Matrix Methods for the Dynamic Range Optimization of Continuous-TimeGm-CFilters

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 55, NO. 9, OCTOBER 2008 2525 Ma ix Me hods o he Dynamic Range Op imiza ion o Con inuous-Time Gm - C Fil e s Juan F. Fe nández-Boo ello, Manuel Delgado-Res i u o, Membe , IEEE, and Angel Rod íguez-Vázquez, Fellow, IEEE Abs ac —This pape p esen s a syn hesis p ocedu e o he op- imiza ion o he dynamic ange o con inuous- ime ully di e en- ial - il e s. Such p ocedu e builds up on a gene al ex ended s a e-space sys em ep esen a ion which p o ides simple ma ix algeb a mechanisms o e alua e he noise and dis o ion pe o - mances o il e s, as well as, he e ec o ampli ude and impedance scaling ope a ions. Using hese me hods, an analy ical echnique o he dynamic ange op imiza ion o weakly nonlinea - il- e s unde powe dissipa ion cons ain s is p esen ed. The p oce- du e is i s explained o gene al il e s uc u es and hen illus- a ed wi h a simple biquad a ic sec ion. Index Te ms— - il e s, il e syn hesis, dynamic ange op- imiza ion, noise analysis, dis o ion analysis, low powe . I. INTRODUCTION FULLY DIFFERENTIAL - ci cui s and echniques a e widely employed o design in eg a ed con inuous- ime il e s. O e he yea s signi ican con ibu ions ha e been made ega ding he p oposal o ansconduc o opologies wi h enhanced noise and dis o ion pe o mance. Based on such ansconduc o s, il e s wi h enla ged dynamic ange (DR)1can be buil o di e en p ac ical applica ions. Howe e , despi e he a ailabili y o high-pe o mance ansconduc o s, dynamic ange op imiza ion may be hampe ed due o inaccu a e e alu- a ion o he impac o noise and dis o ion on he o e all il e pe o mance. Rega ding he impac o noise, one o he mos signi ican ea ly con ibu ions was due o G oenewold [1]. He employed Manusc ip ecei ed Oc obe 22, 2007; e ised Janua y 11, 2008. Fi s published Ap il 18, 2008; cu en e sion published Oc obe 29, 2008. This wo k was suppo ed by he Spanish Minis y o Educa ion & Science unde G an TEC2006-03022, and he Jun a de Andalucía unde G an TIC-02818. This pape was ecommended by Associa e Edi o T. B. Ta im. J. F. Fe nández-Boo ello and A. Rod íguez-Vázquez a e wi h AnaFocus, Se illa 41092, Spain. M. Delgado-Res i u o is wi h he Ins i u o de Mic oelec Ónica de Se illa, Cen o Nacional de Mic oelec ónica, Consejo Supe io de In es igaciones Cien í icas (CSIC), and Uni e sidad de Se illa, 41012 Se illa, Spain (e-mail: [email p o ec ed]). Digi al Objec Iden i ie 10.1109/TCSI.2008.921048 1DR is de ined as he a io be ween he maximum and minimum signal ha can be p ocessed by he il e . The la e is limi ed by he o al in eg a ed ou pu - e e ed noise powe o he il e , U , and he o me is limi ed by he maximum ou pu dis o ion le el which can be ole a ed. Assuming a single- one exci a ion and le ing be he maximum signal le el a he ou pu o he il e , he dynamic ange can be exp essed as DR =(1 = 2) 1 (  =U )  P =U s a e-space desc ip ions o e alua e noise pe o mance h ough simple ma ix manipula ions. Simila app oaches o noise e al- ua ion ha e been used in [2], [3]. In his pape , we adop hese noise e alua ion echniques. Also, we adop he gene al - s uc u e p oposed in [3]. Rega ding he impac o dis o ion, di e en echniques ha e been epo ed; o ins ance, hose p esen ed in [4]–[9]. P oposals in [4], [5] a e based on equency-domain calcula ion using Vol e a se ies which, al hough powe ul, end o be e y cumbe some as he il e o de inc eases. In [6], [7] nonlinea i- ies o indi idual ansconduc o s a e p opaga ed by means o pa ial ans e unc ions o he ou pu , whe e he con ibu ions a e summed o es ima e he o e all dis o ion beha io o he il e . Finally, [8], [9] use ime-domain analysis and s a e-space modelling o he e alua ion o ha monic and in e modula ion dis o ion in il e s wi h no loa ing capaci o s. None o hese app oaches p o ides he simple, gene al and sys ema ic ma ix manipula ion echniques which a e a ailable o noise. Conse- quen ly, no sys ema ic