Mathematical Needs for Behavioral Modeling of Telecommunication Circuits and Systems (First Part Model Formulation)

Transcription

1 Mathematical Need for Behavioral Modelig of Telecommuicatio Circuit ad Sytem Firt Part Model Formulatio Joé Carlo Pedro Itituto de Telecomuicaçõe Uiveridade de Aveiro Portugal Mathematical Techiue ad Problem i Telecommuicatio Sept. 006 J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep. 006

2 Preetatio Outlie. Itroductio to Behavioral Modelig. Geeral Noliear Behavior of Wirele Sytem 3. Baic o Sytem Idetificatio Theory 4. Noliear Behavioral Modelig of Wirele Sytem 5. Cocluio J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep. 006

3 . Itroductio to Behavioral Modelig Noliear Modelig of RF Noliear Device - Model are eceary for CAD of microwave circuit ad ytem - Noliear CAD of Circuit i already i a mature tate; However, it ca ot upport large ad heterogeeou circuit uch a complete telecommuicatio ytem Reduced Compleity Higher Level of Hierarchical Decriptio Circuit Level CAD/Modelig Sytem Level CAD/Modelig J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

4 J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep Itroductio to Behavioral Modelig Phyical Modelig v. Behavioral Modelig Phyic Baed Modelig: [ ] = a d N N p ε ψ p p p p p R G p D p p t p R G D t = = ψ µ ψ µ ψ µ ψ µ DS D GS G S V y V y y = = =,,, 0,, ψ ψ ψ

5 . Itroductio to Behavioral Modelig Phyical Modelig v. Behavioral Modelig Empirical, Behavioral or Black Bo Modelig: i t i [v t,v t,i t,i t ] i t v t v t - - i [v t,v t,i t,i t ] where v t = [ v t, v& t, v& &,...],,,, t [ i t, i& t, & i,...] i, t =,,, t J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

6 . Itroductio to Behavioral Modelig Phyical Modelig v. Behavioral Modelig Euivalet Circuit Modelig ca be ee a Behavioral Modelig uig a-priori phyic kowledge of the topology: J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

7 . Itroductio to Behavioral Modelig Phyical Modelig v. Behavioral Modelig Euivalet Circuit Modelig ca be ee a Behavioral Modelig uig a-priori phyic kowledge of the topology: G Lg Rg Ccf Rd Ld D Ccg Cpg Cgd Cg Id Ri Cd Cpd Ccd R S L J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

8 . Itroductio to Behavioral Modelig Phyical Modelig v. Behavioral Modelig t yt t H yt Phyical Model ca be deduced from the phyic of the device Behavioral Model are Empirical i Nature: They rely o iput-output Behavioral obervatio They eed to compeate the lack of kowledge of device cotitutio Black-Bo Model with obervatio data J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

9 Preetatio Outlie. Itroductio to Behavioral Modelig. Geeral Noliear Behavior of Wirele Sytem 3. Baic o Sytem Idetificatio Theory 4. Noliear Behavioral Modelig of Wirele Sytem 5. Cocluio J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

10 V Tik. Geeral Noliear Behavior of Wirele Sytem Geeral Ditortio Behavior of a Microwave PA A wirele power amplifier iclude, ot oly hort-term RF memory effect, a log-term evelope memory caued by bia circuitry, chargecarrier trap ad elf-heatig. G R g C g C gd C RF R d D G D R i i DS R C d R RF C Th R Th V Th I Th S Heat-Sik S L Heat-Sik Z 0 IT-UA 003 v RF t Z 0 V GG V DD J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

11 . Geeral Noliear Behavior of Wirele Sytem v RF τ, t Geeral Ditortio Behavior of a Microwave PA Thee diparate origi of memory effect iteract to create a very comple dyamic feedback behavior. Z 0 Liear Dyamic Matchig Network M i Ω, v GS τ, t Z L Ω, Liear i DS Dyamic v DSτ, t Matchig v O τ, t Network M o Ω, Z 0 i DS τ, i DS t τ, i= DS t fτ, NL = [v t f NL GS = [v τ, f GS NL t, τ, [v GS vt, DS τ, τ, v DS t] t, τ, Tτ] t] v DS τ, t = L[i DS τ, t, Z L Ω, ] Tτ = L TH [i DS τ, t, Z TH Ω] J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep. 006

12 J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep. 006 O D a S = H F D a = Noliear Dyamic Model of a Feedback Wirele Sytem. Geeral Noliear Behavior of Wirele Sytem = 3,, D F D F D F a a D O D D D H H H S F t yt Liear / Dyamic Noliear / Memoryle et a eta et a 3 et 3 O Liear / Dyamic H Liear / Dyamic A wirele ytem will the preet Liear ad Noliear Dyamic Effect.

13 J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep Noliear Dyamic Model of a Feedback Wirele Sytem. Geeral Noliear Behavior of Wirele Sytem So, oliear dyamic caot be repreeted by ay Filter-Noliearity Wieer model = 3,, D F D F D F a a D O D D D H H H S F t yt Liear / Dyamic Noliear / Memoryle et a eta et a 3 et 3 O Liear / Dyamic H Liear / Dyamic 3 3 a S Γ = Γ

14 J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep Geeral Noliear Behavior of Wirele Sytem Noliear Dyamic Model of a Feedback Wirele Sytem or ay Noliearity-Filter Hammertei model cacade! 3 3 Γ a S = F t yt Liear / Dyamic Noliear / Memoryle et a eta et a 3 et 3 O Liear / Dyamic H Liear / Dyamic = 3,, D F D F D F a a D O D D D H H H S

15 Preetatio Outlie. Itroductio to Behavioral Modelig. Geeral Noliear Behavior of Wirele Sytem 3. Baic o Sytem Idetificatio Theory 4. Noliear Behavioral Modelig of Wirele Sytem 5. Cocluio J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

16 3. Baic o Sytem Idetificatio Theory The Behavioral Modelig Problem I it poible to produce a behavioral model with predictive capabilitie i.e. decribig the whole ytem, from a fiite et of obervatio? Ye, it i!! Cotrary to ome a-priori ituitio ad may heuritic behavioral model, ytem idetificatio theory how that our wirele ytem ca be decribed by a mathematical operator a fuctio of fuctio that map a fuctio of time t the iput igal oto aother fuctio of time yt the output: S[ft] t yt = S[t] J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

