Continuous-time cascaded ΣΔ modulators for VDSL: A comparative study

* [email p o ec ed]; phone +34955056666; ax +34955056686; www.imse.cnm.es Con inuous-Time Cascaded Σ∆Modula o s o VDSL: A Compa a i e S udy Ramón To osa, José M. de la Rosa*, Angel Rod íguez-Vázquez and F ancisco V. Fe nández Ins i u o de Mic oelec ónica de Se illa, IMSE-CNM (CSIC) Edi icio CICA-CNM, A da Reina Me cedes s/n, 41012-Se illa, SPAIN ABSTRACT This pape desc ibes new cascaded con inuous- ime Σ∆ modula o s in ended o cope wi h e y high- a e digi al subsc ibe line speci ica ions, i.e 12-bi esolu ion wi hin a 20-MHz signal bandwid h. These modula o s ha e been syn hesized using a new me hodology ha is based on he di ec syn hesis o he whole cascaded a chi ec u e in he con inuous- ime domain ins ead o using a disc e e- o-con inuous ime ans o ma ion as has been done in p e ious app oaches. This me hod allows o place he ze oes/poles o he loop- il e ans e unc ion in an op imal way and o educe he numbe o analog componen s, namely: ansconduc o s and/o ampli ie s, esis o s, capaci o s and digi al- o-analog con e e s. This leads o mo e e icien opologies in e ms o ci cui y complexi y, powe consump ion and obus ness wi h espec o ci cui non-ideali ies. A compa ison s udy o he syn hesized a chi ec u es is done conside ing hei sensi i i y o mos c i ical ci cui e o mechanisms. Time-domain beha io al simula ions a e shown o alida e he p esen ed app oach. Keywo ds: Analog- o-digi al con e e s, sigma-del a modula o s, con inuous- ime ci cui s. 1. INTRODUCTION Con inuous-Time (CT) Sigma-Del a Modula o s (Σ∆Μs)ha e demons a ed o be an a ac i e solu ion o he imple- men a ion o Analog- o-Digi al (A/D) in e aces in sys ems-on-chip in eg a ed in deep-submic on s anda d CMOS ech- nologies 1. Al hough mos epo ed Σ∆Ms ha e been implemen ed using Disc e e-Time (DT) ci cui s, he inc easing demand o b oadband da a communica ion sys ems has mo i a ed he use o CT ci cui echniques. In addi ion o show an in insic an ialiasing il e ing, CT Σ∆Ms p o ide po en ially as e ope a ion wi h lowe powe consump ion han hei DT coun e pa s 2,3. In spi e o hei men ioned ad an ages, CT Σ∆Ms a e mo e sensi i e han DT Σ∆Ms o some ci cui e o s, namely: clock ji e , excess loop delay and echnology pa ame e a ia ions 2,3. The la e a e specially c i ical o he ealiza ion o cascaded a chi ec u es. This has o ced he use o single-loop opologies in mos epo ed silicon p o o ypes e en hough low o e sampling a ios ( ) a e needed 4,5, whe eas e y ew cascaded CT Σ∆M In eg a ed Ci cui s (ICs) ha e been epo ed 6. Howe e , he need o achie e medium-high esolu ions ( ) wi hin high signal bandwid hs ( ) while gua an eeing s abili y, has p omp ed he in e es in p ope me hods o he syn hesis o high-o de cascaded CT Σ∆Ms 7- 9. These me hods a e based on applying a DT- o-CT ans o ma ion o an equi alen DT opology ha ul ils he equi ed speci ica ions. In mos cases, he use o such a ans o ma ion is no mally ansla ed in o an inc ease o he analog ci cui complexi y wi h he subsequen penal y in silicon a ea, powe consump ion and sensi i i y o pa ame e ole ances. This pape p esen s a di ec syn hesis me hod o