Volterra Behavioral Model for Wideband RF Amplifiers

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. XX, NO. Y, MONTH 2006 1 Vol e a Beha io al Model o Wideband RF Ampli ie s Ca los C espo-Cadenas, Associa e, IEEE, Ja ie Reina-Tosina, Associa e, IEEE and Ma ´ıa J. Made o-Ayo a Abs ac — This pape p oposes a beha io al modeling ap- p oach o he desc ip ion o nonlinea i ies in wideband wi eless communica ion ci cui s wi h memo y. The model is o mally de i ed exploi ing he dependence on equency o he am- pli ie nonlinea ans e unc ions and educe he numbe o pa ame e s in a gene al Vol e a-based beha io al model. To alida e he p oposed app oach, a comme cial ampli ie a 915 MHz, exhibi ing nonlinea memo y e ec s, has been widely cha ac e ized using di e en s imuli, including wo ones, QPSK-WCDMA, and 16-QAM signals wi h ec angula and oo - aised cosine con o ming pulses. The heo e ical esul s ha e been compa ed wi h expe imen al da a demons a ing ha he model pe o mance is compa able o he well-es ablished memo y polynomial model. Calcula ed and measu ed baseband wa e o ms, signal cons ella ion, spec al eg ow h and ACPR a e igh ly coinciden in all cases, emphasizing he ele ance o he p oposed model. Index Te ms— Beha io al models, mic owa e ampli ie s, Vol e a se ies, nonlinea memo y e ec s. I. INTRODUCTION As a undamen al block in wi eless communica ions sys- ems, he powe ampli ie (PA) has unde gone exhaus i e s udy o i s cha ac e is ics, in pa icula hose ela ed wi h nonlinea memo y e ec s. Many e o s ha e been de o ed o ob ain beha io al models o mic owa e PAs, o which he ou pu o his black-box me hod is p edic ed wi hou knowledge o he nonlinea de ice in e nal s uc u e. The goals o hese app oaches a e, on he one hand, educ ion o complexi y main aining an accu acy compa able o he esul s ob ained wi h ci cui -le el simula ions and, on he o he hand, a simple me hod o ex ac he model pa ame e s. An indica o o he impo ance o hese app oaches is he aluable wo k p esen ed in he las yea s, o example [1]-[3], and he ecen publica ion o an ex ensi e e ision ela ed o his opic [4]. Exploi ing he bandlimi ed cha ac e o wi eless signals, PA desc ip ion can be ansla ed in o an en elope ep esen- a ion and equen ly has been deduced as Vol e a se ies, a p ocedu e ha ea s his p oblem in an s ic ly and o de ly way. Howe e , one di icul y o ampli ie modeling using his Vol e a app oach is i s high compu a ional complexi y [5]. The g ea numbe o coe icien s equi ed o he desc ip ion o sys ems wi h a s ong nonlinea i y and long memo y has s ee ed he wo k o many esea che s in o de o educe he This wo k was comple ed wi h he suppo o he Spanish Na ional Boa d o Scien i ic and Technological Resea ch (CICYT) wi hin he p ojec TEC2004- 06451-C05-03. The au ho s a e wi h he Depa amen o de Teo ´ıa de la Se˜nal y Comunica- ciones, Escuela Supe io de Ingenie os, Uni e sidad de Se illa, Spain. E-mail: cc e[email p o ec ed], j ein[email p o ec ed], mjmade [email protected] numbe o model pa ame e s. P obably, he mos manageable solu ion o educe he numbe o coe icien s is he memo y polynomial model p oposed in [1], he s uc u e o which p esen s a no able unca ion in he numbe o pa ame e s. Al hough he educ ion ob ained wi h his simpli ied Vol e a model is impo an , he numbe o coe icien s emains high, pa icula ly in he case o ampli ie s wi h long memo y. To achie e a u he educ ion, an ex ension o he memo y poly- nomial model wi h spa se delay ap s uc u e was p oposed in [2]. I is no clea how he pa ame e educ ion o hese pa i- cula s uc u es a ec s he a ainable accu acy o he model owing o he possible impo ance o o he unde es ima ed e ms. Speci ically, he need o conside ing hose neglec ed e ms was he pu pose o he no el s uc u e epo ed in [3]. Tha new app oach is based on he p uning o edundan ke nels in he ull Vol e a se ies model so ha he coe icien s wi h less e ec on he ou pu signal a e disca ded ollowing an a pos e io i p ocedu e. Despi e he signi icance o he ci ed models, i is desi able o de elop an app oach wi h an op imized numbe o coe icien s, sus ained on heo e ical p inciples and wi h no need o a p e ious empi ical selec ion. Tha was he aim o he au ho ’s ini ial s udy o an ampli ie wi h one FET based on i s simpli ied equi alen ci cui . A hi d-o de model was alida ed wi h expe imen al da a and pa ial esul s we e p esen ed in [6]. In his pape he au ho s in oduce he demons a ion o a i h o de Vol e a model o a gene al ampli ie wi h bandwid h la ge han he RF signal band. The app oach allows he analysis o nonlinea memo y e ec s om a model based on he ans e unc ions wi h no es ic ions in he numbe o kind o nonlinea de ices composing he ampli ie . In he nex sec ion he s udy o a wideband ampli ie and he comple ion o i s beha io al model based on a Vol e a se ies app oach is desc ibed. The equency independence o he ampli ie