17 3. Baic o Sytem Idetificatio Theory The Behavioral Modelig Problem Thi iput-output map ca be repreeted by the followig forced oliear differetial euatio: f p d y t,..., p d t d y t, d t y t, r d t,..., r d t d t, t d t = 0 J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

18 3. Baic o Sytem Idetificatio Theory Recurive ad No-Recurive Model I a digital computer, time i a ucceio of uiform time ample: t yt y o, our oliear differetial euatio become a differece euatio. The olutio of thi oliear differece euatio ca be epreed i the followig Recurive Form Noliear IIR Filter: [ y,..., y p,...,,,...,,...] y = fr r J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

19 3. Baic o Sytem Idetificatio Theory Recurive ad No-Recurive Model If the ytem i caual, table ad of fadig memory it ca alo be repreeted by a No-Recurive,, or Direct, Form, where the relevat iput pat i retricted to {0,,,,}, the ytem Memory Spa Noliear FIR Filter: [,,..., ] y = f D J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

20 3. Baic o Sytem Idetificatio Theory Recurive ad No-Recurive Model t -to- [,,..., ] y = f D The S[t] operator i replaced by a -to- liear mappig, z - z - -to- Multi - Dimeioal Memoryle Fuctio yt followed by a -to- oliear f D [.,...,.] tatic fuctio. z - Noliear Memoryle Dyamic Liear Network J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

21 3. Baic o Sytem Idetificatio Theory Polyomial Filter ad Artificial Neural Network The multi-dimeioal fuctio f R. ad f D., have bee epreed i two differet form, leadig to: f R. ad f D. are approimated by Polyomial Polyomial Filter f R. ad f D. are approimated by Artificial Neural Network J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep. 006

22 J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep Baic o Sytem Idetificatio Theory Noliear IIR Filter A a Recurive Polyomial Filter, f R. i replaced by a multidimeioal polyomial approimatio: [ ] ,...,,,...,...,,,,,...,,,..., N N N N N N N N N N R y y a a y a y y a a y a y y P y N N N N = = = = = = = = = = = = = = L L

23 3. Baic o Sytem Idetificatio Theory Noliear IIR Filter E.g., a Recurive Biliear Filter d order IIR i implemeted a: y = = a 0 a0 y a, y = 0 = = 0 z - z -... z -... a 0,0 a 0, a 0, a 0, a,0 a,... a, y a 0, a 0, a 0, z -... z -... z - J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

24 J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep Baic o Sytem Idetificatio Theory Noliear FIR Filter [ ] = = = = = = = ,...,...,,...,, N N N D N a a a P y L A a Direct Polyomial Filter, f D. i replaced by a multi-dimeioal polyomial approimatio:

25 3. Baic o Sytem Idetificatio Theory Noliear FIR Filter If f D. i approimated by a Taylor erie, the thi oliear FIR filter i kow a the Volterra Serie or Volterra Filter Optimal approimatio i uiform error ee ear the epaio poit, provide: Good modelig propertie of mall-igal or mildly oliear regime Catatrophic degradatio uder trog oliear operatio J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

26 3. Baic o Sytem Idetificatio Theory Noliear FIR Filter But f D. ca alo be approimated by ay other Multi-Dimeioal Orthogoal Polyomial, geeratig a Geeral Noliear FIR Filter Optimal approimatio i mea uare error ee i the viciity of a certai operatig power level, ad for a particular type of iput, provide: Good modelig propertie of trog oliear regime A optimum a the iput igal tatitic are cloe to thoe of the timulu for which the polyomial i orthogoal J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

27 3. Baic o Sytem Idetificatio Theory Noliear FIR Filter E.g., a Direct t Order Polyomial Filter Liear FIR i implemeted a: y = a = 0 z - z -... z - a,0 a, a, a,... y J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

28 3. Baic o Sytem Idetificatio Theory Noliear FIR Filter while a Direct 3rd Order Polyomial Filter 3rd Order FIR would be: y 3 = a,, 3 3 = 0 = 0 = z - z - z a 3,000 a 3,00... a 3,00 a 3,0...a 3, a 3, a 3, a 3, a 3, a 3, y 3 J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

29 J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep Baic o Sytem Idetificatio Theory Dyamic Feedforward Artificial Neural Network [ ] = = = = K k k k a o o D w b f k w b f y 0,,, K i which b o, b k, w o k ad w k are the model parameter, ad f a [.] i a oe-dimeioal oliearity typically a igmoid kow a the activatio fuctio.

30 3. Baic o Sytem Idetificatio Theory Dyamic Feedforward Artificial Neural Network w k [,,, ] y = f D K b k f[u k ] z- z-... z- u k u k w o k b o y J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

31 3. Baic o Sytem Idetificatio Theory Dyamic Recurive Artificial Neural Network [ y,..., y,,...,, ] y = fr z - b k wy k f[u k ] z - wy o k... z - z - z - u k u k b o y... z - w k J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

32 Preetatio Outlie. Itroductio to Behavioral Modelig. Geeral Noliear Behavior of Wirele Sytem 3. Baic o Sytem Idetificatio Theory 4. Noliear Behavioral Modelig of Wirele Sytem 5. Cocluio J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

33 4. Noliear Behavioral Modelig of Microwave PA Bad-Pa ad Low-Pa Euivalet Behavioral Model [Pedro ad Maa, IEEE T-MTT 005] Coiderig that a geeral wirele ytem procee a modulated RF carrier, it bad-pa iput ad output igal ca be epreed a: { j[ ]} 0 t φ t r t e = r t co[ t φ t ] t = Re 0 whoe RF carrier i: t c t = co 0 ad low-pa euivalet comple evelope i: ~ jφ t t = r t e J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

34 4. Noliear Behavioral Modelig of Microwave PA Bad-Pa ad Low-Pa Euivalet Behavioral Model Therefore, we may coceive a Low-pa Euivalet Behavioral Model to hadle oly the comple bae-bad evelope, t: ~ Re[ τ ~ ] Im[ τ ~ ] Evelope Time τ Re[ yτ ~ ] Im[ ~ yτ ] ~ ~ XΩ YΩ 0 Hz 0 Hz Evelope Freuecy Ω J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