cascaded CT Σ∆Ms which, dispensing wi h he DT- o-CT equi a- lence, make i possible o educe he analog ci cui y complexi y and place he ze oes/poles o he quan iza ion noise ans- e unc ion in an op imal way, hus yielding o mo e obus a chi ec u es han using a DT- o-CT ans o ma ion. As an applica ion, he p oposed me hodology is used o ind op imum CT Σ∆Ms o Ve y high- a e Digi al Subsc ibe Line (VDSL). Th ee i h-o de cascaded opologies a e syn hesized: 2-1-1-1, 2-2 and 3-2. These modula o s a e designed o 12-bi @20-MHz speci ica ions and hei pe o mances a e compa ed in e ms o ime-domain simula ions ha ake in o accoun c i ical e o mechanisms like misma ch and clock ji e e o . 12< 12bi s>20MHz> VLSI Ci cui s and Sys ems II, edi ed by José Fco. López, F ancisco V. Fe nández, José Ma ía López-Villegas, José M. de la Rosa, P oceedings o SPIE Vol. 5837 (SPIE, Bellingham, WA, 2005) 0277-786X/05/$15 · doi: 10.1117/12.607923 59 Downloaded F om: h ps://www.spiedigi allib a y.o g/con e ence-p oceedings-o -spie on 24 Jan 2020 Te ms o Use: h ps://www.spiedigi allib a y.o g/ e ms-o -use 2. CASCADED CONTINUOUS-TIME Σ∆ MODULATORS Fig.1 shows he concep ual block diag am o a cascaded CT Σ∆M. Each s age, consis ing o a single-quan- ize CT Σ∆M, e-modula es a signal con aining he quan iza ion e o gene a ed in he p e ious s age. Once in he digi al domain, he ou pu s, , o he s ages a e p ope ly p ocessed and combined (by he cancella ion logic) in o de o cancel ou he quan iza ion e o s o all he s ages, bu he las one in he cascade. This la e e o appea s a he o e all modula o ou pu shaped by a unc ion o o de equal o he summa ion o he o de s o all he s ages. Cascaded CT Σ∆Ms a e no mally syn hesized om equi alen (well-known) DT sys ems and use he same digi al can- cella ion logic 8. This DT/CT equi alence can be gua an eed because he o e all open loop ans e unc ion o each s age in Fig.1 is in ac a DT sys em 2. Thus, in he case o a ec angula impulsi e esponse o he Digi al- o-Analog Con e e (DAC), i can be shown ha he equi alen DT loop il e ans e unc ion is gi en by 10,11: (1) whe e is he sampling equency; ; ; and a e espec i ely he ime delay and pulse wid h o he DAC wa e o m; a e he poles o and s ands o he esidue o . In o de o ge a unc ional CT Σ∆M while keeping he cancella ion logic o he o iginal DT Σ∆M, e e y s a e a iable and DAC ou pu mus be connec ed o he in eg a o inpu o la e s ages 8. This inc eases he numbe o analog compo- nen s, i.e ansconduc o s, ampli ie s and DACs. As an illus a ion, Fig.2(a) shows a cascaded CT Σ∆M ob ained om an exis ing DT Σ∆M12. No e ha a leas eigh scaling coe icien s ( ) and hei co esponding signal pa hs a e needed o connec he di e en s ages o he modula o . The numbe o in eg a ing pa hs can be educed −as shown in Fig.2(b) − i he whole cascaded Σ∆M is di ec ly syn hesized in he CT domain as p oposed in he nex sec ion. 3. PROPOSED SYNTHESIS METHODOLOGY The idea o dispensing wi h he DT- o-CT ans o ma ion was p e iously epo ed in 3 o single-loop a chi ec u es. Howe e , in he case o cascaded a chi ec u es, he cancella ion logic unc ions (no p esen in single-loop Σ∆Ms) mus be included in he syn hesis p ocedu e o ge