or rotatio-ivariat. For these systems, groups of real umbers ad complex umbers, respectively, are usually assumed as domais for the defiitio of iput ad Received July 06, 2011 Ackowledgemet. The research reported i this paper is partly supported by the Miistry of Educatio ad Sciece of the Republic of Serbia, project OI ( ).

2 60 D. B. GAJIĆ, R. S. STANKOVIĆ output sigals. There are, however, areas where systems o groups differet from these groups are more suitable as mathematical models. I particular, dyadic systems, i.e., systems modeled o dyadic groups (defied bellow), are ofte met i commuicatios, computig, ad cryptography. Dyadic systems were itroduced by Weiser i [36], ad their theory was further cotributed by Harmuth [16, 17], based o the backgroud mathematical theory by Polyak ad Shreider [29]. The dyadic systems have bee itesively studied by Pichler i a series of publicatios [22, 23, 24, 25, 26, 27, 28], ad by several other authors [5, 7]. Recetly, logic etworks, which ca be viewed as facets of digital systems modeled o dyadic groups, were studied from the system theory poit of view [34]. I study ad practical applicatios of dyadic systems, computatio of the dyadic covolutio is frequetly required. Efficiecy of these computatios ofte determies the applicability of such mathematical models i egieerig practice. With that motivatio, i this paper we preset a efficiet techique for a accelerated calculatio of the dyadic covolutio, through a parallel implemetatio of the fast algorithm derived from the covolutio theorem ad processed o the Graphics Processig Uit (GPU). Due to the covolutio theorem, computatio of the dyadic covolutio coverts ito performig two direct ad a iverse Walsh trasform, which ca be doe by the correspodig FFT-like algorithms, i.e., the Fast Walsh Trasform FWT [4, 8, 18]. The proposed implemetatio is developed usig the Ope Computig Laguage (OpeCL) ad processed o a GPU. Experimetal results ad comparisos with the classical implemetatio cofirm that the proposed method leads to a sigificat computatioal speedup. This paper is based o a shorter prelimiary versio preseted at the coferece ICEST 2011 [11], ad it is orgaized as follows. After a itroductio to the ecessary backgroud theory i Sectio 2, Sectio 3 presets a brief review of related work. Sectio 4 is devoted to the descriptio of the mappig of the fast algorithm for computig the dyadic covolutio to the GPU architecture ad the desig of the correspodig OpeCL implemetatio. I Sectio 5, we preset the experimetal eviromet used to evaluate the method, ad report experimetal results. The closig Sectio 6 offers some coclusios draw from the preseted research Dyadic Covolutio 2. BACKGROUND THEORY Covolutio is a mathematical operatio that expresses relatioships betwee values of two sigals (modeled by fuctios f ad g) i poits at a fixed distace. The covolutio C = f gis a fuctio that resembles either the fuctio f or the fuctio g, modified by the other. Whe the fiite dyadic group is used as the algebraic structure o which the covolutio is defied, we use the term dyadic covolutio [4, 18, 30, 32]. The fiite dyadic group of order is defied as C2 C2, where C 2 = ({0,1}, ), ad i1 stads for the additio modulo 2, ad is the direct (Cartesia) product.

3 Computatio of Dyadic Covolutio o GPU for Efficiet Modelig of Dyadic LTI Systems 61 For two fuctios f, g : C2, where is the field of ratioal umbers, the dyadic covolutio is defied as C f g 2 1 ( ) f( x) g( x ), x0 1 0,1,..., 2. (1) I biary otatio, x = (x 1, x 2,, x ), ad = ( 1, 2,, ), where x i, i {0,1}. A direct calculatio of the dyadic covolutio from (1) has the expoetial complexity i the umber of iputs ad is ufeasible i practice for large sigals. Therefore, algorithms for the fast computatio of covolutio are derived by usig the covolutio theorem [18] o the correspodig algebraic structure Walsh Trasform The Walsh trasform [18] is defied by the Walsh matrix W( ) W (1), (2) i1 where deotes the Kroecker product ad 1 1 W (1) 1 1 (3) is the basic Walsh matrix. From (2), W() is geerated by the Kroecker product ad, therefore, the Walsh fuctios (rows of W()) appear i the Hadamard order, see for istace [17]. Sice W() is a self-iverse matrix up to the scalar 2 -, the iverse Walsh trasform is defied as 1 W ( ) 2 W ( ). (4) It follows that both the direct ad the iverse Walsh trasforms ca be computed by usig the same algorithm. The Walsh spectrum S f, h = [S f,h (0), S f,h (1),, S f,h (2-1)] T of a fuctio f : C2, specified by the fuctio vector F = [f (0), f (1),, f (2-1)] T, is defied as S, W( ) F. (5) f h The spectral coefficiets appear i Hadamard orderig, which is idicated by the idex h i S f,h. The fuctio f is recostructed from the Walsh spectrum as F W S (6) 1 2 ( ) f, h, ad (5) ad (6) form the Walsh trasform pair. Example 1. If f : C is specified by the fuctio vector F = [1, 0, 1, 1] T, the 2 2 the correspodig Walsh spectrum S f,h = [S f,h (0), S f,h (1), S f,h (2), S f,h (3)] T ca be calculated directly by usig its defiitio (5)

