Non-ideal quantization noise shaping in switched-current bandpass ΣΔ modulators

NON-IDEAL QUANTIZATION NOISE SHAPING IN SWITCHED-CURRENT BANDPASS EA MODULATORS J.M. de la Rosa, B. Pe' ez-Ve dh, E Medei o, R. del Rio and A. Rod iguez-Vbzquez (IEEE Fellow) Ins i u o de Mic oelec 6nica de Se illa, IMSE-CNM (CSIC) Edi icio CICA-CNM, C/Ta ia s/n, 41012-Se illa, SPAIN Phone: +34 95 4239923; FAX: +34 95 4231832 E-mail: [email p o ec ed] ABSTRACT This pape p esen s a sys ema ic analysis o he majo swi ched-cu en (SI) e o s and hei in luence on he quan iza- ion noise shaping o SI BandPass ZA Modula o s (BPXAMs). Closed o m equa ions a e p o ided o he deg ada ion o he sig- nal- o-noise a io and o he change o he no ch equency posi- ion in o de o acili a e he design o his class o modula o s. All hese esul s ha e been alida ed by non-ideal ime-domain be- ha io al simula ions.(*) 1. INTRODUCTION A wide a ie y o CMOS analog sampled-da a EA-modula o s (EAMs) ha e been epo ed in li e a u e o applica ions anging om ins umen a ion o elecom [I]. Mos o hem use swi ched-capaci o (SC) ci cui s. Howe e , he end owa ds he combina ion o analog and digi al ci cui y in mixed-signal chips wi h low p oduc ion cos has mo i a ed explo ing design ech- niques such as swi ched-cu en (SI), which can be manu ac u ed in s anda d VLSI single-poly echnologies wi h no ex a p ocess- ing s eps equi ed [2]. Se e al SI lowpass XAM ICs o di e en applica ions ha e al eady been epo ed [3][4]. Recen ly, we ha e also epo ed a SI BPEAM ealized in a single-poly 0.8 pm CMOS echnology [5]. Unlike SC EAMs, whose non-ideali ies ha e been desc ibed and modeled [I], hus allowing gene a ion o sys ema ic op-down design me hodologies [6][7], un il now he e is no such equi a- len s udy o SI ZAMs. This pape p esen s a sys ema ic analysis o he majo SI e o mechanisms which deg ade he quan iza ion noise shaping o 4 h-o de BPZAMs. The aim o his s udy is o iden i y he in luence o each e o and o p o ide equa ions which acili a e he design o his class o modula o s. 2. QUANTIZATION NOISE SHAPING IN BANDPASS ZA MODULATORS Fig. 1 (a) shows he block diag am o a single-loop BPZAM. As- suming ha he quan ize can be modeled as an addi i e, whi e noise sou ce, he z- ans o m o he modula o ou pu can be ex- p essed as ollows: Y(Z) = &F(Z)X(Z) + NTF(Z)E(Z) (1) whe e X(z) is he inpu signal and E(z) ep esen s he quan iza ion e o . Typically, BPZAM a chi ec u es a e ob ained om he co e- ma ion; his keeps he s abili y p ope ies o he o iginal a chi ec u e and allows us o ake ad an age o he a ailable knowl- edge on he p ope ies o lowpass modula o s [I]. Thus, he ideal quan iza ion Noise T ans e Func ion (NTF) is o he bands op (*)This wo k has been suppo ed by he Spanish CICYT P ojec sponding lowpass p o o ypes by applying a z- I + -z -2 ans o - TIC 97-OS80 2L h-o de BP Fil e N-bi Quan ize -2 -2 1 Z 1 +z Resona o 1 Resona o 2 Compa a o Figu e 1. Concep ual Block Diag am o a BandPass ZA Modula- o , a) N-bi 2L h-O de . b) 1-bi 4 h-O de . ype7 (2) -2 L N,&) = (1 + z 1 whe e 2L is he loop o de . This la e unc ion has L ansmission ze oes a i4, whe e& s ands o he sampling equency. Assuming ha he N-bi quan ize in Fig.l(a) has a quan iza ion s ep equal o q , i s quan iza ion noise powe spec al densi y is NQ q /(12 s). As a consequence o he noise shaping, he in-band quan iza ion noise powe esul s in 2 ,/4 i B,/2 [A/(2N- 1)I2xzL ,/4 - B,/2 12(2L+ 1)MZL+' (3) PQ = j 2NQlNTF( )12d whe e A ep esen s ull-scale ange o he quan ize ; i.e. A = q(2 - 1) , B, is he signal bandwid h