A Sensitive Method to Measure the Integral Non-Linearity of a Digital-to-Time Converter based on Phase Modulation

Transcription

1 A Sensiive Mehod o Measure he Inegral Non-ineariy of a Digial-o-Time Converer based on Phase Modulaion Claudia Palaella, Suden Member, IEEE, Eric A. M. Klumperink, Senior Member, IEEE, Jiayun (Zhiyu) u, Member, IEEE, and Bram Naua, Fellow, IEEE Absrac A Digial-o-Time Converer (DTC) produces a ime delay based on a digial code. ike for daa converers, lineariy is a key meric for a DTC and i can be characerized by is Inegral Non-ineariy (IN). However, measuring he IN of a sub-ps resoluion DTC is problemaic even when using he bes available high-speed oscilloscopes. In his paper, we propose a new mehod o measure he IN of a DTC, by applying digial phase modulaion and measuring he oupu specrum wih a specrum analyzer. The frequency seleciviy of his mehod allows for improved measuremen resoluion down o a few fs and allows o measure a IN below fs. The proposed mehod is verified by behavioral simulaions and employed o measure he IN of a high-resoluion DTC realized in 65 nm CMOS, wih ime-resoluion of 25 fs and sandard deviaion of 27 fs. I. INTODUCTION T ime or clock generaion wih high fideliy is a he hear of numerous elecronic sysems. The rapid developmen in Time-o-Digial Converers (TDCs) and Digial-o-Time Converers (DTCs) [], ha are increasingly used in Phase- ocked oops (Ps) [2] [4], pushes he required ime resoluion o well below ps. This paper arges he measuremen of such small iming seps. The principal insrumens radiionally employed for such measuremens are he nework analyzer or he oscilloscope. Nework-analyzer-based measuremen mehods quanify he phase difference beween wo sinusoidal signals, generaed by he same source and passing hrough wo differen pahs [5]. This phase difference is ranslaed ino a ime difference, assuming accurae knowledge of he carrier frequency. Processing algorihms applied o he deeced phase difference allow o achieve more han ps ime accuracy wih hese mehods [6]. However, hey are no suiable o measure ime differences beween non-sinusoidal digial signals. Oscilloscope-based ime measuremens are applicable o digial signals. The achievable ime measuremen resoluion depends amongs ohers on he bandwidh and he accuracy of he oscilloscope s sampling clock. The laes commercially available oscilloscopes can provide a sample clock jier of 75 fs, wih a dela-ime measuremen accuracy in he order of 5 fs for rail-o-rail digial signals [7]. This is jus enough o measure.25 ps resoluion of he sae-of-he-ar TDC [8], [9], C. Palaella, E.A.M. Klumperink and B. Naua are wih Universiy of Twene. J. Z. u also was, and is currenly wih Qualcomm. or 55 fs resoluion of he laes DTC [], bu i is insufficien o measure ime delays in he order of fs or below. Aiming o overcome he oscilloscope s resoluion and accuracy limis, in his paper we propose a new mehod for ime measuremens ha uses a specrum analyzer as principal insrumen. The proposed mehod is specially devised for a DTC and is based on digial phase modulaion, while observing he oupu specrum. A DTC produces a delayed version of is clock, conrolled by a digial inpu code. I has gained renewed ineres especially in he P research field [2] [4], because i can be used inside a P o relax he requiremens of he TDC. Similarly o daa converers, Inegral Non-ineariy (IN) is an essenial meric also for ime converers (DTCs and TDCs). The radiional way o measure he IN of a DTC is oscilloscope-based: he oscilloscope deecs he ime difference beween he hreshold-crossing poins of he delayed oupu edges. Throughou his paper, we will refer o his procedure as he direc mehod. Alernaively, our proposed approach is an indirec mehod: insead of an oscilloscope measuring direcly a delay, we use a specrum analyzer o measure a deliberaely generaed spur, whose heigh is in a one-o-one correspondence wih he delay o be measured; hen, he delay is deduced and employed o calculae he IN. The frequency seleciviy of his approach permis o achieve a ime resoluion up o a few fs. This paper is organized as follows. Secion II explains he main idea behind he proposed mehod. Then, he mehod is verified by behavioral