35 4. Noliear Behavioral Modelig of Microwave PA Low-Pa Euivalet Behavioral Model [Pedro ad Maa, IEEE T-MTT 005; Iako et al., IEEE T-MTT 006] Simple AM-AM / AM-PM Model Noliear / Memoryle t ~ r t r y [r t]coφ y [r t] r y [r t]e jφ y[r t] r y [r t]iφ y [r t] Noliear / Memoryle t ~ j ~ yt t ~ t ~ e jθt Memoryle AM-AM/AM-PM Saleh Model [Saleh., IEEE T-COM 98] J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

36 4. Noliear Behavioral Modelig of Microwave PA Low-Pa Euivalet Behavioral Model Two-Bo ad Three-Bo Model are implified Noliear FIR Filter where memory effect are eparated from Noliearity. They aume a Filter-Noliearity Wieer Model, a Noliearity-Filter Hammertei Model or Filter-Noliearity-Filter Wieer-Hammertei Model tructure: t H Noliear / Memoryle AM-AM / AM-PM O yt Liear / Dyamic Liear / Dyamic J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

37 4. Noliear Behavioral Modelig of Microwave PA Low-Pa Euivalet Behavioral Model Modificatio of thee Two or Three-Bo model have alo bee ued W z f [z ] w k b k f[u k ] z k W k... z W K K... f k [z k ]... f K [z K t] y z- z-... z- u k u k w o k b o Parallel Wieer Model [Ku, Mckiley ad Keey, IEEE T-MTT 00] or No-Recurret ANN J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

38 J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep Noliear Behavioral Modelig of Microwave PA Low-Pa Euivalet Behavioral Model I a effort to reach a more ytematic tructure for the model a recurive ANN [O Brie et al., IMS 006], but motly everal Polyomial FIR Filter have bee tried. The complete Volterra Model ha bee tried by Zhu, Wre ad Brazil. [Zhu, Wre ad Brazil, IEEE IMS 003] = = = = = = M M N N N M M M N h h h y ,...,..., L

39 4. Noliear Behavioral Modelig of Microwave PA Low-Pa Euivalet Behavioral Model Ufortuately, the compleity of the parameter etractio poe evere retrictio o both the order N degree of oliearity ad the umber of delay M dyamic behavior. Thi demaded the ue of everal alterative for pruig the Volterra coefficiet [Zhu & Brazil, IEEE MWCL-004; Zhu & Brazil, IEEE IMS 005, Zhu et al., IEEE IMS 006; Dooley et al., IEEE IMS 006; Iako et al., IMS 006] or to etract them, oe by oe, i a orthogoal eparable way [Lavrador et al., APMC 005; Pedro et al., IEEE IMS 006] J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

40 4. Noliear Behavioral Modelig of Microwave PA Low-Pa Euivalet Behavioral Model A oe-dimeioal alterative wa recetly propoed: ~ =0 ~~ a,0 z - = ~~ a, z - ~ y z - = ~~ a, Oe-Dimeioal Volterra Model [Ku, Mckiley ad Keey, IEEE IMS 003] which ca be how to be imilar to the Noliear Itegral Model. J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

41 4. Noliear Behavioral Modelig of Microwave PA Low-Pa Euivalet Behavioral Model Suppoig the ytem may be trogly oliear, but memoryle, while howig approimately liear dyamic effect, allow the applicatio of the Noliear Itegral Model [Filicori et al., IEEE T-MTT 99]: ~ y = = 0 ~ f [ ~, ] ~ Comple Evelope Noliear Itegral Model [Mirri et al., IMTC/99], [Soury et al., IEEE IMS 003, Zhu et al., IMS 006] i which f [.] i a oliear impule repoe, defied at a certai itataeou ecitatio level. J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

42 4. Noliear Behavioral Modelig of Microwave PA Low-Pa Euivalet Behavioral Model Accordigly, the NIM could be implemeted a: ~ =0 f 0 [] ~ ~ y = = 0 ~ f [ ~, ] ~ = z - z f [] ~... y ~ y = z - f [] ~ J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

43 4. Noliear Behavioral Modelig of Microwave PA Low-Pa Euivalet Behavioral Model or, i a lightly differet form, a a geeralizatio of a lier FIR filter: ~ y = = 0 h ~ ~ y = = 0 ~ f [ ~, ] ~ ~ =0 h 0 ~ =0 f 0 [] ~ z - z - = h ~ y = z - z - f [-] ~ ~ y z - z - = h = f [-] ~ J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

44 Preetatio Outlie. Itroductio to Behavioral Modelig. Geeral Noliear Behavior of Wirele Sytem 3. Baic o Sytem Idetificatio Theory 4. Noliear Behavioral Modelig of Wirele Sytem 5. Cocluio J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

45 5. Cocluio. Noliear behavior of wirele ytem how igificat dyamic effect caued by the iteractio betwee the tuig etwork, bia circuit ad active device low-freuecy diperio.. Sytem idetificatio how that may modelig activitie ca be framed ito a mall et of caoic behavioral model tructure. 3. Behavioral modelig of wirele ytem ha bee directed to comple evelope low-pa euivalet that proce the amplitude ad phae data. 4. Although formal tructure a the Polyomial FIR Filter, or Feedforward ANN, provide guarateed predictive capabilitie, they ivolve a large umber of parameter ad are very difficult to etract. J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

46 Referece S. Boyd, Y. Tag ad L. Chua, Meaurig Volterra Kerel, IEEE Tra. o Circuit ad Sytem, Vol. CAS. 30, No. 8, pp , 983. G. Chriiko, C. Clark, A. Moulthrop, M. Muha ad C. Silva, A Noliear ARMA Model for Simulatig Power Amplifier, IEEE MTT-S It. Microwave Symp. Dig., pp , Baltimore, Ju L. Chua ad Y. Liao, Meaurig Volterra Kerel II, It. J. Circuit Theory ad Applicatio, Vol. 7, No., pp.5-90, Apr L. Chua ad Y. Liao, Meaurig Volterra Kerel III: How to Etimate the Highet Sigificat Order, It. J. Circuit Theory ad Applicatio, Vol. 9, No., pp.89-09, Mar.-Apr. 99. J. Dooley et al., Behavioral Modelig of RF Power Amplifier Uig Adaptive Recurive Polyomial Fuctio, IEEE It. Microwave Sympoium Dig., pp , Sa Fracico, Ju C. Eva et al., Periodic Sigal for Meaurig Volterra Kerel, IEEE Tra. o Itrumetatio ad Meauremet, Vol. IM-45, No., pp.36-37, Apr J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