me hod is ye a ailable o e alua e he impac o bo h noise and dis o ion on a gene al - il e s uc u e, he eby limi ing he abili y o designe s o maximize he dynamic ange o p ac ical il e implemen a ions. This pape p esen s me hods and echniques o e alua e he impac o bo h noise and dis o ion on he - il e s uc u e o [3] h ough simple and sys ema ic ma ix algeb a. Fo dis- o ion e alua ion we combine s a e-space and Vol e a Se ies ep esen a ions [10] in such a way ha hey can be easily em- bedded in ma ix o m. The pape also add esses he issue o scaling. The op i- miza ion o DR h ough modi ica ions o he ampli ude and impedance le els a he in e nal il e nodes is co e ed o bo h biquad a ic sec ions and gene al - il e s. In he case o biquad a ic sec ions, he pape epo s compac DR exp essions which p o ide mo e accu a e es ima ions o he in luence o he quali y ac o , , han hose in [1]. Fo gene al il e s, he echniques p esen ed in his pape o e come he lack o uni ocal solu ions obse ed in he app oach p esen ed in [6]. The pape is s uc u ed as ollows. Sec ion II desc ibes he s a e-space ep esen a ion o gene al - il e s, and p o ides ela ionships among he s a e-space ma ix and a se o in e nal ans e unc ions which a e essen ial o he o egoing analysis. Sec ions III and IV p esen he sys ema ic echniques o he e alua ion o noise and dis o ion, espec i ely. Sec ion V deals wi h he scaling o - il e s. Sec ion VI add esses he dy- namic ange op imiza ion o gene ic - il e s and he p o- cedu e is illus a ed in Sec ion VII o biquad s uc u es. Finally, Sec ion VIII concludes he pape . 1549-8328/$25.00 © 2008 IEEE 2526 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 55, NO. 9, OCTOBER 2008 Fig. 1. Simpli ied schema ic o a ully balanced G - C il e . II. SPACE-STATE REPRESENTATION OF - FILTERS Fig. 1 shows he concep ual schema ic o a gene ic con in- uous- ime ully balanced - il e wi h in eg a ion nodes. We assume ha each in eg a ion node can be connec ed o he inpu and o all he emaining in eg a ion nodes; we u he as- sume ha hese connec ions can be ealized wi h ei he capac- i o s o ansconduc o s. I is illus a ed in Fig. 1 o a gene ic in eg a ion node . Capaci i e connec ions o o he in eg a ion nodes a e ealized wi h capaci- o s ; ansconduc ance connec ions o o he in eg a ion nodes a e ealized wi h ansconduc o s ; connec ions o he inpu a e ealized wi h capaci o s and ansconduc o s , espec i ely. This gene al diag am con empla es also he gene al case whe e he il e ou pu is ob- ained as he linea combina ion o he inpu and an a bi a y numbe o in e nal node ol ages.Using an ex ended s a e-space no a ion, he il e is desc ibed as [3] (1) which is ac ually a gene aliza ion o he ep esen a ion gi en in [1]. In he exp ession abo e: •and a e, espec i ely, he inpu and ou pu ol ages o he il e and is a s a e ec o which ga he s all he in eg a ion node ol ages. Ope a o deno es ma ix anspose. •is an ma ix whose elemen ep esen s he ansconduc ance o he ansconduc o connec ed om node o node . •is an ec o gi en by (2) whe e and ep esen , espec i ely, he ansconduc- ance and capaci ance be ween he il e inpu node and he in eg a ion node ; is o med by ansconduc ances and is composed by capaci ances. •in (1) is a ec o whose elemen deno es he ol age ampli ica ion be ween node and he ou pu ; e- alized by a ansconduc o wi h inpu and ansconduc- ance , loaded wi h a esis o o esis ance implemen ed by a eedback ansconduc o [see Fig. 1(a)]. Pa ame e is he ol age gain o he o wa d pa h om he inpu o he ou pu o he il e ; ealized by a ansconduc o wi h gain loaded by he same eedback anscon- duc o as be o e.2 • Finally, is an ma ix composed by capaci ances wi h he ollowing s uc u e: (3) whe e he nega i e and posi i e componen s o he in e- g a ion nodes a e chosen such ha all he ou -o -diagonal en ies a e nega i e. In con en ional - s uc u es, he an i-diagonal en ies o , namely and , coincide as hey ep esen he same ca- paci o . Howe e , he e a e p ac ical opologies which lead o non-symme