simulaions, as described in secion III. In Secion IV, experimenal resuls on a high resoluion DTC are presened, while conclusions are drawn in Secion V. A. The concep II. POPOSED METHOD The main goal of he proposed mehod is o measure a DTC- IN which is oo small o be measured reliably by an oscilloscope. An oscilloscope used for direc delay measuremens needs wide bandwidh o avoid affecing he observed rise/fall imes of he es signal, leading o high vulnerabiliy o noise and inerferences in a broad band of frequencies. In conras, he mehod o be proposed relies on phase modulaion of he DTC-oupu by means of a digially conrolled periodic delaysep, ha generaes a narrowband spur relaed o he size of ha delay-sep. This spur can be measured by a specrum analyzer

2 2 b h+ b h a h+ a h b h 2 n a h DTC n bis + ider CK a h bh+ b h a h+ Specrum analyzer f f f + f f f DT C ou φ h π Fig.. Block diagram of he phase modulaion seup for DTC-IN measuremens, expeced specrum wih lef () and righ () sidebands, and waveforms in ime-domain. wih small resoluion bandwidh, hus avoiding disurbances a all oher frequencies. The concep can be explained using he block diagram and he waveforms shown in Fig.. The Delay Word ( ) is he code a he digial inpu of he DTC and is periodically swiched beween wo values: a h and b h (red waveform). Subscrip h will be used o idenify a paricular saring code used for experimen h. A he DTC oupu, he rising edges are delayed by a ime ha depends on he inpu code. The periodic swiching beween a h and b h produces a jump of he rising edge of he DTC oupu beween wo deermined posiions, as shown in Fig.. As only he edge conrolled by he DTC should be deeced, and no he oher, a 2 frequency divider is insered beween he DTC oupu and he specrum analyzer. In his way, a phase modulaion of he signal is achieved. Noe ha he modulaing signal (wih frequency f ) is a code waveform (in DTC SBs), because of he digial naure of he DTC inpu. The phase modulaion appears, in he frequency domain, as a couple of sidebands, shifed by an offse frequency f (and is harmonics) from he carrier frequency f. These sidebands can be measured using a specrum analyzer. B. Analysis The solid waveforms DT C ou and on he righ-hand side of Fig. are he unmodulaed signals ha will occur if he code applied a he DTC inpu is consanly equal o b h. The doed waveforms are he modulaed signals: hey coincide wih he solid waveforms when b h is applied, bu hey are shifed o he dashed edges as long as a h is applied. The frequency f of he code waveform is chosen so ha he resuling sidebands are locaed in an inerferencefree porion of he specrum, and far enough from he carrier frequency f CK o no be affeced by is phase noise, including /f noise. Afer inerference measuremens, we chose f CK = 2f, as shown in Fig.. However, he following analysis is independen of he choice of f. The waveform φ h on he boom righ in Fig. represens he phase difference beween he unmodulaed signal (consan ) and is modulaed form ( square wave wih heigh ), sampled a every rising edge of he signal. The waveform φ h is a square wave wih he same frequency as signal; i has 5% duy cycle and is heigh is / (2 ) 2π = / π, where is he clock period and is he delay-sep produced by he code-sep = b h a h, as shown in Fig.. The firs harmonic Φ h of he phase difference is given by: Φ h = 2 () and can be reaed as in sandard phase modulaion heory [], [2], leading o a spur level relaive o he carrier [dbc] given by 2log ( Φ h /2) and, herefore: spur h (f div ± f ) = 2log ( τh ) [dbc] (2) Equaion (2) allows he applicaion of a specrum-analyzerbased ime measuremen, because i provides he link beween he frequency-domain, in which he measuremens are acually done, and he ime-domain. From he spur measuremen, hrough equaion (2), we can deduce he measured delay-sep associaed wih. Nex, he associaed Differenial Nonlineariy (DN), expressed in seconds, can be calculaed wih he following: DN (h) = τ id (3) where τ id is he ideal delay-sep produced by he code difference, evaluaed as he average of all he values obained. Finally, he IN can be compued from he cumulaive sum of he DN: h IN (h) = DN (k) (4) C. Sensiiviy and esoluion k= To incorporae all DTC-codes in he IN es, he bes value o assign o is SB. However, his choice can resul in very slow measuremens o ge a complee IN plo, because he number of required poins is 2 n for an n bi DTC, while a measuremen wih narrow resoluion bandwidh wih a specrum analyzer akes considerable ime. If he IN-behavior of a DTC is raher smooh, is lineariy can also be described wih a subse of IN poins, using a