4 62 D. B. GAJIĆ, R. S. STANKOVIĆ S f, h (1) (2) W. i1 F W F (7) Fast Walsh Trasform The computatio of the Walsh trasform based o its defiitio (5) is iefficiet, sice it expresses the O(N 2 ) time complexity, where N = 2 is the size of the iput vector. Fortuately, a more efficiet algorithm, the Fast Walsh trasform (FWT) with the time complexity of O(NlogN) exist, see for istace [4, 18]. The Fast Walsh trasform ca be defied by usig the followig factorizatio [18] W( ) C w ( ) i i, (8) i1 where j j W(1), i j, Cw ( ) C (1), (1) i w C 1 i w (9) j i I(1), i j. The matrix C ( w ) defies the partial Walsh trasform ad correspods to the i-th step i of the FWT. This particular factorizatio leads to a FFT-like algorithm of the Cooley- Tukey type [12, 18]. 2 Example 2. For the fuctio f : C2 i Example 1, the Walsh spectrum ca be calculated with 4 additios ad 4 subtractios by applyig the FWT algorithm, istead of 16 multiplicatios ad 12 additios i the direct computatio usig the defiitio. This algorithm is derived from the followig factorizatio of the matrix W(2) where S f, h W(2) F ( CC 1 2) F, (10) C1 W(1) I (1), (11) C2 I(1) W (1) (12) Computatios i C 1 ad C 2 ca be performed by the algorithms whose flow-graphs are show i Fig. 1 (a) ad (b). I this figure, the solid ad dashed lies deote additio ad subtractio, respectively.

5 Computatio of Dyadic Covolutio o GPU for Efficiet Modelig of Dyadic LTI Systems 63 (a) Fig. 1 Flow-graphs of the FWT for = 2, (a) First step derived from C 1, (b) Secod step derived from C 2. (b) Therefore, computig the Walsh spectrum goes as follows f (0) S f, h(0) f (1) S f, h(1) F S f, h. (13) f (2) S f, h(2) f (3) S f, h(3) 2.4. Covolutio Theorem I classical Fourier aalysis, the covolutio theorem states that the Fourier trasform of the covolutio fuctio C = f g is the compoetwise product of the Fourier trasforms of f ad g, thus S f g = S f S g, see for istace [4, 8, 18]. I other words, a rather complex operatio i the origial domai coverts ito a simple operatio i the spectral domai. For fuctios o the fiite dyadic group C 2, the computatio of the dyadic covolutio through the applicatio of the covolutio theorem is doe as follows -1 Cf g 2 W ( )(( W( ) F)( W( ) G )). (14) Therefore, a efficiet algorithm for the computatio of the dyadic covolutio ca be developed i terms of the FWT, as illustrated i Fig. 2 for = 2, N = 4. Fig. 2 Computatio of the dyadic covolutio through the applicatio of the covolutio theorem for = 2, N = 4