esponse inside he RF bandwid h is exploi ed o educe he o de model and o e eal he dependence o coe icien s ex ac ion on he sampling a e. Sec ion III is dedica ed o desc ibe he p ocedu e o pa ame e ex ac ion, which has been simpli ied because o o de educ ion in he model s uc u e. Applica ion o he me hod o a comme cial ampli ie and compa ison wi h he memo y polynomial model, and also wi h expe imen al da a using di e en ypes o inpu signals, is p esen ed o alida e he demons a ed heo e ical esul s. Finally, a gene aliza ion o he model o any o de is p oposed and some ele an s a emen s a e commen ed. 2 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. XX, NO. Y, MONTH 2006 INPUT ASSOCIATED LINEAR NETWORK OUTPUT Ys i u (1) (1) (1) ( , ) i u (2) (2) (2) ( , ) ... Fig. 1. Gene al schema ic o a nonlinea ci cui . II. VOLTERRA MODEL FOR A WIDEBAND AMPLIFIER A. Vol e a-based beha io al model backg ound Le a gene al ampli ie be ep esen ed by he ci cui shown in Fig. 1. Nonlinea i ies a e cons i u ed by hei linea compo- nen s, included in he associa ed linea ci cui , and nonlinea sou ces which a e assumed o be dependen on wo con ol ol ages (a)and u(a). Le he inpu ˜x( )be an RF cu en exci a ion and ˜y( ) he ou pu ol age co esponding o he undamen al equency zone, cen e ed a ωc. Making use o he ac ha a wi eless signal commonly has a bandwid h B negligible wi h espec o he ca ie equency c, he disc e e ime-domain complex en elope Vol e a model o his gene al nonlinea sys em can be exp essed as y(k) = X q1 h1(q1)x(k−q1)+ +X q3 h3(q3)x(k−q1)x(k−q2)x∗(k−q3)+ +X q5 h5(q5)x(k−q1)x(k−q2)x(k−q3)× ×x∗(k−q4)x∗(k−q5) + ··· (1) whe e x(k)and y(k)a e complex en elope samples o he inpu and ou pu RF signals, espec i ely, hn(qn) ep esen s he disc e e Vol e a ke nels o o de nand qnis an n- dimensional ec o composed o he in ege - alued delays qi (i= 1,··· , n) [3]. Al hough a sampling a e equal o he inpu signal RF bandwid h is su icien o memo yless nonlinea sys em iden i ica ion [7], i should be inc eased acco ding o he b oadening o he ou pu bandwid h Boi aliasing has o be a oided. As a consequence, o an adequa e ep esen a ion o he ou pu y( ), he sampling ime has o be educed co espondingly o s= 1/Bo. Equa ion (1) is a disc e e- ime Vol e a se ies de i ed om he ep esen a ion using he mul idimensional nonlinea ans- e unc ions (NLTF) ˜ Hn(ω)[8]. In con inuous- ime o m, he n- h o de e m o he ou pu signal can be w i en as yn( ) = 2 (4π)nn mZ∞ −∞ ˜ Hn(ωcn +ω) m+1 Y i=1 X(ωi)× × n Y i=m+2 X∗(ωi) exp(jω′ )dω,(2) o n= 2m+ 1. In (2), ωcn is a ec o wi h i s i s m+ 1 componen s equal o ωcand he emaining mcomponen s a e equal o −ωc,ω= (ω1,··· , ωn)′is a column ec o ha ing nbaseband equency componen s, ω′is he anspose o ω and is a column ec o wi h i s ncomponen s equal o . The bandlimi ed condi ion o he inpu signal allows o neglec he in eg al ou side wo n-dimensional boxes wi h leng h Boand cen e ed a ±ωcn. Consequen ly, i is pos- sible o subs i u e ˜ Hn(ξ)wi h an equi alen ans e unc- ion bandlimi ed in o hese n-dimensional hype cubes, ˆ Hn(ξ), om which he bandlimi ed equi alen Vol e a ke nels a e ob ained making use o a mul idimensional in e se Fou ie ans o m. This equi alen ans e unc ion and he disc e e- ime co esponding ke nel ˆ hn(qn)sa is y he ela ion ˆ Hn(ωn) = X qn ˆ hn(qn) Bn o exp(−jω′ nqn s).(3) Subs i u ion in (2) allows o sepa a e he in eg als and o ob ain he ou pu componen yn( ). The e o e, a e sampling a ins an s =k s, exp ession (1) is immedia ely de i ed. The Vol e a model (1) is a e y gene al esul bu i has a high deg ee o di icul y due o he la ge numbe o pa ame e s and nume ical ope a ions in ol ed [5]. The complexi y o he p oblem is e ealed in he ac ha he ke nels hn(qn) o m a n-dimensional g id de ined by he disc e e delays in each axis o he mul idimensional space q1,··· , qn, hence i is desi able o educe he numbe o hese delays. One o he mos ex ended me hods p oposed o achie e a mo e manageable numbe o pa ame e s is he memo y poly- nomial model desc ibed in [1]. Fo his model he educ ion in he numbe o coe icien s is ob ained by selec ing only he delays posi ioned in he diagonal, i.e. he delays along he di ec ion de ined by q1=q2=···=qn. Mo eo e , i a spa se delay ap s uc u e is adop ed and only he mos signi ican delays a e e ained ollowing an a pos e io i p ocedu e, a u he impo an cu back in he numbe o coe icien s can be p ocu ed [2]. Howe e , he model p ecision can be diminished due o he possible impo ance o non-diagonal e ms. Follow- ing a mo e elaxed p uning app oach, which also e ains he e ms nea he diagonal, a mo e ecen model was p oposed wi h a consequen imp o emen in p ecision a he expense o a mode a e inc ease in he numbe o coe icien s [3]. Al hough he memo y polynomial model has p o en o be e ec i e and he educ ion o coe icien s is conside able, he lack o a heo e ical jus i ica ion o igina es