and M = s/(2Bw) is he o e sampling a io. This pape ocuses on I-bi (N = 1) modula o s wi h L = 2. These modula o s a e easy o unde s and and ela i ely simple o design. They a e capable o p o ide high esolu ion oge he wi h la ge ole ance o impe ec ions and obus , s able ope a ion. Thus, we will s a om a 2nd-o de lowpass modula o , which a - e he z-' -+ -z-* ans o ma ion, esul s in he associa ed band- pass a chi ec u e shown in Fig.l(b). The esona o in Fig.l(b) can be ealized using di e en al e na i es [8]. Fig.2(a) shows one based on wo LDI in eg a o s in a eedback loop. This implemen- a ion is adop ed because i emains s able unde changes o he eedback coe icien due o ci cui pa asi ics. Fig.2(b) illus a es his by plo ing he mo emen o he esona o poles due o chang- es o he scaling coe icien AFA, a ound i s nominal alue o 2. I can be seen ha hey emain a he uni y ci cle. Fig.2(c) shows he a chi ec u e o a 4 h-o de bandpass EA N 0-7803-5471 -0/99/$10.0001999 IEEE 11-476 Resona o 1 Re's6nZ o 2 Compa a o LmJAGJ ADACI Figu e 2. a) LDI Loop Resona o . b) Mo emen o Poles A ound he Uni y Ci cle Due o Changes in he Feedback Gain. c) Block Diag am o a BPCAM Based on he Resona o o Fig.2(a). modula o based on he esona o o Fig.2(a). The scaling ac o s ha e been op imized o ob ain simila dynamic ange o bo h es- ona o s [SI, gi ing ARES2 = ADAC2 = -ADAcl = 1 and AREsl = 1/2. By making N = 1 and L = 2 in (3), we ob ain he quan iza- ion noise powe Pp o his modula o . On he o he hand, as- suming ha he inpu signal o he modula o is a sinewa e o am- pli ude A < A/2, we ob ain he ollowing exp essions o he Sig- nal- o-Noise Ra io (SNR) and he Dynamic Range (DR) [I] as i can be de i ed om (4), he esolu ion o his modula o inc eases wi h M a a a e o abou 2.5 bi s pe oc a e. Howe e , as we will see in he ollowing sec ion, e o sou ces in p ac ical SI ci cui s cause he quan iza ion noise o inc ease in he signal band, hus lowe ing he eached esolu ion. 3. SI BUILDING BLOCKS AND ASOCIATED ERRORS As shown in [2], he majo e o sou ces o he memo y cell a e: jini e d ain conduc ance, incomple e se ling, and swi ch cha ge injec ion. Thei isola ed in luence on he memo y cell and on he in eg a o ans e unc ion we e analysed in [2]. Howe e , o ou s udy we a e in e es ed on hei cumula i e in luence on he deg ada ion o he quan iza ion noise shaping in BPZAMs. Conside he cascaded memo y cells shown in Fig.3(a). Du ing clock phase Cell1 is in hold mode while Cell2 is in sampling mode. The small-signal equi alen ci cui o such a con igu a ion is shown in Fig.3 (b). The s a iona y cu en o he memo y an- sis o in Ce112, iD(n) , is gi en by (5) whe e E, = 2go /gm, wi h g, and gou being he small-signal inpu and ou pu conduc ance o Ce112, espec i ely. Exp ession (5) is also alid o o he memo y cells by subs i u ing g, by he co - esponding inpu ansconduc ance. Howe e , i applies only i he memo y cell eaches he s eady s a e be o e he end o he sampling phase. O he wise, an addi ional e o is gene a ed as a consequence o he incomple e se ling o he ol age a he ga e o he memo y iD(n) = -(1 -E )i (n) K DI - C) &- & a) Cascaded Memo y Cells and Clock Phases. b) Figu e 3. Equi alen Ci cui Du ing Phase & and c) Du ing Phase $2. ansis o . Sol ing he equi alen ci cui o Fig.3(b) o V, wi h V (n- 1) = i; em(n- l)/gm as he ini ial condi ion o he ga e-sou ce capaci o C,, yields K (6) whe e E,$ = e~p[-T,g,/(2C~,~)] s ands o he incomple e se - ling e o and iEem( i - 1) ep esen s he memo ized d ain cu - en in he p e ious sampling phase. When he memo y swi