Analysis and Optimization of Monolithic RF Downconversion Receivers

Transcription

1 Analyi and Optimizatin Mnlithic RF Dwncnverin Receiver hritpher D. Hull Electrical Engineering and mputer Science Univerity alirnia at Berkeley echnical Reprt N. UB/EES April 6, 009

2 pyright 009, y the authr. All right reerved. Permiin t make digital r hard cpie all r part thi wrk r pernal r clarm ue i granted withut ee prvided that cpie are nt made r ditriuted r prit r cmmercial advantage and that cpie ear thi ntice and the ull citatin n the irt page. cpy therwie, t repulih, t pt n erver r t reditriute t lit, require prir peciic permiin.

3 Analyi and Optimizatin Mnlithic RF Dwncnverin Receiver y hritpher D. Hull B.S. Univerity alirnia at San Dieg 987 M.S. Univerity alirnia at Berkeley 989 A diertatin umitted in partial atiactin the requirement r the degree Dctr Philphy in Engineering-Electrical Engineering and mputer Science in the GRADUAE DSON the UNERSY OF ALFORNA at BERKELEY mmittee in charge: Prer Rert G. Meyer, hair Prer Edward A Lee Prer Heinz O rde 99

4 he diertatin hritpher D. Hull i apprved: hair Date Date Date Univerity alirnia at Berkeley 99

5 Analyi and Optimizatin Mnlithic RF Dwncnverin Receiver y hritpher D. Hull Dctr Philphy in Engineering-Electrical Engineering and mputer Science Univerity O alirnia at Berkeley Prer Rert G. Meyer, hair Deign cnideratin r the rnt-end radi-requency receiver are preented. Emphai i n ilicn iplar technlgy r receiver in the -3 GHz requency range, thugh theretical principle derived apply ver a rad range requencie. Baic mixer and ampliier tplgie are preented and their perrmance characteritic are analyzed. Analytic exprein r nie and ditrtin in linear ampliier are preented. he perrmance dierent tplgie are cmpared. A new methd nie analyi r mixer i preented. he nie analyi i applied t the emitter-cupled pair mixer ver a wide range parameter variatin t allw the deigner t undertand hw nie perrmance change with parameter variatin. Reult ditrtin imulatin ver a range parameter value are al preented. he mechanim that create the ditrtin are explained, and the imulatin reult are preented in a way that allw an intuitive link etween the imulated value the ditrtin and the mechanim that create that ditrtin. Fr veriicatin the methdlgy preented, the analyi technique are applied t a peciic circuit and cmpared t meaured value. mputed value are cle t the meaured ne. Atract Apprved: hei hairman

6 hapter : ntrductin Wirele cmmunicatin i a cnvenient way t tranmit vice r data rm pint t pint, and i eential r mile cmmunicatin. mmercial applicatin include cellular telephny, glalpitining atellite, direct-radcat atellite, and wirele cmputing. A lck diagram a radirequency tranceiver tructure ued r wirele cmmunicatin i hwn in Figure. LNA MXNG DEEON BASEBAND SGNAL PROESSNG DAA OU POWER AMP MODULAON DAA N Figure : Lw-Pwer ranceiver Architecture he mdulatr and pwer ampliier lck rm the tranmitter. he LNA lw-nie ampliier, mixer, detectin circuitry, and aeand ignal prcer rm the receiver. he receiver rnt-end cnit the LNA and mixing lck. he purpe thee lck i t ampliy the weak ignal received rm the antenna and cnvert the carrier requency dwn t a range that i mre eaily prceed. Detectin and aeand ignal prceing technique are dependent n the type tranmiin mdulatin e.g. AM,FM,PSK. he rnt-end the receiver will e the cu thi diertatin..: Sytem Requirement r mmercial RF Receiver Amng the imprtant deign cnideratin are pwer cnumptin, ct, phyical ize, reliaility, electivity and dynamic range. Selectivity i the aility a receiver t elect the deired ignal and rect the unwanted ignal. Dynamic range i the rati the maximum ignal level the receiver can tlerate with an acceptale level ditrtin ver the minimum ignal level ere nie make detectin impile.

7 n addreing the deign cnideratin, ne mut cnider the technlgie availale. urrent technlgy chice are mnlithic circuit v. dicrete circuit, ilicn v. gallium-arenide, and iplar unctin tranitr v. ield-eect tranitr. Mnlithic technlgy er the advantage cmpact ize, higher reliaility, and lwer aemly ct. Hwever, dicrete deign are eaier t adut. Mnlithic implementatin invlve cniderale tart-up ct, and thu are apprpriate r high-vlume cmmercial applicatin. Dicrete implementatin are mre apprpriate r cutm deign. t huld e nted that mt ytem ue a cminatin dicrete and mnlithic element. While GaA technlgy er tate--the-art perrmance and i widely ued r military applicatin, it high ct and lw yield make it apprpriate where perrmance i paramunt imprtance. he relatively lw ct and high yield ilicn technlgy make large cale integratin practical. hi give ilicn a utantial advantage r high-vlume cmmercial applicatin. n ilicn technlgy, iplar tranitr er higher perrmance than FE device. While FE have cmparale device gain-andwidth prduct, they require utantially higher gate-urce perating vltage than the ae-emitter perating vltage a iplar tranitr. Aciated with thi i a much lwer trancnductance-t-current rati. Fr lw-pwer applicatin th lw current and lw vltage the BJ er cnideraly etter perrmance. An alternative r FE i t perate them at lw gate-urce vltage. While ue lw gate-urce vltage imprve the trancnductance-t-current rati, the high-requency current gain and drp cnideraly, and the paraitic capacitance ecme quite large. A FE ize cale dwn, FE may ecme practical alternative t iplar tranitr in the lw GHz range. Hwever, in the current 0.8 micrn technlgy, the perrmance FE uer dratically eynd a ew hundred MHz. One the mar advantage FE technlgy i the aility t integrate with MOS digital circuitry. Hwever, with the advent BiMOS technlgy, it i nt neceary t acriice perrmance r integratin. t huld e nted that PMOS tranitr give ar etter perrmance than the paraitic PNP availale in many iplar and BiMOS prcee. hee may e quite ueul r active lad and iaing.

