Rapid Intermodulation Distortion Estimation in Fully Balanced Weakly Nonlinear Gm-C Filters using State-Space Modeling

Transcription

1 Rapid Itermodulatio Distortio Estimatio i Fully Balaced Weakly Noliear Gm-C Filters usig State-Space Modelig Paul Sotiriadis Electrical ad Computer Egieerig Johs Hopkis Uiversity pps@jhuedu Abdullah Celik Electrical ad Computer Egieerig Johs Hopkis Uiversity abdullahcelik@gmailcom Zhaoia Zhag Electrical ad Computer Egieerig Johs Hopkis Uiversity zz@jhuedu ABSTRACT State-space modelig of fully differetial Gm-C filters with weak oliearities is used to develop a fast algorithm for itermodulatio distortio estimatio It results i simple aalytic formulas that apply directly to Gm-C filters of ay order ad ay fully balaced topology The algorithm has bee verified usig SpectreS (SPICE) ad Simulik simulatio Theory ad simulatio results are foud i good agreemet Categories ad Subject Descriptors B7 [Itegrated Circuits ]: Geeral Terms Algorithms, Desig, Theory, Performace Keywords G m-c filters, harmoic distortio, weak oliearity, circuit aalysis, fully balaced, fully differetial, distortio model, perturbatio, state space, fast algorithm 1 INTRODUCTION Over the past few decades, cotiuous-time G m C filters have become oe of the most popular classes of active filters used i a vast variety of applicatios [1] Sigificat efforts have also bee devoted to aalyzig ad optimizig all aspects of their performace Some of the early G m C filters were implemeted i bipolar techology [2], [3] However, the advaces i CMOS semicoductor techologies provided a ideal groud for them [4] Amog the may types of circuit topologies for G m C filters, the fully differetial oes are almost always preferred due to their sigificatly higher liearity Permissio to make digital or hard copies of all or part of this work for persoal or classroom use is grated without fee provided that copies are ot made or distributed for profit or commercial advatage ad that copies bear this otice ad the full citatio o the first page To copy otherwise, to republish, to post o servers or to redistribute to lists, requires prior specific permissio ad/or a fee GLSVLSI 6, April 3 May 2, 26, Philadelphia, PA, USA Copyright 26 ACM /6/4 $5 Although the methodology preseted i this paper ca be applied to a large variety of liear circuits, the focus is o distortio estimatio i bad-pass G m C filters Bad-pass filters are importat i may applicatios, icludig wireless commuicatio systems, ad their oliearity (alog with oise) determies the performace of the whole system By their ature, (arrow) bad-pass filters suffer maily from itermodulatio distortio (IMD), ad i most practical cases by third order IMD, (IM 3) The stadard test sigal for itermodulatio distortio estimatio is the two-beat sigal u = k 1 si(2πf 1)+k 2 si(2πf 2) I bad-pass filters f 1 ad f 2 are chose to be close to each other ad withi the pass-bad The domiat itermodulatio products that usually appear withi the pass-bad are at frequecies (m + 1)f 1 mf 2 ad (m + 1)f 2 mf 1, m = 1, 2, Also, i weakly oliear filters the magitude of the IMD products drops rapidly with their order So, i practice, IMD aalysis i bad-pass filters focuses o the 3 rd order IMD compoet (IM 3) at frequecy 2f 1 f 2 [5], [6] The most popular measure of IM 3 