3 3 Spur [dbc] Noise floor SA c_sep=64 c_sep=6 c_sep=2 ow sensiiviy [SB] Fig. 2. Plo of spur as a funcion of, described in equaion (6), for he case n =, τ F S = ps, and = 2 ns. coarser delay-sep. However, he logarihmic relaion beween spur and delay in equaion (2) suggess ha a very coarse delay-sep may lead o reduced sensiiviy. This is because a specrum analyzer has a limied resoluion and accuracy and a some poin he variaion of spur srengh may be oo small o be deeced. In his secion, we will focus on his rade-off beween noise-limiaions and he limied sensiiviy. The relaion beween inpu codes and delays produced by he DTC is linear: = τ F S 2 n (5) where is he delay-sep resuling from he applicaion of he code difference a he DTC s inpu, n is he DTC s number of bis, and τ F S is he DTC s full-scale delay (i.e. he delay corresponding o he inpu code going from o 2 n ). Equaion (2) can be rewrien in he following form: ( τf S 2 spur h (f div ± f ) = 2log n ) [dbc] (6) where he logarihmic dependence of spur h as a funcion of is eviden, as shown in Fig. 2, for he case n =, τ F S = ps, and = 2 ns. The sensiiviy of he mehod can be quanified as he variaion in spur srengh due o he change of he delaysep, evaluaed a a cerain nominal delay-sep value. We can calculae i by aking he derivaive of equaion (2), or as a funcion of (in SBs), using equaion (5), obaining: spur h = [2log (e)] 8.69 Equaion (7) can be used o undersand he limis associaed wih he choice, highlighed in he spur curve in Fig. 2. For a coarse (around 2 SB), he limiaion is he flaness of he spur curve, i.e. a low sensiiviy. As an example, wih he values used for he plo in Fig. 2, for = 2 a deviaion of 3 SB (non-lineariy) in he DTC s delay would resul in only.3 db spur change, which is hardly disinguishable from oher environmenal sources of variaion (he experimenally observed uncerainy was.2 db in he PXA-SA [3]). For a equal o one or a few SBs, more IN poins are available, bu he main limi is he noise floor of he specrum analyzer ha can preven i from disinguishing he low spur. However, he specrum analyzer s noise floor can (7) be reduced wih a narrow resoluion bandwidh BW (up o 55 dbm noise floor wih BW= Hz in he PXA-SA [3]). A good value for is, herefore, he minimum value needed o disinguish he spur from he noise floor. For example, wih BW= 2 khz, he specrum analyzer s noise floor is 2 dbm for [3]; wih a 3 dbm carrier and equaion (6), he value = 2 SB would produce a disinguishable spur of dbc= 97 dbm, wih a resuling ime resoluion of 96 fs. An average beween muliple measuremens is needed o reduce he variabiliy due o noise. By pushing he noise floor o he minimum, i becomes possible o deec a 4 dbc spur, corresponding (from equaion (2)) o only 2 fs ime delay. This value is far beyond he hundreds of fs values achievable wih op-class oscilloscopes commercially available [7]. D. Algorihm Aiming o deec sub-ps IN, he essence of he proposed measuremen mehod is o always generae he same delay by changing he inpu waveform. In his way, he mehod s sensiiviy in equaion (7) is he same during he measuremens, and oher variaions are minimized. Depending on he spur srengh, he measured delay can be higher or lower han is nominal value and his deermines he polariy of he IN curve. The algorihm of he proposed mehod is based on hese consideraions ogeher wih equaions (2)-(4). I consiss of he following seps, parly shown in Fig. : ) divide he overall code range ino equal inervals of heigh, covering codes from up o 2 n ; 2) sar wih index h = 3) apply he waveform h and measure spur h ; 4) calculae by applying equaion (2); 5) increase h and repea seps 3 o 4, unil all values of h have been considered; 6) evaluae τ id as he average of all he values ; 7) calculae DN using equaion (3) and IN using equaion (4). Noice ha in sep