8 .: Frnt-end Receiver Architecture Figure hw a lck diagram an RF receiver rnt end. mage rectin, F intermediate requency, and lp ilter are chip, ince high inductr are nt currently availale in mnlithic technlgy. he LNA utput and mixer input mut e matched t the impedance the image ilter ten 50 Ω. he O, divider, XO crytal cntrlled cillatr and lp ilter rm a requency-multiplying phae-lck lp. An external crytal prvide a tale reerence requency. he requency divider in the eedack lp the phae-lcked lp yield a requency multiplicatin the reerence. Oten the requency divider i cntrlled digitally, allwing t e varied y digital cntrl. hi i cnvenient when the receiver i t e ued t decde a numer input channel that are eparated in requency requency-dmain multiplexing. mage Filter F Filter Mixer RF N LNA r - r F Amp F OU O n* re Divider _. n Phae Detectr re XO Lp Filter Figure : Simpliied RF Frnt End Receiver Architecture

9 hapter : ircuit tplgy r RF Ampliier and Mixer nput Filter mage Filter Mixer F Filter RF N LNA r - r F OU LO N Figure : RF Ampliier and Mixer Figure hw the arrangement the RF ampliier and the mixer, which tgether with the lcal cillatr, rm the rnt end the receiver. An input ilter i neceary t prevent verlad the LNA rm ut and ignal cming rm the antenna, and al imprve image rectin. Since the ampliier and mixer take their input and utput rm chip, they mut have matched impedance at th the input and the utput. mpedance matching netwrk may e ued r thi purpe. an image-rectin mixer i ued, then ue an image ilter etween the preamp and mixer i unneceary, and hence, the utput impedance the LNA and input impedance the mixer need nt e matched. mage-rectin mixer require twice the hardware and pwer cnumptin an equivalent mixer that de nt rect the image requency. hu there i a trade etween the advantage gained rm the increaed level integratin an image-rectin mixer and the increaed pwer cnumptin..: Lw-Nie Ampliier niguratin O the three aic cniguratin cmmn-emitter, cmmn-ae, and cmmn cllectr, the cmmn-emitter r cmmn-urce r FE i the nly ne ering th current and vltage gain. hi i quite advantageu r nie purpe. Hence, the irt tage any lw nie ampliier i almt alway a cmmn-emitter. At high requencie the cmmn-emitter ha a lw input and utput impedance making it uitale r matching t the lwer impedance typically een in RF ilter ytem, cale, and antenna.

10 mmn-ae tage er lw input impedance, ut very high utput impedance, and a matching netwrk i neceary at the utput a cmmn-ae tage. Shunt eedack may al e ued t reduce the utput impedance, ut ha limited applicaility at high requencie, a the exce pen-lp gain required t give adequate lp gain i nt readily availale. Multiple tage may e ued t tain the required lp gain, ut taility iue generally limit the numer tage t tw r three. mmn-cllectr tage are cmmnly ued at lw requencie; hwever, at high requencie, the utput impedance i quite inductive and the cniguratin i prne t paraitic cillatin. Fr thee rean, cmmn-emitter ampliier are preerred r high-requency matched impedance applicatin that are narrw-and. A typical tw tage cniguratin i depicted in Figure. Matching netwrk huld e made reactive element t avid adding additinal nie urce t the circuit. n particular, "rute rce" matching with erie r hunt reitr huld e avided, a thi degrade the nie perrmance the ampliier utantially. he ia netwrk ue negative eedack t tailize the dc perating pint the tranitr. Output Matching Netwrk R nput Matching Netwrk O F FLER - Bia Feedack ircuit Figure : w-tage lw-nie ampliier

11 cc R cc R - R in - R e R L R e - R L Figure 3: Wideand Matched-mpedance Ampliier An alternative matching technique i t ue eedack. An example thi circuit i hwn in Figure 3. he advantage thi technique i that matching ccur ver a wide range requencie. hi i deirale r general purpe ampliier. Hwever, eedack ampliier generally have prer nie epecially at high requencie cmpared t nn-eedack ampliier. hi diertatin will cu n tplgie that d nt ue eedack..: Mixer niguratin A wide variety mixer cniguratin are pile. Fundamentally, all mixer rely n peridic witching the ignal r dwn cnverin. hi i hwn chematically in Figure 4. LO SWH RF N - F Out Figure 4: Fundamental Dwn nverin Prce

12 F OU - LO N - 3 RF Figure 5: Emitter-upled Pair Mixer n iplar technlgy the witch i uually implemented uing an emitter-cupled pair a hwn in Figure 5. Nte that an input ignal in the rm a current i required. hi implie that the witch huld e driven with a high urce impedance. Since the impedance lking ack int the F ilter tend t e lw, a vltage t current cnverin tage i neceary. hee tage mut e matched at the input and have a high utput impedance. O the three aic circuit cniguratin, th the cmmn-ae and the cmmnemitter have the deired prpertie. Figure 6 hw a cmmn-ae driver r the emitter-cupled pair mixer. Reitr R m matche the circuit and linearize the circuit, ut al increae the nie the circuit. n additin, the cmmn-ae tage lack current gain and thu the current nie rm the emitter-cupled pair mixer i reerred ack t the input withut reductin. An alternative i t ue an active matching netwrk at the input. hi will increae the current gain and reduce the nie, ut the ditrtin will al increae. he cmmn-emitter cniguratin in Figure 7 ha the advantage etter nie perrmance and higher gain than the cmmn-ae. At lw requencie the linearity i quite pr. Hwever, in the GHz

13 range, the linearity a well deigned cmmn-emitter ampliier may e quite gd ee hapter 3. Stale iaing i tained y generating a reerence BE uing a dide. RF BAS R m 50 R ia - Figure 6: mmn-bae Driver RF 50 Matching Netwrk - Bia Netwrk BE re Figure 7: mmn-emitter Driver

14 RF 50 R L ut - R e R R e Figure 8: urrent-feedack Pair Driver A with preamp, the driver tage a mixer may ue eedack t generate matching ver a wide range requencie. he current-eedack pair cniguratin hwn in Figure 8 give a cntrlled lwimpedance at the input and a high impedance at the utput. he nie perrmance penalty i minimal. Hwever, the tw tage give mewhat higher gain than deired and cnume additinal pwer. ncreaing the degeneratin reitr, R e, t reduce the gain will degrade the nie perrmance. While FE mixer may e uilt uing circuit directly analgu t the iplar circuit preented ave, an alternative exit r FE that de nt exit r iplar tranitr. With iplar tranitr, i the cllectr-emitter ptential i drpped elw aut 0., the cllectr-ae unctin ecme rward iaed, and the ae i lded with charge aturatin. t take a utantial amunt time r the tranitr t recver rm thi cnditin. Hwever, FE d nt exhiit thi ehavir. hu, a FE can e witched n and y changing it drain-urce ptential. A imple circuit cniguratin that achieve thi i hwn in Figure 9. he gate J i cntrlled y the LO, and thi in turn cntrl the drain-urce ptential J. hi cniguratin i very advantageu ince the drain regin J and the urce regin J may e cmined int a ingle regin. N external cntact t thi regin i neceary. hi decreae