ical ma ices. This happens, o ins ance, in he so-called - opamp s uc u es whe e he in eg a ing capac- i o is connec ed in eedback con igu a ion a ound one opamp [11]. F om he ep esen a ion in (1) he inpu –ou pu ans e unc- ion o he il e can be calcula ed as (4) In addi ion o his o e all ans e unc ion, o he unc ions need o be de ined. On he one hand, le be he ans e unc ion om o he in eg a ion node . I can be shown ha (5) whe e (6) 2Fo he sake o gene ali y he ma hema ical o mula ion in he pape con- side s he gene al ep esen a ion abo e. In many p ac ical si ua ions, he e is no o wa d pa h om he inpu so ha d =0 . Also, in many cases he ou pu o he il e is aken om a single in e nal node so ha only one en y o ec o C is non-ze o (i.e., C =[0 ; ... ;c ; ... ; 0] ). Mo eo e , i no ou pu ol age ampli ica ion is equi ed, hen ec o C is uni a y ( he only non-ze o en y has uni y alue) and he ou pu summing ne wo k can be supp essed. FERNÁNDEZ-BOOTELLO e al.: MATRIX METHODS FOR THE DYNAMIC RANGE OPTIMIZATION OF CONTINUOUS-TIME -FILTERS 2527 On he o he hand, le be he ans e unc ion om a cu en sou ce connec ed o in eg a ion node (be ween he ou pu e minals o ansconduc o s wi h gain o ) o he il e ou pu o . I all he ans e unc ions , a e collec ed in o a ec o (7) i is easy o show ha (8) As will be shown a e wa ds, ec o s and signi ican ly sim- pli y he e alua ion o il e dis o ion. Also, hey a e use ul o calcula e he ela i e sensi i i ies o he il e ans e unc ion wi h espec o he ma ices in he ep esen a ion (1). They can be easily ob ained using he Hadama d p oduc as3 (9) whe e he sensi i i ies a e de ined as . Up o now, i has been assumed ha ansconduc ances emain cons an wi h equency. In a mo e gene al case, he ep esen a ion in (1) mus be ex ended o also cope wi h e- quency-dependen ansconduc ances [7]. This can be done a he p ice o a mo e complex s a e-space ma ix desc ip ion. As an example, le us assume ha ansconduc ance exhibi s a one-pole oll-o wi h ime cons an . This can be modelled by eplacing ansconduc o by he ne wo k a he igh o Fig. 2(a), whe e and he low- equency ansconduc ance is gi en by . No e ha his model adds a new node o he opology (la- belled ) pe ansconduc o . Hence, ma ices o he ex ended s a e-space ep esen a ion mus be ebuil . This is indica ed in Fig. 2(b) which shows he ows and columns ha mus be added o accoun o he new il e node, . Fo consis ency, he low equency beha io mus be made equal o he o iginal e- quency-independen en y in (1). III. NOISE IN - FILTERS We assume noise is only con ibu ed by ansconduc o s and ha equency is high enough so ha noise beha io is dom- ina ed by he mal con ibu ions. The mal noise con ibu ions om di e en ansconduc o s a e assumed un-co ela ed ( his is suppo ed by he ac ha ansconduc o s a e di e en phys- ical en i ies), and a e modelled by ou pu cu en noise sou ces wi h double-sided powe spec al densi y (PSD) whe e is he ansconduc ance, is Bol zmann’s cons an , 3The Hadama d p oduc o wo m 2 n ma ices A and B , deno ed by A  B , is an m 2 n ma ix gi en by ( A  B ) = a b [4]. Fig. 2. (a) Ci cui eplacemen o adding one-pole oll-o ansconduc ance cha ac e is ics o he gene ic schema ic o Fig. 1(a). (b) Requi ed modi ica ions on he s a e-space ma ices. is he absolu e empe a u e and is he noise excess ac o o he ansconduc o — he alue o his la e pa ame e depends on he ac ual ansconduc o implemen a ion [12]. Using he ans e unc ions de ined in he p e ious sec ion [see (7) and (8)], he o al ou pu - e e ed noise ol age PSD o he il e can be exp essed as (10) He e, he i s e m accoun s o he noise con ibu ions o all ansconduc o s in he il e co e. The second e m co esponds o he ou pu summing s uc u e. F om now on we will assume ha his second e m is ei he negligible (which occu s o la ge enough alues o as i happens in p ac ice), o null (which co esponds o he case whe e he il e ou pu is simply aken om a single in e nal node). Wi h his assump ion, he o al ou pu noise o he il e is app oxima ely gi en by (11) which ep esen s an uppe -limi alue. In o de o e alua e he in eg al in (11) i is wo h no ing ha ma ix , de ined as [1], [13]4 w(12) can be algeb aically ob ained om he ollowing gene alized Lyapuno equa ion [14]: (13) 4 W is ela ed o he obse abili y g ammian o he sys em, W ,as W = EWE [15]. 