47 Referece F. Filicori, G. Vaii ad V. Moaco, A Noliear Itegral Model of Electro Device for HB Circuit Aalyi, IEEE Tra. o Microwave Theory ad Tech., vol. MTT-40, No. 7, pp , Jul. 99. M. Iako et al., "A Comparative Aalyi of Behavioral Model for RF Power Amplifier", IEEE Tra. o Microwave Theory ad Tech., vol. MTT-54, No., pp , Ja H. Ku, M. Mckiley ad J. S. Keey, uatifyig Memory Effect i RF Power Amplifier, IEEE Tra. o Microwave Theory ad Tech., vol. MTT-50, No., pp , Dec. 00. H. Ku ad J. Keey, Behavioral Modelig of RF Power Amplifier Coiderig IMD ad Spectral Regrowth Aymmetrie, IEEE MTT-S It. Microwave Symp. Dig., Philadelphia, Ju P. Lavrador et al., A New Volterra Serie Baed Orthogoal Behavioral Model for Power Amplifier, Aia-Pacific Microwave Coferece Dig., pp., Suzhou, Dec J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

48 Referece D. Mirri, F. Filicori, G. Iuculao, G. Paii, A No-Liear Dyamic Model for Performace Aalyi of Large-Sigal Amplifier i Commuicatio Sytem, IMTC/99 IEEE Itrumetatio ad Meauremet Techology Cof. Dig., pp , Veice, May 999. B. O Brie et al., RF Power Amplifier Behavioral Modelig uig a Globally Recurret Neural Network, IEEE It. Microwave Sympoium Dig., pp , Sa Fracico, Ju J. Pedro ad N Carvalho, Deigig Bad-Pa Multiie Ecitatio for Microwave Behavioral Model Idetificatio, IEEE It. Microwave Sympoium Dig. CDROM, Fort Worth, Ju J. Pedro ad N. Carvalho, "Deigig Multiie Ecitatio for Noliear Model Tetig", IEEE Tra. o Microwave Theory ad Tech., Vol. MTT- 53, No., pp.45-54, Ja J. Pedro ad S. Maa, A Comparative Overview of Microwave ad Wirele Power-Amplifier Behavioral Modelig Approache, IEEE Tra. o Microwave Theory ad Tech., vol. MTT-53, pp.50-63, Apr J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

49 Referece J. Pedro et al., A Formal Procedure for Microwave Power Amplifier Behavioral Modelig, IEEE It. Microwave Sympoium Dig., pp , Sa Fracico, Ju R. Pitelo ad J. Schouke, Sytem Idetificatio A Freuecy Domai Approach, IEEE Pre, New York 00. K. Remley, Multiie Ecitatio for ACPR Meauremet, IEEE MTT-S It. Microwave Symp. Dig., Philadelphia, Jue 003. A. Saleh, Freuecy-Idepedet ad Freuecy-Depedet Noliear Model of TWT Amplifier, IEEE Tra. o Commuicatio, Vol. COM- 9, No., pp.75-70, 98. M. Schetze, The Volterra ad Wieer Theorie of Noliear Sytem, Joh Wiley & So, New York, 980. D. Schreur, M. Myliki ad K. A. Remley, RF Behavioural Modellig from Multiie Meauremet: Ifluece of Ecitatio Type, 33rd Europea Microwave Cof. Dig., Muich, Oct J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

50 Referece A. Soury, et. al., A New Behavioral Model Takig ito Accout Noliear Memory Effect ad Traiet Behavior i Widebad SSPA, IEEE MTT-S It. Microwave Symp. Dig., pp , Seattle, Ju. 00. A. Soury et al., Meauremet Baed Modelig of Power Amplifier for Reliable Deig of Moder Commuicatio Sytem, IEEE MTT-S It. Microwave Symp. Dig., Philadelphia, Ju F. Verbeyt ad M. V. Boche, VIOMAP, the S-parameter Euivalet for Weakly Noliear RF ad Microwave Device, IEEE Tra. o Microwave Theory ad Tech., vol. MTT-4, pp , 994. T. Wag ad T. Brazil, Volterra-Mappig-Baed Behavioral Modelig of Noliear Circuit ad Sytem for High Freuecie, IEEE Tra. o Microwave Theory ad Tech., vol. MTT-5, No. 5, pp , May 003. D. Weier ad G. Naditch, A Scatterig Variable Approach to the Volterra Aalyi of Noliear Sytem, IEEE Tra. o Microwave Theory ad Tech., vol. MTT-4, pp.4-433, 976. J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

51 Referece J. Wood ad D. Root, The Behavioral Modelig of Microwave/RF IC uig No-Liear Time Serie Aalyi, IEEE MTT-S It. Microwave Symp. Dig., Philadelphia, Ju Zhag, Artificial Neural Network, Workhop o Fudametal of Noliear Behavioral Modelig: Foudatio ad Applicatio, IEEE MTT-S It. Microwave Symp., Philadelphia, Ju A. Zhu, M. Wre ad T. Brazil, A Efficiet Volterra-Baed Behavioral Model for Widebad RF Power Amplifier, IEEE MTT-S It. Microwave Symp. Dig., Philadelphia, Ju A. Zhu ad T. Brazil, Behavioral Modelig of RF Power Amplifier Baed o Prued Volterra Serie, IEEE Microwave ad Wirele Compoet Lett., vol. 4, No., pp , Dec J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

52 Mathematical Need for Behavioral Modelig of Telecommuicatio Circuit ad Sytem Secod Part Ecitatio Deig for Model Etractio ad Validatio Joé Carlo Pedro Itituto de Telecomuicaçõe Uiveridade de Aveiro Portugal Mathematical Techiue ad Problem i Telecommuicatio Sept. 006 J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep. 006

53 Preetatio Outlie. Ecitatio Deig for Behavioral Model Etractio. The Role of Ecitatio Statitic o Noliear Dyamic Sytem Repoe 3. Multiie Deig for Behavioral Model Validatio 4. Cocluio J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep. 006