15 the paraitic capacitance aciated with that nde the circuit. When thee regin area are cmined a new ur terminal device knwn a a dual-gate FE i rmed. Dual-gate FE mixer are requently ued in GaA technlgy. F OU R Matching Netwrk LO J J - Figure 9: mplete dual-gate FE Mixer.3: Dule-Balanced Mixer All the ave mixer are either ingle-alanced r unalanced. A ingle-alanced mixer allw either the RF r LO ignal t pa t the utput with little attenuatin. A dule-alanced mixer rect th the RF and LO requencie at the utput. he undamental cniguratin a dule-alance mixer i hwn in Figure 0. he RF, LO, and F prt all have alanced ignal. he tw witche perate in ppite plarity. LO SWH RF N F Out - - Figure 0: Fundamental cniguratin a dule-alanced mixer

16 Figure hw an implementatin the dule-alanced mixer uing three emitter-cupled pair. w emitter-cupled pair 3 6 are ued t d the witching and ne - i ued r vltage t current cnverin. he vltage t current driver i degenerated t imprve it linearity. hi mixer i ten incrrectly reerred t a a "Gilert ell Mixer". he Gilert ell add pre-ditrtin technique t achieve linear multiplicatin the tw input ignal wherea the circuit in Figure i nnlinear with repect t the LO input. While analg multiplicatin reduce puriu utput ignal, the nie perrmance a Gilert ell analg multiplier i prer. Hencerth, the dule-alanced emitter cupled pair mixer withut pre-ditrtin will e reerred t a the "uad" mixer ince ur tranitr are ued t perrm the witching peratin. he input t the mixer in Figure are nt matched, and a matching netwrk i required. Oten "rute rce" matching i ued in the rm a reitr t grund. hi i diadvantageu rm the pint view nie perrmance, ut it i ten the implet way t match the RF and LO input prt. ut LO LO 5 6 LO r - R e R e Figure : Dule-Balanced EP Mixer

17 .4: mage-rectin Mixer While dule-alanced mixer prevent RF and LO ignal rm reaching the utput, puriu ignal till exit. Even a mixer which perrm ideal multiplicatin allw tw dierent requencie t e cnverted t the intermediate-requency. Fr example, i the LO requency i GHz, the input requency i 900MHz, and the intermediate-requency i 00MHz, then ignal at.ghz will al e cnverted dwn t the intermediate-requency. hi extra requency that i cnverted dwn t the F i knwn a the image requency. n mt mixer deign, the image requency i iltered ut with a harp andpa ilter centered arund the ignal requency. Hwever, a cminatin tw mixer and tw 90 degree phae hiter can e cmined t rm a mixer that rect image. A lck diagram an image-rectin mixer i hwn in Figure. LO N RF N 90 degree Σ F OU phae hiter 90 degree phae hiter Figure : mage-rectin Mixer

18 hapter 3: Lw-Nie Ampliier Randm nie i generated y all reitr and active device within a circuit. he dminant mechanim are randm thermal nie in reitr, and ht nie thrugh p-n unctin. deal reactive element d nt generate nie, thugh they may aect the verall nie perrmance in a circuit. deal eedack de nt add nie; hwever, reitive eedack de add additinal nie urce. Fr thi rean, reitive eedack i t e avided in lw-nie ampliier. Since eedack i cmmnly ued t reduce ditrtin in ampliier, deigning withut eedack require that attentin e paid t linearity iue. areul deign i required t tain lw nie and acceptale linearity. Reitive eedack i al cmmnly ued t tailize the gain and terminal impedance ver wide andwidth; hwever, r lw nie it i neceary t ue ther technique. Reactive impedance matching netwrk r reactive eedack may e ued t tain matching ver narrw andwidth. Generally, thee technique will nt achieve a wideand match, and it i therere neceary t have a peciic requency range in mind when deigning lw-nie ampliier. 3.: Nie Figure in Ampliier he mt cmmn meaure nie perrmance i the nie igure an ampliier. he nie igure i deined a i : F S / N S / N in ut S/N i the ignal-t-nie rati. he nie igure i thu a meaure the amunt y which the ignal-tnie rati i degraded. A nie igure unity r 0 db indicate a niele ampliier. When tw ampliier are cacaded, the verall nie igure i given y: G i the pwer gain the irt tage. y: F F F G he nie igure an ampliier i given in term it equivalent input vltage and current nie

19 where vn in Z F 3 v v 4 K R Equatin 3 i quite general, and include the eect crrelatin etween vltage and current nie at the input. 3.: Phyical Nie Surce Biplar and FE tranitr have imilar mall ignal mdel at high requencie. he mall-ignal mdel with nie urce included i hwn in Figure. deal eedack de nt aect the equivalent input nie generatr ii ; hence, eedack rm c de nt aect the nie igure. Hwever, the lading c at the input de aect the nie igure mewhat. A gd irt rder apprximatin r nie calculatin i t add the value c t. v n r B π c i n i cn β * Figure : ranitr Mdel ncluding Nie Surce E he variance the nie urce r iplar tranitr are given y iii : n i v n Fr FE, the variance the nie urce are: 4 K r gm icn 4 K 4 gm icn K β β 4a 4 4c

20 r K v g n 4 5a g K i m cn n i 5c he equivalent input nie urce are expreed in term the three phyical nie urce a: m cn n n m cn n n g i v r i r g i v v β 6a β i i i cn n n 6 Fr iplar tranitr, all three nie urce play a igniicant rle; hwever, r FE, cn i dminate. Flicker nie ha een neglected in Equatin 4-6 ince it i rarely a actr at RF and micrwave requencie; hwever, elw 00MHz GaA MESFE exhiit igniicant licker nie. 3.3: Nie Figure in a Single Stage Ampliier Equatin 3 and 6 may e cmined t determine the nie igure a ingle tage ampliier in term phyical nie urce and urce impedance. he nie igure i: n m cn n v r Z i r Z g i v F β 7a the cmplex urce impedance i written S X R Z, and term that are n the rder / β are neglected, then the nie igure i given y: [ ] S S S m cn S n n v r R X g i X r R i v F β β 7

21 he relatin: n i cn i β / i true r FE a well a BJ i ne interpret G D / β r FE. herere, Equatin 7 may e written: S S S S m cn n v X r R r R X g i v F β β β 7c By dierentiating Equatin 7c, the ptimum value r urce impedance may e derived. Uing the relatinhip / in g m β, the ptimum urce reitance i given y: β β g X i v X r R m in cn n pt 8 he ptimum urce reactance i: in m pt g X β β β β β 9 n the cae that β β <<, Equatin 9 indicate that urce reactance i uch that it cancel the input reactance. Fr FE the ptimal urce reactance i alway equal and ppite t the input reactance. Oten it i cnvenient t realize the urce reactance, S X, with an inductr. he ptimal value thi inductance i then given y: in m in pt g L β 0 3.4: Nie Figure Fr Biplar ranitr: Uing Equatin 7c and Equatin 4a-c, the nie igure a ingle-tage BJ ampliier ecme:

22 [ ] [ ] [ ] S in S in m S m r R X R g R X r R g R r F β he ptimum urce reitance and trancnductance are given y: β β g X g r X r R m in m S pt [ ] S pt m r R X X r R g β τ 3 tain the ptimum nie perrmance, Equatin and 3 mut e lved imultaneuly. Since an analytic lutin de nt exit, iteratin r me ther numerical technique mut e applied. An analytic lutin exit r β. n that cae: in pt X 4a [ ] m S in pt R g r R R r X F 4 β r g r R m pt 4c pt m g τ 4d 4 pt r r F τ τ 4e When β β β >> << the limiting value given y Equatin 4a-e are cle t the exact lutin Equatin and 3. Equatin 4a-d make a gd tarting "gue" r numerical ptimizatin the nie perrmance.