is the ratio of the amplitude of the parasitic sigal at frequecy 2f 1 f 2 over the amplitude of the sigal at f 1, ie IM 3 is defied relatively to the beat at f 1, [5], [7] Other measures of IM 3 are the 3 rd order itercept poit (IP 3), the 1-dB compressio poit (P 1dB ), ad the Spurious-Free Dyamic Rage(SFDR), eg [5], [8], [9] ad [1] Itermodulatio distortio estimatio ca be doe at the trasistor or the filter level Basic trasistor amplifyig uits ca be treated as iput-output static (memoryless) fuctios [6], [7], [11], [12], while filters are dyamical systems (they have memory) which makes their IMD estimatio a more complicated problem Volterra series has bee the most popular tool to address it eg [13], [14], [15] However, derivig aalytical results is practically limited to low order filters, eg [16], [17] I most cases the Volterra series method leads to complicated algebraic expressios A alterative approach was itroduced i [18] where the total IMD product of the filter was approximated by the sum of the IMD products of the idividual trascoductors liearly propagated to the output of the filter through the correspodig partial trasfer fuctios I cotrast to existig techiques this paper itroduces a method for very fast IMD estimatio that is based o state-space modelig ad mathematical treatmet of the filter [19]-[2] The proposed method leads to aalytic ex-

2 pressios that explicitly deped o the structural matrices of the filter ad the compoets values providig a simple ad geeral tool for IMD estimatio The derived formulas are valid for G m C filters of ay order ad ay fully differetial topology To evaluate the developed algorithm, a Tow-Thomas G m C biquad was desiged i a 5µm stadard CMOS process usig Cadece ad was simulated i SpectreS (SPICE) ad Simulik 2 FULLY DIFFERENTIAL WEAKLY NON- LINEAR TRANSCONDUCTORS Fully differetial trascoductors are preferred whe low distortio is required i liear circuits such as G m C filters, [1], [3], [1], [4], [21] Because of their balaced structure, they exhibit maily odd order oliearity Moreover, i most practical cases the third order oliear term is domiat ad higher order terms ca be safely igored i I x j,i i g j,i Figure 1: Trascoductor model Let I j,i be tracoductor s output curret flowig ito ode j whe its iput is coected to ode i as show i Figure 1 (sigle-eded otatio is used for otatioal coveiece) Followig the discussio above, I j,i = g j,ix i+e j,ix 3 i Moreover, i may practical cases, e j,i, is proportioal to g j,i implyig that I j,i = g j,i x i + α g j,i x 3 i (1) These assumptios are typical i estimatig the distortio of filters with fully differetial trascoductors [14], [18] ad are adopted here The trascoductace g j,i ad the (small) costat α, which has uits of Volt 2, relate directly to the trasistorlevel desig of the trascoductor They ca be derived aalytically or, umerically by fittig a third order polyomial to the I V characteristic of the trascoductor 3 STATE-SPACE MODEL OF G M C FIL- TERS WITH WEAKLY NONLINEAR FULLY DIFFERENTIAL TRANSCONDUCTORS The state-space model is itroduced usig the followig example Cosider the secod order G m C filter i Figure 2 u(t) g 1,I x 1 x 2 g 1,1 g 2,1 g 1,2 C 1 C2 j g O,1 I out (t) ẋ1 ẋ2 = g 1,1 g 1,2 g 1,1 g 1,2 C 1 C g 1 x1 C 2,1 + α 1 C 1 x 3 1 g C x 2,1 2 2 C x g1,i C 