an op imum a chi ec u e. Le us conside he mo e gene al case o he cascaded CT Σ∆M shown in Fig.1. The o e all ou pu , , is gi en by: (2) Figu e 1. Concep ual block diag am o a cascaded CT Σ∆M. CT Σ∆M 1 CT Σ∆M 2 CL 1 CL 2 CL m . . . . . .. . . x(s) y 2 (z) y 1 (z) y m (z) y o (z) E 1 (z) E 2 (z) E m (z) Cancella ion Logic CT Σ∆M m + DAC To nex s age E 2 (z) y 2 (s) x 2 (s)F(s) y 2 (z) m-s age yi Fz() Re Fs() s ----------- em1TSs⋅ ze TSs⋅ – -------------------- ⋅    pi ∑Re Fs() s ----------- em2TSs⋅ ze TSs⋅ – -------------------- ⋅    pi ∑ –= s1T s ⁄ = m11 dTs ⁄ –= m21 dτ + ()Ts ⁄ –= dτ piFs()s⁄Re x() x 2-1-1 kg29– m-s age yo yoz() ykz()CLkz() k1= m ∑ = 60 P oc. o SPIE Vol. 5837 Downloaded F om: h ps://www.spiedigi allib a y.o g/con e ence-p oceedings-o -spie on 24 Jan 2020 Te ms o Use: h ps://www.spiedigi allib a y.o g/ e ms-o -use whe e and ep esen espec i ely he ou pu and pa ial cancella ion logic ans e unc ion o he s age. I he modula o inpu , , is se o ze o, he ou pu o each s age can be w i en as: (3) whe e s ands o he , is he in e se Laplace ans o m, is he ans e unc ion o he DAC, and (4) ep esen s he ans e unc ion om o he inpu o quan ize . Using he ollowing no a ion (5) he ou pu o each s age is gi en by: (6) and he ou pu o he modula o can be w i en as: (7) Figu e 2. Cascaded 2-1-1 CT Σ∆M a chi ec u e ob ained (a) om an equi alen DT Σ∆M (b) using he p oposed me hod. DAC CL 1 CL 3 1 T s s DAC CL 2 k g1 1 T s s k b2 1 T s s k g5 k g9 k g8 k g7 k g6 k b4 x(s) y o (z) (a) DAC DAC CL 1 CL 3 DAC CL 2 E 2 (z) E 1 (z) E 3 (z) y 1 (z) y 2 (z) y 3 (z)y 3 (s) y 2 (s) y 1 (s) y o (z) x(s) (b) E 1 (z) E 2 (z) E 3 (z) y 1 (s) y 2 (s) y 3 (s) y 1 (z) y 2 (z) y 3 (z) k in1 1 T s s k b1 DAC k g2 k g3 k g4 k b3 k in1 1 T s s k b1 k in3 1 T s s k b4 k g1 1 T s s k b2 k in2 1 T s s k b3 ykz() CLkz() k- h x () ykz() Ekz() ZL 1–HDFik [] nTs    yiz() i1= k1 – ∑ + 1ZL 1–HDFkk [] nTs    – ------------------------------------------------------------------------------------------- = ZZ- ans o m L1 – HDHDAC s()≡ Fij Fij s()≡Inpu Quan ize j yis() ----------------------------------------- = yis() j- h Z km ZL 1–HDFkm () nTs   ≡ ykz() Ekz() 1Zkk – -----------------Zik yiz() 1Zkk – -------------------- i1= k1– ∑ += yoykCLk k1= m ∑Ek 1Zkk – -----------------1 1Zkk – -----------------Zik yi i1= k1– ∑ +    CLk k1= m ∑ == P oc. o SPIE Vol. 5837 61 Downloaded F om: h ps://www.spiedigi allib a y.o g/con e ence-p oceedings-o -spie on 24 Jan 2020 Te ms o Use: h ps://www.spiedigi allib a y.o g/ e ms-o -use The pa ial cancella ion logic ans e unc ions ( ) can be calcula ed by imposing he cancella ion o he ans e unc ion o he i s quan iza ion e o s in (7). This gi es: (8) whe e he pa ial cancella ion logic ans e unc ion o he las s age, , can be chosen o be he simples o m ha p ese es he equi ed noise shaping. No e ha he design equa ions (2)-(8) do no only ake in o accoun he single-s age loop il e ans e unc ions ( ), bu also he in e -s age loop il e ans e unc ions ( ). The la e a e con inuous- ime in eg a ing pa hs appea ing only when he modula o s ages a e connec ed o o m he cascaded Σ∆M and mus be included in he syn hesis me hod- ology o ob ain