23 3.5: Lw-Nie Ampliier Realizatin Output Matching Netwrk R L O F FLER - L e Bia Feedack ircuit Figure : Single-Stage mmn-emitter Ampliier A lw-nie cmmn-emitter ampliier i hwn in Figure. nput matching i achieved thrugh the ue package nd-wire inductance. Output matching may e achieved either thrugh an external matching netwrk r with capacitive hunt eedack. the cllectr-ae unctin capacitance i neglected, then the input impedance i given y: Z in r β Le r Le L Le g m π hu L e give a reitive cmpnent t the input impedance withut adding nie. hi allw r lwnie deign, while maintaining gd matching at the input. he value emitter nd-wire inductance required r matching i given y: where: L e R r τ τ t τ g 5 m t Fr nie analyi purpe, the ae and emitter nd-wire inductance cntriute t the urce reactance. Uing Equatin, the nie igure thi ampliier i:

24 [ R r L ] [ L R r ] r gm F π π S R β R g R m S 6 L L L e he ptimum ttal inductance i given y Equatin 0. While Equatin and 3 may e ued r deign ptimizatin, quite ten it i eaier t minimize the nie igure thrugh direct ue an ptimizatin package. n additin, it i ten mre cnvenient t keep the urce reitance cntant and allw the device area t vary. he eect device area n current i manieted thrugh r and nider a prce with minimum ize device having parameter: device area A relative t the minimum ize device ha: π r g τ Equatin 6 i then minimized with repect t A and cnider the cae where β A m r A A r r and.. hen a c. get a tarting value r the ptimizatin,. Equatin 4a-d can e tranrmed t yield: r pt R c pt r τ 7 A τ he irt rder eect inite β may e taken int accunt via the apprximatin: A β τ 8 c pt 9 τ Equatin 7 and 9 uually give value quite cle t the actual ptimum. Since the nie igure i nt very enitive t thee parameter, it may e uicient t ue the value tained rm Equatin 7 and 9 r an actual deign.

25 3.6: Ditrtin in Single-Stage Ampliier At High Frequencie: lterra Serie technique will e applied t the cmmn-emitter/cmmn-ae tage t determine ditrtin characteritic. n particular, the third-rder intermdulatin ditrtin intercept i accurately predicted uing lterra technique. nider the ampliier cniguratin hwn in Figure 3. Z - Z e Figure 3: mmn Emitter/mmn-Bae Ampliier niguratin n a lterra Serie, the cllectr current i expreed in term the urce vltage a: i c a 3 v a, v a3,, 3 v where the peratr indicate that the amplitude and phae all inuid in the magnitude and phae a,,...,. n n n v are t e mdiied y the cllectr-ae unctin capacitance i neglected, then lterra Serie analyi give: a Z e Z e τ 0a

26 [ ] [ ], e Z a a a a 0 [ ] [ ] ,, e Z a a a a a a a 0c where: Z Z Z e e 3 3,,, a a a a a a a a Ditrtin i meaured in term the rati the puriu ignal generated and the deired ignal at the utput. Spuriu ignal generated at the utput are prprtinal t: 3 3 3,, v v v a, where v i i the input amplitude the i'th input ignal. he deired utput ignal i given y v a i. herere: v v v v a a Ditrtin ,, nider the cae where 3 and v v 3. hat i, ne the three input ignal prducing the ditrtin al prduce the deired utput ignal. 3 3,, v v a a Ditrtin n thi cae the ditrtin i prprtinal t the ignal trength the tw undeired ignal. Nw cnider the cae where all three ignal generate utput that are deired. he utput ignal level r the three deired nn-puriu ignal are: v a i v a i v a i he cnventinal way t meaure intermdulatin ditrtin i with equal utput amplitude: i i i i 3

27 n thi cae: ,, i a a a a Ditrtin Uing Equatin 0a-c, it i und that: 3,, κ κ κ κ a a a a where e e e Z Z Z τ κ hird rder intermdulatin ditrtin i deined a the ditrtin generated y the cuic nnlinearity with tw input ignal. O thee ignal, ten the mt imprtant ne i the ne with utput requency given y. Fr <<, the ditrtin' utput requency i near the tw ignal requencie. Fr a cntant utput level r tw input requencie, the third-rder dierence intermdulatinditrtin i given y: 3 3,, 4 3, i a a a DM 3 4, i DM κ κ κ he uual ituatin interet i when, <<. he ditrtin requency i, which ten all in the ignal andwidth, and cannt e iltered ut. he ditrtin r thi cae i given y: 3 4, i DM κ κ κ Uing the triangle inequality, an upper und n the ditrtin can e placed:

28 3 4, i DM κ κ κ 3 Nw : κ π Z Z Z Z e e e e where: τ Fr κ κ << the ditrtin i given y: 3 8, i DM κ κ 4 Fr κ κ >> the ditrtin i given y: 3 4, i DM κ κ 5 nider the cmmn-emitter ampliier in Figure. Fr thi ampliier: [ ] [ ] e e e r R L L L r R L L τ κ 6a π π κ L L R R L L L R R L e e e e 6 Nte that r uiciently mall, κ κ << and Equatin 4 applie. Fr increaing, κ will increae. When κ κ > the ditrtin will egin t increae utantially. Hence, it i neceary t have harp iltering at the input that ut--and

29 ignal which may have a wide requency pread d nt intermdulate t prduce ditrtin that i inand. he exprein r κ i a tw-ple, tw-zer traner unctin, and may e characterized y the renance requency and r the ple and zer. z L L 7a p z p R r L R r e L L L e π L L e π e Le R r 7 7c 7d n general Fr large p < z and p z Fr mall < and κ, κ κ 3 0 reache a minimum near z. DM and the minimum ditrtin ccur near: z md 8a L Le, the intermdulatin ditrtin can e written DM, κ 3 4, where i me unctin. Auming i held cntant, the requency minimum ditrtin i given y: md z 8 L L e Fr th mall and large Fr a lw-nie deign, the minimum ditrtin ccur etween 70-00% z. Equatin 0, the minimum ptimum inductance i given y: g m τ ee Equatin 4d. herere π. Frm L pt π

30 the impedance i matched at the input and lw-nie deign: r << RS then rm Equatin 5, Le R z τ. hu r a hat i, the zer requency ccur a actr mall Equatin 8a implie that: ave the requency that nie wa ptimized r. Fr md hat i, the minimum ditrtin ccur near the requency ued r nie ptimizatin. hi i mar advantage the cmmn-emitter ampliier at high requencie. t i the nly cniguratin that tain lw ditrtin and lw nie imultaneuly.