ch opens, in he beginning o phase q2, he cha ge injec ed by he swi ch ansis o in C,,$ in oduces an addi ional e o e m ( AV4 )[2], so ha VK(n + 1/2) = V,(n) + AVq = (1 - E,)V,(n) (7) Fig.3(c) shows he equi alen ci cui o he Cell2 in he hold mode. Conside ing all e o s abo e, he z- ans o m o iEem is mem V (n) = [( 1 - E,)iD(n) + (n - 1 )]/gm Le conside now he concep ual SI ealiza ion o he esona o block o Fig.2(a) shown in Fig.4. In addi ion o he memo y cell e o s (E~, E! and E$), he e a e some e o s due o non-ideali ies in he connec ion o he in eg a o s. We de ine hese e o s as ol- lows: 'F, FB = 'ongoF, FB ;&gF,.. = (goF, FB/gm)(' -&F, FE) (9) whe e ion is he s ee ing swi ch on- esis ance and go,, goFB a e he ou pu conduc ances o he cu en mi o s, as s a ed in Fig.4. Using he equi alen ci cui shown in Fig.3 o he memo y cells which o m he in eg a o s, he ollowing ans e unc ion is ob ained o he non-ideal esona o : Figu e 4. Schema ic o he Resona o Including SI E o s. 11-477 whe e Subs i u ing (1 0) in he ans e unc ion o he esona o in Fig.2(c), he e oneous quan iza ion noise ans e unc ion NYi is app oxima ely gi en by [l +~lz-l+(l -C2)z-212 NTi(z) s (12) 1 +251z-1+[5:-2(52-P*)1z-2 As (12) shows, he ze oes o NYi a e shi ed om hei nomi- nal posi ions a ;/4, hus deg ading he il e ing pe o med by he esona o s and making he quan iza ion noise loo o inc ease in he signal band, and co espondingly, he SNR o dec ease. We can g oup he men ioned e o s in h ee di e en amilies a ending o he way hey deg ade he ze oes o NY;(z), which map in o di e en inc eases o he quan iza ion noise powe PQ. Table I shows he non-ideal quan iza ion noise powe o each amily o e o s. This able p o ides insigh on he in luence o each e o sou ce. Assuming ypical a ia ions o he e o pa a- me e s be ween 0.1 % and 1 %, se e al conclusions a e d awn om (1 2) and Table I: *The only e ec o e o s and E~, deno ed by pa ame e E, in Table I, consis s in educing he Q- ac o o he esona o ans e unc ion, hus lowe ing he bands op a enua ion o he modula o bandpass il e ing. Fo hese e o s, he de ia ion in he quan iza ion noise powe is domina ed by he e m (E M)’ up o E,M = 0.6 ; beyond his limi he e m (E,M)~ domina es, hus p ac ically des oying all he bene i s o he o e sampling. *The e o s and ep esen ed by pa ame e in Table I, jus change he posi ion o signal band cen e equency, o en called no ch equency. Howe e , he quan iza ion noise powe does no signi ican ly inc ease. Fo his amily o e o s, he e m (E@)* domina es up o E~M = 2.2 . E~~~, ep esen ed h ough pa ame e s espec i ely In Table I, deg ade he posi ion o he a *The e o s E, and and Table I: Non-Ideal Quan iza ion Noise Powe E o I Quan iza ion Noise Powe , Po I 0.2 0.4 0.6 0.8 1 &(%lo) 70b . . - ’ I 64 128 256 512 16 32 Figu e 5. DR Deg ada ion wi h E o s. a) DR s. E o o M = 128 . b) DR s. M o E o Equal o 0.5%. no ch equency and inc ease he quan iza ion noise powe in he signal band. Fo E,, he e m (E~ M)’ domina es up o M = 0.6, while o (E~SM)’ domina es up o E~:M = 2.6 . *Fo simila alues o pa ame e s E,, ED,, E..,~ and E+, he se ling e o p oduces la ge de ia ions in he noise ans e unc ion han he es o e o s - illus a ed in Fig.S(a). This o ces using la ge o e sampling a ios o achie e he ideal SNR le el as is shown in FigS(b) by plo ing DR deg aded by he di e en e o s as a unc ion o M. O he scaling e o s a e ound a he modula o le el. Howe e , hey do no in luence he esona o ans e unc ion bu he eso- na o and DAC gains. Fo ypical alues o hese e o s (1 %), he inc ease in he quan iza ion noise powe is negligible. All hese esul s ha e been alida ed by simula ion using a ime-domain SI beha io al simula o [9]. As an illus a ion, Fig.6 shows se e al simula ed modula o ou pu spec a ob ained o a sinusoidal inpu signal o ampli ude A = A/4 and cen e ed a 0 -20 - - 5 -60 e -80 ,- .