2528 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 55, NO. 9, OCTOBER 2008 Fig. 3. Concep ual schema ic a he i h in eg a ion node o he il e including nonlinea i ies. and, he e o e, he o al noise o he il e can be w i en as (14) whe e w, ep esen he elemen s a he diag- onal o ma ix is he sum o all he ansconduc ances d i ing node . In he igh -hand side o his equa ion, ep esen s he noise con ibu ed o he ou pu om he h in eg a ion node. IV. DISTORTION IN - FILTERS We ocus he e on he impac o nonlinea i ies on he inpu –ou pu ol age- o-cu en ans o ma ion. Assuming weakly nonlinea ope a ion condi ions and ha ansconduc o s a e ully balanced; hei inpu –ou pu cha ac e is ic can be app oxima ed by (15) whe e is he hi d-o de nonlinea i y coe icien and is he inpu ol age. This simpli ied model, whe e e en-o de non- linea coe icien s a e null because o he balanced s uc u e, su ices o mos p ac ical ansconduc o s [11], [12]. Le us i s conside ha he ou pu summing s uc u e o Fig. 1(a) does no gene a e dis o ion; he in luence o his s uc- u e will be compu ed in la e on. Using (15), he ex ended s a e-space ep esen a ion in (1) becomes (16) whe e is he Hadama d cube o x(i is ob ained by h ee consecu i e Hadama d p oduc s—see oo - no e 3). Fo illus a ion pu poses, Fig. 3 shows a concep ual schema ic (single-ended o simplici y o he d awing) which displays he componen s which con ibu e o he h in eg a ion node acco ding o (16). In his igu e, he linea and nonlinea componen s a e clea ly sepa a ed. Se e al app oaches a e ound in li e a u e o he analysis o he nonlinea beha io desc ibed by (16), [4]–[9]. He e we use Vol e a’s se ies expansions [10]. This me hod consis s in de- composing he in e nal nodes a iables in ope a o s, acco ding o [16] (17) whe e is he inpu signal o he sys em, is an a bi a y ampli ude scaling ac o (18) is e e ed o as he h o de Vol e a ope a o and is he h Vol e a ke nel [10]. The Laplace ans o m o his mul idi- mensional ke nel is de ined as (19) whe e is he -dimensional Laplace a iable. Func ion desc ibes in equency-domain he h o de dis o ion pe o mance o he sys em. Hence, desc ibes he linea beha io o he sys em, accoun s o he hi d-o de nonlinea beha io , and so on. The ele an ea u e o Vol e a’s se ies expansions app oach is ha , o weakly nonlinea sys ems and low alues o , se ies (17) apidly con e ges and i can be app oxima ed by he i s ew e ms. The e o e, i he dis o ion beha io o a ully bal- anced - il e is domina ed by he hi d-o de nonlinea i- ies o he ansconduc o s, can be simply app oxima ed by . Replacing his exp ession in (16) and g ouping e ms wi h he same powe o , he ol- lowing wo linea sys ems in and a e ob ained (20) and (21) whe e . The i s equa ions o hese wo sys ems can be mapped in o he i s - and hi d-o de ci cui s shown in Fig. 4(a) and (b), espec i ely. Sol ing bo h ci cui s, he i s - and hi d-o de ans o med ke nels o he il e a e espec i ely gi en by (22) and (23) F om hese exp essions, he hi d-o de ha monic dis o ion o he il e and i s in e modula ion pe o mance can be es i- ma ed, wi h no ansien analysis needed, by [16] (24) whe e is he ampli ude o he inpu ones applied o he sys em. FERNÁNDEZ-BOOTELLO e al.: MATRIX METHODS FOR THE DYNAMIC RANGE OPTIMIZATION OF CONTINUOUS-TIME -FILTERS 2529 Fig. 4. (a) Fi s -o de and (b) hi d-o de ci cui s a node i . Equa ion (24) can be ela ed o ec o s and , de ined in (6) and (7), espec i ely, as shown in (25) a he bo om o he page. I e eals ha he dis o ion e alua ion o a - il e can be easily accomplished by simple ma ix algeb a. These closed- o m exp essions compa e a ou ably o o he s in he li - e a u e, demanding a much mo e cumbe some o mula ion [4], [5]. Le us now conside