POST-CORRECTION OF ADC NON-LINEARITY USING INTEGRAL NON-LINEATITY CURVE

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1 Vldimir Hsz, Dvid Slepic, Petr Suce / Computig,, Vol., Issue, 56-6 computig@computigolie.et ISSN Itertiol Jourl of Computig POST-CORRECTION OF ADC NON-LINEARITY USING INTEGRAL NON-LINEATITY CURVE Vldimir Hsz, Dvid Slepic, Petr Suce Czec Tecicl Uiversity i Prgue, Fculty of electricl Egieerig, Tecic, 667 Pr 6, Czec Republic, sz@fel.cvut.cz, slepicd@fel.cvut.cz, ttp://mesure.feld.cvut.cz Evolvig systems cosultig s.r.o., Čs. rmády 4, 6 Pr 6, Czec Republic, petr.suce@evolvsys.cz Abstrct: Te ccurcy of AD coversio c be improved usig te post-correctio of digitizer o-lierity. I priciple two metods could be pplied loo-up tble or lyticl iverse fuctio of itegrl o-lierity curve (INL(. Loo-up tble c be esily implemeted but it demds uge memory spce prticulrly for ig resolutio ADCs. Iverse fuctio offers flexible solutio for prmeteriztio (e.g. frequecy depedece but it lso requires fst DSP for rel-time correctio. Te dt or coefficiets for bot metods re frequetly determied from istogrm of cquired pure siusoidl sigl. No-lierity curve c lso be gied by oter procedure demdig sigifictly less smples pproximtio from frequecy spectrum. Te correctio of ADC olierity by mes of iverse fuctio of INL( curve is lyzed i tis pper d te results re preseted. Keywords: log-to-digitl coverter, ADC o-lierity, INL, trsfer fuctio, pproximtio, o-lierity correctio, simultios, experimetl verifictio.. INTRODUCTION Te ADC o-lierity is ieretly described by te Itegrl No-lierity curve INL( wic is defied s te differece of ADC output d iput s te fuctio of te iput level. INL( c be directly determied usig istogrm metod [], but tis metod demds uge umber of smples i record, tus it is time cosumig. However, olierity cuses lso distortio i te digitized sigl d te frequecy spectrum c provide similr iformtio s te INL( i te code domi. Te INL( curve c be split ito its low code frequecy compoet (LCF d te ig code frequecy compoet (HCF. Te LCF (te roug curve of te INL( see Fig., dotted curve is resposible for rmoic distortio t lower rmoic compoets [,, 4], usully te strogest re d d te rd oes. If pproximtio of te INL( curve usig polyomils is pplied, te tird order polyomil is mostly sufficiet for te followig itegrl o-lierity correctio.. APPROXIMATION OF INL(N CURVE Usig polyomils te INL( is pproximted by INL (LSB Fig. A exmple of INL( curve d its low code frequecy compoet (dotted curve 56

2 Vldimir Hsz, Dvid Slepic, Petr Suce / Computig,, Vol., Issue, 56-6 INL( H mx x ( ( were re te olierity coefficiets up to te mximum order H mx, wic is te igest rmoic compoet cosidered, is te ormlized ADC code wit bipolr rge, d x te ADC iput. Hvig te coefficiets d cosequetly te pproximtio of INL( curve, te o-lierity of digitizer c be corrected. Te pproximted trsfer fuctio TF s to be clculted by ddig strigt lie to te INL(, suc tt TF ( + INL( ( were is te ADC code d fter te substitutio it c be expressed s Hmx ( TF( + x ( If te trsfer fuctio TF is mootoicl, its iverse exists. For tis cse, let s propose tt te pproximtio of te iverted trsfer fuctio TF - will lso be polyomil of te sme order (K mx H mx defied s K mx TF ( y b y (4 were y is te ADC output. Substitutig y TF( from ( ito (4 TF K mx ( TF ( b H mx K mx b [ TF ( ] (5 Te distortio of te d d te rd rmoic compoet is usully te most importt for mjority of digitizers d te iger compoets re usully egligible. Terefore te K mx H mx will be te ito ccout for te followig solutio. I tis cse te geerl expressio (5 cges to TF ( TF( b ( + Cosiderig + b ( + + b ( + b (6 TF ( TF( (7 d comprig te coefficiets of te sme powers of over determied equtio system rises ( uows vribles b, b, b, 9 equtios. Two metods for te determitio of coefficiet b re preseted i tis pper. I te first metod te coefficiets b, b, b re determied from loworder equtios, te equtios wit polyomils l, l > re eglected. Te coefficiets re give by b, b, b (8 5 4 Te secod metod is more sopisticted. Sice equtio (7 c rdly be fulfilled completely te error fuctio ( ( ( e TF TF (9 is miimized (lest squre. Let s itegrte te error fuctio over te fullscle of te ADC to obti te re below te error fuctio. Te full-scle rge of te ADC sps over ;+ itervl becuse of ormliztio. ( ( ( + I( b, b, b TF TF d d fter te substitutio from (6 + I( b, b, b ( b ( + + b ( + + b ( d ( ( Itegrtio ( elimites te vrible d oly vribles b remi. Let s te te prtil derivtives of te I(b,b,b fuctio wit respect to tree vribles d equl tem to zero. 57