a unc ional modula o wi h minimum numbe o in e -s age pa hs. The e o e, he ollowing p ocedu e can be used in a sys ema ic me hodology o he syn hesis o cascaded CT Σ∆Ms††: • Fi s , he poles o single-s age ans e unc ions ( ) a e op imally placed in he signal bandwid h o gi en speci ica ions. This p ocess is ca ied ou en i ely in he CT domain and no equi alence o an exis ing DT modu- la o needs o be imposed. • Second, once he indi idual s ages a e designed and op imized, cancella ion logics a e calcula ed using (8). Fo illus a i e pu poses, he 2-1-1 CT Σ∆M o Fig.2(b) was syn hesized using (2)-(8) o achie e esolu ion in a bandwid h, wi h a sampling equency o 48MHz (o e sampling a io, ) 12.Fo simplici y, in o de o acili a e he compa ison o he pe o mance o bo h modula o s in Fig.2, he coe icien s o he i s s age ( ) a e aken o be equal in bo h sys ems and a e ob ained om a DT- o-CT ans o ma ion o he i s s age o a DT Σ∆Min12. The es o coe icien s in Fig.2(b) a e aken such ha he ime cons an o he in eg a o s is he in e se o he sampling equency ( ): (9) Hence, he single-loop and in e -s age ans e unc ions a e gi en by: (10) and he pa ial cancella ion logic ans e unc ions can be calcula ed using (8)-(10). Conside ing a Non-Re u n- o-Ze o (NRZ) DAC, he ollowing cancella ion logics a e de i ed: (11) †† In his p ocedu e, he modula o o de , o e sampling a io and numbe o bi s o in e nal quan ize s a e as- sumed o be de e mined o gi en speci ica ions om well-known exp essions 1. CLk m1 – Ekz() C Lkz() ZkmCLm – 1Zmm – ------------------------ ZL 1–HDFkm [] nTs    CLmz( ) – 1ZL 1–HDFmm [] nTs    – ------------------------------------------------------------------------ - == CLmz() Fii F ij ij≠, Fii s() 16-bi 750-kHz M32 = kin1kg1k b1k b2 ,, , T s1 s ⁄ = kin1k b1 – 14⁄ ;== k b238⁄ –= kg1kin2k– b3kin3k– b41== == = F11 3Ts 8 ---------s1 4 --- +   – sTs () 2 ------------------------------ =F22 F33 1– sTs -------- == F13 3Ts 8 ---------s1 4 --- +   – sTs () 4 ------------------------------ =F23 1– sTs () 2 --------------- = CL1 z1– 48 -------729+z1–7z2– –5z3– –()= CL2z1–1z1– +()1z1– –() 2 = CL321 z1– –() 3 = 62 P oc. o SPIE Vol. 5837 Downloaded F om: h ps://www.spiedigi allib a y.o g/con e ence-p oceedings-o -spie on 24 Jan 2020 Te ms o Use: h ps://www.spiedigi allib a y.o g/ e ms-o -use whe e is chosen o ha e h ee ze oes a DC, co esponding o he ze oes con ibu ed by he i s h ee in eg a o s. In o de o compa e he obus ness o bo h modula o s in Fig.2, he e ec o misma ch on he Signal- o-Noise Ra io ( ) was also simula ed using SIMSIDES, a SIMULINK-based ime-domain beha io al simula o o Σ∆Ms 13.Fo his pu pose, maximum alues o misma ch we e es ima ed o a 0.13 µm CMOS echnology and bo h modula o s in Fig.2 we e simula ed conside ing a Gm-C implemen a ion. The esul s a e shown in Fig.3, whe e he loss is ep esen ed as a unc ion o he s anda d de ia ion o he ansconduc ances ( ) and capaci ances ( ). Fo each poin o hese su aces, 150 simula ions we e ca ied ou using andom a ia ions wi h he s anda d de ia ion gi en in he diag ams. The alue o loss ep esen ed in Fig.3 s ands o he di e ence be ween he ideal , i.e wi h no pa ame e a ia ion, and