DYNAMICAL SYSTEMS DETERMINABLE BY DISCRETE SAMPLES

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1 SISOM ad Sessio of the Commissio of Acoustics Bucharest 5-6 May DYNAMICAL SYSTEMS DETERMINABLE BY DISCRETE SAMPLES Ovidiu Ilie ŞANDRU Luige VLĂDĂREANU Alexadra ŞANDRU 3 Departmet of Mathematics II Politehica Uiversity of Bucharest 33 Splaiul Idepedeţei 64 Bucharest ROMANIA oisadru@yahoo.com Istitute of Solid Mechaics of the Romaia Academy C-ti Mille 5 Bucharest ROMANIA luigiv@arexim.ro 3 Departmet of Electroic Techology ad Reliability Politehica Uiversity of Bucharest Bd. Iuliu Maiu -3 Bucharest 6 ROMANIA alexadra_sadru@yahoo.com I this paper we shall defie ad study a importat class of dyamic systems which allow to effectively determie their mathematical model exclusively o experimetal basis. The usefuless of these results of mathematical ature obtaied by extedig the Whittaer-Shao samplig theory will be highlighted through a applied example from the field of optoelectroics. Keywords: Recostructible dyamical systems Whittaer-Shao samplig theory.. INTRODUCTION The cocept of dyamic system allows the mathematical modellig of a very large rage of pheomea occurrig i ature from which those cosidered as belogig to atural scieces are studied withi physics biology ad chemistry where as those belogig to humaities are studied withi social scieces or ecoomics etc. Thus it is o eed i isistig upo the importace of this cocept. However for those who wish to iitiate themselves or who wat to bolster owledge i this directio we recommed the moograph [] as a iitial study. Those who wat to elighte themselves o the potetial the otio of dyamic system offers whe mathematically modelig pheomea from ature we recommed for example the papers [ ] where they ca fid some iterestig applicatios of the cocept of dyamic system others tha the ow classical oes. Also for those iterested i ews of theoretical ature from the field of dyamic systems we recommed paper [3 7]. I this paper we shall study the geeral case of the fiite dimesioal dyamical systems of iput-output type ituitively defied by the diagram represeted i fig.. Iput Blac-box dyamical system Output Figure. Coceptual model of a abstract iput-output dyamical system. If by x.. x we deote the iputs ad by y.. ym the outputs the such a system ca be mathematically abstracted through a set of m equatios

2 Ovidiu Ilie ŞANDRU Luige VLĂDĂREANU Alexadra ŞANDRU 374 ( ) (.. ) y s f x x s Σ :... () ym( sm) fm( x.. x sm) where the set of parameters { s.. s m} expresses the possible states of the system or the operatig mode i which it fids itself whe receivig the impulses x.. x. Each parameter s.. m ca be i part a scalar a vector or a fuctio depedig o the ature of the system that characterizes it. Remars :) It is very importat to eep i mid that the problem we would lie to approach i this paper has i view the geeral ad very abstract case of dyamical systems used for modelig processes ad f f x.. x s pheomea occurrig i ature that is especially those cases i which the fuctios ( ) f f ( x.. x s are ot explicitly ow. For example the moetary policies redoud upo the ecoomical systems through the effects they produce but this fact does ot allow us to ow the exact iteral mechaisms they trigger (formally expressed by a array of fuctios f.. f m ) although the existece of these mechaisms is assumed. m m m). THE COMPATIBILITY CONDITIONS OF THE PROBLEM.. m m m have a certai form for istace they are polyomials or badlimited that is fuctios that admit the Fourier trasform ad whose trasforms have a compact support the the problem of their determiatio by virtue of s resulted from practical f f x.. x s.. If the fuctios f f ( x.. x s ).. f f ( x x s ) experimets is possible. Ideed whe we ow a priori that the uow fuctios ( ) fm fm ( x.. x s m ) are polyomials their determiatio from the correspodig represetatios ca be easily doe with the help of the iterpolatig polyomials. Hereiafter we shall focus o solvig this problem for the case of badlimited fuctios because the systems belogig to this class have umerous applicatios i optoelectroics. 3. THE RESOLUTION OF THE PROBLEM FOR THE CASE OF BANDLIMITED FUNCTIONS.. m m m are ot ow a priori we ca still determie i a experimetal way the ext -rectagular lattice of s Although the fuctios f f ( x.. x s ).. f f ( x x s ) x x f ( x.. x s) comb comb f( x.. x s).. m () h h where comb( x) δ ( x) relatios () ca be brought to the equivalet form δ beig the so-called Dirac fuctio. Ideed let us otice first that the x x f ( x.. x s) δ j δ j f( x.. x s).. m j h j h (3)