3 Vldimir Hsz, Dvid Slepic, Petr Suce / Computig,, Vol., Issue, 56-6 δi δi δi,, ( δb δb δb Te system wit tree equtios of tree vribles is obtied cb + cb + cb d cb 4 + cb 5 + cb 6 d cb+ cb + cb d ( were c i d d j re collected terms resultig from te prtil derivtios (, e.g. d c ( d + 5 (5 Te equtios ( c be rewritte ito mtrix form C b d (6 d te coefficiets b, b, b c be clculted by pplyig te Crmer s rule bsed o determits det( C det( C det(, C b, b b det( C det( C det( C Te solutio is (7 dcc 5 9+ dcc 8 + ccd 6 ccd 5 cdc 9 dcc 6 8 b ccc 5 9+ ccc ccc 6 7 ccc 5 7 ccc 4 9 ccc 6 8 cdc 9+ cdc 4 + dcc 6 7 cdc 7 dcc 4 9 ccd 6 b ccc 5 9+ ccc ccc 6 7 ccc 5 7 ccc 4 9 ccc 6 8 ccd 5 + ccd cd c7 dcc 5 7 ccd 4 cdc 8 b ccc 5 9+ ccc ccc 6 7 ccc 5 7 ccc 4 9 ccc 6 8 (8. SIMULATION OF NON-LINEARITY CORRECTION Te correctio of te simulted ADC olierity ws performed i te secod step. Te coefficiets of te polyomils of te pproximted o-lierity INL( curve ( were computed from te istogrm metod [] mesured by te digitizer NI PXI 5. Spectrlly pure (filtered testig sigl (THD < - db ws used for tis purpose [5]. Te pproximtio of te iverted trsfer fuctio ws foud pplyig te polyomils. Oly te most domit coefficiets i (te d d te rd order were cosidered. Te levels of rmoic compoets of te simulted output sigl d of te sme sigl fter te correctio re preseted i Tble. Te performce of bot metods metioed bove is sow. Tble. Results of Correctio Hrmoic compoet Digitl output before correctio Digitl output fter correctio Direct LSE iversio miimiztio 77dB db 4dB 8dB 6dB 55dB Te modeled iput d correspodig output sigl were i very good greemet wit te rel sigls. Te correctio pplied o tis sigl sowed to be very effective. However, te correctio o rel output dt did ot improve te sigl s expected. Te reso seemed to be i oter ADC imperfectios (dditive oise, jitter i smplig, o-zero smpled sigl pse d ysteretic bevior wic were ot te ito ccout i te simultio. To fid te source of te worse correctio results i te cse of rel dt furter simultios were executed. Te ifluece of te icoeretly smpled sigl (wit o-zero α ws suppressed by pplyig te Blcm-Hrris widow of te 7 t order to te recorded dt. Te simulted distorted ADC output ws geerted s ADC output y sig ( ( ( bs ADC ( iput (9 were [dc_full_scle, 8, ] (dc_full_scle for te simulted -bit ADC d te umbers 8 d re te coefficiets of te d d te rd o-lierity order. No roudig (qutiztio i mplitude of te ADC output ws used i order to better observe te performce of te correctio. Te followig ADC imperfectios were dded to te modeled output sigl: dditive wite oise, b smplig jitter, c ifluece of o-zero smpled sigl pse, d ysteresis. Te result correspodig to rmoic distortio oly ws used s te referece oe (Fig.. Te ifluece of te dditive oise is preseted i Fig.. 58