6 64 D. B. GAJIĆ, R. S. STANKOVIĆ Example 3. For two fuctios f, g : 2 C2, give by their respective fuctio vectors F = [1, 0, 1, 1] T ad G = [0, 1, 0, 1] T, performig the FWT o each of them produces their Walsh spectra S f,h = [3, 1, -1, 1] T ad S g,h = [2, -2, 0, 0] T, respectively. The result of the compoetwise multiplicatio of these two spectra is S f.g,h = [6, -2, 0, 0] T. After the iverse FWT is performed o S f.g,h, followed by multiplicatio with the scalig factor 2-2, we get the dyadic covolutio coefficiets C [1,2,1,2] T. f g 3. BACKGROUND WORK The fast algorithm for the dyadic covolutio is based o the applicatio of the Walsh trasform which is the Fourier trasform o fiite dyadic groups. The implemetatio of various Fast Fourier Trasforms (FFTs) o differet techological platforms is a widely cosidered topic, see for istace [4, 8, 11, 15] ad refereces therei. The calculatio of dyadic covolutio o classical Cetral Processig Uits (CPUs) through the applicatio of the covolutio theorem, both o vectors ad decisio diagrams, is preseted i [30]. Referece [4] presets a applicatio of the dyadic covolutio for the fast multiplicatio of hyper-complex umbers. I recet years, the techique of performig Geeral Purpose computatios o the GPU (GPGPU) has prove to be a suitable approach i solvig may computatioallyitesive tasks [2, 3, 12, 15, 19]. I particular, the GPU-accelerated calculatio of FFT algorithms usig CUDA is described i [15, 20]. Referece [12] presets a OpeCL implemetatio of the FWT that uses the GPU hardware ad leads to sigificat speedups over traditioal CPU processig. However, i our best kowledge, there are o publicatios except the prelimiary versio of this work [11] that discuss either CUDA or OpeCL GPU implemetatios of the fast algorithm for the computatio of dyadic covolutio based o the covolutio theorem. This fact, together with the iteded applicatio of the dyadic covolutio for the modelig of dyadic LTI systems, was the motivatio for the research reported i this paper. 4. MAPPING OF THE ALGORITHM AND IMPLEMENTATION DETAILS The covolutio theorem, expressed i (14), leads to the followig fast algorithm for the calculatio of the dyadic covolutio, illustrated by the example i Fig. 2: Step 1. Perform the FWT o f ad g ad compute their Walsh spectra S f ad S g. Step 2. Perform the compoetwise multiplicatio of S f ad S g. Step 3. Perform the iverse FWT over S f S g to obtai C fg.

7 Computatio of Dyadic Covolutio o GPU for Efficiet Modelig of Dyadic LTI Systems 65 Algorithm 1 FAST CALCULATION OF DYADIC CONVOLUTION C = f * g 1 Allocate buffers buffer1 ad buffer2 i the global memory of the GPU device. 2 Trasfer iput vectors f ad g from the host CPU memory to GPU buffers buffer1 ad buffer2, respectively. 3 Perform the Walsh trasform o vectors stored i buffer1 ad buffer2 usig the followig i-place OpeCL implemetatio of the Cooley-Tukey algorithm for the FWT: a. For each step of the FWT, from step 0 to step (log 2 N) - 1, call the OpeCL kerel for the FWT with iput parameters beig the appropriate buffer i the GPU s global memory ad the value of the curret step 2 step. The kerel is executed by N/2 threads i parallel o the GPU. Each thread reads two elemets, determied by (15) ad (16), from the buffer, performs the operatios defied by the Walsh matrix W (1) ad stores back the results i the same locatios. 4 After computig the FWT of both vectors, execute the OpeCL kerel for the compoetwise multiplicatio of the two Walsh spectra with N threads executed i parallel. The resultig vector is stored i buffer1. 5 Perform the iverse FWT o buffer1 usig the same kerel as for the FWT. 6 Scale the cotets of buffer1 with the factor 2 - usig the OpeCL kerel with N threads executed i parallel. 7 Trasfer the cotets of the GPU buffer buffer1, which holds the resultig dyadic covolutio coefficiets, back to the host CPU memory. Sice both multiplicatio ad compoetwise multiplicatio of vectors ca be doe fast o moder CPUs ad GPUs, the key issue i developig a efficiet implemetatio of the above algorithm is performig the FWT ad the iverse FWT, required i steps 3 ad 5 of Algorithm 1, i a efficiet maer. Therefore, we developed a kerel cotaiig a OpeCL i-place implemetatio of the Cooley-Tukey algorithm for the FWT [12, 18]. As i all FFT-like algorithms, steps of the algorithm are executed sequetially ad parallelism is used oly withi the steps. I each step, N/2 threads are executed i parallel. This large umber of threads permits to overcome the data access latecy to the GPU global memory [2, 3]. Each thread reads two elemets from the GPU buffer with idices op1 ad op2 calculated as op1 thread_id mod step + 2 step (thread_id/step), (15) op2 op1 + step. (16) Parameters thread_id ad step are the global idetifier of the thread ad the idetifier of the curret step of the algorithm, respectively. All threads execute the elemetary butterfly operatio defied by the basic Walsh trasform matrix W(1) ad store the results back i the same locatios i the GPU memory, as i other implemetatios of the iplace FWT algorithms. The compoetwise multiplicatio of vectors is also performed by the correspodig OpeCL kerel which is executed by N threads i parallel, with each thread multiplyig the two correspodig elemets of the iput vectors. After the multiplicatio, the iverse FWT is performed with the same kerel that is used for the direct trasform, followed by scalig with 2 -. The scalig is also performed i parallel through the executio of the correspodig OpeCL kerel.