he wi h 90% o he 150 simula ions abo e i . I is shown ha he lowe analog componen coun in Fig.2(b) is e lec ed in a lowe a iance o he modula o coe icien s, leading o a be e beha io in e ms o sensi i i y o misma ch. 4. APPLICATION TO VDSL As an applica ion o he p oposed me hodology, h ee 5 h-o de cascaded CT Σ∆Ms, shown in Fig.4, we e syn hesized o cope wi h VDSL speci ica ions: 12-bi esolu ion wi hin a 20-MHz signal bandwid h. In o de o ul il hese speci ica- ions wi hou being limi ed by he clock ji e e o , he sampling equency, , and he numbe o bi s o he in e nal quan- ize s (and DACs), , mus be p ope ly chosen. In he case o a 5 h-o de modula o s like hose shown in Fig.4, he in- band ji e noise powe is minimized o and 14. Ano he c i ical sou ce o e o in CT Σ∆Ms is he excess loop delay. As shown in 15 his e o can be compensa ed by adding an ex a eedback b anch be ween he ou pu and he inpu o he quan ize (DAC2in Fig.4) and wo D-la ches. By adding his ex a b anch wi h he app op ia e gain, he loop impulse esponse is exac ly he same as ha o he o iginal. This ex a eedback e m can be easily included in he calcula ion o he cancella ion logic. In a p ac ical implemen a ion i could be ad an ageous o make DAC2p og ammable 5. Conside ing he ac o s abo e, he CT Σ∆Ms in Fig.4 we e syn hesized using he me hodology desc ibed in Sec ion 3, aking in o accoun he ollowing conside a ions: • The i s s age o he 2-1-1-1 a chi ec u e (Fig.4(a)) is o med by a esona o which has i s poles placed a , in o de o minimize he quan iza ion Noise T ans e Func ion (NTF) in he signal bandwid h, . Resis o a ia ions can be uned ou using a combina ion o a disc e e ough uning o he esis o s ( , and ) and a con inuous ine uning o he ansconduc o s and . This uning can be also used o cancel he e ec o ini e Gain-Bandwid h p oduc ( ) o he on -end opamp, due o he ac ha his e o can be mod- elled as an in eg a o gain e o 5. All he o he ansconduc o s could be uned in o de o keep he ime cons an unchanged o e a ia ions. • An addi ional esona o has been used in he 2-2-1 a chi ec u e (Fig.4(b)), in o de o op imally dis ibu e he poles o NTF 16. CL3 SNR Figu e 3. E ec o misma ch on he SNR o a cascaded 2-1-1 CT Σ∆M ob ained om: (a) an equi alen DT Σ∆M; (b) p opose d me hod. (b) SNR Loss (dB) SNR Loss (dB) (a) SNR σgm σC SNR SNR SNR s B s240MHz = B4 = ωp2π79⁄Bw = BwRin R b R k kg1 GB Cg m ⁄C P oc. o SPIE Vol. 5837 63 Downloaded F om: h ps://www.spiedigi allib a y.o g/con e ence-p oceedings-o -spie on 24 Jan 2020 Te ms o Use: h ps://www.spiedigi allib a y.o g/ e ms-o -use Figu e 4. Cascaded CT Σ∆Ms syn hesized o VDSL: (a) 2-1-1-1; (b) 2-2-1; (c) 3-2. + − D La ch DAC 2 D La ch DAC 1 DAC 3 DAC 4 DAC 5 CLK CLK CLK CLK CLK CLK CL 1 CL 2 CL 4 CL 3 C 1 C 2 C 3 C 4 C 5 R in k R b R k g1 k in3 k in4 k g5 k g2 k in2 k g3 k g4 k b2 k b3 k b4 y o (z) x(s) (b)(a) (c) + − D La ch DAC 2 D La ch DAC 1 DAC 4 CLK CLK CLK CLK CLK CL 1 CL 2 CL 3 C 1 C 2 C 4 C 5 R in k 1 R b R k g1 k in3 k g2 k in2 k g4 k g5 k b2 k b3 DAC 2 D La ch DAC 1 CLK CLK D La ch k 2 k 2 k g3 C 3 y o (z) x(s) + − D La ch DAC 2 D La ch DAC 1 CLK CLK CLK CLK CL 1 CL 2 C 1 C 5 R in k 2 R b k g2 k g3 k in2 k g5 k b2 