he case whe e he ou pu summing s uc- u e o Fig. 1(a) also con ibu es dis o ion. In his case, he con- cep ual schema ic o he il e ou pu akes he o m in Fig. 5, simila o ha in Fig. 3 o he il e co e. As abo e, he analysis o his ci cui encompasses decomposi ion in o a i s - and a hi d-o de schema ics (shown, espec i ely, in Fig. 6(a) and (b)) which a e sol ed one a e he o he o gi e (26) Fig. 5. Concep ual schema ic o he ou pu s uc u e including nonlinea i ies. Fig. 6. Fi s -o de (a) and (b) hi d-o de (b) ci cui s o e alua e he dis o ion o he ou pu s age. whe e and a e ob ained om (22) and (23). In hese ep esen s he linea pa o he il e esponse, and he hi d-o de nonlinea con ibu ion. A e some algeb a, he hi d-o de ha monic dis o ion o he il e and i s in e modula- ion pe o mance ake he o m in (27), shown a he bo om o he page, whe e e ms and , de ined in (25), a e due o he co e o he il e . As an illus a ion o he p oposed dis o ion e alua ion me hod, Fig. 7 shows in solid lines he calcula ed hi d-o de in e modula ion and ha monic dis o ion componen s o a ully balanced se en h-o de low-pass Chebyshe il e wi h 10 MHz cu -o equency. The il e uses a s anda d leap- og s uc u e (25) (27) 2530 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 55, NO. 9, OCTOBER 2008 Fig. 7. Es ima ions o IM (2 ! 0 ! ) and HD ( ! ) . using ansien analysis and he p oposed me hod. and i is assumed ha all ansconduc o s exhibi a hi d-o de nonlinea e m . Calcula ions a e made assuming ha he il e is d i en by inpu ones o 125 mV a di e en equencies. The hi d-o de in e modula ion es ima ion assumes a 100 kHz o se equency be ween he inpu ones. Fo compa ison pu poses, Fig. 7 in- cludes also en ies o he hi d-o de in e modula ion (as e - isks) and ha monic dis o ion (ci cles) calcula ed a di e en inpu equencies by means o con en ional Fou ie analysis o a ansien esponse. As can be seen, bo h me hods gi e app ox- ima ely he same esul s, howe e , whe eas he compu a ional cos o he ansien app oach is 135 s cpu ime, ha o he p oposed me hod is o only 0.6 s (da a using a 1.8 GHz Pen- ium Mobile p ocesso ), i.e., mo e han wo o de s o magni ude lowe . V. SCALING OF - FILTERS This sec ion p esen s analy ic exp essions o accoun o he impac o scaling on il e noise and dis o ion. This is wo h doing because scaling is cus oma ily employed o il e de- sign [6], [12], [17], [18]. We conside only scaling ope a ions which e ain he equency esponse, , equi ed by he ap- plica ion. This excludes equency scaling ope a ions which a e easily implemen ed by mul iplying all il e capaci ances by he same ac o [ his makes and, he ea e , ] o by scaling all il e anscon- duc ances by [ his makes and, he ea e , ]. In he o egoing analysis, il e s a e g ouped in o wo ca e- go ies. On he one hand, il e s wi h all capaci o s g ounded ex- cep ing, pe haps, hose connec ing he in eg a ion nodes wi h he il e inpu ( hey a e cha ac e ized by a diagonal ma ix ). On he o he hand, il e s which include loa ing capaci o s be- ween in eg a ion nodes. Unless o he wise s a ed, i deno es a gi en a iable in he p o o ype sys em, ep esen s he co esponding ans o med a iable. A. Fil e s Wi hou Floa ing Capaci o s Table I summa izes esul s o he h ee ypes o scaling con- side ed he ein. The second column shows he basic and de i ed ma ix equa ions o each ype o scaling. The hi d column in- cludes commen s ega ding he e ec s o co esponding ans- o ma ion on he noise and dis o ion pe o mance o he il e . A i s scaling app oach consis s o mul iplying each ow o he s a e equa ion in (1) by a co esponding posi i e numbe , . This ans o ma ion, deno ed as noise scaling in Table I, mod- i ies he local impedance a each node o he il e wi hou al- e ing hei ol age swings. Hence, i does no a ec he dis o - ion beha io o he il e -in e es ing p ope y ha will be ex- ploi ed la e on. The noise con ibu ed o he ou pu om he h in eg a ion node in he scaled il e becomes , whe eas he sum o all ansconduc ances d i ing he h node o he il e