he need o hese empi ical-based me hods. Addi ionally, impo an issues as he adequa e sampling a e o he dependence o he disc e e ke nels on his sampling a e in (1), should be add essed by a beha io al model. B. Nonlinea T ans e Func ions o a Wideband Ampli ie A o mal educ ion in he numbe o coe icien s in (1) can be ob ained unde he only assump ion o a wideband ampli ie , i.e. an ampli ie wi h a pass band la ge han he RF signal bandwid h. This supposi ion does no in oduce any impo an loss o gene ali y since many wi eless am- pli ie s p esen ing nonlinea memo y e ec s, ha e equency CRESPO-CADENAS e al.: VOLTERRA BEHAVIORAL MODEL FOR WIDEBAND RF AMPLIFIERS 3 Fig. 2. Illus a i e example o show equency dependence o he nonlinea ans e unc ions. (a) Elemen a y nonlinea ne wo k. (b) Associa ed linea ne wo k exci ed by app op ia e nonlinea cu en s and spec um o he gene a ed second-o de ans e unc ions. (c) The same associa ed linea ne wo k p oducing he hi d-o de ans e unc ions and hei ela ed spec um. esponses essen ially cons an in hei espec i e RF signal bands, see o example [2]. Unde his assump ion, comple- ion o he pa icula equency dependence o he ans e unc ions ˜ Hn(ξ) o a ypical ci cui can be accomplished by a combina ion o he P obing Me hod and he Nonlinea Cu en s Me hod, a widesp ead p ocedu e [10], [11]. Fo he gene al ci cui o Fig. 1, he ec o o med wi h he nonlinea ans e unc ions o o de n ela ing he ol ages o he independen po s o he associa ed linea ne wo k can be ob ained by using he ollowing equa ion ˜ Hn(ξ) = −Y−1(ξ1+ξ2+···+ξn)˜ Fn(in)(4) whe e Y(ξ)is he admi ance ma ix o he associa ed linea ne wo k and ˜ Fn(in)is a ec o wi h he spec al componen s o he nonlinea cu en s exci ing he independen po s. The bandlimi ed condi ion o he wi eless signal allows o ex end he wideband ampli ie assump ion o all he ha monic zones so ha he admi ance ma ix can be app oxima ed by Y(lωc+ ω)≈Y(lωc) o l= 1,2,···, a all ele an alues o ω. Fo an n h-o de app oxima ion he band o in e es in he i s ha monic zone should be he e supposed o be nB. Fo cla i y, le conside only ˜ Kn(ξ), he ans e unc ions ela ing he inpu wi h he ol age a he po o one nonlinea - i y, ypically a ol age con olled cu en sou ce, a conduc ance o a capaci ance. To illus a e he p ocedu e, an elemen a y ci cui wi h a nonlinea cu en sou ce is shown in Fig. 2 and a ske ched summa y o he me hod is a ached. In he example, a nonlinea cu en sou ce dependen on ol age is conside ed as he main nonlinea i y. Fo each o de he associa ed linea ne wo k is exci ed by app op ia e nonlinea cu en s in o de o ob ain he ans e unc ions ela ing ol age wi h he inpu . Spec a show he p e alen componen s o o de s 2 and 3. The wideband condi ion o he ampli ie allows o app ox- ima e he linea unc ion ˜ K1(ωc)as a coe icien independen o baseband equencies. The same is ue o he ans e unc ion ela ing he linea pa o he ou pu wi h he inpu , ˜ H1(ωc). The spec al componen s unc ion ˜ F2(ξ)can depend on he sum o baseband equencies ωi+ωj, as is he case o a non- linea capaci ance, o is independen o baseband equencies o o he nonlinea i ies. In any case, since he componen s o he admi ance ma ix ha e his same dependence, he second- o de nonlinea ans e unc ion ˜ K20(ωi+ωj)depends on ωi+ωj(wi h ξi=ωc+ωiand ξj=−ωc+ωj) in he dc zone, and can be conside ed as a cons an coe icien in he second-ha monic zone (see Table I). No ice ha i o he nonlinea i ies a e p esen , hei con ibu ion is supe posed a he same equencies and he e o e, he ype o dependence emains unchanged. Bo h in he undamen al equency and in he hi d- ha monic zones, he admi ance ma ix is a unc ion only o he ca ie equency, because he baseband equency dependence o he hi d-o de ans e unc ion ˜ K3(ξ)is due o he spec al componen s o ˜ F3(ξ). Taking in o accoun ha his dependence comes om ˜ K2(ξ), in he undamen al equency zone i is possible o de e mine wo ypes o e ms: a baseband equency independen e m ˜ K(1) 31 , and e ms o he o m ˜ K(2) 31 (ωi+ωj). In he hi d-ha monic zone his ans e unc ion does no p esen dependence on baseband equencies. The ype o dependence desc ibed abo e is in ag eemen wi h p e iously published esul s [12]-[16]. Al hough he ex ension o he analysis ocused o he 4 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. XX, NO. Y, MONTH 2006 TABLE I TYPES OF BASEBAND FREQUENCY DEPENDENCE FOR ˜ Kn(ξ) F equency zone ˜ K1(ξ)˜ K3(ξ) 1s . ha monic ˜ K1˜ K(1) 31 ˜ K(2) 31 (ωi+ωj) 3 d. ha monic −˜ K33 F eq. zone ˜ K2(ξ)˜ K4(ξ) dc ˜ K20(ωi+ωj)˜ K(1) 40 (¯ω) ˜ K(2) 40 =α(¯ω)κ(2) 40 (ωi+ωj) ˜ K(3) 40 =α(¯ω)κ(3) 40 (ωi+ωj)× ×κ(3) 40 (ωi′+ωj′) 2nd. ˜ K22 ˜ K(1) 42 ha monic ˜ K(2) 42 (ωi+ωj) 4 h. ha m. −˜ K44 The ollowing de ini ions ha e been used: ¯ω=ω1+ω2+ω3+ω4, ξi,i′=ωc+ωi,i′and ξj,j′=−ωc+ωj,j′,i6=i′and j6=j′. deduc ion o closed o m exp essions o highe o de ans e unc ions is almos beyond he bounds o possibili y, keeping