DAC 2 D La ch DAC 1 CLK CLK D La ch k 3 k 2 k g4 C 4 C 2 k 1 k 1 k g1 C 3 64 P oc. o SPIE Vol. 5837 Downloaded F om: h ps://www.spiedigi allib a y.o g/con e ence-p oceedings-o -spie on 24 Jan 2020 Te ms o Use: h ps://www.spiedigi allib a y.o g/ e ms-o -use • The 3-2 modula o shown in Fig.4(c) includes a i s s age which consis s o an in eg a o and a esona o . This opology allows he same op imum pole posi ioning as in he 2-2-1 modula o wi h one less s age. Howe e , s a- bili y p oblems migh a ise ha comp omise he modula o pe o mance. Table 1 shows he single-loop and in e -s age ans e unc ions ( ) o he di e en a chi ec u es in Fig.4 as a unc- ion o he loop il e coe icien s . The exp essions o , ob ained om (8) and (11), a e also shown. I is impo an o no e ha a e ound om an i e a i e simula ion-based p ocess ha op imizes he i s s age o he modula o in o de o maximize while keeping s abili y. The ou come o he op imiza ion p ocess −en i ely done in he CT domain −is summa ized in Table 2.This able includes he alues o loop il e coe icien s, (implemen ed as ansconduc ances) as well as he capaci ances, , and esis ances, ob ained om he op imiza ion p ocess. The modula o s in Fig.4 we e simula ed using SIMSIDES 13. Fig.5 shows he ideal ou pu spec a o he modula o s when clocked a . I can be obse ed he e ec o he esona o s poles dis ibu ed wi hin he signal band- wid h. The impac on he in-band noise powe is be e app ecia ed in Fig.6 ha ep esen s he Signal- o-(Noise+Dis o - ion) Ra io (SNDR) s inpu ampli ude. No e ha , al hough bo h he 2-2-1 and 3-2 a chi ec u es ha e he same loca ion o he ze oes o he NTF, he 3-2 modula o achie es a wo se esolu ion. This is due o he ac ha he op imiza ion p ocess applied o ha a chi ec u e was mo e conse a i e as a consequence o he s abili y cons ains imposed by he 3 d-o de s age. In addi ion o he ideal pe o mance desc ibed abo e, he e ec o mos c i ical limi ing ac o s has been aken in o accoun in he high-le el design. Fig.7 shows he SNR loss caused by clock ji e e o . No e ha he 2-1-1-1 a chi ec u e seems o be less sensi i e o his e o han he o he a chi ec u es. Howe e , i is impo an o no e ha he ideal SNR o his modula o is lowe han he o he s. The e o e, he e is a highe componen o quan iza ion noise masking he e ec o clock ji e . Two c i ical limi ing ac o s in cascaded Σ∆Ms, and pa icula ly in hei CT implemen a ion, a e ci cui ole ances and componen misma ch. The i s one can be con olled by using uning o ime cons an s 4,5 o digi al calib a ion 6. Howe e misma ch e o s ill emains. In o de o e alua e he impac o his e o on he pe o mance o he modula o s in Fig.4, maximum alues o misma ch we e es ima ed o a 0.13 µm CMOS echnology conside ing a Gm-C implemen a ion. The Fij bij CLi bij SNR kiCi Ri Figu e 5. Ideal ou pu spec a o he cascaded CT Σ∆Ms in Fig.4: (a) 2-1-1-1. (b) 2-2-1. (c) 3-2. (a) (b) (c) s240 MHz= P oc. o SPIE Vol. 5837 65 Downloaded F om: h ps://www.spiedigi allib a y.o g/con e ence-p oceedings-o -spie on 24 Jan 2020 Te ms o Use: h ps://www.spiedigi allib a y.o g/ e ms-o -use Table 1: T ans e unc ions and cancella ion logic unc ions o he modula o s in Fig.4 Modula o T ans e unc ions