54 J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep Ecitatio Deig for Behavioral Model Etractio Cotrary to other modelig techiue, uch a the Artificial Neural Network, Fudametal of Behavioral Model Etractio = = = K k k k a o o w b f k w b y 0 which are oliear i their parameter, [b k ad w k ], ad thu reuire oliear optimizatio techiue for parameter etractio,

55 J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep Polyomial Filter are liear i the parameter the Volterra Kerel, Fudametal of Behavioral Model Etractio = = = = = = ,...,..., h h h y K L therefore eablig the ue of tadard liear regreio method. Thi alo eaie the gatherig of kowledge for ecitatio deig.. Ecitatio Deig for Behavioral Model Etractio

56 J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep I fact, i the ame way the Impule Repoe Fuctio, h, of a Liear Dyamic Sytem ca be etimated olvig the followig liear regreio ytem : = = h y 0 = h h h y y y M M L L M M M L L M M M L L M M provided the iput i ufficietly rich i cotet Fudametal of Behavioral Model Etractio. Ecitatio Deig for Behavioral Model Etractio

57 J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep the th order Noliear Impule Repoe Fuctio, h,,, of the Volterra-Wieer Model L K L L = = = = = i i h h y 0 0 0,, Fudametal of Behavioral Model Etractio. Ecitatio Deig for Behavioral Model Etractio

58 J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep ca be etimated olvig the followig liear regreio ytem : =,,,,,0 0, h h h y y y i i i i i i K M K M K L L M M M L L M M M L L M M provided the multi-iput Π- i i agai ufficietly rich i cotet. Fudametal of Behavioral Model Etractio. Ecitatio Deig for Behavioral Model Etractio

59 . Ecitatio Deig for Behavioral Model Etractio Fudametal of Behavioral Model Etractio Or, i the ame way the Trafer Fuctio, H, of a Liear Dyamic Sytem ca be etimated from the iput-output cro-correlatio, S y, ad i iput auto-correlatio S : H = S S y = Y. X X X provided the iput X i ufficietly rich i cotet J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

60 J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep the th order Noliear Trafer Fuctio of a Noliear Dyamic Sytem, H,,, L L L = = = = = K k K k j i k k k K k j k k k k i k e X H e X H y...,..., Fudametal of Behavioral Model Etractio. Ecitatio Deig for Behavioral Model Etractio

61 J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep = =..,,,,,, y X X X X X X Y S S H K K K K K K K K K ca be etimated from the higher-order iput-output cro-correlatio, S y,,, ad i higher-order iput auto-correlatio S,, : provided the multi-iput ΠX i i agai ufficietly rich i cotet. Fudametal of Behavioral Model Etractio. Ecitatio Deig for Behavioral Model Etractio

62 . Ecitatio Deig for Behavioral Model Etractio Fudametal of Behavioral Model Etractio Thi how that, o matter the domai time or freuecy, or the type of timulu ued for the tet, the model etractio procedure will be ucceful a log a the timulu ecite all wirele ytem tate: -, -,, - -,.-,,.-, -,, -.-,, - - 3,.-,,.-, - 3,, -.-,, ,...- -,, - J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep. 006

63 . Ecitatio Deig for Behavioral Model Etractio Fudametal of Behavioral Model Etractio or: - X, X,, X K - X, X.X,, X.X K,, X K.X,, X K - X 3, X.X,, X.X K,, X K.X,, X K X, X X -,, X K J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep. 006

64 . Ecitatio Deig for Behavioral Model Etractio Some Hitorical Step Toward Behavioral Model Etractio I the 40 Wieer proved that White Gauia Noie wa rich eough to ecite a Volterra ytem. The, i the 60, Schezte ad Lee ued that ecitatio to etract the Wieer Model a polyomial filter orthogoal to thi iput. The th order Wieer fuctioal wa obtaied by time-domai correlatio betwee the output, yt, ad a th order delayed verio of the iput, tτ.t-τ t-τ. I the 80, Boyd, Tag ad Chua ad the Chua ad Liao propoed method for etractig Volterra kerel i the freuecy-domai, uig Sparely Ditributed Harmoically Related Siuoid. J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

65 . Ecitatio Deig for Behavioral Model Etractio Some Hitorical Step Toward Behavioral Model Etractio More recetly, much of the effort ha bee directed to obtai ueful data from Multiie or eve time-domai Real Modulated Wirele ecitatio. However, the traditioal freuecy-domai Siuoidal Ecitatio ad the time-domai Step Stimulu have alo bee eteively tried. J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

66 . Ecitatio Deig for Behavioral Model Etractio Noliear Sytem Idetificatio Theory May Help Agai Recogizig that both digitally ytheized Multiie, Peudo-Radom Noie Seuece or Fiite Modulatio Seuece are dicrete periodic fuctio i time ad freuecy domai, they ca be related by the Dicrete Fourier Serie: N X k = T 0 0 e N =N jk T K T = X k0 k =K jk T e 0 which how that they are imply two ditict way of etractig ame type of iformatio. J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

67 . Ecitatio Deig for Behavioral Model Etractio Noliear Sytem Idetificatio Theory May Help Agai Furthermore, ice the repoe of a ytem ecited with variou realizatio of: - Radom Multiie - Multiie with radomized phae - Periodic Noie - Multiie with radomized amplitude ad phae the repoe of that ame ytem whe ecited with - Bad-Limited White Gauia Noie Volterra-Wieer theorie prove that both of thee timuli could be ued to etract a behavioral model of, at leat, a Noliear Sytem of Fadig Memory! J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

68 Preetatio Outlie. Ecitatio Deig for Behavioral Model Etractio. The Role of Ecitatio Statitic o Noliear Dyamic Sytem Repoe 3. Multiie Deig for Behavioral Model Validatio 4. Cocluio J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

69 . The Role of Ecitatio Statitic o Noliear Dyamic Sytem Repoe Why do ecitatio with imilar PSD ad itegrated power produced o ditict olier repoe? Output Power [dbm] AWGN Out W-CDMA Out AWGN I W-CDMA I Freuecy-900MHz [KHz] J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