8 66 D. B. GAJIĆ, R. S. STANKOVIĆ Before executig ay of the kerels, both iput vectors are trasferred from the mai memory to the GPU global memory. After the calculatios, the resultig covolutio coefficiets are copied back to the host. These memory operatios take a sigificat share of the total GPU ruig times as reported i Sectio EXPERIMENTAL ENVIRONMENT AND RESULTS The test platform used to perform the experimets is a HP Pavilio dv7-4060us otebook computer (see Table 1). The OpeCL kerels are developed usig MS Visual Studio 2010 Ultimate ad ATI APP SDK 2.3 [1]. ATI Stream Profiler 2.1 is used for GPU performace aalysis, i accordace with istructios provided i [2]. The source code is compiled for the x64 platform. The referet C/C++ implemetatio is compiled with the maximum level of performace-orieted compiler optimizatios. As i all FFT implemetatios over vectors, the resultig performace is idepedet of the fuctio values. Therefore, we perform the experimets usig radomly geerated biary vectors. All values i Table 2 are a average for 10 program executios. We show the GPU calculatio time ad, also, the time for trasfer of data to/from the GPU which offers a more complete perspective to the reader. To estimate the performace of the proposed method, we performed a comparative aalysis with respect to the classical C/C++ implemetatio of the same algorithm processed o the CPU. Table 1. Specificatio of the experimetal platform. CPU RAM OS GPU - egie speed - global memory - compute uits - processig elemets - price AMD Pheom II N830 triple-core (2.1 GHz) 4GB DDR3 Widows 7 (64-bit) ATI Mobility Radeo MHz 1 GB DDR3 800 MHz ~ 100$ The results of the experimets are preseted i Table 2 ad Fig. 3. The OpeCL implemetatio processed o a iexpesive commodity GPU (Table 1) clearly outperforms the referet CPU implemetatio, by a factor of up to 5.5 whe oly calculatio times are compared, ad by a factor of up to 4.5 whe total times, icludig memory trasfers to/from GPU, are take ito accout. The executio of the same kerels o a more powerful GPU (with more stream cores ad a larger memory badwidth) would directly lead to much larger speedups, which would ot be the case if a more powerful CPU was used for the referet C/C++ implemetatio [3, 12].

9 Computatio of Dyadic Covolutio o GPU for Efficiet Modelig of Dyadic LTI Systems 67 Table 2. Ruig times for the referet CPU ad the proposed GPU implemetatio Time [ms] = log 2 N GPU CPU Computatio Memory time time Number of iput variables CPU Referet C/C++ implemetatio o the CPU GPU Proposed OpeCL C implemetatio o the GPU Fig. 3. Computatio times for the referet CPU ad the proposed GPU implemetatio 6. CONCLUSIONS The dyadic systems are a particular class of liear traslatio ivariat systems o fiite groups modeled i terms of the dyadic covolutio. Efficiet computatio of the dyadic covolutio is therefore essetial i practical applicatios of such systems.