1 u + α g1,i C 1 u 3 ẋ = A x + αa x 3 + bu + αbu 3 ad the output algebraic equatio (y = I out) y = g o,1 x1 + α g x o,1 x x 3 2 y = c T x + αc T x 3 where Hadamard s product a b = (a 1b 1, a 2b 2,, a k b k ) T ad power a ρ = (a 1 ρ, a 2 ρ,, a k ρ ) T were used Note that i may G m C filters the output is the voltage across a capacitor If x 2, for example, was the output sigal i Figure 2 the there would be o output trascoductor stage ad the output equatio would be y = x 2, which is liear The state-space model for the geeral th order G m C filter with weak 3 rd order oliearity is ẋ(t) = Ax(t) + αax 3 (t) + bu(t) + αbu 3 (t) (2) y(t) = c T x(t) + αc T x 3 (3) where A R, x,b,c R 1 ad u(t), y(t) R Agai, stads for Hadamard product or power Fially, the iput u cosidered i this work is the stadard two-beat sigal used i IMD estimatio u(t) = k 1si(w 1t) + k 2si (w 2t) (4) 4 FILTER S CASCADE STRUCTURAL DE- COMPOSITION S 2 S 3 u 3 w x x b()+αb() c()+αc T 3 () 3 A()+αA() Figure 3: Block diagram of the weakly oliear G m C filter The state-space represetatio of the weakly oliear G m C filter, Eqs (2) ad (3), is show i the block diagram of Figure 3 The filter is a cascade of three stages; the iput stage, represeted by the sigal operator, the filter core stage - operator S 2 ad the output stage - operator S 3 It is w = (u), x = S 2(w) ad y = S 3(x) y Figure 2: Secod order badpass G m C filter Usig equatio (1) the state-space formulatio of the filter is give by the followig system of differetial equatios The total respose of the system is y = (S 3 S 2 ) (u) Two clarificatios are i order regardig, S 2 ad S 3 1) Sice we are iterested i IMD estimatio ad we use the two-beats iput sigal (4), oly the steady state behavior of the filter (ad therefore of each of the stages) is take ito accout To this ed, w, x ad y are the steady state

3 resposes of the stages 1, ad operators, S 2 ad S 3 are the correspodig mappigs betwee them (ad iput u) 2) The iput ad output stages, ie ad S 3 are static fuctios Distortio due to static oliearities has bee studied extesively The filter core however, S 2, has dyamics makig the IMD estimatio challegig To deal with operator S 2 regular perturbatio theory was employed [22] ad the mathematical details ca be foud i [2] Each of the three stages 2 is aturally decomposed ito two parts, the liear oe (which is the ideal ad desirable oe) ad the oliear oe (which represets the oliearities i the stage) We write: S i = S l i + S i, i = 1, 2, 3 (5) The decompositio of the iput,, ad output, S 3, stages is implied directly from Figure 3 S l 1(u) b u S 1 (u) αbu 3 S l 3(x) c T x S 3 (x) αc T x 3 (6) The liear part, S l 2, of S 2 is the steady state respose of the (asymptotically stable) liear system ẋ = Ax + w, ie S l 2(w) = x (steady state) Operator S 2 is defied by S 2 S 2 S l 2 ie S 2 (w) = x d = x x is the differece betwee the steady state resposes of the oliear ad the liear systems show i Figure 4 w x x o A()+αA()3 A() x x o + - x d S 2 (w)=x d Figure 4: Defiitio of operator S 2 The decompositio of the stages, Eqs (5), trasforms system s block diagram i Figure 3 ito that i Figure 5 The cascade of parallel pairs i Figure 5 is equivalet to eight S l 3 S 2 S 1 3 S l 2 S 1 3 S 2 S l 1 (u) + S 3 S 2 S 1 (u) Sice we cosider weakly oliear filters, it