31 3.7: Deign Example nider the circuit hwn in igure. Suppe that the minimum ize tranitr availale in a given prce ha the llwing parameter: 400 r, 33 F, τ p, β 00. he deign requency i GHz. Applying Equatin 7 and 9 give the reult: A pt 7 c pt 3.mA hen Equatin 0 give: L pt N pt 5.3nH.0dB Direct numerical ptimizatin Equatin 0 yield: c pt L pt.9ma A nH hi repreent a variatin nly 7%. Further, the calculated nie igure r thee tw deign dier y nly 0.00dB. Equatin 5 give the emitter nd-wire inductance r impedance matching t e: L e.6nh then: L 3.7nH Figure 4 hw the intermdulatin ditrtin v. requency with the requency eparatin kept ixed at 0MHz, and the utput mdulatin i / at 00%. Nte that the actual ditrtin r 00% c mdulatin will nt e equal t that given in Figure 4 ince there are higher rder term in the lterra Serie. Hwever, 00% mdulatin i a cnvenient numer r reerence. Fr example the ditrtin r 0% mdulatin will e 40dB elw the level hwn in Figure 4. Ntice that the minimum ditrtin ccur near the GHz deign requency. Figure 5 hw the intermdulatin ditrtin v. requency eparatin with kept cntant at GHz. he ditrtin increae igniicantly r requency eparatin greater than 00MHz.

32 5 0 5 M3 db Frequency GHz Figure 4: Ditrtin v. Frequency r 0 MHzi c / -0-5 < M3 db > Frequency GHz Figure 5: Ditrtin v. r GHz

33

34 hapter 4: Mnlithic BJ Mixer Deign Figure and are cmplete chematic r a mnlithic BJ mixer. n Figure, a cmmn-ae driver i ued. n Figure, a cmmn-emitter driver i ued. n hapter, Figure, a dule-alanced mixer uing an emitter-cupled pair driver i hwn. Oten the linearity and nie perrmance a mixer are cntrained y the driver deign. he driver deign al ha the mt igniicant eect n the mixer cnverin gain. hu, a lgical irt tep in a mixer deign i the electin the driver tplgy. n thi chapter the mt cmmn driver tplgie are analyzed r gain, nie, and ditrtin perrmance. F OU LO N - 3 RF BAS R m R R ia - Figure : Emitter-upled Pair Mixer with mmn-bae Drive

35 F OU LO N Ω L - BE re KΩ Le 0 pf Figure : Emitter-upled Pair Mixer with mmn-emitter Drive 4.: mmn-emitter Driver mmn-emitter driver have the advantage prviding lw nie and high gain. Al, at high requencie, the linearity perrmance the cmmn-emitter i quite gd. he linearity the cmmn-emitter driver i identical t the cmmn-emitter ampliier analyzed in hapter 3. he current gain a cmmn-emitter ampliier i given y: β ai β τ

36 the cllectr-ae unctin capacitance i neglected, then the input impedance i: L L L r g L r Z e e m e in π β A in hapter 3, an input match i tained when the emitter inductance i: t e r R L τ and the ttal inductance i given y: t L e L τ 4.: mmn-bae Driver mmn-ae driver are advantageu when wideand peratin i required. mmn-ae tage prvide a nearly cntant input impedance and gain. he input impedance the cmmn-ae tage in Figure i: m m in r g R Z τ τ he current gain i given y: i a τ Nte that i m r g, the input impedance i a cntant reitance that i independent requency. Fr maller device which have a larger r the input impedance will have an inductive cmpnent with r L τ. Bradand impedance matching i achieved when m m R g R /. he ditrtin the cmmn-ae ampliier i tained y applying Equatin rm hapter 3 with m e r R R Z and m e R R Z. herere: 3 4, c i DM κ κ κ 3 π κ r R R R R r R R m m m

37 y: << and R R / > then κ << and the ditrtin can e apprximated m DM r ic 4 R Rm 3 4 [ R Rm / ] he irt the tw term in Equatin 4 i due t the expnential relatinhip vltage and current in the iplar device. he latter term repreent a ditrtin mechanim which give ditrtin that increae linearly with requency. At lw requencie the ditrtin in the cmmn-ae tage i quite lw, a the nnlinear trancnductance tend t e canceled y the nnlinear input impedance. At high requencie the ditrtin rie, ecaue the input impedance i linearized y nnlinear. he ditrtin i 3dB ave it lw requency value when: R R r m while the trancnductance remain 5 he ditrtin in the cmmn-ae i independent, the eparatin the tw input requencie. Ditrtin in the cmmn ae rie mntnically with increaing requency. At lw requencie the ditrtin i quite uperir t an undegenerated cmmn-emitter cniguratin. Hwever, ditrtin in the cmmn-emitter tage tend t decreae with requency ee hap. 3, Figure 4 while ditrtin in the cmmn-ae increae. t i intereting t erve the requency at which the tw cniguratin have equal ditrtin. Fr thi calculatin, nd wire are neglected. herere: R κ Re κ i the parameter ued in hapter 3. Fr the cmmn-ae, e π e R e R i the emitter i ed rm an ideal current urce and i << then: κ c

38 and M 3 c ic 4 6 Fr the cmmn-emitter withut degeneratin: κ At high requencie thi can e apprximated y: And uing Equatin rm hapter 3: M he tw tage have equal ditrtin r: 3 π κ R R π r r i c ce 8 7 π Fr typical lw-nie deign π and the ditrtin are equal r: 4 8τ π 8 hat i, the ditrtin the cmmn-ae and cmmn-emitter are aut equal at 5% the actual device' r.5% the typical device' peak. Fr a mdern ilicn iplar prce with τ ps, the ditrtin the tw tage are equal at.8ghz. n cmmn-emitter tage, the nd-wire inductance will reduce the ditrtin igniicantly elw that predicted y Equatin 7 r requencie near / L L. Hwever, in cmmn-ae tage the nd-wire inductance ha little eect n the ditrtin. herere, nd-wire may allw the cmmnemitter tage t exhiit lwer ditrtin than the cmmn ae r an ctave r tw elw the requency given y Equatin 8. t huld al e nted that the ditrtin the tw tage wa cmpared r cntant utput current level. Since a cmmn-emitter tage ha current gain, it input intercept will e utantially lwer than it utput intercept. A cmmn-ae tage ha unity current-gain, thu it input intercept when e π