ack o hei equency dependence is a mo e easible ex- e cise. Nex , his s udy is widened o highe o de nonlinea ans e unc ions. The cause o equency dependence in ˜ K4(ξ)is wo old. On he one hand, he admi ance ma ix can be app oxima ed by Y(ω1+ω2+ω3+ω4)in he dc zone, and by a equency independen unc ion in he second-ha monic and ou h- ha monic zones. On he o he hand, he componen ˜ F4(ξ) p esen s se e al e ms wi h p oduc s o he ans e unc ions ˜ K1,˜ K2and ˜ K3. Summa izing, in he dc zone he ou h-o de ans e unc- ion is composed by: one e m ha can be exp essed by ˜ K(1) 40 (ω1+ω2+ω3+ω4), a second ype o e ms ˜ K(2) 40 = α(ω1+ω2+ω3+ω4)κ(2) 40 (ωi+ωj), and a hi d ype o e ms wi h he o m ˜ K(3) 40 =α(ω1+ω2+ω3+ω4)κ(3) 40 (ωi+ ωj)κ(3) 40 (ωi′+ωj′). I has been conside ed ha ξi,i′=ωc+ωi,i′ and ξj,j′=−ωc+ωj,j′, o i= 1,2and j= 3,4wi h i6=i′ and j6=j′(see Table I). No e ha ˜ K(2) 40 is ep esen ed as he p oduc o wo sepa a ed unc ions, αand κ(2) 40 , and in he same o m, ˜ K(3) 40 is deno ed by he p oduc o he unc ions α and κ(3) 40 . In hese exp essions, αis a gene ic unc ion wi h he speci ic dependence on he sum o ou baseband equencies and κ(2,3) 40 a e gene ic unc ions dependen on he sum o wo baseband equencies. The signi icance o his pa icula equency dependence is discussed below. In he second-ha monic zone he e a e wo ype o e ms, ˜ K(1) 42 and ˜ K(2) 42 (ωi+ωj), and in he ou h-ha monic zone he e is only one ( equency independen ) e m, ˜ K44. The explici dependence has been omi ed in he componen s ha a e only unc ions o he ca ie equency. These a gumen s a e su icien o deduce he equency beha io o he ele an ou pu ans e unc ions ˜ H3(ξ)and ˜ H5(ξ). The ans e unc ion ˜ H3(ξ)has a equency dependence simila o ˜ K3(ξ), discussed abo e. In he case o ˜ H5(ξ), TABLE II TYPES OF BASEBAND FREQUENCY DEPENDENCE FOR ˜ H5(ξ) Type ˜ H5(ξ) 1˜ H(1) 51 2˜ H(2) 51 (ωi+ωj) 3˜ H(3) 51 =η(3) 51 (ωi+ωj)η(3) 51 (ωi′+ωj′) 4˜ H(4) 51 (ωi+ωj+ωi′+ωj′) 5˜ H(5) 51 =α(ω1+ω2+ω3+ω4)η(5) 51 (ωi+ωj) 6˜ H(6) 51 =α(ω1+ω2+ω3+ω4)η(6) 51 (ωi+ωj)η(6) 51 (ωi′+ωj′) The gene ic unc ions η51 ha e he speci ic dependence on he sum o wo baseband equencies. conside ing ha he admi ance ma ix does no in oduce any equency dependence, i s beha io is de e mined by ˜ F5(ξ), o equi alen ly, by he unc ions ˜ K1(ξ) o ˜ K4(ξ). Recalling ha he zone o in e es is he undamen al equency zone, he ele an ans e unc ions can be ep esen ed by he 6 ypes o e ms shown in Table II. Obse e ha now he e a e ou di e en ypes o e ms wi h a comple e dependence on all he equencies ω1 o ω4. These esul s can be exploi ed o educe he numbe o pa ame e s in he beha io al model (1) wi hou he addi ion o any o he es ic ion. C. Ke nels o he educed-o de beha io al model. 1) Thi d o de ke nel: Based on he p e ious deduc ions, he hi d-o de e m o he ou pu ol age is ob ained by subs i u ing in (2) he co esponding componen s ˜ H(1) 31 and ˜ H(2) 31 (ω1+ω2). The i s ype gene a es he memo yless e m, and he second ype gi es ise o he gene ic exp ession y(2) 3( ) = 3 4(2π)2x( )Z∞ −∞ ˜ H(2) 31 (ω1+ω2)× ×X(ω1)X∗(ω2) exp(jω1 +jω2 )dω1dω2.(5) Al hough his is a double in eg al, he ans e unc ion is a one-dimensional unc ion, a ac ha can be explici ly displayed wi h a change o he new a iables ω=ω1+ω2 and ξ= (ω2−ω1)/2, o which dω1dω2=dωdξ, so ha y(2) 3( ) = 3 4 1 (2π)2x( )Z∞ −∞ ˜ H(2) 31 (ω)× ×X(ω/2−ξ)X∗(ω/2 + ξ) exp(jω )dωdξ. (6) Relying on he bandlimi ed assump ion o x( ), he in eg al in ωis negligible ou side any bandwid h Bo≥2B, allowing he de ini ion o an equi alen ans e unc ion ˆ H3(ω)con ined o his band. Making use o he Fou ie ans o m and o equencies inside Bo,ˆ H3(ω)can be exp essed in e ms o i s disc e e impulse esponse ˆ H3(ω) = 1 BoX q ˆ h3(q) exp(−jqω s),(7) so ha he sampling ime s= 1/Boshould be, a leas , hal he symbol pe iod. Subs i u ing in (6) and changing now o CRESPO-CADENAS e al.: VOLTERRA BEHAVIORAL MODEL FOR WIDEBAND RF AMPLIFIERS 5 he o iginal a iables, he wo in eg als become sepa able y(2) 3( ) = 3 4Bo x( )X q ˆ h3(q)1 (2π)2× ×Z∞ −∞ X(ω1)X∗(ω2) exp[jω1( −q s) + jω2( −q s)]dω= =3 4Bo x( )X q ˆ h3(q)|x( −q s)|2.(8) Finally, a e adding he memo yless pa and sampling a ins an s =k s, he hi d-o de e m o he ou pu complex en elope can be exp essed in a disc e e- ime o m y3(k) = X q h3(q)|x(k−q)|2x(k).(9) When he memo yless e m is included in (9), we ob ain he ollowing exp ession o he hi d-o de coe icien s h3(q) = (3 4[˜ H(1) 31 +1 Bo ˆ h3(0)], o q= 0 3 4Bo ˆ h3(q), o q6= 0.(10) 2) Fi h o de ke nel: Acco ding o he p e ious discussion, he i h-o de ans e unc ion is composed by 6 di e en ypes o e ms which p oduce a pa icula se o ke nels a e subs i u ion in (2). Like in he hi d-o de ans e unc ion, one o he i e in eg als in ol ed gi es ise again o x( ), emaining in his case a quad uple in eg al, o which he i s ype o e ms can be compu ed di ec ly o con ibu e only o memo yless nonlinea e ec s. The second ype o e ms depends on ωi+ωj, so ha only wo in eg als can be compu ed di ec ly and he o he wo can be handled as in he hi d-o de case p oducing one-dimensional ke nels o he ype y(2) 5(k) = X q h(2) 5(q)|x(k−q)|2|x(k)|2x(k).