3 375 Dyamic systems determiable by discrete s ad the by choosig o the x axis the step h o the x axis the step h ad o the x axis the step the values of the fuctios f f x.. x s f.. m fm x x sm i the poits x h j Z x h j j Z x h j j Z ca be experimetally determied eve if the h ( ).. ( ) j aalytical expressios of these fuctios are ot ow. Oe amedmet must be made o this occasio that of cosiderig all the fuctios f f ( x.. x s ).. (.. ) fm fm x x sm defied o R. This requisite is however easy to fulfill uder the hypothesis we wor because we ca cosider that ay of these fuctios is equal to zero i the poits i which these fuctios are ot defied. By carryig out the otatios { } (.. ) (.. ) F ω ω s F f x x s { } ( ) ( ) F ω.. ω s F f x.. x s.. m where F represets the Fourier trasform ad by allowig for the relatios x x F ( ω.. ω s) F comb comb F(.. s).. h h ω ω m x x ( ω ) ( ω ) F comb comb h h comb h comb h h h j j δ ω.. ω j j h h we get j j F ( ω.. ω s) F ω.. ω.. j j h h m. (4) Sice the fuctio f is badlimited by hypothesis its spectrum F is o-zero oly o a certai fiite regio R ( F has the support R bouded) of the space of frequecies ω.. ω. where From (4) we deduce that the support of the d fuctio T j j j.. h h h the parameters F T ( R ) j j j j.. j h h h j j j.. h h h represets the vector traslatio h.. h is i the space of the frequecies. The smaller (the steps of the divisios) will be (that is whe the s are sufficietly close

4 Ovidiu Ilie ŞANDRU Luige VLĂDĂREANU Alexadra ŞANDRU 376 T j j j.. h h h together) the larger the distace betwee the differet spectral sub-regios ( ) we ca suppose that by coveietly choosig these parameters the regios T ( ) j j j.. h h h F R will be so that R j.. j Z will be disjoit two by two. Due to this the exact recovery of the origial spectrum from ca be accomplished by passig this d fuctio through a liear filter that trasmits the term correspodig to j j from (4) without distortio while perfectly excludig all other terms. Thus at the output of this filter we fid a exact replica of the origial spectrum data ad cosequetly of the origial data f. I order to determie the ecessary spacig betwee the s let F b..b F be the dimesios i the directio of the axes ω.. ω of the smallest -parallelepiped which icludes the regio R. As the differet sub-regios T ( R ) are situated at the distaces j j j.. h i the directio of the ω axis h h h h i the directio of the ω axis ad h i the directio of the ω axis respectively oe from aother the separatio betwee these spectral regios is esured if h.. h b. b At this poit the oly thig left is to idicate the trasfer fuctio (the filter) through which the data of the used ( ) must pass through. The filter searched has the expressio F ω ω Φ ( ω.. ω) rect rect b. b Cosequetly i order to rebuild from F we oly have to apply the filter H to that is F we have the relatio ω ω F( ω.. ω s) F ( ω.. ω s) rect rect. b b F By applyig to this relatio the iverse Fourier trasform F we fid that where x x f( x.. x s) comb comb f( x.. x s) φ( x.. x).. m (5) h h ( ) φ x.. x F { Φ ( ω.. ω) } b b sic( b x) sic ( b x )