39 expreed a a current i identical t it utput intercept. Hwever, ince the cmmn-ae ha n current gain, a high-gain preamp i neceary r adequate verall rnt-end gain. a cmmn-emitter driver i ued, a lwer-gain preamp r n preamp at all i deirale in rder t maintain an adequate third rder intercept pint r the rnt-end. he nie igure a cmmn-ae driver i given y: F v n v mn i cn g m Z r R β v m i n Z r R m 9 Equatin 9 i almt identical t Equatin 7a hapter 3. he rean r thi i that the equivalent input nie generatr are identical r all three the aic tranitr cniguratin cmmnemitter, cmmn-ae, and cmmn-cllectr. iv here i an additinal term due t the nie the emitter erie reitr ued r matching. he equatin in hapter 3 hld r the cmmn-ae lng a r i replaced y r R. m Hwever, ditrtin and matching cnideratin are dierent r the cmmn-ae than r the >> cmmn-emitter. Fr lw ditrtin in the cmmn-ae it i neceary that g R R and / <<. Matching require that R m / gm R m. gether thee cnditin mandate m that Rm R, and mut e mall. hee requirement are in direct ppitin t the requirement r lw nie. A cmprmie mut e made when ching etween a large device which er minimum nie and a minimum ize device which er minimum ditrtin. A reanale chice i a device ize that make the tw term in Equatin 4 apprximately equal. hat i: R 0 Fr thi value the ditrtin i apprximately: Ditrtin i reduced y increaing the ia current. ic DM 3 4 R

40 4.3: Deign Example Fr mmn Bae Driver Suppe that the minimum ize tranitr availale in a given prce ha the llwing parameter: r 400, 33F, τ p, β 00. he deign requency i GHz. Applying Equatin 0: 796F he device area relative t a minimum ize device i then: A Nie Figure db c Figure 3: Nie Figure v. Bia urrent r a mmn-ae tage he nie igure v. ia current i pltted in Figure 3. Nte that the nie igure, while till airly gd, i nt a lw a the cmmn-emitter tage dicued in hapter 3. hi i nt urpriing, ince the cmmn-emitter tage wa ptimized r nie perrmance. he nie igure in the cmmn-ae increae mntnically with increaing ia current.

41 he third rder intercept pint v. ia current i pltted in Figure 4. Nte that the intercept pint increae rapidly r increaing ia current. hu, there i a trade etween linearity, pwer, and nie. he ia current huld e chen t e the minimum value that give adequate linearity. he eect the driver' intercept pint n the ytem depend n the preamp gain. A the preamp gain i increaed, a higher mixer ia current i required r an adequate ytem intercept pint. herere, the preamp gain huld nt e et t high. Sytem gain and nie cnideratin generally et the minimum gain the preamp. ypically the preamp will have 5-0 db gain hird Order ntercept dbm c Figure 4: hird Order ntercept v. Bia urrent r a mmn-ae tage 4.4: Emitter-upled Pair Driver An emitter-cupled pair driver i hwn in Figure 5. Emitter-cupled pair are cmmnly ued t drive dule-alanced mixer uch a the ne hwn in hapter, Figure. Emitter-cupled pair eaily cnvert unalanced ignal t alanced ignal r vice-vera. Unrtunately, the emitter-cupled pair ha a

42 greater numer nie urce cmpared t a ingle ended ampliier; hence, it tend t have prer nie perrmance. make matter wre, it i diicult t match the emitter-cupled pair' high input impedance t a typical urce impedance. t i cmmn practice t ue a hunt reitr at the input t tain a match. Unrtunately, uch a "rute-rce" apprach urther degrade nie perrmance. r R e R e R m Figure 5: Emitter-upled Pair Driver quantiy the eect "rute-rce" matching, uppe that the ampliier withut the hunt reitr ha equivalent input vltage and current nie urce v n and i n. Auming that X 0, the nie igure withut matching reitr i: F vn in R v Auming that Where: R R, the nie igure with matching reitr i: m n vn in R F 3 v i i v v v v v cn cn n n en en g m

43 β i i i cn n n R K v e en 4 hat i, the eect the vltage nie ha duled, and even i the ampliier itel i niele, the ptimum nie igure i 3dB. n general, matching with a hunt reitr degrade ptimum nie perrmance y 3dB, and may degrade nie perrmance y a much a 6 db, i vltage nie dminate. make matter wre, the equivalent input vltage nie in an emitter-cupled pair i twice a much a the equivalent cmmn-emitter ampliier. he nie igure the emitter-cupled driver hwn in Figure 5 auming m R R i given y: 4 8 β β R g R g R R r F m m e 4 he lterra ceicient r the emitter-cupled pair are: [ ] e t e e t m e R r R r R r g R a τ τ π g g a a a m m, [ ] ,, a a a a a R r a a e where: 3 3,,, a a a a a a a a he third rder dierence intermdulatin ditrtin i then: 4, 3 d i DM κ 5

44 κ e g m R e r R [ r Re π gm Re r ] Re π r S lng a r << R and R << e e, the ditrtin in the emitter-cupled pair i apprximately independent the device ize chen. 4.5: Deign Example Fr Emitter-upled Pair Driver nider an emitter-cupled pair driver that ue the ame prce a the cmmn-ae example. n Figure 6 the ditrtin i pltted v. device area r ur dierent value 6.3mA. Since R e. he ttal ia current i R e repreent lcal eedack, it eect n ditrtin i a unctin the lp gain. Fr thi eedack cniguratin, the lp gain i given y: g R m e Ntice that r device ize etween 0x and 00x, the ditrtin i relatively independent area. here appear t e tw way t achieve lw ditrtin: ue a mall device, r ue utantial degeneratin. While th apprache degrade nie perrmance, the latter appear t e a mre viale lutin.

45 0 5 M3 db Area Figure 6: Ditrtin. Device Area r an Emitter-upled Pair clariy thi pint, the nie igure i pltted againt area in Figure 7. Fr mall area the nie igure i very pr. hi i a reult the vltage nie multiplicatin the matching reitr and the inherently higher vltage nie the emitter-cupled pair. he ptimum device area i arund 50-00x, and i apprximately independent. he ptimum nie igure i etween 5-0 db depending n the amunt degeneratin. A degeneratin actr near 3 eem t e a gd cmprmie, ince a larger value degeneratin de nt imprve the ditrtin much, ut increae the nie igure utantially. t i intereting t cmpare the perrmance the emitter-cupled pair tage and the cmmn-ae tage. Auming equal ttal current 6.3mA, the cmmn-ae tage ha a nie igure 4.5dB ee Figure 3 and the ditrtin i dwn 40dB r 00% mdulatin ee Figure 4. Uing an emitter-cupled pair area 00x relative t the minimum ize device and degeneratin actr 3, the nie igure i aut 7.0dB Figure 7, and the ditrtin i dwn nly 4dB Figure 6. hu the dynamic range the emittercupled pair i 0.5dB le than the cmmn-ae.