scales as . I means ha o educe he noise con ibu ion a node by a ac o , he o al ansconduc ance mus be inc eased by he same ac o . This inc eases he a ea occupa ion o he il e as well, because capaci ances a e also scaled by . Fo a gi en alue o he o al ansconduc ance, , he e is an op imum se o scaling co- e icien s , which minimizes he o al noise con ibu ed by he il e . A e some calcula ions (de ailed in Appendix I) i is ound ha such op imum se is ob ained when all he diagonal elemen s o he ans o med ma ix a e iden- ical, i.e., w w w (28) which gi es w w (29) In his case, he o al noise o he il e can be exp essed as (30) A special case o noise scaling is powe scaling in which all he mul iplying ac o s ake on he same alue and, hence, all capaci o s and ansconduc o s o he il e co e a e scaled by . In his case, he o al ou pu noise alue o he il e is ans o med acco ding o , wi hou a ec ing he dis o ion beha io . This ac will be used in Sec ion VI o ela e he o al noise o he il e wi h i s powe consump ion. Conside now ha scaling is made by mul iplying column en ies ins ead o ow en ies. This ans o ma ion is labelled dis o ion scaling in Table I whe e scaling ac o s a e called . I a ec s he ampli ude le el o he ol ages a he in e nal nodes o he il e and, he e o e, modi ies i s dis- o ion beha io . Ac ually, dis o ion imp o es o scaling ac- o s . Howe e , his ope a ion a ec s he noise beha io and de ines a ade-o be ween noise and dis o ion. I is wo h no ing ha ma ix emains unal e ed a e dis o ion scaling and so w w . This ac will be exploi ed in he nex sec ion. FERNÁNDEZ-BOOTELLO e al.: MATRIX METHODS FOR THE DYNAMIC RANGE OPTIMIZATION OF CONTINUOUS-TIME -FILTERS 2531 TABLE I BASIC SCALING TRANSFORMATIONS A las op ion o il e scaling consis s o applying a sim- ila i y ans o ma ion on he s a e-space ep esen a ion in (1). A ypical case, illus a ed in Table I, co esponds o a diagonal ans o ming ma ix wi h coe icien s .Asin he p e ious case, his mapping ca ies on bo h noise and dis o ion modi ica ions on he p o o ype il e , and a ade-o can be es ablished as well: scaling ac o s dec ease he noise con ibu ion bu wo sen dis o ion. This ype o ans o ma ion is ypically used o equalize and maximize he peak ol ages a he inpu o he ansconduc o s in he il e — his is done by means o a diagonal ma ix wi h coe icien s , whe e is he maximum inpu signal ange o ansconduc o s and a e he peak alues o each de ined in (5) ( hey usually occu nea he passband o he il e ) [17]. B. Fil e s Wi h Floa ing Capaci o s The p esence o loa ing capaci o s be ween he in eg a ion nodes makes he ma ix non-diagonal. As a consequence, p e ious scaling ope a ions canno be applied in s and-alone manne . O he wise he symme y o [see (3)] would be los , he eedback and o wa d pa hs o he loa ing connec ions would be di e en and, consequen ly, hey would be un ealiz- able wi h simple capaci o s. To o e come his si ua ion, scaling ope a ions should be pai -wise applied so ha symme y o is always es o ed a e scaling.5 As an example, le us assume ha a p o o ype il e wi h loa ing capaci o s is scaled by a diagonal simila i y ans o - ma ion . The non-diagonal componen s o ma ix change as , hus, b eaking he symme y o . In o de o es o e his p ope y and, hence, allow he use o loa ing capaci o s as in he p o o ype il e , one possi- bili y is o noise scale he il e (see Table I), in such a way ha he ela ionship is me o , wha gua an ees ha . A simila p ocedu e can be en i- 5Rigo ously speaking, scaling could be pe o med in a single s ep by means o a simila i y ans o ma ion wi h an o hogonal T ma ix ( T = T ) . Howe e , om a syn hesis pe spec i e, i is mo e simple and in ui i e o use wo consecu i e scaling ope a ions. 2532 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 55, NO. 9, OCTOBER 2008 sioned i , ins ead o noise scaling, a dis o ion scaling is applied o es o ing symme y. VI. DYNAMIC RANGE OPTIMIZATION OF GENERAL - FILTERS We ha e seen ha o a gi en o al ansconduc ance he noise o