- w g -100 -120 0.24 0.25 0.26 0.27 F equency / Sumplbig F equency Figu e 6. In luence o Di e en SI E o s on NTF 11-478 ,/4 . This igu e compa es he deg ada ion o NTF in he p es- ence o E~,~, E~~, E,, and E~~, ob ained wi h he heo e ical model (solid line) wi h ha ob ained h ough simula ion. A good ag ee- men can be obse ed be ween bo h app oaches. 5. GUIDELINES FOR THE DESIGN OF SI 4 h-ORDER BANDPASS XA MODULATORS In p ac ical applica ions, he designe should conside he cu- mula i e in luence o e o s on he modula o pe o mance. F om (3) and (12), he quan iza ion noise powe deg aded by all e o s can be de i ed as ollows Making all e o s equal o an e o bound named E he ollowing quan iza ion noise powe is ob ained whe e k, = 4960/3 and k2 = 438080. This equa ion allows us o exp ess he quan iza ion noise powe deg ada ion in e ms o E. Thus, o cing all SI e o s in he modula o o be smalle han E bounds he quan iza ion noise powe wi h (14). Fig.7 plo s he hal -scale SNR (ob ained o a sinusoidal inpu signal o ampli ude A = A/4 ) as a unc ion o E , o M = 64, M = 128 and M = 256. Simula ion esul s ma ch e y well wi h he da a calcula ed om (14). In BPZAMs, he con ol o he no ch equency ( , ) posi ion is also c i ical. In p ac ice, his equency is ixed by he sys em e- qui emen s, so ha i s posi ion should no be signi ican ly shi ed om J4. We can de i e , by sol ing (12) o he equency o he ze oes. Assuming ha C,, C2 << 1 he e o in , , deno ed as G ,,isgi enby F om (1 1) i is seen ha c1 5 0 and he e o e , 5 ,/4 o all SI BPZAMs. On he o he hand, conside ing ha he quan iza ion noise powe is minimum a , , we can de ine a maximum e o IS , oll 5 Bw/2 . F om (15) and assuming all SI e o s o be equa o E, his condi ion is sa is ied i E 5 n/( 10M). Fo ins ance, i M = 128, i ob ains E 2 0.25 %. Fig.8 illus a es his by plo ing h ee ou pu spec a co esponding o di e en alues o E. I is seen ha he p edic ions o (15) ag ees wi h simula ion da a. 100 7s- x % -+-- ‘Simula ion, M = 64 OSimula ?on, M = 128 ASimula ion, M = 256 - - Model 1 40b.01 . . . . . . ... 0.1 1 E(%) Figu e 7. SNR Deg ada ion s. E o an Inpu Signal o A/4 Ampli ude. M = 64 ,U = 128, U = 256. . . . . . . . . . V - , p edic ed by ( 15 -20 “I M=128 F equency /Sumpling F equency Figu e 8. No ch F equency Deg ada ion wi h SI E o s. 6. CONCLUSIONS The s udy p esen ed in his pape classi ies he SI non-ideali ies a ending o he way hey deg ade he quan iza ion noise shaping in bandpass ZA modula o s. Closed- o m equa ions ha e been de- i ed o he in luence o SI e o s on he modula o pe o mance. As a esul o such an analysis, some p ac ical guidelines o he design o BPZAMs using SI ci cui s a e gi en. This app oach has been alida ed by de ailed ime domain beha io al simula ion. REFERENCES S.R. No swo hy, R. Sch eie , G.C. Temes (edi o s), Del a-Sigma Con e e s. Theo y, Design and Simula ion. New Yo k, IEEE P ess, 1997. C.Toumazou, J.B.Hughes, and N.C. Ba e sby (edi o s), Swi ched-Cu en s: An Analogue Technique o digi al echnology, London, Pe e Pe eg inus L d.,l993. N. Tan and S. E iksson, “A Low-Vol age Swi ched- Cu en Del a-Sigma Modula o ”, IEEE Jou nal o Solid-s a e Ci cui s, pp. 599-603, May 1995. N. Moeneclaey and A. 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