he waveforms differ only by heir lower ( a h ) and upper ( b h ) values. Therefore, each waveform is idenified by he index h. The upper value of one waveform mus coincide wih he lower value of he nex one, ha is b h = a h+ ; his allows us o obain IN as cumulaive sum of DN. Ideally, for all he values h, he measured spur, and herefore he delay-sep, will be always he same. However, due o he circui nonlineariy, he values are dependen on he index h. III. SIMUATIONS We esed he proposed mehod by running behavioral simulaions. The goal is o verify he equaions presened in secion II and o check wheher he convenional direc and our indirec mehod lead o he same IN resuls. To compare resuls boh in ime and in frequency domain, he behavioral simulaions need o have enough ime resoluion, o disinguish he delay of he DTC in he ime

4 4 Power [dbm] 5 c_sep=64 c_sep= Frequency [MHz] Fig. 3. Simulaed specrum of signal, wih = 2 SB and = 64 SB applied a he DTC inpu. Delay Error [fs] 2 4 c_sep 2 6 c_sep 6 no-noise, indirec no-noise, direc DAC code Fig. 4. Simulaed IN, for differen values of and for direc and indirec mehods. The values of used are highlighed on Fig. 2. domain, bu also long simulaion ime, o produce a specrum wih enough frequency resoluion. For hese simulaions, we chose a ime resoluion of 2 fs and a simulaion ime of 4µs, corresponding o a frequency resoluion of 25 khz. The simulaed DTC has n = bis, a full-scale delay τ F S = ps, and a clock wih period = 2 ns. A quadraic nonlineariy is insered on purpose in he model wih maximum IN of 25 fs. The frequency of he waveform is se o f = 2.5 MHz, he signal has f = 25 MHz. Whie noise has been added o model a 2 dbm specrum analyzer s noise floor. Fig. 3 shows he simulaed specrum of he divider oupu waveform, as i would appear on he screen of a specrum analyzer, for wo quie differen values of wihin he limis discussed in secion II-C, o clearly show he spur differences. As expeced, he specrum exhibis he carrier one a f = 25 MHz and wo sidebands a f ± f (firs harmonics), namely 22.5 MHz and 27.5 MHz. For = 64 SB, he figure also shows he hird harmonics a f ± 3f, due o he square wave shape of he modulaing signal; hese higher harmonics do no add more informaion and hey are no considered for he measuremens. The simulaed firs-harmonic sidebands are.6 dbc for = 2 SB and 7.4 dbc for = 64 SB, maching he values obained from equaion (6). Fig. 4 aims o compare he IN using boh he direc and he indirec mehods. The figure also shows he effec on he IN curve by wo values in he rade-off range discussed in secion II-C. For simpliciy, he wo mehods are compared in he noiseless case, wih = 6 SB. The shape of he noiseless IN curve (circles) exhibis he quadraic nonlineariy insered on-purpose in he behavioral model. The algorihm of secion II-D was used o calculae i, where one specrum for each poin of he IN curve has been obained. The small-doed curve in Fig. 4 refers o he direc mehod. I has been derived by ploing he DTC oupu as a funcion of ime, hen evaluaing he ime insans k where he rising edges cross a 7 mv volage hreshold and finally using he sandard formulas from [4]. The IN from he wo mehods differ by a mos. fs, due presumably o numerical noise, i.e. only % of DTC SB. By adding 2 dbm whie noise in he simulaions, we can invesigae he effec of differen code-sep choices on he IN. For = 2 SB, he firs spur is close o he simulaed noise floor, as shown in Fig. 3, resuling in a deailed (5 poins) bu noisy IN plo in Fig. 4. The drif due o high-frequency noise can be reduced by obaining muliple IN plos and averaging hem a each code. However, as simulaions already ake muliple days, no averages for any value of have been done here. Insead a higher value of = 6 SB was used, resuling in a less noisy resul closer o he rue IN. A larger produces less IN poins bu is more robus o noise. IV. MEASUEMENTS We acually developed his IN-measuremen mehod o allow for measuring he IN of a record high-resoluion DTC, implemened in CMOS 65 nm echnology ha explois a consan slope principle [5]. Measuring he DTC wih onchip DAC using he direc mehod failed, because of he DTC resoluion in he order of ens of fs. Fig. 5 shows he block scheme of he measuremen