Lecture 12. Aperture and Noise. Jaeha Kim Mixed-Signal IC and System Group (MICS) Seoul National University

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1 Lecure. Aperure and Noie Analyi of locked omparaor Jaeha Kim Mixed-Signal I and Syem Group MIS Seoul Naional Univeriy jaeha@ieee.org

2 locked omparaor a.k.a. regeneraive amplifier, ene-amplifier, flip-flop, lach, ec. A every clock edge, ample he inpu coninuou and decide wheher i i or binary Therefore, i inherenly nonlinear operaion

3 omparaor haraceriic Offe and hyerei Sampling aperure, iming reoluion, uncerainy window Regeneraion gain, volage eniiviy, meaabiliy Random deciion error, inpu-referred noie an be analyzed and imulaed baed on a linear, ime- varying LTV model of he comparaor 3

4 locked omparaor Operaion 4 operaing phae: ree, ample, regeneraion & deciion Sampling & regeneraion phae can be modeled a LTV 4

5 An Ideal omparaor Model V k = V i o +kt, o+kt V i D k V i D k Sampling and deciion Infiniely-fa racking of V i A realiic comparaor ac on a filered verion of V i 5

6 LTV Model for locked omparaor ob +kt V i D k V i V o Noiy Nonlinear Filer V k = V o ob +kt D k LTV mall ignal model v i =h, n o v o Aume a noiy, nonlinear filer before he ampling The filer mall-ignal ignal repone i modeled wih ISF * J. Kim, e al., Simulaion and Analyi of Random Deciion Error in locked omparaor, IEEE TAS-I, 8/9. 6

7 ISF for Ocillaor Impule eniiviy funcion ISF i defined a: = he final hif in he ocillaor phae due o a uni impule arriving a ime = * A. Hajimiri and T. H. Lee, A General Theory of Phae Noie in Elecrical Ocillaor, IEEE JSS, Feb

8 ISF for Ocillaor ISF decribe he ime-varying repone of a ocillaor Repone o each impule add up via uperpoiion For arbirary noie inpu n, he reuling phae hif i: n d ISF led o ome key ocillaor deign idiom: Sharpen he clock edge o lower ISF i.e. minimize RMS Align noie even wihin low-isf period Balance ISF i.e. D = o preven /f-noie up-converion 8

9 ISF for Sampler and omparaor For ample-and-hold circui, he ampled volage V can be expreed via a ampling funcion f: V f V i d * H. O. Johanon,. Svenon, Time Reoluion of NMOS Sampling Swiche Ued on Low-Swing Signal, JSS, Feb For clocked comparaor, we imply add he deciion : D k gn V gn Γ V d k * P. Nuzzo, e al., Noie Analyi of Regeneraive omparaor for Reconfigurable AD Archiecure, TAS-I, July 8. i 9

10 ISF for locked omparaor ISF how ampling aperure, i.e. iming reoluion In frequency domain, i how ampling gain and BW ISF F.T. { - }

11 Generalized ISF In general, ISF i a ube of a o-called ime-varying impule repone h, for LTV yem*: y h, x d h, : he yem repone a o a uni impule arriving a For LTI yem, h, = h- convoluion ISF = h, : he ime a which he yem repone i oberved For ocillaor, = + For comparaor, i before he deciion i made more laer * L. Zadeh, Frequency Analyi of Variable Nework, Proc. I.R.E. Mar. 95.

12 Noie in LTV Syem If he inpu x o an LTV yem i a noie proce, hen he oupu y i a ime-varying noie in general Expreion become very complex cyclo-aionary a be We can keep hing imple if we are inereed in he noie only a one ime poin in our cae: = ob +kt

13 LTV Oupu Noie a = y y y y dv v x v h du u x u h,, dv du v h u h v x u x,,,, dv du v h u h v u R R xx u, v i he auo-correlaion of he inpu noie x du dv v h u h v u R xx,,, R xx u, v i he auo correlaion of he inpu noie x 3

14 Repone o Whie and /f Noie If he inpu x i whie noie, i.e. R xx u, v = x u-v: d Γ du u h x x y, If he inpu x i /f noie, i.e. R xx u, v = x : d, Γ du u h x x y Agree wih Hajimiri/Lee low-noie deign idiom: To minimize conribuion of whie noie, minimize RMS T i i i ib i f /f i k To minimize conribuion of /f noie, make D = 4

15 Random Deciion Error Probabiliy If we have muliple noie ource, heir conribuion add up via RMS um auming hey are independen: y, oal o y, j j If he comparaor ha ignal V o and noie no n,o a ob,, he deciion error probabiliy Perror can be eimaed a: o SNR V o ob n, o ob SNR P error Q exp x / SNR dx 5

16 ircui Analyi Example A varian of SrongARM comparaor When clk i low, he comparaor i in ree ou+/- are a V dd X/X are ~V dd -V TN When clk rie ay =, he comparaor goe hru: Sampling phae ~ Regeneraion phae ~ X X 6

17 . Sampling Phae = ~ While ou+/- remain high: M pair dicharge X/X M-pair dicharge X/X M-pair dicharge ou+/- S S ranfer from v o v : S.S. ranfer from v in o v ou : / x ou x ou m x ou m m in ou g g g v v / x ou m m x ou x ou m x ou in g g g v The ISF w.r.. v in i: G R 7 R

18 . Sampling Phae = ~ S.S. repone o M noie: v ou n i g m ou x n g m G S.S. S repone o M noie: vou i n ou n ou G R R 8

19 . Regeneraion Phae = ~ We aume X/X ~ V and M-pair i in linear region The circui i no longer eniive o v in ISF= ro-coupled inverer amplify ignal via poiive-feedback: G R R exp R The ISF w.r.. noie i: ou / g m, r g m3, n r, r exp ou R 9