46 learly the emitter-cupled pair prvide inerir perrmance t th the cmmn-ae and cmmn-emitter ampliier. he nie perrmance i the wrt the three tage. he linearity at et equal the cmmn-ae when heavy degeneratin i ued. When the degeneratin i reduced ecaue nie cnideratin, the linearity i much wre than the cmmn-ae. Nnethele, the emitter-cupled pair i widely ued r dule-alanced mixer ince it make ue external BALUNS unneceary. 0 5 Nie Figure db Area Figure 7: Nie Figure v. Device Area r an Emitter-upled Pair 6.3mA

47 hapter 5: Nie Analyi Nnlinear ircuit: Active mixer are widely ued r dwn cnverin in UHF and micrwave receiver. n cntrat t paive mixer, active mixer prvide gain a well a requency cnverin. A mixer i hwn chematically in Figure. he mixer ha an RF radi-requency and LO lcal-cillatr input prt and an F intermediate requency utput prt. deally the mixer huld prduce nly a caled verin the prduct the tw input ignal. Hwever, real mixer add puriu ignal and randm nie t the deired utput ignal. Mixer RF nput F Ouput Lcal Ocillatr nput Figure : Baic Mixer Structure t i deirale t e ale t predict the nie perrmance a given mixer deign. Ampliier nie analyi technique d nt apply t mixer, ecaue the preence a large LO ignal caue utantial change in the active device' perating pint ver a perid. echnique that have een previuly preented have the diadvantage that they are nn-ytematic, and numerically ill-cnditined. v,vi Additinally, thee methd ail r ht nie in the aence a high- tuned circuit. n thi chapter a methd i preented that i numerically eicient and well cnditined, ytematic, and accurate. A igniicant advantage thi technique i that ne imulatin yield inrmatin n the mixer perrmance r all RF and F input requencie. Previuly preented analyi technique required a eparate imulatin r each RF input and F utput requency interet. vii

48 5.: State Equatin r Mixer: t i a aic reult circuit thery that any circuit made up element that are either current cntrlled r vltage cntrlled can e decried y a ytem tate equatin the rm: viii r d dt S ut r v F v, a r r i the vectr tate variale, r i the vectr ignal vltage applied t the circuit, and S ut i the utput ignal. State variale are made up capacitr vltage r charge and inductr current r lux. n iplar tranitr, the tate variale crrepnding t the vltage acr cllectr current thrugh the algeraic tranrmatin: c π e may e replaced y the π An alternative rmulatin knwn a mdiied ndal analyi ue nde vltage and inductr current. hen r i the vectr nde vltage and inductr current. he relatinhip etween mdiied ndal analyi equatin MNA and tate variale equatin i quite imple. Mdiied ndal analyi prduce ne redundant equatin r each nde that ha n capacitive element attached t it. Depite the large matrix tructure created, MNA i currently implemented in many AD package e.g., SPE and uch a rmulatin i deirale r integratin int the cmputer cde uch package. All mixer perate y ue a large LO ignal that mdulate the perating pint the active device r dide r paive mixer in the mixer. n the aence RF verlad, the LO i the nly large ignal applied t the mixer. Nie urce in the mixer can e thught a mall ignal applied t an therwie niele mixer circuit. Becaue the large LO ignal, linear nie analyi mixer aed n a ixed perating pint i nt pile. Analyi mixer uing availale nn-linear technique i numerically ill-cnditined, ince a mall numerical errr relative t the LO amplitude may e quite large

49 relative t ther ignal in the circuit. Hence, it i deirale t tain a methd that wrk independently n the large and mall ignal. Such a methd i nw preented. Auming a large LO ignal and a mall RF ignal, the tate equatin r mixer can e written: r d t dt S ut r v F [ t, t, v t ] a LO r [ ] t t r Nrmally, the tate the mixer i determined primarily y the LO, with the RF ignal cauing nly r a mall perturatin. Suppe t t a the quiecent tate vectr. hat i, r r d i the tate vectr in the aence an RF ignal hencerth reerred dt t t i the lutin t: r v F [ t, t,0] 3 LO hen the tate vectr with the RF ignal included i: r r r r t t i t dit r v r r r where: F[ t i t, t, v t ] F[ t, t,0] dt LO r LO Uing a irt rder aylr Expanin F aut the quiecent tate give: r dit dt r r G t i t h t v t 4a dfi G i, d r r df h t dv where r t r t

50 he ntatin t i ued t mean that the derivative i evaluated at the quiecent tate. A imilar analyi tarting with Equatin give the mall-ignal utput a: ut where and " " indicate matrix multiplicatin. r r t c t i t 4 r c t d v d Secnd rder aylr expanin term are generally negligile i the RF ignal vltage r nie vltage i uiciently mall that nnlinearitie the circuit are nt igniicantly excited. Becaue the RF ignal vltage and internal nie vltage in the mixer are mall, uperpitin applie, and each ne can e analyzed eparately. Equatin 4a& are linear time-varying equatin. he ceicient vary with time in a manner determined y the applied LO ignal and the circuit cniguratin. the LO ignal i peridic a i uually the cae, the ceicient in Equatin 4a& ecme peridic and the ytem equatin i a linear peridically time-varying ytem r LP. A preented in thi chapter, Equatin 4a& are derived rm dierentiatin the tate equatin the ytem. Hwever, thee equatin may e tained directly rm the circuit y replacing each element the nnlinear circuit y it linear time-varying equivalent circuit. hu, the mixer circuit equatin are lved in tw tep: t Step : Slve the large-ignal ytem equatin in Equatin 3. he RF and nie urce are turned nly the LO urce i let n, and all the tate variale are lved a a unctin time r ne LO perid.