(11) Fo he o he ou ypes i is possible o w i e a gene ic i h- o de ans e unc ion wi h a equency dependence gi en by ˜ H5(ω1+ω3, ω2+ω4), whe e ω1,2and ω3,4a e de ined a ound ωcand −ωc, espec i ely. Le conside he di e en p ope ies o symme y ha his unc ion can p esen , beginning wi h he mo e gene al condi ion co esponding o he i h and six h ypes o e ms. A a i s glance, he mul iple in eg al in (2) is negligible ou side a ou -dimensional cube o leng h 4B. Howe e , he pa icula symme y o ˜ H5in ol es a bidimensional equency dependence ha is exhibi ed clea ly a e he change o a iables ξ1=ω1+ω3,ξ2=ω2+ω4, ξ3= (ω3−ω1)/2and ξ4= (ω4−ω2)/2: y(5) 5( ) = 5 8 1 (2π)4x( )Z∞ −∞ dξ3dξ4··· ···Z∞ −∞ ˜ H5(ξ1, ξ2)X(ξ1/2−ξ3)X(ξ2/2−ξ4)× ×X∗(ξ1/2 + ξ3)X∗(ξ2/2 + ξ4) exp(jξ1 +jξ2 )dξ1dξ2. (12) The e o e, ˜ H5(ξ1, ξ2)is negligible ou side any squa e o leng h Bo≥2Band can be subs i u ed by i s equi alen unc ion. I is possible o exp ess his bandlimi ed unc ion as a ela ion be ween he co esponding disc e e- ime ke nels ˆ H(5) 5(ξ2) = 1 B2 oX q2 ˆ h(5) 5(q2) exp(−jq′ 2ξ2 s).(13) Subs i u ing in (12) and changing o he o iginal a iables, he ou in eg als a e now sepa able and, sampling a ins an s k s, he ou pu in disc e e- ime o m can be w i en as y(5) 5(k) = X q1,q2 h(5) 5(q1, q2)|x(k−q1)|2|x(k−q2)|2x(k)(14) wi h h(5) 5(q1, q2) = 5 8B2 o ˆ h(5) 5(q1, q2).(15) I is immedia e o no e ha h(3) 5, ela ed wi h he hi d ype ˜ H(3) 51 , is a pa icula case o his esul in which he ans e unc ion is di ec ly sepa able. Mo e e ealing is he ou h ype ˜ H(4) 51 , which in ol es he sum o all equencies so ha i p esen s he highes deg ee o symme y and p oduces e ms gi en by y(4) 5(k) = X q h(4) 5(q)|x(k−q)|4x(k)(16) wi h h(4) 5(q) = 5 8Bo ˆ h(4) 5(q).(17) No ice ha o his pa icula coe icien s, he sampling a e should be a leas ou imes he symbol a e. As a conclusion o his sec ion, le obse e ha he demon- s a ed Vol e a beha io al model inco po a es a subs an ial educ ion in he numbe o pa ame e s, when compa ed o (1), wi h he only assump ion o a wideband ampli ie . Su p is- ingly, he desc ibed ep esen a ion exhibi s an exclusi ely “ou o diagonal” s uc u e, di e en o o he well-known published beha io al models [1],[2]. To co obo a e hese new esul s, an ampli ie has been es ed and he model pa ame e s ha e been ex ac ed om expe imen al da a, as discussed in he nex sec ion. III. MODEL PARAMETERS EXTRACTION AND VALIDATION The comme cial ampli ie MAX2430 manu ac u ed by MAXIM In eg a ed P oduc s Inc. (Sunny ale, CA), has been modeled wi h he p esen s uc u e. I is a wideband ampli ie a 915 MHz, howe e , in he expe imen al cha ac e iza ion wi h wo ones sepa a ed 2 MHz, he ampli ie exhibi ed an asymme y in he IMD p oduc s, a clea indica ion o he exis ence o nonlinea memo y e ec s. The measu emen se up used in his s udy is basically he same as ha p esen ed in [15]. Howe e , he exci a ions aken in o accoun a e di e se in o de o es he p oposed model wi h a wide a ie y o signals. In pa icula , s anda d wo- one as well as digi ally modula ed signals like QPSK-WCDMA, 16-QAM wi h ec angula pulses and oo - aised cosine pulses ha e been used as inpu s imuli. These signals ha e been loaded in he in e nal memo y o an SMIQ02B signal gene a o wi h buil -in a bi a y wa e o m acili y and he E4407B spec um analyze wi h a modula ion analysis op ion has been used o acqui e he baseband signal a he ampli ie ’s ou pu . 6 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. XX, NO. Y, MONTH 2006 −14 −13.5 −13 −12.5 −12 −11.5 −11 −10.5 −10 0 0.1 0.2 0.3 0.4 0.5 Inpu signal le el (dBm) NMSE (%) Memo yless VBW, 3 d. o de Q3 = 30 MP model 7 delays VBW, 5 h. o de Q3 = Q5 = 30 Fig. 3. No malized e o as a unc ion o he inpu le el. P oposed VBW model (squa es and solid line) and memo y polynomial (MP) model (ci cles and do -dash line). A. 16-QAM signal wi h ec angula pulses As was demons a ed in he p e ious sec ion, i is necessa y o use a sampling a e o abou ou imes he symbol a e i pa ame e s up o he i h-o de ha e o be ex ac ed. In he case o oo - aised cosine modula ing pulses, each sample is dependen on p e ious and u u e samples, included hose many symbols away. E en in he case o a memo yless ampli ie he ou pu will display memo y and i is he e o e easonable o use o nonlinea cha ac e iza ion modula ing pulses wi hou in e symbol in e e ence, i.e. wi h leng h no longe han a symbol pe iod. Consequen ly, he use o an RF signal modula ed wi h ec angula pulses as an inpu s imulus gua an ees ha memo y e ec s, i p esen in he ou pu , ha e been caused by he nonlinea memo y o he de ice. The modula ion o ma is also ele an because i is well known ha