70 . The Role of Ecitatio Statitic o Noliear Dyamic Sytem Repoe Probability Deity, pdf, a a Weightig Fuctio Sice typical laboratory data like Output Power, Power Spectrum, etc., i averaged i ature: P { } = i E pdf = d P out { } y = y pdf y y dy = = E f pdf d NL it i ituitive to epect that, more importat tha the trajectory of amplitude value aumed by the ecitatio, t, hould be the Probability with which each value i reached, i.e., the Ecitatio pdf. J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

71 . The Role of Ecitatio Statitic o Noliear Dyamic Sytem Repoe Probability Deity, pdf, a a Weightig Fuctio To epoe the role of the pdf we teted a tatic oliear ytem with three igal of eual itegrated power but ditict amplitude ditributio: Probability Deity Gauia Uiform 0.0 Two-Toe Amplitude J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

72 . The Role of Ecitatio Statitic o Noliear Dyamic Sytem Repoe Memoryle Noliear Sytem Model The elected oliear ytem wa a imple igmoid fuctio: y[t] y t = f NL [ t ] = ta t 0 N = k t ice it liear regio, followed by a moothly aturatig behavior, i may time ued to repreet practical memoryle oliearitie. J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep. 006

73 . The Role of Ecitatio Statitic o Noliear Dyamic Sytem Repoe Multiie for Memoryle Noliear Sytem The ecitatio ued were two evely paced cotat amplitude Bad-Pa Multiie: t K = A k= k co t φ ; A = A : k k k k whoe phae, φ k, were deiged for Gauia ad Uiform pdf uig a pecially coceived algorithm. J. Pedro ad N. Carvalho, IEEE IMS 004 J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep. 006

74 . The Role of Ecitatio Statitic o Noliear Dyamic Sytem Repoe Multiie for Memoryle Noliear Sytem Uiform Ditributio Gauia Ditributio ACPR=38dB -0 ACPR=3dB Amplitude Amplitude Freuecy 00MHz a b Power pectrum of the repoe of a igmoid memoryle ytem ecited by the uiformly ditributed - a ad Gauia ditributed multiie - b. -00 Freuecy 00MHz J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

75 J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep Noliear Dyamic Sytem Model A dyamic ytem i oe whoe output i depedet o the preet iput ad o it pat: [ ],...,, f y D = A oliear FIR filter approimatio would be: [ ] L K K L L = = = = h h P y 0 0 0,,,...,, The Role of Ecitatio Statitic o Noliear Dyamic Sytem Repoe.

76 . The Role of Ecitatio Statitic o Noliear Dyamic Sytem Repoe Noliear Dyamic Sytem Model If y = f [ ] D [,,..., ] y = fd the ytem output hould o loger be oly depedet o the iput tatitic, pdf, but o the Joit Statitic of the iput ad it pat ample, pdf - [,,-]. So, the th order repoe of a oliear dyamic ytem to a certai timulu ow deped o the Memory Spa,, of h,, ad o the correlatio betwee ad all other -,,-. Not oly the amplitude ditributio i importat a i the igal evolutio with time, i.e., it Time-Domai Waveform or Freuecy- Domai Spectrum. J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

77 . The Role of Ecitatio Statitic o Noliear Dyamic Sytem Repoe A a eample, coider a Hammertei Model compoed of a memoryle oliearity, followed by a liear filter: Noliear Dyamic Sytem Model N y = k = y = h y = 0 t y t k t = y t h t H y t Noliear / Memoryle Liear / Dyamic The ytem repoe ow deped o the memory pa of h,. J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

78 . The Role of Ecitatio Statitic o Noliear Dyamic Sytem Repoe Short Memory Spa, Noliear Dyamic Sytem Model y = h = Log Memory Spa, a Liear po-filter ued i the ditortio imulatio. a Filter with hort memory pa. b Filter with log memory pa. J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep b

79 . The Role of Ecitatio Statitic o Noliear Dyamic Sytem Repoe Short Memory Spa, Noliear Dyamic Sytem Model y = h = 0 Log Memory Spa, time a time Liear po-filter ued i the ditortio imulatio. a Filter with hort memory pa. b Filter with log memory pa. J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep b

80 . The Role of Ecitatio Statitic o Noliear Dyamic Sytem Repoe Repoe of the Hammertei Model to Multiie of Eual pdf 0.06 K t = A k= k co t φ ; A = A : k k Gauia Ditributio k k Gauia pdf of the two multiie of eual power pectrum. J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

81 5. The Role of Ecitatio Statitic o Noliear Dyamic Sytem Repoe Repoe of the Hammertei Model to Multiie of Eual pdf K t = A Gauia Ditributio k= k co t φ ; A = A : k k k 30 k Gauia Ditributio Amplitude Amplitude Time a b Time-domai waveform of the two multiie of eual pdf ad power pectrum. J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep Time

82 . The Role of Ecitatio Statitic o Noliear Dyamic Sytem Repoe Repoe of the Hammertei Model to Multiie of Ditict pdf Although differet phae arragemet produce ditict power pectra i a memoryle oliearity, they geerate eual itegrated ACPR value. 0 0 ACPR=8dB 0 0 ACPR=8dB a t = k t y t y t b Noliear / Memoryle Li J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

83 . The Role of Ecitatio Statitic o Noliear Dyamic Sytem Repoe Now, although the wide-bad output filter would keep the etire pectra, the arrow-bad liear bad-pa filter ca rehape the pectrum ide lobe, ad thu geerate uite differet itegrated output ACPR value. 0 0 ACPR=8dB 0 0 ACPR=3dB a t = k t y t y t h t H y t b Noliear / Memoryle Liear / Dyamic J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

84 Preetatio Outlie. Ecitatio Deig for Behavioral Model Etractio. The Role of Ecitatio Statitic o Noliear Dyamic Sytem Repoe 3. Multiie Deig for Behavioral Model Validatio 4. Cocluio J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

85 3. Multiie Deig For Behavioral Model Validatio Repoe of Geeral Noliear Dyamic Sytem to Arbitrary Sigal The reao for thee dicrepacie ca be traced to the way the oliearity geerate the pectral regrowth. Sice the multiie i periodic, there i a commo freuecy eparatio betwee the may differet toe. So, the geeral multiie epreio: K t = A k= k co t φ ; A = A : k implie that all toe freuecie ca be epreed by: k J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep k k k = 0 ad o, ay e.g., 3rd order output pectral lie at will be give by all miig product verifyig: = k i which = k k k 3 k k3 k