seup. The DTC realizes a delay using a ramp waveform wih a fixed slope, saring from an iniial volage defined by a Digial-o-Analog Converer (DAC). The delay is conrollable by he DAC volage, which can eiher be on-chip or exernal. The digial inerface of his chip is no fas enough o suppor MHz modulaion of he DAC code. Insead, an exernal DAC (Agilen M89A Arbirary Waveform Generaor) was used in hese experimens o produce a square wave (V ex ) ha periodically swiches beween wo volage levels, effecively implemening he delay-sep, and he chip acs as a Volage-o-Time Converer (VTC). The procedure is he same as described in secion II; compared o he scheme in Fig., he only difference here is ha he waveforms wih heigh are now convered ino V ex square waves wih heigh v sep, hrough he exernal DAC. The V ex jier did no affec he sideband posiion during he measuremen. We used he MSB-bis of he Agilen M89A 4-bi DAC. We checked is IN was below +/.5 SB (.5% referring o -bi full-scale) so i is no he boleneck in our DTC-IN measuremen. Daa averaging can reduce he DAC hermal noise, however some /f noise remains. The measuremens are done wih f CK = 5 MHz, f = 25 MHz, and using 4 values for he square waves V ex wih f Vex = f = 2.5 MHz and heigh v sep =.98 mv each. The choice of v sep is equivalen o of abou 25 SB in a -bi full scale, on he same order as he choices in secion III. A smaller v sep or would resul in more

5 5 Exernal DAC CK V ex v sep VTC 2 SA is less han he sandard deviaion 27.3 fs of he ime sep measuremen. Therefore, in his case, eiher he lef or he righ spurs produce a sufficienly precise IN plo. However, depending on he implemenaion of he DTC, if here is also a coexising ampliude modulaion, he delay measured using he lef spur would be higher han he one for he righ spur. In such a case, averaging beween lef and righ spurs produces he proper resul. Fig. 5. Seup used o measure he IN of he DTC [5] (acing as a VTC), wih he proposed mehod. Sep Delay [fs] mean mean Max STD = 27.3fs Volage [mv] Fig. 6. Measured delay-sep produced by he DTC [5], as a funcion of he upper volage of he modulaing square waves; 5 repeiions for each delay. IN [ fs ] Delay [ps] Fig. 7. Measured IN, produced by he DTC [5], se wih full-scale 9 ps, using he proposed mehod (V ex = o 33.6mV wih seps of.98mv). measured IN poins bu is less robus o noise and requires longer measuremen ime. The algorihm in secion II-D is repeaed 5 imes, leading o 5 sweeps hrough he se of 4 square wave volages. The DTC in measuremen has a unable full-scale delay 9 89 ps, and we use i here a is minimum delay o apply he mehod in he mos challenging case. The specrum analyzer s resoluion bandwidh is se o khz, leading o 5 dbm noise floor and 8 dbc minimum deecable spur (assuming SN= db and 3 dbm carrier), and resuling in 25 fs ime resoluion from equaion (2). Fig. 6 shows he 4 measured delays and indeed shows an average of abou 475 fs (9 ps/4). The delays have been calculaed separaely for he lef () and he righ () sideband. The maximum sandard deviaion is 27.3 fs. The resuling averaged IN curve is shown in Fig. 7. The wo y-axes refer o he absolue IN in fs, and is normalized value wih respec o he DTC full-scale delay, respecively. The maximum IN value is 64 fs and corresponds o.34% in he normalized scale. The difference beween lef and righ sidebands produces a maximum IN difference of 2 fs, ha IN / full scale V. CONCUSION In his paper, we have presened a very sensiive mehod o measure he IN of DTCs, based on phase modulaion, and capable o achieve a ime resoluion of a few fs, which is -2 orders of magniude beer han wha is achievable wih high-speed sampling oscilloscope. The new mehod has been verified wih behavioral simulaions, and used o measure he IN of a high-resoluion DTC wih 9 ps full-scale. An IN in he order of 5 fs was measured, wih a sandard deviaion of 27.3 fs. EFEENCES [] G. obers and M. Ali-Bakhshian, A brief inroducion o ime-odigial and digial-o-ime converers, Circuis and Sysems II: Express Briefs, IEEE Transacions on, vol. 57, no. 3, pp , March 2. [2] V. Chillara, Y.-H. iu, B. Wang, A. Ba, M. Vidojkovic, K. Philips, H. de Groo, and. 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