Cancella ion Logic 2-1-1-1 2-2-1 3-2 Cancella ion Logic Coe icien s 2-1-1-1 2-2-1 F14 b10 s3s2ωp 2 +() --------------------------- = F 24 1– Ts 3s3 ----------- =F34 1– Ts 2s2 ----------- = F44 1– Tss -------- = C L1z1 – n14 n+13z1 – n12z2 – n11z3 – n10z4 – +++()= C L2 1 6 ---z1–14z1–z2– –+()12 Tsωp ()cos z1– –z2– +()= C L3 1 2 ---z1–1z1– +()1z1– –()12 Tsωp ()cos z1– –z2– +()= C L41z1– –() 212 Tsωp ()cos z1– –z2– +()= F13 b10 ss 2ωp1 2 +()s2ωp2 2 +() ----------------------------------------------------- = F 23 b20 ss 2ωp2 2 +() ---------------------------- =F44 1– Tss -------- = C L1z1–n14 n+13z1 – n12z2 – n11z3–n10z4– +++()= C L2z1–n22 n21z1–n20z2– ++()12 Tsωp1 ()cos z1– –z2– +()= C L312 Tsωp1 ()cos z1– –z2– +()12 Tsωp2 ()cos z1– –z2– +( ) = F12 b11sb 10 + ss 2ωp1 2 +()s2ωp2 2 +() ----------------------------------------------------- = F 22 b21sb 20 + s2ωp2 2 + ----------------------- eTSs–kceTSs–2⁄ += C L1z2–n14 n+13z1 – n12z2 – n11z3 – n10z4 – +++()= C L212 Tsωp1 ()cos z1– –z2– +()1z1– –()⋅= 12 Tsωp2 ()kc –cos()z1– –1n21 kc2Tsωp2 ()cos–+()z2–n20 k+c ()z3– ++ () n 10 n14 b – 10 Ts 3ωp 5 ------------- Tsωp ()sin Tsωp –1 6 ---Tsωp () 3 +== n 11 n13 b–10 Ts 3ωp 5 ------------- [Tsωp () 32Tsωp ()cos– 3 ----------------------------------- 4Tsωp ()2TsωpTsωp ()1+cos()+sin–== n 12 b–10 Ts 3ωp 5 ------------- [Tsωp () 314 Tsωp ()cos– 3 ---------------------------------------6Tsωp ()sin 2Tsωp12 Tsωp ()cos+()]–+= n 10 n14 b – 10 ωp1 3ωp2 3ωp2 2ωp1 2 –() ------------------------------------------------ Tsωp1ωp2 3ωp1 3ωp2 –()ω p1 3Tsωp2 ()ω p2 3Tsωp1 ()sin–sin+[== n 11 n13 2b10 – ωp1 3ωp2 3ωp2 2ωp1 2 –() ------------------------------------------------ [Tsωp2ωp1 3ωp2 3ωp1 –()Tsωp1 () Tsωp2 ()cos+cos()()+== ωp2 3Tsωp1 ()1Tsωp2 ()cos+()ω p1 3Tsωp2 ()1Tsωp1 ()cos+()]sin–sin+ n 12 2b10 – ωp1 3ωp2 3ωp2 2ωp1 2 –() ------------------------------------------------ [Tsωp1ωp2 3ωp1 3ωp2 –()12 Tsωp1 ()Tsωp2 ()coscos+()+= ωp1 3Tsωp2 ()12 Tsωp1 ()cos+()ω p2 3Tsωp1 ()12 Tsωp2 ()cos+()]sin–sin+ n 20 n22 b20 – ωp2 3 ---------- Tsωp2Tsωp2 ()sin–[]== n21 2b20 – ωp2 3 --------------Tsωp2 ()Tsωp2Tsωp2 ()cos–sin[ ] = 66 P oc. o SPIE Vol. 5837 Downloaded F om: h ps://www.spiedigi allib a y.o g/con e ence-p oceedings-o -spie on 24 Jan 2020 Te ms o Use: h ps://www.spiedigi allib a y.o g/ e ms-o -use esul s o his analysis a e shown in Fig.8 whe e he SNR is ep esen ed as a unc ion o he s anda d de ia ion o he ansconduc ances ( ) and capaci ances ( ). Fo each poin o hese su aces, a Mon eCa lo analysis o 150 simula- ions was ca ied ou . The alue o he SNR ep esen ed in he e ical axis o Fig.8 is ob ained by 90% o he simula ions o each case o and . No e ha e en in he wo s -case misma ch, he esolu ion is abo e he speci ied ( ). Finally, he modula o s in Fig.4 we e high-le el sized, i.e, he sys em-le el speci ica ions ( ) we e mapped on o building-block speci ica ion using s a is ical op imiza ion o design pa ame e selec ion, and beha io al simula ion o e alua ion. The esul s o his sizing p ocess a e summa ized in Table 3 and Table 4 showing he maximum (minimum) alues o he ci cui e o mechanisms ha can be ole a ed in o de o ul il he equi ed modula o pe o m- ance. As an illus a ion, Fig.9 shows he ou pu spec a o he modula o s aking in o accoun he non-ideali ies lis ed in hese ables. The e ec i e esolu ion is and o he modula o s in Fig.4(b) and (c), espec i ely. Table 1: T ans e unc ions and cancella ion logic unc ions o he modula o s in Fig.4.