o PSK signals, a hi d-o de model can cap u e some o he highe o de nonlinea cha ac e is ics and p oduce degene a ion in he pa ame e ex ac ion p ocess [9]. Acco ding o he abo e conside a ions, a 915 MHz ca ie modula ed wi h a andom ain o ec angula pulses a 2 Msymb/s and a 16-QAM o ma , was selec ed as he i s sounding signal. The a bi a y wa e o m gene a o can handle up o 40 Msa/s so ha he shape o he pulses was a he ec angula . Since in he eco e y pa he se up has a sampling a e o 15 Msa/s, he acqui ed ou pu signals we e sampled a 7.5 samples pe symbol, amply sui able o signal ep e- sen a ion and i h-o de pa ame e s ex ac ion. A i h-o de Vol e a beha io al model o wideband ampli ie s (VBW) was ex ac ed om he acqui ed ou pu complex en elope samples h ough he minimiza ion o he a e age no malized mean squa e e o (NMSE) be ween measu ed and modeled ou pu s. A i s esul is shown in Fig. 3, whe e he no malized e o is plo ed as a unc ion o he inpu le el. The e o is ep esen ed in do ed line o a memo yless model and he dashed line co esponds o he esul s o a hi d-o de VBW model wi h a memo y o Q3= 30 samples. In bo h cases he e o g ows up as he ampli ie en e s in a mo e nonlinea condi ion indica ing 20 20.5 21 21.5 22 22.5 23 23.5 24 24.5 25 −1 −0.5 0 0.5 1 Time (µs) In−phase componen 20 20.5 21 21.5 22 22.5 23 23.5 24 24.5 25 −1 −0.5 0 0.5 1 Time (µs) In−phase componen Acqui ed da a VBW model MP model Fig. 4. No malized complex en elope in-phase componen o a 16-QAM signal. a) Rec angula pulses and b) Roo - aised cosine pulses. Acqui ed da a: do s. VBW model: solid line. MP model: do -dash line. −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 In−phase componen Quad a u e componen Fig. 5. Vec o ep esen a ion o he 16-QAM signal wi h ec angula pulses. Inpu signal: squa es. Acqui ed da a: c osses. VBW model: do s. in he i s case he p esence o nonlinea memo y and, in he second case, ha no all he nonlinea memo y coe icien s ha e been ex ac ed. On he con a y, he i h-o de VBW model depic ed in solid line shows a e y low e o , cons an in all he ange o inpu le els, which is an e iden con i ma ion o a co ec pa ame e s iden i ica ion o he memo y nonlinea model. To co obo a e he alidi y o he p esen esul s, he well-es ablished memo y polynomial model (MP) desc ibed in [2] was ex ac ed wi h a gene ous numbe o delays, and aken as a eliable e e ence. The e o o his model wi h se en delays is ep esen ed in he same igu e (dash-do line) demons a ing a simila pe o mance wi h espec o he VBW model. Fu he mo e, in he ange o highe le els, he mos ele an o his con ex , he VBW model ou pe o ms he MP model. The co esponding ime-domain in-phase componen o an inpu le el o −11 dBm is shown no malized in Fig. 4a). The acquisi ions (do s), he p edic ion o he MP model (dash-do CRESPO-CADENAS e al.: VOLTERRA BEHAVIORAL MODEL FOR WIDEBAND RF AMPLIFIERS 7 −14 −13.5 −13 −12.5 −12 −11.5 −11 −10.5 −10 −20 −10 0 10 20 Inpu le el pe one (dBm) Pou (dBm) −14 −13.5 −13 −12.5 −12 −11.5 −11 −10.5 −10 2 4 6 8 10 Inpu le el pe one (dBm) IM3 asymme y (dB) Measu emen s VBW model Uppe IM3 Lowe IM3 Fig. 6. Measu ed and simula ed esul s o IM3 when one spacing is 2 MHz. Acqui ed da a: iangles. VBW model: solid line. line) and he p edic ion o he p esen VBW model (solid line) ha e been ep esen ed. The a e age NMSE o he VBW model is -30.5 dB, ep esen ing 1 dB o imp o emen compa ed o he MP model. The ec o ep esen a ion o he modeled complex en elope is plo ed wi h do s in Fig. 5 and compa ed wi h he inpu en elope (squa es) and he acqui ed ou pu (c osses). B. Two one signal Ano he se o expe imen s was pe o med using an inpu signal o med by wo ones o equal magni ude and phase, wi h a equency sepa a ion o 2 MHz, and measu ing he uppe and lowe hi d-o de IM p oduc s. When he ampli ie is wo king in linea mode he IM p oduc s a e negligible compa ed o he non-sys ema ic e o s o he measu emen se up, so ha model pa ame e s ex ac ed om acquisi ions a low signal le els a e i egula . This is no specially incon enien , because he in e es is in he ange o high signal le els, whe e he nonlinea e ec s a e mo e ele an and model pa ame e s can be eliably ex ac ed. Fo ha eason, a e ejec ion o meaningless da a, he expe imen al poin s ep esen ed in Fig. 6 belong o le els nea he 1 dB comp ession poin . In Fig. 6a) he measu ed ou pu o he undamen al ones and he hi d- o de in e modula ion p oduc s (IM3) a e ep esen ed wi h ma ks. In he same igu e he esul s o he ex ac ed model a e depic ed in solid line, e lec ing a ema kable co espondence wi h he acqui ed da a. Acco ding o he p e ious discussion, he di e ence be ween measu ed and calcula ed lowe IM3 a −14 dBm is caused by se up limi a ion a low inpu le els. On he con a y, he signi ican coincidence inside he ai h ul ange is also e ealed when measu ed and p edic ed asymme ies a e compa ed, as i is shown in Fig. 