is expected that operators S 1, S 2 ad S 3 have a mior ifluece o the sigal compared to that of the correspodig liear oes Moreover, compositios of two or more oliear operators S 1, S 2 ad S 3 i a sigal path should result i egligible sigal compoets Therefore, keepig oly the sigals paths that are liear or iclude oly oe oliear operator results i a good approximatio of filter s behavior, ie y S3 l S2 l S1 l + S3 l S2 l S1 + S3 l S2 S1 l + S3 S2 l S1 l (u) The derivatio of (S l 3 S l 2 S l 1)(u), (S l 3 S l 2 S 1 )(u), (S l 3 S 2 S l 1)(u) ad (S 3 S l 2 S l 1)(u) leads to the rapid IMD estimatio method for G m C filters of ay order ad ay topology that is summarized i the ext sectio The derivatio of the term (S l 3 S 2 S l 1)(u) is challegig sice operator S 2 is realized by the oliear dyamical system of Figure 4 To this ed regular perturbatio techiques were used More details of this methodology ca be foud i [2] 5 STEPS TO DERIVE IM 3 The results of this work are summarized i the followig procedure for estimatig the third order itermodulatio distortio referred to iput sigal at w 1, formally defied i (7) where A 2w2 w 1 ad A w1 are the amplitudes of the output sigal s (y) compoets at frequecies 2w 2 w 1 ad w 1 respectively [6] IM 3 A 2w1 w 2 /A w1 (7) 1 Derive the state-space parameters of the G m C filter : Fid matrices A,b ad c Derive oliearity parameter α of the trascoductor aalytically or by fittig a third-order polyomial to its I V characteristic 2 Form iput sigal u : Choose amplitudes k 1, k 2 ad frequecies w 1, w 2 of the iput sigal u = k 1 si(w 1t)+ k 2 si(w 2t) u l S 2 l S 3 l y 3 Calculate h 1, p 1, h 2 ad p 2 S 2 S 3 Figure 5: The filter as a cascade of decomposed stages parallel sigals paths from the iput u to the output y, each ivolvig either a liear or a oliear operator from every stage, amely: y = (S 3 S 2 ) (u) = S l 3 S l 2 S l 1 (u) + S l 3 S l 2 S 1 l 3 S 2 S l 1 3 S l η S l 1 (u) + 1 The (liear) filter is assumed asymptotically stable by desig (ie all eigevalues of matrix A have egative real parts) 2 Each stage is idetified with its operator ad the terms stage ad operator are used idistiguishably h 1 = w1i 2 + A 2 1 Ab, h 2 = w2i 2 + A 2 1 Ab, 4 Calculate s 2, 1 ad c 2, 1 p 1 = w1i 2 + A 2 1 w 1b p 2 = w2i 2 + A 2 1 w 2b s 2, 1 = h 1 2 h h 1 p 1 p 2 p 1 2 h 2 c 2, 1 = p 1 2 p h 1 p 1 h 2 h 1 2 p 2 5 Calculate F (2w 1 w 2) 2 I + A 2 1, S2, 1 ad C 2, 1 Remark: Terms correspodig to liear or o-existig stages of the filter must be removed from S 2, 1 ad

4 C 2, 1 S 2, 1 = c T FAb iput +c T s 2, 1 output c T F A 2 s 2, 1 (2w 1 w 2)A c 2, 1 filter core C 2, 1 = (2w 1 w 2)c T Fb iput c T F A 2 c 2, 1 + (2w 1 w 2)A s 2, 1 +c T c 2, 1 filter core output 6 Calculate : J (w 1, w 2) = Õ (S 2, 1) 2 + (C 2, 1) 2 c T (jw 1I A) 1 b 7 Calculate IM 3 relatively to frequecy compoet at w 1 IM 3 α 3 k1k2 4 J (w 1, w 2) (8) Note that fuctio J (w 1, w 2) depeds oly o the model parameters of the ideal filter, ie matrices A,b ad c, ad o frequecies w 1, w 2; it is idepedet of the amplitudes k 1, k 2 It is worth comparig expressio (8) to that of IM 3 itroduced by a static weakly oliear fuctio of the form v = f(u) = a 1u + a 2u 2 + a 3u 3 + where u is give by (4) I this case IM 3 referred to iput sigal at w 1, is IM3 static = a3 a 1 3k1k2, eg [5], [6] The