5 377 Dyamic systems determiable by discrete s As the relatio (5) becomes x x ( ) comb comb f x.. x s h h (.. ) δ (.. ) h h f jh j h s x jh x j h j j ( ) f.. x x s ( ) ( ) ( ). b b f jh.. j h s sic b x jh sic b x j h j j Cosiderig h.. h b b the last relatio ca be writte as ( ) f.. x x s j j j j f.. s sic b x sic b x j j b b b b or j... j j j f( x.. x s) r sic b x sic b x j j b b where j... j j j r f.. s.. m. b b 4. AN IMPORTANT SUBCLASS OF BANDLIMITED FUNCTIONS ( ) ) Let.. s s x x ad t t( x.. x be two absolutely itegrable fuctios o R from which at least oe of them is badlimited. The the fuctio f s t where ( )(.. ) (.. ) (.. ) s t x x s ξ ξ t x ξ x ξ dξ dξ R represets the covolutio product of fuctios s ad t is also a badlimited fuctio because ( f ) ( s) ( t) F F F

Ovidiu Ilie ŞANDRU Luige VLĂDĂREANU Alexadra ŞANDRU 378 where just as earlier F symbolizes the -dimesioal Fourier trasform ad from the trasforms F ( s) or F ( t) at least oe has a compact support.

6 Ovidiu Ilie ŞANDRU Luige VLĂDĂREANU Alexadra ŞANDRU 378 where just as earlier F symbolizes the -dimesioal Fourier trasform ad from the trasforms F ( s) or F ( t) at least oe has a compact support. This property allows us to put forward the subclass of dyamical systems of the form () for which the fuctios f.. f m have the particular form ( ) ( )( ) f x.. x s s t x.. x.. m where each pair of fuctios s ad t.. m verify properties similar to the fuctios s ad t whereof we have discussed earlier. Remar: The dyamical systems of this subclass are importat due to their applicatios i differet scietific areas. The proof of this statemet will be discussed hereiafter. 5.ONE ELOQUENT EXAMPLE OF THE USE OF THE MATHEMATICAL APPARATUS DEVELOPPED IN THIS PAPER I order to illustrate the usefuless of the mathematical apparatus developed i this paper we shall study hereiafter a problem from the field of optoelectroics uderliig the iheret fact that the area of applicability of the preseted theorem does ot limit itself oly to the field metioed i the chose example. I [] the mathematical modelig of the diffractio pheomeo at boudaries is doe by relatig a physical system composed of a moochromatic light source S a opaque scree (OpS) edued with a aperture (A) through which the light ca pass ad a observatio scree (ObS) o which the light sigal emitted by the source S will be aalyzed to a three orthogoal axes frame. If xyz is the frame chose for the represetatio of this system the i order to mae a choice we shall place the scree OpS i the plae z so that the ceter of the aperture A coicides with the ceter of the axes of coordiates. The light source is cosidered to be placed behid this scree (somewhere o the egative semi-axis of the z axis) ad the observatio scree ObS is placed i the plae z z z >. See Figure. Figure. The coordiate system used for modelig the diffractio at boudaries.