6b). C. 16-QAM signal wi h oo - aised cosine pulses Ano he ype o sounding signal employed in he ex ac ion p ocess has been a ca ie a 915 MHz modula ed in a 16-QAM o ma wi h a 2 Msymb/s ain o symbols us- ing oo - aised cosine pulses. The acqui ed no malized in- phase componen and he co esponding wa e o m ob ained 910 912 914 916 918 920 −50 −40 −30 −20 −10 0 10 20 F equency (MHz) Ou pu powe (dBm) Measu emen s VBW, 5 h. o de Q3 = Q5 = 30 16−QAM oo − aised cosine Symbol a e: 2 Msym/s Pin = − 11 dBm Resolu ion bandwid h: 100 kHz Fig. 7. Ou pu spec um o a 16-QAM signal wi h 2 Msymb./s. Roo - aised con o ming pulses. Pin =−11 dBm. Spec um analyze ace (do s) and VBW model p edic ion (solid line). −14 −13.5 −13 −12.5 −12 −11.5 −11 −10.5 −10 −30 −25 −20 −15 −10 −5 0 5 10 15 20 Inpu signal le el (dBm) Co−channel and adjacen channel powe (dBm) Uppe adjacen channel Lowe adjacen channel Fig. 8. In-channel and adjacen channel powe . Acqui ed da a: iangles. VBW model: solid lines. MP model: do -dash lines wi h he ex ac ed model a e ep esen ed in Fig. 4b) (do s and solid line, espec i ely). Fo compa ison pu poses, he wa e o m ob ained wi h he memo y polynomial model is also depic ed in he same igu e (dash-do line). As a u he es , he ex ac ed model was used o p edic he spec um o he signal and he adjacen -channel powe (ACP) in o de o be compa ed wi h o he al e na i e measu emen s using he con en ional spec um analyze wi hou he acquisi ion acili y. The esul s a e plo ed in Figs. 7 and 8 using ma ks o he expe imen al da a and solid lines o he modeled ou pu . I is wo h o no e ha he model was i s ex ac ed om an expe imen al acquisi ion o baseband samples and se ed o p edic he ou pu signal spec um. Al hough he ma ks ep esen a measu emen p ocess independen o he acquisi ion, he p edic ion is a o ably compa ed in Fig. 7. The second igu e co obo a es his ou come showing a good ma ch be ween he adjacen channel powe measu ed and calcula ed. The p edic ion is also able o es ima e adequa ely 8 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. XX, NO. Y, MONTH 2006 910 912 914 916 918 920 −50 −40 −30 −20 −10 0 10 20 F equency (MHz) Ou pu powe (dBm) VBW, 5 h. o de Q3 = Q5 = 30 Measu emen s QPSK−WCDMA Symbol a e: 3.84 Msym/s Pin = −10 dBm Resolu ion bandwid h: 100 kHz Fig. 9. Spec al eg ow h o a W-CDMA signal wi h 3.84 Msymb./s. Pin = −10 dBm. Spec a o acqui ed da a (do s) and model ou pu (solid line). −14 −13.5 −13 −12.5 −12 −11.5 −11 −10.5 −10 −25 −20 −15 −10 −5 0 5 10 15 20 Inpu signal le el (dBm) Co−channel and adjacen channel powe (dBm) Uppe adjacen channel Lowe adjacen channel Fig. 10. ACP measu ed and calcula ed wi h he p oposed me hod. Acqui ed da a: iangles. VBW model: solid lines. he ACP asymme y be ween uppe and lowe channels. As a e e ence, he dash-do line ep esen s he esul s o he memo y polynomial model. D. QPSK-WCDMA signal Finally, a ca ie a 915 MHz modula ed wi h a WCDMA signal complian wi h he UMTS s anda d was employed as inpu signal. In his case, he condi ion o ou imes he symbol a e o co ec ly iden i y i h-o de pa ame e s, 4×3.84 Msymb/s, jus exceeds he sampling a e, 15 Msa/s.1 Howe e , he model has been able o ex ac adequa ely he pa ame e s, as can be e i ied by compa ing he acqui ed and modeled spec a, which a e shown in Fig. 9 wi h do s o he expe imen al da a and solid line o he calcula ions wi h he ex ac ed model. Again, he p esen app oach allows 1Conside ing ha wi h espec o he modula ion o ma he e m chip ul ima ely co esponds o a symbol, in his con ex we use he e m symbol ins ead o chip. TABLE III ACPR FOR A QPSK-WCDMA SIGNAL AT 3.84 MSPS. Pin =−10 dBm. Channel Measu ed (dBc) MP model (dBc) VBW model (dBc) Uppe -21.6 -20.4 -21.0 Lowe -24.7 -25.2 -26.1 a eliable p edic ion o ACP and i s asymme y om he acqui ed spec a, as can be con i med in Fig. 10, in which he measu ed da a ( iangles) and he model esul s (solid lines) a e ep esen ed. An al e na i e me hod was used o measu e he ACP wi h an inpu le el o −10 dBm and he measu ed da a a e p esen ed in Table III o he wo adjacen channels. In he same able he calcula ions wi h he p oposed VBW model a e shown. In spi e o he ac ha he acquisi ions we e accomplished beyond he limi s o he heo e ical accu acy, a sa is ac o y ag eemen is e ealed. Fo sake o compa ison, he e e ence MP model is also included, a aining equi alen esul s. IV. FINAL DISCUSSION AND CONCLUSIONS This wo k demons a es a new Vol e a app oach o model wideband ampli ie s wi h nonlinea memo y. The main cha - ac e is ic o he p esen beha io al model is ha i has been o mally de i ed s a ing om a con en ional nonlinea ci cui analysis and makes possible o p opose he ex ension o i s s uc u e o gi e he ollowing equa ion y(k) = h1x(k)+ ∞ X m=1 X qm h2m+1(qm) m Y p=1 |x(k−qp)|2x(k). (18) This exp ession has a ema kable di e ence wi h espec o he memo y polynomial model consis ing in