10 68 D. B. GAJIĆ, R. S. STANKOVIĆ The OpeCL parallel algorithm implemetatio for computig the dyadic covolutio, which is processed o the GPU, provides cosiderable speedups, as documeted by experimets over radomly geerated biary fuctios ad a compariso with the referet C/C++ CPU implemetatio of the algorithm. The same OpeCL implemetatio of the dyadic covolutio ca also be used for the fast calculatio of the dyadic autocorrelatio. The proposed method could, therefore, exted the area of applicatios of these operatios to problems where algorithm ruig time is a essetial ad, ofte, limitig factor. REFERENCES 1. AMD Accelerated Parallel Processig SDK, AMD Ic., website last visited o 10/06/ ATI Stream Computig OpeCL Programmig Guide, AMD Ic., T. M. Aamodt, Architectig graphics processors for o-graphics compute acceleratio, i Proc. of the 2009 IEEE Pacific Rim Cof. o Commuicatios, Computers ad Sigal Processig, Special Sessio o Computer Architecture, Victoria, BC, Caada, August 23-26, J. Ardt, Matters Computatioal: Ideas, Algorithms, Source Code, Spriger, S. Coh-Sfetcu, O the theory of liear dyadic ivariat systems, i Proc. Symp. Theory ad Applic. Walsh ad other No-sius. Fuctios, Hatfield, Eglad, S. Coh-Sfetcu, Topics o geeralized covolutio ad Fourier trasform: Theory ad applicatios i digital sigal processig ad system theory, Ph.D. dissertatio, McMaster Uiversity, Hamilto, Otario, Caada, S. Coh-Sfetcu ad S. T. Nichols, O the idetificatio of liear dyadic ivariat systems, IEEE Trasactios o Electromagetic Compatibility, Vol. EMC-17, No. 2, May 1975, J. W. Cooley ad J. W. Tukey, A algorithm for the machie calculatio of complex Fourier series, Mathematics of Computatio, No. 90, 1965, Y. Edow, Walsh harmoizable processes i liear system theory, Cyberetics ad Systems, 1996, Y. Edow ad R. S. Staković, Gibbs derivatives i liear system theory, Cyberetic ad Systems, Vol. 26, 1995, D. B. Gajić ad R. S. Staković, Calculatio of dyadic covolutio usig graphics processig uits ad OpeCL, paper accepted for presetatio at the ICEST 2011 Coferece, Niš, Serbia, Jue 29 - July 1, D. B. Gajić ad R. S. Staković, Computig fast spectral trasforms o graphics processig uits usig OpeCL, i Proc. of the Reed-Muller 2011 Workshop, Tuusula, Filad, May 25-26, 2011, T. H. Frak, Implemetatio of dyadic correlatio, IEEE Tras. Electromagetic Compatibility, Vol. EMC-13, No. 3, 1971, J. E. Gibbs, J. E. Marshall, ad F. R. Pichler, Electrical measuremets i the light of system theory, i IEEE Coferece o the Use of Computers i Measuremet, Uiversity of York, Sept , 1973, ii N. K. Govidaraju et al., High performace discrete Fourier trasforms o graphics processors, i Proc. of the 2008 ACM/IEEE Cof. o Supercomputig, IEEE Press, Austi, Texas, USA, November 15-21, H. F. Harmuth, Die Orthogoalteilug als Verallgemeierug der Zeit-ud Frequezteilug, Archiv der Elektrische Übertragug, vol. 18, 1964, H. F. Harmuth, Sequecy Theory - Foudatios ad Applicatios, Academic Press, New York 1977; Russia editio: MIR, Moscow 1980; reprited i PR Chia - Chiese editio: Publishig House of the People's Post ad Telecommuicatios, Beijig M. G. Karpovsky, R. S. Staković, ad J. T. Astola, Spectral Logic ad Its Applicatios for the Desig of Digital Devices, Wiley-Itersciece, NVidia CUDA: Compute Uified Device Architecture, gpucomputig.html, NVIDIA Corp., website last visited o 21/06/ NVidia CUDA CUFFT Library, NVIDIA Corp., 2007.