7 379 Dyamic systems determiable by discrete s Kowig that the complex field across the aperture A is represeted by U ( x y) we wat to determie the value of this field i ay poit ( x y z ) of the observatio scree OpS. I the theory of diffractio at boudaries developed i [] the value of the field U i oe poit ( x yz ) of the plae z z Withi this expressio the fuctio is described as the respose of a spatial dyamical system of the form ( ) ( ξ η ) ( ξ η ) U x yz U hx y z dξdη. (6) R U( (the boudary coditios imposed to the system) the coordiates ) describes the state of the system subjected to the experimet x yz wat to evaluate the value of the field U represet the iputs of the system z > of the poit i which we h( ) represets the trasfer fuctio of the system ad the value U ( x yz ) represets the respose (the output) of the system to the imposed boudary coditios (i.e. the iputs). Remar: I order to facilitate the coectio betwee the otios preseted i this paragraph ad the theory euciated i [] that we ow refer to i this paragraph we have chaged the otatio adopted earlier i favor of the oe from []. By taig the results obtaied i [] uder the hypothesis that the distace betwee the plaes havig the equatios z ad z z is at least several wavelegths loger (tha the wavelegth of sigal emitted by the source S) so that the evaescet waves which occur i the immediate eighborhood of the slit A may be eglected the determiatio of the trasfer fuctio h is doe by the relatio below where the fuctio H is described by the relatio i α β π x+ y λ λ α β α β h( x y z) H z e d d λ λ λ λ R z H z πi α β λ α β e α + β < z λ (7) λ λ otherwise from which the parameters α β γ α β α + β symbolize the directios cosies of a hypothetical directio of propagatio of the light sigal ad λ is the wavelegth of this sigal. Due to the relatio (7) the trasfer fuctio h is badlimited. Hece the optical system (6) itegrates itself i the class of systems defied i the previous paragraph. From this remar we coclude that pheomea of diffractio ca be etirely recostructed by virtue of some discrete s which ca be determied experimetally. This result has a tremedous importace i practice because it also allows modelig the diffractio pheomea i the case of diffractio gratigs for which the theoretical apparatus is uow or whose determiatio is very laborious provided that the gratig are bouded (i.e. there exists a strictly positive umber R so that the etire etwor ca be icluded i the disc of radius R). This latter coditio is imposed by the requiremet that the trasfer fuctios of the correspodig dyamic systems to be badlimited.

8 Ovidiu Ilie ŞANDRU Luige VLĂDĂREANU Alexadra ŞANDRU 38 REFERENCES. GOODMAN J.W. Itroductio to Fourier Optics McGRAW-HILL Ic KALMAN R. E. FALB P. L. ARBIB M. A. Topics i mathematical system theory McGraw-Hill New Yor ŞANDRU O. I. VLADAREANU L. ŞANDRU A. A ew method of aproachig the problems of optimal cotrol Proceedigs of th WSEAS It. Cof. o MATHEMATICAL ad COMPUTATIONAL METHODS i SCIENCE ad ENGINEERING (MACMESE'8) Bucharest Romaia November Recet Advaces i Mathematical ad Computatioal Methods i Sciece ad Egieerig Part II ISSN: ŞANDRU O. I. ŞANDRU I. M. D. Regardig maretig problems as dyamic system theory problems Proceedigs of the Iteratioal Coferece o MANAGEMENT MARKETING ad FINANCES (MMF '9) WSEAS Iteratioal Cofereces Uiversity of Housto USA April 3-May ISSN: ISBN: ŞANDRU O. I. ŞANDRU I. M. D. Maagig competitio betwee uiversities by icreasig the quality of the educatioal process Aals of DAAAM for 9 & Proceedigs of th DAAAM Iteratioal Symposium ISBN ISSN ŞANDRU O. I. ŞANDRU I. M. D. Regardig educatio quality maagemet problems as dyamic system theory problems Aals of DAAAM for 9 & Proceedigs of th DAAAM Iteratioal Symposium ISBN ISSN ŞANDRU O. I. VLADAREANU L. ŞANDRU A. Ivertible dyamic systems Proceedigs of the Europea Computig Coferece ECC WSEAS Iteratioal Cofereces Paris Frace April 8-3 ISBN: VLADAREANU L. ION I. ŞANDRU O. I. VELEA L. M. ad MUNTEANU M. S. The Actuators Cotrol by Probabilistic Mathematical Modellig WSEAS TRANSACTIONS o SYSTEMS ad CONTROL Issue 6 Volume 3 Jue ISSN: VLADAREANU L. ŞANDRU O. I. L. VELEA L. M. YU H. Actuator cotrol i cotiuous flux usig Wier filters PROCEEDINGS OF THE ROMANIAN ACADEMY Series A Volume Number /9 ISSN :

3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,

3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense, 3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [