International Association of Scientific Innovation and Research (IASIR) (An Association Unifying the Sciences, Engineering, and Applied Research)

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1 Ieraioal Associaio of Scieific Iovaio ad Research (IASIR (A Associaio Uifyig he Scieces, Egieerig, ad Applied Research ISSN (Pri: 79- ISSN (Olie: Ieraioal Joural of Egieerig, Busiess ad Eerprise Applicaios (IJEBEA Modelig of Power Amplifier usig Volerra Series Ahlawa Ami [], Dhamija Ashuosh [],Gill Amadeep [3] Deparme of Elecroics & Commuicaio Egieerig, GITM Bilaspur, Haryaa,Idia [], Deparme of Elecroics & Commuicaio Egieerig, JMIT, Radaur, Haryaa, Idia [], Deparme of Elecroics& Commuicaio Egieerig, JMIETI, Radaur, Haryaa, Idia [3], Absrac: Coveioal radio-frequecy (RF power amplifiers operaig wih widebad sigals are paricularly suscepible o he geeraio of ou-of-bad spurious emissios, also referred o as specral regrowh. The power amplifier s olieariy broades he ipu sigal s badwidh. This is kow as Specral regrowh ad is udesirable. Specral regrowh causes ierferece wih adjace chaels ad icreases he probabiliy of violaios of he ou of bad emissio requiremes madaed by regulaory bodies. To aalyze power amplifier for fuure commuicaio sysem i is impora o model he behavior of PA olieariy ad memory effec accuraely. Noliear amplifier behavioral models ca be divided Io hree ypes : memoryless (saic, quasi memoryless ad models wih memory. I his paper we model he power amplifiers wih Volerra series filers. I. Iroducio : The umerous mehods for modelig he oliear behavior of a device are classically divided io hree caegories: physical, empirical ad able-based oes. Some may be difficul o caegorize i his way, however, ad herefore models[] are divided here simply io physical ad empirical oes, ad io blackbox ad circui-level oes, The Volerra series, a basis of empirical model ha is sill able o provide circui-level iformaio, is cosidered i his paper. Noliear Volerra heory was developed i he 88s by Vio Volerra. The heory quickly received a grea deal of aeio i he field of elecrical egieerig, ad he laer i he biological field, as a powerful approach o he modelig of oliear sysem behavior. Volerra heory is a geeralizaio of he liear covoluio iegral approach ofe applied o liear, imeivaria sysems. The heory saes ha ay ime-ivaria, oliear sysem ca be modeled as a ifiie sum of mulidimesioal covoluio iegrals of icreasig order. This is represeed symbolically by he series of iegrals, y( h h ( u( d h (, u( u( d d... which is kow as he Volerra series. Here, u( represes he dyamic sysem ipu while y( represes he sysem respose. Volerra heory is based o dyamic daa, ad as such he average values of all ipu ad respose daa ses are removed. Each of he covoluio iegrals coais a kerel, eiher liear (h or oliear (h,..,h, which represes he behavior of he sysem. Kowledge of hese kerels allows he predicio of a sysem s respose o ay arbirary ipu, ad as such is criical o oliear Volerra modelig. The firs erm of he series represes he liear covoluio iegral. The firs order erm is cosidered o represe he mea of he sysem respose. The erm weakly oliear merely implies ha a sysem is well represeed by he firs wo or hree erms of a Volerra series. All higher-order erms i his siuaio are see o quickly ed oward zero, ad are herefore egligible i he sysem represeaio. While Volerra heory has a srog foudaio i boh he biological ad elecrical egieerig fields, i has received lile aeio i he field of aerodyamics uil recely []. Liear respose models have ofe bee assumed sufficie for represeaio of oliear aerodyamic sysems whe excied by small perurbaios. This assumpio derives from he fac ha highly oliear pheomea have a egligible impac o he e effec of various resposes uder codiios such as small perurbaio exciaio. I addiio, he lack of aeio is due i large par o he ihere difficuly of ideifyig Volerra kerels. Time is discreized wih a se of ime seps of equivale size. Discree ime icremes are idexed from IJEBEA 3-38; 3, IJEBEA All Righs Reserved Page 9

2 Ami e al., Ieraioal Joural of Egieerig, Busiess ad Eerprise Applicaios, 4(, March-May, 3, pp (ime o (ime, ad he evaluaio of y a ime is deoed by y[]. The covoluio i discree ime is N N N y[ ] h h [ k] u[ k] h [ k, k ] u[ k ] u[ k ]... k k k Where N is he oal ime record of ieres. Volerra kerels are he backboe of ay Volerra series. Kowledge of a sysem s behavior is coaied wihi hese kerels, ad give ay arbirary ipu he Volerra series ca predic he respose of he sysem.[3] Volerra kerels, boh liear ad oliear, are ipu depede. As a example of his cosider he case where he respose of a liear sysem o a arbirary ipu is desired. Here, he ui impulse respose of he sysem o ha ype of ipu mus firs be defied. The firs order kerel, h, represes he liear ui impulse respose of he sysem. This erm is comparable o he basic frequecy respose fucio (FRF of a liear sysem, rasformed io he ime domai. However, he kerel h gives a more accurae porrayal of a sysem s liear respose ha does he FRF. This is because h exiss wih he kowledge of higher-order, oliear erms while he FRF assumes a compleely liear respose. The secod order kerel, h, is a wo-dimesioal fucio of ime. I represes he respose of he sysem o wo separae ui impulses applied a wo varyig pois i ime. Therefore he kerel is a fucio of boh ime ad ime lag. Similarly, h 3 is a hree-dimesioal fucio of ime, represeig he respose of he sysem o hree separae ui impulses applied a hree varyig pois i ime. Here he kerel is a fucio of ime ad wo disic ime lags. I is hrough hese ime lags ha oliear kerels represe he effec of a previous respose as i is carried hrough ime i he sysem. Volerra kerels ca be rewrie i several ways simply by reorderig he variables of iegraio. Because of his, more ha oe kerel ca geerally be used o describe a give sysem, ad i is herefore ecessary o impose uiqueess upo he kerels. This is accomplished by workig wih resriced forms of he kerels [4]. The rucaed secod-order Volerra series ca be wrie as: N N N y( h h ( k u( k h ( k, k u( k u( k k k k y( h h ( u( d h (, u( u( d d Or k y u( s ( s ( k u( k u( k u( s (, k k s ( k, k u( k u( k u( k u( k y Where s ad s are he firs- ad secod-order kerels, respecively, is a geeral imedepede ipu sigal, ad is he sysem's respose. Silva showed ha he oliear Navier-Sokes equaios ca be cosidered weakly oliear, i.e., ca be accuraely represeed by a rucaed secod order Volerra series, eglecig higher kerels. u IJEBEA 3-38; 3, IJEBEA All Righs Reserved Page 3

3 Ami e al., Ieraioal Joural of Egieerig, Busiess ad Eerprise Applicaios, 4(, March-May, 3, pp II. Firs/Secod Order of Volerra Kerel Ideificaio Le where ad u ( ( ( T is he Dirac s dela fucio. The, he resposes due o he ipus will be expressed as If we assume ha, y ( h ( h (, Sice, Therefore, The secod order of he Volerra kerel will be deermied as Now, le s assume ha If we pu u ( ( ( y ( h h ( ( d h (, ( ( d d. The, y ( h h ( ( ( ( T d u ( ( as a ipu. The respose will be h (, ( ( ( T ( ( ( T d d h y ( h ( h ( T h (, h (, T h ( T, T h (, T h ( T, y ( y ( h ( T h (, T h ( T, T y ( T h ( T h ( T, T y ( y ( y ( T h (, T h (, T y( y( y( T T h (, y( y( y( y ( h h ( ( d h (, ( ( d d h ( 4 h (, Sice h (, y( y( y( IJEBEA 3-38; 3, IJEBEA All Righs Reserved Page 3

4 Ami e al., Ieraioal Joural of Egieerig, Busiess ad Eerprise Applicaios, 4(, March-May, 3, pp y ( h ( 4 h (, From he above equaio, he firs order of he Volerra kerel will be h ( y( y( y ( ( y ( ( where is he respose due o ad is he respose due o. III. Firs Order Series Approximaio Raveh, e al furher assumed ha he respose ca be evaluaed oly wih he firs erm i Volerra kerel. The respose of he sysem ca he be evaluaed by covolvig he impulse respose h ( wih he ipu sigal: y( h( h( k u( k,,, k Or he respose of he sysem o a arbirary ipu is compued by covoluio of he sep respose s( wih he derivaive of ipu sigal u(, accordig o Duhamel's iegral: y( u( s( s( k u( k u( k k u( s( s( k u( k,,, k IV. Resul: Frequecy plos for power amplifiers wihou usig volera modelig ad wih volera modelig: Fig V. Coclusio: I his paper we have compare he power amplifier frequecy plos wih ad wihou usig usig volera modelig.he resuls wih usig volers filers are far beer compared o he resuls wihou volera filers. VI. Refereces: []. A Approximaio of Volerra Series Usig Delay Evelopes, Applied o Digial Predisorio of RF Power Amplifiers Wih Memory Effecs, Nima Safari, Sude Member, IEEE, Terje Røse, Member, IEEE, Pavlo Fedoreko, Sude Member, IEEE, ad J. Seveso Keey, Seior Member, IEEE, VOL. 8, NO., FEBRUARY 8 []. Eus, C.; Powers, E.J.: A ew Volerra predisorer based o he idirec learig archiecure. IEEE Tra. O Sigal Processig 45 (997, 3-7. IJEBEA 3-38; 3, IJEBEA All Righs Reserved Page 3

5 Ami e al., Ieraioal Joural of Egieerig, Busiess ad Eerprise Applicaios, 4(, March-May, 3, pp [3]. Zhu, A.; Brazil, T.J.: A adapive Volerra predisorer for he liearizaio of RF high power amplifiers. MTT-IMS (, [4]. Sapleo, S. P. ad Cavers, J. K., A ew echique for adapaio of liearizig predisorers, i Proc. IEEE Veh. Techol. Cof., pp , IJEBEA 3-38; 3, IJEBEA All Righs Reserved Page 33