86 3. Multiie Deig For Behavioral Model Validatio Repoe of Geeral Noliear Dyamic Sytem to Arbitrary Multiie Each of thee miig product ha a phae of: φ = φk φ k φ k3 ad o, the reultig voltage wie additio deped o the poible correlatio betwee thee φ. φ φ φ 3 φ 3 φ -φ φ 3 φ 3 -φ -K K J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

87 3. Multiie Deig For Behavioral Model Validatio Repoe of Geeral Noliear Dyamic Sytem to Arbitrary Multiie Each of thee miig product ha a phae of: φ = φk φ k φ k3 ad o, the reultig voltage wie additio deped o the poible correlatio betwee thee φ. Thu, i the cotet of oliear dyamic ytem, igal ecitatio ca o loger be completely pecified by their momet pdf, a i memoryle ytem: m { } = E pdf = d J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

88 3. Multiie Deig For Behavioral Model Validatio Repoe of Geeral Noliear Dyamic Sytem to Arbitrary Multiie If igal ecitatio ca ot be completely pecified by their momet pdf, they ca t be either pecified by the ecod order joit tatitic a i liear dyamic ytem: R τ = E{ t t τ } { } = E X X S becaue thi igal metric i blid to the igal phae! J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

89 3. Multiie Deig For Behavioral Model Validatio Repoe of Geeral Noliear Dyamic Sytem to Arbitrary Multiie A epected from the polyomial tructure of the Volterra-Wieer model, multiie mut ow be pecified by their higher-order joit igal tatitic: R τ = E{ t t τ } { } = E X X { t t τ } R3 τ τ, τ = E t S { } X X X S3, = E R { t t τ... t τ } τ,..., τ = E S { } X... X X...,..., E = J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

90 3. Multiie Deig For Behavioral Model Validatio Repoe of Geeral Noliear Dyamic Sytem to Arbitrary Multiie So, the deired multiie mut meet the pdf, the PSD: S { } X X = E ad the higher order tatitic, e.g.: S [ ] X X X X 3 3 3,, = E Thi guaratee that the pectral regrowth i approimated at leat for the order of the tatitic coidered. J. Pedro ad N. Carvalho, IEEE T-MTT 005 J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

91 3. Multiie Deig For Behavioral Model Validatio S [ ] X... X X...,..., = E... Dicretized i K toe, thee higher-order tatitic reuire - dimeioal matri approimatio of K poit, which ca ot be obtaied by a igle multiie of K toe K phae. A Multiie of K toe ha ot eough umber of Degree of Freedom! Two poibilitie: A Eemble of variou Multie of K toe each. A igle Multiie with may more toe. J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

92 3. Multiie Deig For Behavioral Model Validatio If a igle multiie i the goal, the the umber of toe mut be icreaed, but the S,, error evaluated for a maller umber of bi. Freuecy Bi Deiged Multiie -Bw/ 0 Bw/ J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

93 3. Multiie Deig For Behavioral Model Validatio Multiie Deig Eample for Noliear Dyamic Sytem Ecitatio Coider the oliear dyamic ytem, which ca be tued to preet certai propertie idepedetly: t k yt k 3 Hj Liear Filter Log-term, oliear, memory J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

94 3. Multiie Deig For Behavioral Model Validatio Multiie Deig Eample for Noliear Dyamic Sytem Ecitatio Coider alo a iput igal with a pre-determied higher order tatitic, for which the multiie i to be ytheized. Iput Power [dbm] Freuecy-GHz [MHz] Amplitude [V] Time [u] J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

95 3. Multiie Deig For Behavioral Model Validatio Multiie Deig Eample for Noliear Dyamic Sytem Ecitatio t Cae A Noliear Memoryle Sytem: -0-0 Output Power [dbm] Freuecy-GHz [MHz] J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

96 3. Multiie Deig For Behavioral Model Validatio Multiie Deig Eample for Noliear Dyamic Sytem Ecitatio d Cae A Noliear Dyamic Sytem with Memory at the Bae-Bad: -0-0 Output Power [dbm] Freuecy-GHz [MHz] J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

97 3. Multiie Deig For Behavioral Model Validatio Multiie Deig Eample for Noliear Dyamic Sytem Ecitatio 3rd Cae A Noliear Dyamic Sytem with Memory at the Bae-Bad ad d Harmoic: -0-0 Output Power [dbm] Freuecy-GHz [MHz] J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

98 Preetatio Outlie. Ecitatio Deig for Behavioral Model Etractio. The Role of Ecitatio Statitic o Noliear Dyamic Sytem Repoe 3. Multiie Deig for Behavioral Model Validatio 4. Cocluio J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

99 5. Cocluio. Oly very comple tet igal like White Gauia Noie ca provide all the eeded iformatio about the ytem.. Multiie are very promiig timuli, but reuire deep care i electig the toe pacig ad the amplitude ad phae et. 3. Sigal pdf play a determiat role a a igal metric, ad thu for ecitatio deig. 4. For oliear dyamic ytem, the igal pdf i icomplete. Joit pdf of the preet iput ad pat ample withi the memory pa are eceary to uiuely determie the repoe. J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

100 5. Cocluio 5. Complete ecitatio iformatio ca be idetified by the higher order tatitic - Higher Order Auto-Correlatio ad Power Spectral Deity Fuctio - which provide iformatio of both the amplitude ad the phae. 6. I the particular cotet of oliear behavioral model etractio or validatio with bad-limited white Gauia oie, either a eemble of K-toe evely paced multiie of radomized phae, a igle S-toe S>>K evely paced multiie of radomized phae or eve a igle K-toe multiie of u-commeurate freuecie ca be ued. J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