a - il e can be minimized by making all diagonal e ms o ma ix iden ical. We ha e also seen ha , by ap- plying dis o ion scaling, he ha monic dis o ion and in e mod- ula ion pe o mance can be imp o ed while keeping he e ms unal e ed. Hence, an op imum noise-scaled il e will e- ain his ea u e a e dis o ion scaling. This obse a ion is a he co e o a ecen ly p oposed p ocedu e o dynamic ange op imiza ion o weakly nonlinea - il e s unde powe dissipa ion cons ain s [6]. I consis s o he consecu i e appli- ca ion o noise and dis o ion op imiza ions on he p o o ype il e . Howe e , his p ocedu e p esen s wo limi a ions. Fi s , he noise and dis o ion con ibu ions, as well as powe con- sump ion, o ansconduc o s no accoun ed in ma ix a e ne- glec ed. Indeed, ansconduc o s assessed in ma ix a e e- placed by ideal cu en sou ces. This simpli ica ion p ecludes ob aining uni ocal solu ions om he op imiza ion p ocedu e, as will be illus a ed in he nex sec ion by means o an example. Second, dis o ion op imiza ion is cons ained by he condi ion ha he noise gene a ed a he il e co e keeps unal e ed. This condi ion educes he design space o il e pa ame e s and, con- sequen ly, may p eclude ha a global op imum is eached. These d awbacks a e o e come by he p ocedu e p esen ed in his sec ion. Wi hou loss o gene ali y and o keep ma he- ma ics as simple as possible, le us assume ha all ansconduc- o s in he il e sha e he same opology and linea ange, and exhibi he same cu en e iciency, ; whe e his la e pa am- e e is de ined as he a io be ween he ansconduc ance and he biasing cu en o he cell [18], [19]. In his case, he o al powe consump ion o he il e , , is p opo ional o he sum o all i s ansconduc ances, ,as , whe e is he powe supply ol age o he il e [18]. A p ac- ical way o ul illing his assump ion is by implemen ing all ansconduc ance alues h ough he pa allel connec ion o uni- a y ansconduc o s.6 Le us assume ha a gene ic ully balanced - il e wi h no ou pu ne wo k (see Fig. 1(a)) has been, i s , op imally noise scaled and, hen, dis o ion and powe scaled. In his case, aking in o accoun Table I and applying a powe scaling ac o , he o al noise o he il e o a o al powe consump ion becomes (31) 6This is, indeed, a common p ac ice among in eg a ed ci cui designe s. By using mul iple ins ances o a gi en ansconduc o , he design complexi y is no ably educed and he obus ness o he il e agains a ia ions o he ech- nological p ocess is imp o ed. Using his s a egy in combina ion wi h p ope layou echniques, ma ching be ween ansconduc o s is la gely a ou ed and he unabili y o he il e , simpli ied [20], [21]. whe e and a e he coe icien s o he ma ices and a e op imum noise scaling, and is he sum o all he ansconduc ances o he il e a e dis o ion scaling. Use ul o he o egoing analysis, can be also exp essed as (32) whe e ( espec i ely, ) is he sum o he ansconduc- ances o all he ansconduc o s wi h inpu a he h node ( e- spec i ely, il e inpu ) o he op imum noise-scaled il e .7 Le us u he assume, wi hou loss o gene ali y, ha he dis- o ion pe o mance o he il e is e alua ed by he hi d-o de in e modula ion. F om Table I and assuming ha inpu ones a e close oge he , a can be exp essed as (33) whe e , see de ini ion in (7), is ob ained a e op imum noise scaling. Equa ion (33) can be also w i en as (34) whe e coe icien s and , bo h independen o , a e de- ined as (35) The maximum powe a he ou pu o he il e , , o a peak in e modula ion dis o ion alue, , can be ob ained om (34) as (36) om whe e he dynamic ange o he il e (see oo no e 1) can be calcula ed, using (31), as (37) This exp ession can be ecas as (38) whe e is an adimensional numbe , which depends on he pa icula ansconduc o implemen a ion used in he il e , and depends on he il e s uc u e. Pa ame e gi es a measu e on how la ge he dynamic ange o a il e can be o a gi en powe dissipa ion 7In his sec ion and he ollowing, i x deno es a a iable o coe icien o he o iginal p o o ype, ~ x e e s o he co esponding a iable o coe icien