he absence o he so- called “diagonal e ms”. Al hough only he assump ion o a equency independen esponse has been necessa y o ob ain he new model, he huge numbe o coe icien s associa ed o he gene al disc e e- ime Vol e a se ies has been d as ically educed. The pa ame e o de educ ion can be quan i ied o an example in which a i h-o de model wi h Q= 3 delays is conside ed. Taking in o accoun symme y conside a ions, 244 coe icien s a e necessa y wi h he gene al Vol e a model. As a e e ence, ecall ha in he MP model a o al o 12 diagonal coe icien s a e needed, and a o al o 54 o 133 coe icien s o m he model wi h he “nea -diagonali y” s uc u al es ic- ion l= 1 o l= 2, espec i ely [3]. Ins ead, he p esen model would need 21 coe icien s. I is wo h o obse e ha a ai compa ison be ween he p e ious models and he p esen VBW would be only possible i some p ocedu e o p uning o op imize he numbe o coe icien s we e also included. In ela ion o memo yless nonlinea sys ems, he in oduced analysis s a es ha iden i ica ion may be achie ed by sampling a he symbol a e, in acco dance wi h p e iously published esul s [17]. Howe e , i is also demons a ed ha in he case o sys ems wi h nonlinea memo y, an inc ease o n−1in he sampling a e is necessa y o adequa ely iden i y n h-o de pa ame e s. CRESPO-CADENAS e al.: VOLTERRA BEHAVIORAL MODEL FOR WIDEBAND RF AMPLIFIERS 9 Ano he no able concep ual issue is he exhibi ed ela ion o he model coe icien s, hn(q), wi h espec o he adop ed sampling a e, e ealing a somewha in ol ed s uc u e wi h e ms o he same o de showing di e en dependence on his pa ame e , o example eqs. (15) and (17). To con as wi h he heo e ical esul s, a comme cial am- pli ie was cha ac e ized using ou di e en ypes o wa e- o ms and he coe icien s o he model we e ex ac ed using an expe imen al se up wi h acquisi ion acili ies. The model p edic ions we e compa ed wi h he well-es ablished memo y polynomial beha io al model and pe o mance is e y simila , indica ing he alidi y o he p esen p ocedu e. Requi ing a sea ch algo i hm o an abundan numbe o delays, he e - e ence beha io al model exhibi s somewha be e p edic ion o he ou pu cha ac e is ics in he low powe ange. On he con a y, he in oduced Vol e a-based wideband beha io al me hod uses mo e delays wi hou equi ing a sea ch p ocess and ou pe o ms he polynomial model in he ange o powe s whe e nonlinea e ec s a e mo e signi ican . An impo an equi emen o a Vol e a-based model is i s abili y o manage di e en ypes o signal. I has been e ealed by obse ing he pe o mance o he demons a ed model wi h inpu s imuli as di e se as a wo- one signal, a 3GPP W- CDMA signal and 16-QAM signal wi h ec angula and oo - aised cosine pulses. The esul s we e e y sa is ac o y in all he cases. Because o space limi a ions, only he mos ele an p elimina y esul s ha e been p esen ed. A he momen , he model is being es ed wi h he expe imen al da a in o de o e alua e consis ence a di e se measu emen condi ions and i s abili y o manage inpu signals wi h di e en bandwid hs and powe le els. ACKNOWLEDGMENT The au ho s wish o acknowledge he help ul assis ance o T. J. B azil and A. Zhu in he discussion o models complexi y, and he use ul commen s ecei ed o imp o e he quali y o he pape . REFERENCES [1] J. Kim and K. Kons an inou, “Digi al p edis o ion o wideband signals based on powe ampli ie model wi h memo y,” Elec onics Le e s, ol. 37, no. 23, pp. 1417-1418, No . 2001. [2] H. Ku and J. S. 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Ca los C espo-Cadenas was bo n in Mad id, Spain. He ecei ed he deg ee in Physics in 1973 and Doc- o deg ee in 1995 om he Poly echnique Uni e - si y o Mad id. Since 1998 he has has been Associa e P o esso and cu en ly he eaches lec u es on Radio Communica ions in he A ea o Signal Theo y and Communica ions, Uni e si y o Se ille. His cu en in e es s a e Nonlinea Analysis applied o Wi eless Digi al Communica ions and o Mic owa e Mono- li hic In eg a ed Ci cui s (MMIC). Ja ie Reina-Tosina was bo n in Se ille, Spain, in May 1973. He ecei ed he Telecommunica ion Enginee ing and Doc o deg ees om he Uni e si y o Se ille, Se ille, Spain, in 1996 and 2003, espec- i ely. Since 1997 he has been wi h he Depa men o Signal Theo y and Communica ions, Uni e si y o Se ille. His cu en esea ch in e es s include MMIC echnology, nonlinea analysis o ac i e mic owa e de ices and in eg a ion o in o ma ion echnologies in biomedicine. Ma ´ıa J. Made o-Ayo a ecei ed he Telecommu- nica ion Enginee ing deg ee om he Uni e si y o Se ille, Se ille, Spain, in 2002. Since 2003 she has been wi h he Depa men o Signal Theo y and Communica ions, Uni e si y o Se ille, and is cu - en ly wo king owa d he Doc o deg ee in Telecom- munica ion Enginee ing. He esea ch in e es s lie in he a ea o nonlinea analysis o ac i e mic owa e de ices and measu emen echniques o nonlinea communica ion sys ems.