101 Referece S. Boyd, Y. Tag ad L. Chua, Meaurig Volterra Kerel, IEEE Tra. o Circuit ad Sytem, Vol. CAS. 30, No. 8, pp , 983. G. Chriiko, C. Clark, A. Moulthrop, M. Muha ad C. Silva, A Noliear ARMA Model for Simulatig Power Amplifier, IEEE MTT-S It. Microwave Symp. Dig., pp , Baltimore, Ju L. Chua ad Y. Liao, Meaurig Volterra Kerel II, It. J. Circuit Theory ad Applicatio, Vol. 7, No., pp.5-90, Apr L. Chua ad Y. Liao, Meaurig Volterra Kerel III: How to Etimate the Highet Sigificat Order, It. J. Circuit Theory ad Applicatio, Vol. 9, No., pp.89-09, Mar.-Apr. 99. J. Dooley et al., Behavioral Modelig of RF Power Amplifier Uig Adaptive Recurive Polyomial Fuctio, IEEE It. Microwave Sympoium Dig., pp , Sa Fracico, Ju C. Eva et al., Periodic Sigal for Meaurig Volterra Kerel, IEEE Tra. o Itrumetatio ad Meauremet, Vol. IM-45, No., pp.36-37, Apr J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

102 Referece F. Filicori, G. Vaii ad V. Moaco, A Noliear Itegral Model of Electro Device for HB Circuit Aalyi, IEEE Tra. o Microwave Theory ad Tech., vol. MTT-40, No. 7, pp , Jul. 99. M. Iako et al., "A Comparative Aalyi of Behavioral Model for RF Power Amplifier", IEEE Tra. o Microwave Theory ad Tech., vol. MTT-54, No., pp , Ja H. Ku, M. Mckiley ad J. S. Keey, uatifyig Memory Effect i RF Power Amplifier, IEEE Tra. o Microwave Theory ad Tech., vol. MTT-50, No., pp , Dec. 00. H. Ku ad J. Keey, Behavioral Modelig of RF Power Amplifier Coiderig IMD ad Spectral Regrowth Aymmetrie, IEEE MTT-S It. Microwave Symp. Dig., Philadelphia, Ju P. Lavrador et al., A New Volterra Serie Baed Orthogoal Behavioral Model for Power Amplifier, Aia-Pacific Microwave Coferece Dig., pp., Suzhou, Dec J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

103 Referece D. Mirri, F. Filicori, G. Iuculao, G. Paii, A No-Liear Dyamic Model for Performace Aalyi of Large-Sigal Amplifier i Commuicatio Sytem, IMTC/99 IEEE Itrumetatio ad Meauremet Techology Cof. Dig., pp , Veice, May 999. B. O Brie et al., RF Power Amplifier Behavioral Modelig uig a Globally Recurret Neural Network, IEEE It. Microwave Sympoium Dig., pp , Sa Fracico, Ju J. Pedro ad N Carvalho, Deigig Bad-Pa Multiie Ecitatio for Microwave Behavioral Model Idetificatio, IEEE It. Microwave Sympoium Dig. CDROM, Fort Worth, Ju J. Pedro ad N. Carvalho, "Deigig Multiie Ecitatio for Noliear Model Tetig", IEEE Tra. o Microwave Theory ad Tech., Vol. MTT- 53, No., pp.45-54, Ja J. Pedro ad S. Maa, A Comparative Overview of Microwave ad Wirele Power-Amplifier Behavioral Modelig Approache, IEEE Tra. o Microwave Theory ad Tech., vol. MTT-53, pp.50-63, Apr J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

104 Referece J. Pedro et al., A Formal Procedure for Microwave Power Amplifier Behavioral Modelig, IEEE It. Microwave Sympoium Dig., pp , Sa Fracico, Ju R. Pitelo ad J. Schouke, Sytem Idetificatio A Freuecy Domai Approach, IEEE Pre, New York 00. K. Remley, Multiie Ecitatio for ACPR Meauremet, IEEE MTT-S It. Microwave Symp. Dig., Philadelphia, Jue 003. A. Saleh, Freuecy-Idepedet ad Freuecy-Depedet Noliear Model of TWT Amplifier, IEEE Tra. o Commuicatio, Vol. COM- 9, No., pp.75-70, 98. M. Schetze, The Volterra ad Wieer Theorie of Noliear Sytem, Joh Wiley & So, New York, 980. D. Schreur, M. Myliki ad K. A. Remley, RF Behavioural Modellig from Multiie Meauremet: Ifluece of Ecitatio Type, 33rd Europea Microwave Cof. Dig., Muich, Oct J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

105 Referece A. Soury, et. al., A New Behavioral Model Takig ito Accout Noliear Memory Effect ad Traiet Behavior i Widebad SSPA, IEEE MTT-S It. Microwave Symp. Dig., pp , Seattle, Ju. 00. A. Soury et al., Meauremet Baed Modelig of Power Amplifier for Reliable Deig of Moder Commuicatio Sytem, IEEE MTT-S It. Microwave Symp. Dig., Philadelphia, Ju F. Verbeyt ad M. V. Boche, VIOMAP, the S-parameter Euivalet for Weakly Noliear RF ad Microwave Device, IEEE Tra. o Microwave Theory ad Tech., vol. MTT-4, pp , 994. T. Wag ad T. Brazil, Volterra-Mappig-Baed Behavioral Modelig of Noliear Circuit ad Sytem for High Freuecie, IEEE Tra. o Microwave Theory ad Tech., vol. MTT-5, No. 5, pp , May 003. D. Weier ad G. Naditch, A Scatterig Variable Approach to the Volterra Aalyi of Noliear Sytem, IEEE Tra. o Microwave Theory ad Tech., vol. MTT-4, pp.4-433, 976. J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep

106 Referece J. Wood ad D. Root, The Behavioral Modelig of Microwave/RF IC uig No-Liear Time Serie Aalyi, IEEE MTT-S It. Microwave Symp. Dig., Philadelphia, Ju Zhag, Artificial Neural Network, Workhop o Fudametal of Noliear Behavioral Modelig: Foudatio ad Applicatio, IEEE MTT-S It. Microwave Symp., Philadelphia, Ju A. Zhu, M. Wre ad T. Brazil, A Efficiet Volterra-Baed Behavioral Model for Widebad RF Power Amplifier, IEEE MTT-S It. Microwave Symp. Dig., Philadelphia, Ju A. Zhu ad T. Brazil, Behavioral Modelig of RF Power Amplifier Baed o Prued Volterra Serie, IEEE Microwave ad Wirele Compoet Lett., vol. 4, No., pp , Dec J.C.Pedro, Mathematic for Behavioral Modelig, MTPT, Leiria, Sep