o he op imally noise scaled il e . FERNÁNDEZ-BOOTELLO e al.: MATRIX METHODS FOR THE DYNAMIC RANGE OPTIMIZATION OF CONTINUOUS-TIME -FILTERS 2533 and dis o ion pe o mance. Hence, i can be used as a igu e o me i o compa ing ansconduc o implemen a ions. Equa ion (38) also shows ha o a gi en powe consump- ion , maximum ole a ed dis o ion and selec ed ansconduc o opology , he op imiza ion o he il e dynamic ange implies maximizing .As e- mains unal e ed a e scaling, he only scaling-dependen ac o in is , which we de ine as a cos unc ion in he dynamic ange op imiza ion p ocedu e (39) Clea ly, o op imize he dynamic ange o a il e , mus be minimized. The minimum o is calcula ed by se ing , which a e some algeb a ob ains he se o equa- ions (40) whe e is he conjuga e o and ex ac s he eal pa o he a gumen . Summing he le -hand e ms o he abo e equa ions, on he one hand, and he igh -hand e ms, on he o he , he ollowing wo ela ionships a e ob ained: (41) om whe e, equa ing he esul s and using (40), he op imum dis o ion-scaling coe icien s can be calcula ed by ecu - si ely sol ing he exp ession (42) using he de ini ion o in (34). Wi h hese alues, he ab- solu e minimum o becomes (43) No e ha i inpu ansconduc o s a e no accoun ed o in he abo e analysis, which is he si ua ion conside ed in [6], coe - icien s and a e null and (43) becomes inde e mina e.8 8By emo ing he e ec s o inpu ansconduc o s on dis o ion and noise, hey a e assumed o pe o m as pe ec ol age- o-cu en con e e s. In essence, hese ideal con e e s play he same ole as he inpu cu en -con olled cu en sou ces used in [6], i.e., o injec signal in o a cu en -inpu il e . Assuming ha B has no capaci i e componen s, i.e., B is a null ec o , he d i ing signal o he cu en -inpu il e akes in ou model he o m B . In he case o [6], such d i ing signal is simply gi en by Bi . Ob iously, he dimensionali y o ma ix B in bo h p ocedu es is di e en bu , om he poin o iew o op imizing he dynamic ange o he cu en -inpu il e , de ined by ma ix A and E , his is i ele an . Fig. 8. Fully di e en ial G - C biquad a ic sec ion. This e eals ha he e is an in ini e numbe o solu ions ha achie e he same dynamic ange o a gi en powe dissipa ion, and no a single solu ion as s a ed in [6]. I is o be unde s ood ha by including he e ec s o inpu ansconduc o s on dis o - ion and noise, he op imiza ion algo i hm inds he necessa y cons ain o achie e a single and uni ocal solu ion. I is also wo h men ioning ha he dynamic ange op imiza- ion may esul in a non-uni y dis o ion-scaling coe icien o he ou pu node o he il e , i.e., i he ou pu is aken om node , coe icien will mo e likely be . This implies ha ec o becomes non-uni a y, howe e , he e is no need o add an ampli ica ion ou pu s age o he il e (see Fig. 1) o gua - an ee an op imum dynamic ange. Ins ead, he ou pu o he il e could be di ec ly aken om node . No e ha any po en ial ou pu ne wo k will ideally scale he ou pu noise and desi ed signal powe by he same ac o while e aining he dis o ion pe o mance o he il e co e. The e o e, he dynamic ange will emain unal e ed a e ampli ica ion. Finally, no e ha he op imiza ion p ocess desc ibed abo e is es ic ed o a single ope a ing equency. The e o e, such e- quency mus be ca e ully chosen so ha i co esponds o he wo s -case dynamic ange o he il e . This can be done by a p e ious analysis on he noise and dis o ion dependence wi h equency h ough (10) and (25). VII. CASE STUDY:BIQUADRATIC SECTIONS As a case o s udy, he p ocedu e in he p e ious sec ion is he ein applied o he biquad a ic sec ion o Fig. 8. In he ollowing analysis, ansconduc o -dependen pa ame e s ha e been de i ed om a simple olded-cascode opology, designed in a 0.13 m CMOS echnology a a powe supply o 3.3 V. They a e and gi ing . Using hese ansconduc o s, a low-pass il e wi h cu -o equency a MHz has been designed. The powe consump ion o his il e is mW. Using he ex ended s a e-space ep esen a ion o Sec ion II, he biquad in Fig. 8 can be desc ibed by he ma ices (44)