20 Puing I All Togeher The overall gain G i: dc i ob o v v G, d / The oal inpu referred R exp The oal inpu-referred noie i: 3,, / kt kt G ou x o n i n Mo of he noie i conribued by M and M pair during he ampling phae ou x ampling phae

21 Deign Trade-Off The inpu-referred noie can be approximaed a: where 3, kt kt ou x i n Tp d m Tp d m ou x V I g V I g, Therefore, noie improve wih larger g m /I d raio and wider ampling aperure - p g p However, ampling bandwidh and/or gain may degrade onrolling he ail urn-on rae i a good way o keep high gain g g y p g g

22 Simulaing Aperure & Noie RF imulaor e.g. SpecreRF can imulae mall-ignal LPTV repone and noie efficienly: Simulae linearized repone around a periodic eady-ae PA analyi give Hj; ; = Fourier ranform of h,*, PNOISE analyi can give he noie PSD a one ime poin The remaining queion i how o chooe ob? We d like o chooe i o mark he end of he regeneraion Since in our LTV model capure ampling + regeneraion * J. Kim, e al., Impule Seniiviy Funcion Analyi of Periodic ircui, IAD 8.

23 omparaor Periodic Seady-Sae PSS Diff Oup pu V Differenial Oupu V Ree Sampling Regeneraion ob Deciion i / ompreion Differenial Oupu 5 5 Time p PSS repone of he comparaor for a mall D inpu Near he clock riing edge; reurn o ree no hown 3

24 omparaor Sampling Aperure PA V V Differenial Oupu Dif ff Oupu D /V* V / V ISF V/ Ree Oupu SNR Differenial Oupu 5 5 Sampling ob Regeneraion Deciion / ompreion Γ h ob, 5 5 Time p V o ob VV i 4

25 omparaor Noie PNOISE Diff Differeni Ou al Oupu V Ree Sampling Regeneraion ob Deciion / ompreion Differenial Oupu RMS Oupu Noie 5 5 Time p Magena line plo he rm oupu noie v. ime, obained by inegraing i he noie PSD a each ime poin Thi i no ranien noie analyi i a ime ample of cyclo-aionary i noie much more efficien i 5

26 omparaor Oupu SNR iff Differenial Oupu u V D R db SNR db SN Ree Oupu SNR Regeneraion Differenial Oupu RMS Oupu Noie 5 5 Sampling Deciion / ompreion Oupu SNR ob 5 5 Time p 6

27 Deciding on ob How o chooe ob ha mark he end of regeneraion Mo of he noie i conribued during he ampling phae Noie ha ener during he ampling phae ee he full gain Noie ha ener laer during he regeneraion phae ee an exponenially decreaing gain wih ime For he purpoe of eimaing deciion error, elecion of ob i no criical a long a i wihin regeneraion phae The SNR and deciion error probabiliy ay ~conan I chooe ob when he comparaor ha he max. mall-ignal gain i.e. before he nonlineariy ar uppreing he gain 7

28 Meauremen Reul V i D k D k+ V i A D 4k,,4k+3 LTI Fron End Receiver A 9nm Receiver B 65nm Boh receiver are baed on SrongARM comparaor Differenial in ~ pf hermal noie from he inpu erminaion reior < Vrm Exce noie facor no pec d by foundrie imulaed ld a muliple lil value 8

29 Receiver A Direc Sampling Fron-End -4 logb BER D Inpu mv Simulaion of he deciion error BER = QV o ob / o ob veru he D inpu level exce noie facor = 9

30 Receiver A Direc Sampling Fron-End -4 logb BER D Inpu mv Meauremen of he deciion error BER baed on he deniy of he wrong oupu veru he D inpu level 3

31 Receiver A Direc Sampling Fron-End -4 BER V i. dc V Q i, dc i i logb BER = -.6mV =.79mV = +3.3mV =.79mV D Inpu mv Fi boh e of poin o he Gauian BER model ompare he eimaed inpu-referred rm noie 3

32 Simulaion v. Meauremen Simulaed mv,rm Meaured mv,rm Receiver = = =3 =4 Po. / Neg. / Avg. A 9nm Direc Sampling Fron-End /.65 /.7 B 65nm w/ Linear Fron-End /.83 /.85 Po. / Neg. / Avg. refer o meauremen reul for poiive V IN, negaive V IN, and heir average 3

33 Noie Filering via Finie Aperure ISF Noi ie PSD Vo ol/qrhz E-6 E-7 E-8 Fron-end oupu noie pecrum Noie pecrum wih ISF filering 65.65mVrm inpu referred.8mvrm inpu referred E Frequency GHz In receiver B, he noie conribued by he linear fron- end i filered by he finie i aperure of he comparaor 33

34 oncluion The linear ime-varying LTV yem model i a good ool for underanding he key characeriic of clocked comparaor Sampling aperure and bandwidh Regeneraion gain and meaabiliy Random deciion error and inpu-referred noie The impule eniiviy funcion ISF ha a cenral role in i: A i did for ocillaor Guide deign rade-off beween noie, bandwidh, gain, ec. The LTV framework i demonraed on he analyi, imulaion, and meauremen of clocked comparaor 34

35 Back-Up Slide

36 Exracing ISF from h, hooe ob a he maximum mall-ignal gain poin ISF: = h ob, Maximum Gain Poin ob ob 36

37 Effec of he Bridging Device Improve hold ime and meaabiliy Increaed Reg. Gain Improved Hold Time 37

38 Effec of Inpu and Oupu Loading Increaed Seup Time Increaed Hold Time 38