ratio a 3/a 1 4 correspods to oliearity parameter α i the model of trascoductors give by (1) Moreover, it ca be show that whe the filter has o dyamics, or, whe w 1, w 2 while w 1/w 2 remais fixed, the J (w 1, w 2) 1 Therefore, i the lack of dyamics, or i relatively very low frequecies, IM 3 derived usig the proposed method equals IM3 static The graph of J (w 1, w 2) is show i Figure 6 for the Tow- Thomas badpass biquad G m C (cetered at 17Mhz) discussed i the simulatio Sectio 6 The maximum appears ear (17MHz, 17MHz) Amplitudes f 1 f 2 2f 1 f 2 MHz Simulik(dB) k 1=1mV Theory(dB) k 2=1mV Cadece(dB) Error(dB) Simulik(dB) k 1=1mV Theory(dB) k 2=1mV Cadece(dB) Error(dB) Simulik(dB) k 1=1mV Theory(dB) k 2=1mV Cadece(dB) Error(dB) Table 1: Simulatio ad theoretical results els Also, the circuit was modeled ad simulated i Simulik (MATLAB) The schematic of the bad-pass filter is as that i Figure 2, but without the output stage, the output is the voltage of the secod capacitor, y = x 2 Sice there is o output stage, the calculatio of S 2, 1 ad C 2, 1 is doe by igorig the last (output) term i their expressios The ceter frequecy of the badpass filter is f o = 17Mhz ad the quality factor is Q = 2 The fully differetial trascoductor used i the filter is show i Figure 7 V Pcas V Ncas V Ndow V i- Out- M 3 M 4 M 5 M 6 V ss V i+ V dd M 12 M 13 M 1 M 2 M 11 M 7 V Pcas V Ncas V Ndow Figure 7: The trascoductor s circuit I bias M 8 M 9 M 1 V ss Out+ J f 1 (MHz) f 2 (MHz) Figure 6: J(w 1, w 2) i liear scale 6 SPICE AND MATLAB SIMULATION A G m C Tow-Thomas badpass biquad was desiged i a stadard 5µm CMOS process usig Cadece ad was simulated i SpectreS (SPICE) usig bsim3v3 trasistor mod- The values of the trascoductaces ad capacitors of the filter are g 1,I = 3126µA/V, g 1,1 = 3126µA/V, g 2,1 = 6252µA/V, g 1,2 = 6252µA/V ad C 1 = C 2 = 9354pF Parameter α was estimated by curve fittig a third order polyomial to the I-V characteristic of the trascoductor, ad α = 535V 2 The test iput sigal used for simulatio ad theoretical estimatio is give by (4) Differet combiatios of amplitude pairs (k 1, k 2) ad frequecy pairs (f 1, f 2), aroud the cetrer frequecy of 17MHz were used The amplitudes of the frequecy compoets at the output, at frequecies f 1, f 2 ad 2f 1 f 2 are show i Tables 1 ad 2, all i logarithmic scale: 2 log 1 (Amplitude i Volts) The error values are the differeces betwee SpectreS ad theory I all cases the theoretical results were very close to those of SpectreS simulatio The largest errors appeared with the smallest amplitudes, i which cases, the itermodulatio sigal at 2f 1 f 2 is about 11dB below the refereced iput at w 1 ad therefore egligible for most applicatios

5 Amplitudes f 1 f 2 2f 1 f 2 MHz Simulik k 1=1mV Theory(dB) k 2=1mV Cadece(dB) Error(dB) Simulik k 1=1mV Theory(dB) k 2=1mV Cadece(dB) Error(dB) Simulik k 1=1mV Theory(dB) k 2=1mV Cadece(dB) Error(dB) Table 2: Simulatio ad theoretical results 7 CONCLUSIONS A fast algorithm to estimate the itermodulatio distortio of bad-pass G m C filters with fully differetial weakly oliear trascoductors has bee itroduced It is based o state-space modelig ad is idepedet of the order or the particular topology of the G m C filter The algorithm has bee verified usig a bad-pass G m C biquad with fully differetial