51 Step Mixer RF nput F Ouput Lcal Ocillatr nput Figure : Firt tep in mixer perrmance calculatin Step : Slve the mall ignal time-varying circuit equatin given y Equatin 4, r rm a linearized circuit mdel r the RF ignal and each nie urce. Becaue the linearity Equatin 4, uperpitin applie t each mall-ignal urce. he lutin tep i quite traight-rward. Many tandard AD package can e ued t tain the teady tate repne t the LO input. he lutin tep i currently nt implemented in any cmmercial AD package. n the remainder thi chapter, tw related technique will e demntrated r lving LP ytem r th determinitic and tchatic input ignal. he irt technique i mre eicient and well cnditined, while the latter i eaily implemented uing availale AD package. 5.: Equatin r Linear-time arying Sytem: Fr an L ytem the input-utput relatin i given y ix : y t h t, u x u du 5 he input-utput relatin Equatin 5 i imilar t the tandard cnvlutin ued in a linear timeinvariant ytem. Hwever, the value the impule repne i a unctin th the launch time the

52 impule, u, and the ervatin time, t. n a time-invariant ytem, the impule repne i nly a unctin the dierence etween the ervatin time and the launch time. h t, u hlti t u Under the ave cnditin, Equatin 5 reduce t the amiliar cnvlutin integral. n an L ytem the impule repne may lk quite dierent r dierent launch time. Fr mixer with peridic LO excitatin, the impule repne i peridic in launch time, and thu can e een a a unctin the launch phae the phae the LO at launch time. n the requency dmain the relatinhip etween the utput and input pectrum i given y: Y H, X d 6 i i r X and Y are the Furier ranrm input and utput ignal, and H i given y: H i, r h t u e π, r r u A derivatin Equatin 6 i given in Appendix A. r du e i t Frm Equatin 6 it i een that r a general linear time-varying ytem, a ingle input requency prduce a cntinuu pectrum utput requencie, nt ut a ingle utput requency a in the cae an L ytem. Fr peridic LO excitatin requency, the requency dmain equatin, which are derived in Appendix B, ecme: dt n Y H X n 7 i n i i where n u v i H n i g v u e du e dv, 8 0

53 g v, u h v u, u n an LP ytem a given input requency prduce a dicrete et utput requencie, eparated in magnitude y. he utput pectrum i a linear uperpitin hited and iltered verin the input pectrum. Fr each hit the requency repne the ytem i given y H, where n i the numer LO requencie that the input pectrum i hited. Anther pint view i that multiple input requencie given y: n ± r n i 9 are all dwn-cnverted t the F utput requency thrugh mdulatin againt the n'th LO harmnic. hi relatinhip i epecially imprtant in mixer nie analyi, ince nie at a numer dierent input requencie may cntriute utput nie at the intermediate requency. Frequencie particular interet are: r i and r ± i crrepnding t n0 and n. he latter tw requencie are the input-ignal requency and the image requency. he exitence the image requency i prlematic in lw nie mixer deign ince the nie rm that requency cntriute t the utput. Oten the nie at the image requency cntriute equally t the nie at the RF ignal requency, degrading the nie igure y 3 db. nput nie at the intermediate requency can e a igniicant prlem in unalanced mixer; hwever, in alanced mixer the nie rm the intermediate requency i ideally canceled at the utput. Fr tatinary nie the input-utput relatin i: n S H S n y i n i 0 x i S x i the input pectral denity and S y i the utput pectral denity. the input nie i white cntant pectral denity, and i the utput requency i much lwer than any time cntant in the ytem, then the utput pectral denity can e apprximated y:

54 S y S x n H n 0 S x 0 g v, u dve n u du hi apprximatin i ten ueul r dwncnverin mixer. he impule repne unctin, h t, u, tgether with Equatin 7 and 8, are uicient t decrie the mall ignal input-utput ehavir the mixer r all pile excitatin. Since the impule repne a mixer depend n the lcatin the input excitatin, a eparate calculatin r each nie urce i neceary. Oten a numer nie urce can e lumped int a ingle urce, thu reducing the numer impule repne that mut e calculated. ircuit ymmetry can al e explited t urther reduce required calculatin. 5.3: Otaining the mpule repne an L ytem A theretical apprach that ue tate equatin t tain the impule repne i preented in thi ectin. hi methd, while eicient and theretically und, i currently nt implemented in any cmmercially availale AD package. Reerring t Equatin 4a&, the value the mall-ignal tate vectr and impule repne at ervatin time ut ater the launch time can e hwn t e: r r i u, u h t r r h u, u c u h u a he ecnd argument the unctin r i crrepnd t the launch time. Fr ervatin time t > u, the dierential equatin i: r dit,u dt r G t i t, u c he impule repne i tained rm the linearized relatin: r r h t, u c t i t, u d

55 Equatin a-d cntitute a hmgenu initial value prlem. hee equatin can e lved y tandard numerical ODE methd uch a the trapezidal methd. he value c r r t, G t, and t are peridic, and depend n the large-ignal ODE lutin Equatin 3. he value thee unctin are calculated ver an LO perid and then tred. 5.4: Furier ranrm Analyi Once the impule repne i calculated r launch time that pan the range all LO phae, the repne mut e prceed y a tw-dimeninal at-furier tranrm t tain the ytem unctin a given in Equatin 8. deally, the impule repne wuld e calculated r all launch time in [0,] and r all ervatin time. Fr caual ytem it i nt neceary t cnider ervatin time prir t the launch time. Since it i nt pile t expre a cled rm lutin the impule repne r even imple mixer circuit, the impule repne value are calculated at inite interval in th ervatin time and launch time. hi dicretizatin intrduce aliaing errr. Further, it i neceary t aume that at ervatin time tmu, r me M, the impule repne decay t a negligile value. Fr accurate reult M mut e chen t e much larger than the larget time cntant in the circuit under wrt cae cnditin. the interval etween ucceive ervatin time pint i chen t e d, and the interval etween ucceive launch time i du, then the ttal numer pint required t decrie the impule h repne i: N M LO du d learly r a ixed value N, there i a trade etween the cnlicting requirement large M, and mall du and d. hing an M that i t mall will caue "lurring" in the requency dmain due t cnvlutin with a inc unctin. he value d huld e chen t e much maller then the invere the F andwidth, and du huld e chen t e much maller than the invere the RF andwidth.

56 hing du r d t large will caue aliaing. t i et t che M, du, and d t alance ut the three errr, that n ingle ne dminate. Oten nly lw utput requencie are interet. n uch a cae a lw pa ilter i placed at the utput, and the ampling interval in the ervatin time, d, may e made utantially larger. Fr imulatin purpe high- F ilter huld e avided, ince they caue the impule repne t ring, and thu require a very large value M much larger than the invere the F andwidth. A three-ple lwpa F ilter at three time the LO requency yield a gd trade etween accuracy and imulatin time. M i uually chen t e an LO perid, and d i chen t e /3 an LO perid. he three-ple ilter reduce pectral cmpnent uiciently t prevent aliaing. Figure 3 illutrate the relatinhip etween the grid chen in the time dmain and a crrepnding grid tained in the requency dmain ater a twdimeninal FF i perrmed. ime Dmain Frequency Dmain P/ Launch ime U du LO Harmnic 0 n 0 0 M d -P/ - *d 0 i /M *d Oervatin ime Output Frequency Figure 3: Grid in ime and Frequency Dmain he tw dimeninal FF i tained y calculating an FF the rw ht,u and then an FF it clumn. are mut e taken t erve the expnential ign and caling actr r each directin