(Con .) Mod. Cancella ion Logic Coe icien s 3-2 10 b–10 Tsωp1ωp2ωp2 2ωp1 2 –()ω p1 3Tsωp2 ()ω p2 3Tsωp1 ()sin–sin+() ωp1 3ωp2 3ωp2 2ωp1 2 –() -------------------------------------------------------------------------------------------------------------------------------------------------------------------- += b–11 ωp1 21Tsωp2 ()cos–()ω p2 21Tsωp1 ()cos–()–() ωp1 2ωp2 2ωp2 2ωp1 2 –() ---------------------------------------------------------------------------------------------------------------------------------- + 11 2b10 –Tsωp1ωp2ωp1 2ωp2 2 –()Tsωp1 ()Tsωp2 ()cos+cos()ω p2 3Tsωp1 ()1Tsωp2 ()cos+()sin+( ) ωp1 3ωp2 3ωp2 2ωp1 2 –() -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+= 2b10 ωp1 3Tsωp2 ()sin 1 Tsωp2 ()cos+()() ωp1 3ωp2 3ωp2 2ωp1 2 –() ----------------------------------------------------------------------------------------------------2b11 ωp2 2Tsωp2 ()cos 1 Tsωp1 ()cos–()ω p1 2Tsωp1 ()cos 1 Tsωp2 ()cos–( ) –( ωp1 2ωp2 2ωp2 2ωp1 2 –() ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- + 12 2b10 –Tsωp1ωp2ωp2 2ωp1 2 –()12 Tsωp1 ()cos Tsωp2 ()cos+()ω p1 3Tsωp2 ()12 Tsωp1 ()cos+()sin+( ) ωp1 3ωp2 3ωp2 2ωp1 2 –() ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+= 2b10 ωp2 3Tsωp1 ()sin 1 2 Tsωp2 ()cos+()() ωp1 3ωp2 3ωp2 2ωp1 2 –() ------------------------------------------------------------------------------------------------------- 13 2b10 –Tsωp1ωp2ωp1 2ωp2 2 –()Tsωp1 ()Tsωp2 ()cos+cos()ω p2 3Tsωp1 ()1Tsωp2 ()cos+()sin+( ) ωp1 3ωp2 3ωp2 2ωp1 2 –() -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+= 2b10 ωp1 3Tsωp2 ()sin 1 Tsωp2 ()cos+()() ωp1 3ωp2 3ωp2 2ωp1 2 –() ----------------------------------------------------------------------------------------------------+2b11 ωp1 2Tsωp1 ()cos 1 Tsωp2 ()cos–()ω p2 2Tsωp2 ()cos 1 Tsωp1 ()cos–()–( ) ωp1 2ωp2 2ωp2 2ωp1 2 –() ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - 14 b–10 Tsωp1ωp2ωp2 2ωp1 2 –()ω p1 3Tsωp2 ()ω p2 3Tsωp1 ()sin–sin+() ωp1 3ωp2 3ωp2 2ωp1 2 –() -------------------------------------------------------------------------------------------------------------------------------------------------------------------- –= b 11 ωp2 21Tsωp1 ()cos–()ω p1 21Tsωp2 ()cos–()–() ωp1 2ωp2 2ωp2 2ωp1 2 –() - ------------------------------------------------------------------------------------------------------------------------------ – 2 0 b21 Tsωp2 ()sin ωp2 -------------------------------------b20 1Tsωp2 ()cos–() ωp2 2 ---------------------------------------------------- –= n21 b–21 Tsωp2 ()sin ωp2 ---------------------------------------- b20 1Tsωp2 ()cos–() ωp2 2 ---------------------------------------------------- –= σgm σC σgm σC72-dB 12-bi @20-MHz 13.4 bi s≅13.2bi s≅ P oc. o SPIE Vol. 5837 67 Downloaded F om: h ps://www.spiedigi allib a y.o g/con e ence-p oceedings-o -spie on 24 Jan 2020 Te ms o Use: h ps://www.spiedigi allib a y.o g/ e ms-o -use