11 Computatio of Dyadic Covolutio o GPU for Efficiet Modelig of Dyadic LTI Systems J. Pearl, Optimal dyadic models of time-ivariat systems, IEEE Tras. Comput., Vol. C-24, 1975, F. Pichler, Walsh fuctios ad liear system theory, i Proc. Applic. Walsh Fuctios, Washigto, D.C., 1970, F. Pichler, J. E. Marshall, ad J. E. Gibbs, A system-theory approach to electrical measuremets, i Colloquium o the Theory ad Applicatios of Walsh Fuctios, The Hatfield Polytechic, Hatfield, Hertfordshire, Jue 28-29, F. Pichler, Fast liear methods for image filterig, i Applicatios of Iformatio ad Cotrol Systems, D. G. Laiiotis ad N. S. Tzaues, (Eds.), D. Reidel Publishig Compay, 1980, F. Pichler, Experimets with 1-D ad 2-D sigals usig Gibbs derivatives, i Theory ad Applicatios of Gibbs Derivatives, P. L. Butzer ad R. S. Staković, (Eds.), Belgrade, Serbia: Matematički istitut, 1990, F. Pichler, Some historical remarks o the theory of Walsh fuctios ad their applicatio i iformatio egieerig, i Theory ad Applicatios of Gibbs Derivatives, P. L. Butzer ad R. S. Staković, (Eds.), Belgrade, Serbia: Matematički istitut, F. Pichler, Realizatios of Priggogie's λ-trasform by dyadic covolutio, i Cyberetics ad Systems Research, R. Trappl ad W. Hor, (Eds.), Austria Society for Cyberetic Studies, 1992, ISBN X. 28. F. Pichler, O state-space descriptio of liear dyadic-ivariat systems, i IEEE Trasactios o Electromagetic Compatibility, Vol. EMC-13, No. 3, 1971, B. T. Polyak ad Y. A. Shreider, Applicatios of Walsh fuctios i approximate calculatios, i Voprosy Teorii Matematichskikh Mashi, 1962, M. M. Radmaović, Calculatio of dyadic covolutio through biary decisio diagrams, Facta Uiversitatis: Automatic Cotrol ad Robotics, Vol. 8, No. 1, 2009, R. S. Staković, M. Bhattacharaya, ad J. T. Astola, Calculatio of dyadic autocorrelatio through decisio diagrams, i Proc. of the Europea Coferece o Circuit Theory ad Desig, August 2001, R. S. Staković, M. S. Staković, ad C. Moraga, Remarks o systems ad differetial operators o groups, Facta Uiversitatis: Electroics ad Eergetics, Vol. 18, No. 3, 2005, E. A. Trachteberg ad M. G. Karpovsky, Optimal varyig dyadic structure models of time ivariat systems, i Proc IEEE It. Symp. Circuits ad Systems, Espoo, Filad, Jue 1988, M. A. Thorto, Spectral aalysis of digital logic circuit etlists, i Proc. of the Iteratioal Coferece o Computer Aided Systems Theory (EUROCAST), February 6-11, 2011, The OpeCL Specificatio 1.1, Khroos OpeCL Workig Group, F. E. Weiser, Walsh fuctio aalysis of istataeous oliear stochastic problems, Ph.D. dissertatio, Polytechic Istitute of Brookly, Brookly, Jue IZRAČUNAVANJE DIJADIČKE KONVOLUCIJE NA GRAFIČKIM PROCESNIM JEDINICAMA ZA EFIKASNO MODELOVANJE DIJADIČKIH LTI SISTEMA Duša B. Gajić, Radomir S. Staković Kovolucioi sistemi predstavljaju važu klasu sistema koji se često koriste u mogim praktičim primeama u različitim oblastima auke i ižejerstva. U slučaju kada su ulazi i izlazi sigali biaro kodirai, običo se kao algebarska struktura a kojoj su defiisai odgovarajući sistemi uzimaju koače dijadičke grupe. Modelovaje i rad sa ovakvim sistemima zahtevaju veoma česta izračuavaja dijadičke koovlucije. U ovom radu predstavlje je metod za efikaso izračuavaje dijadičke kovolucije a grafičkim procesim jediicama (Graphics Processig Uits - GPUs) pri čemu je odgovarajući algoritam implemetira primeom programskog jezika Ope Computig Laguage (OpeCL). Razmotreo je ekoliko veoma začajih pitaja koja se tiču efikasog mapiraja algoritma a arhitekturu grafičkih procesih jediica.

12 70 D. B. GAJIĆ, R. S. STANKOVIĆ Performase predložee implemetacije upoređee su sa referetom C/C++ implemetacijom koja se izvršava a cetraloj procesoj jediici (Cetral Processig Uit - CPU). Eksperimetali rezultati potvrđuju da primea predložeog metoda za izračuavaje dijadičke kovolucije a GPU doosi začaja ubrzaja u odosu a referetu CPU implemetaciju. Ključe reči: Lieari traslacioo ezavisi sistemi, Dijadička kovolucija, Brza Walsh-ova trasformacija, GPU paralelo programiraje, OpeCL