weakly oliear trascoductors desiged i a 5µm stadard CMOS process The theoretical results were foud i good agreemet with simulatio 8 REFERENCES [1] Y Tsividis, Itergrated Cotiuous-Time Filter Desig - A Overview, IEEE J Solid-State Circuits, Vol 29, No 3, Mar 1994, pp [2] K W Mouldig, J R Quartly, P J Raki, R J Thompso, ad G A Wilso, Gyrator video filter IC with automatic tuig, IEEE J Solid-State Circuits, vol SC-15, pp , Dec 198 [3] J Voorma, W H A BN s, ad P J Barth, Itegratio of aalog filters i bipolar process, IEEE J Solid-State Circuits, vol SC-17, pp , Aug 1982 [4] Y Tsividis ad J O Voorma, Itegrated Cotiuous-Time Filters, New York, NY: IEEE Press, 1993 [5] Behzad Razavi, RF Microelectroics, Pretice Hall, 1998 [6] Doald Pederso ad Kartikeya Mayaram, Aalog Itegrated Circuits for Commuicatios, Kluwer academic Publishers 1991 [7] P Wambacq, W Sase, Distortio Aalysis of Aalog Itegrated Circuits, Kluwer Academic Publishers, 1998 [8] Thomas H Lee, The Desig of CMOS Radio-Frequecy Itegrated Circuits, Cambridge Uiversity Press, 23 [9] D A Johs ad K Marti, Aalog Itegrated Circuit Desig, Joh Wiley & Sos, 1997 [1] J E Fraca ad Y Tsividis, Editors, Desig of Aalog-Digital VLSI Circuits for Telecommuicatios ad Sigal Processig, Secod Editio, Pretice Hall, New Jersey, 1994 [11] Tsividis, YP Fraser, DL Harmoic distortio i sigle-chael MOS itegrated circuits,ieee J Solid-State Circuits Vol 16, Issue 6, Dec 1981 pp [12] E Fog ad R Zema, Aalysis of harmoic distortio i sigle-chael MOS itegrated circuits, IEEE J Solid-State Circuits, Vol 17, Issue 1, Feb 1982 pp [13] RA Baki, CBeaiy, MN El-Gamal, Distortio aalysis of high-frequecy log-domai filters usig Volterra series, IEEE Tra Circuits ad Systems II Vol 5, Issue 1, Ja 23 pp 1-11 [14] Cherry, JA Selgrove, WM O the Characterizatio ad Reductio of Distortio i Badpass Filters, IEEE Tra Circuits ad Systems I, Vol 45, Issue 5, May 1998 pp [15] S Narayaa, Applicatio of Volterra series to itermodulatio distortio aalysis of trasistor feedback amplifiers, Circuits ad Systems, IEEE Tra Vol 17, Issue 4, Nov 197 pp [16] P Wambacq, G Giele, P Kiget, ad W Sase, High-Frequecy Distortio Aalysis of Aalog Itegrated Circuits, IEEE Tras Circuits Syst II, vol 46, pp , Mar 1999 [17] G Palumbo ad S Peisi, High-Frequecy Harmoic Distortio i Feedback Amplifiers: Aalysis ad Applicatios, IEEE Tras Circuits Syst I, vol 5, o 3, Mar 23 [18] Y Palaskas, Y Tsividis, Dyamic Rage Optimizatio of Weakly Noliear Fully differetial, Gm-C Filters with Power Dissipatio costraits, IEEE Tra CAS II Vol5, 23, pp [19] Z Zhag, A Celik, P Sotiriadis, State Space Harmoic Distortio Modelig i Weakly Noliear, Fully Balaced Gm-C Filters - A Modular Approach Resultig i Closed Form, IEEE Tras Circuits Syst I, Vol 53, No 1, Ja 26, pp [2] A Celik, Z Zhag, P Sotiriadis, A State-Space Approach to Itermodulatio Distortio Estimatio i Fully Differetial Bad-Pass Gm-C Filters with Weak Noliearities, IEEE Tras Circ Syst I, to appear [21] Z Y Chag, et al, A Highly Liear CMOS Gm-C Badpass Filter for Video Applicatios, IEEE Custom Itegrated Circuits Cof 1996 [22] T Kato, A Short Itroductio to Perturbatio Theory for Liear Operators, Spriger-Verlag, New York, 1982 [23] BSIM3 Versio 3 Maual, Fial Versio BSIM Research Group, Dept Elec Eg Comput Sci, Uiv Calif, Berkeley : bsim3