4 Vldimir Hsz, Dvid Slepic, Petr Suce / Computig,, Vol., Issue, 56-6 SINAD 96 db THD -96 db SNHR 65 db SINAD 6 db THD -69 db SNHR 65 db Frequecy spectrum before correctio b Frequecy spectrum fter correctio Fig. No-lierity correctio of simulted sigl sie-wve sigl wit rmoic distortio SINAD 9 db THD -96 db SNHR 9 db SINAD 9 db THD - db SNHR 9 db Frequecy spectrum before correctio b Frequecy spectrum fter correctio Fig. No-lierity correctio of simulted sigl sie-wve sigl wit wite oise (σ 6 LSB d rmoic distortio Tis simultio proved tt te presece of dditive wite oise does ot cuse oticebly ifluece te results of correctio iger rmoic compoets were suppressed to egligible level below oise. Te sme result ws foud for smplig jitter. Te simultio wit vrible ozero smpled sigl pse sowed lso o ifluece. Te lst imperfectio of rel ADC, wic ws ivestigted, ws te ysteresis. For better observtio te resultig frequecy spectrum s well s te itegrl o-lierity curve (te devitio from te idel trsfer fuctio residul mplitude were clculted (see Fig. 4. Te residul mplitude for fllig d risig slopes ws plotted idividully. Te dsed d dotds lies were recostructed from te fllig d risig slopes of te sigl. INLd is so clled "differetil-mode" compoet wic correspods to ADC ysteresis bevior. Te full lie represets te commo-mode o-lierity compoet INLc [6]. Te ysteresis ws modeled usig equtio yst x y ( x β sig( x X, ( were y yst (x is te dditive cotributio of te ADC ysteresis to its output, β is te proportio fctor, X is te mplitude of te iput sigl, d te sig(x is biry fuctio wit + d output vlues depedig o te slope of te iput sigl x. Fig. 4c sows typicl curves of residul mplitude, wic is sigifictly differet for filig d risig slopes. 59

5 Vldimir Hsz, Dvid Slepic, Petr Suce / Computig,, Vol., Issue, 56-6 SINAD 96 db THD -96 db SNHR 55 db SINAD 5 db THD -5 db SNHR 55 db Frequecy spectrum before correctio b Frequecy spectrum fter correctio c Trsfer fuctio before correctio d Trsfer fuctio fter correctio Fig. 4 No-lierity correctio of simulted sigl sie-wve sigl wit ysteresis α 4 LSB d rmoic distortio Te result of te correctio is preseted i Fig. 4d. Te commo mode o-lierity ws correctly estimted s f(x (8.x (x LSB d removed. Oly te ysteresis remied fter te correctio. It correspods to te premise tt te proposed metod c correct oly pure o-lierity, but ot te ysteresis. 4. EXPERIMENTAL VERIFICATION To verify te simultio results experimetl mesuremets usig two ig qulity digitizers (- bit Digitizer VXI HP E4A d 4-bit Digitizer NI PXI-59 were performed. Hig-qulity ADC testig system t te CTU i Prgue [5] ws pplied for tis purpose. Te iput sigl ws geerted by ultr-low distortio Stford Reserc DS6 geertor d it ws subsequetly filtered by bdpss filter to cieve ig spectrl purity of te sigl. Te record of MS ws divided ito four segmets. Oe of tem ws selected s te referece for clcultig coefficiets of te iverted polyomil. Te oter tree dt segmets were te corrected. I te cse of VXI HP E4A digitizer iput sigl frequecy of.9 Hz ws used. A exmple of te frequecy spectr of te output sigl before d fter te correctio is sow i Fig. 5. Te THD ws improved by bout 5 db by mes of te correctio. Secodly, te 4-bit Digitizer NI PXI-59 ws tested. I tis cse two frequecies of iput sigl were used:.9 Hz d.5 MHz. Te oter coditios remied te sme. Te residuls before d fter te correctio sow more detils t te frequecy spectr i tis cse (see Fig. 6 d 7. I cse of Hz iput sigl te ysteresis sligtly iflueces te result. Te commo mode o-lierity is well suppressed by te correctio but te residul o-lierity of bout 5 LSB cused by ysteresis (differet for fllig d risig slopes remis. Te resultig commo-mode olierity decresed bout times. 6

6 Vldimir Hsz, Dvid Slepic, Petr Suce / Computig,, Vol., Issue, 56-6 Before correctio b After correctio Fig. 5 Frequecy spectr of digitized output sigl (VXI HP E4A x x Before correctio b After correctio Fig. 6 Itegrl o-lierity (NI PXI-59, fip.9 Hz 8 x Before correctio x b After correctio Fig. 7 Itegrl o-lierity (NI PXI-59, fip.5 MHz 6

7 Vldimir Hsz, Dvid Slepic, Petr Suce / Computig,, Vol., Issue, 56-6 However, for MHz iput sigl te ysteresis is bout two udreds times iger te for Hz d te differetil-mode o-lierity cused by te ysteresis is domit. Commo-mode itegrl o-lierity is suppressed well by te correctio ideed but te residul differetil-mode olierity remis d te remiig o-lierity fter te correctio is prcticlly te sme s before. 5. CONCLUSION Two metods for clcultio of coefficiets of te iverted polyomil used for ADC o-lierity correctio were itroduced, derived d compred. Te first oe uses more strigtforwrd derivtio of te coefficiets; te secod oe miimizes te lest squre error. Te first simultio did ot te ADC imperfectios (dditive oise, jitter i smplig, o-zero smpled sigl pse d ysteretic bevior ito ccout. Tey sowed tt te bot metods used for clcultio coefficiets re pplicble. Simultios verified tt te correctio is useble lso i te cses we oter ADC imperfectios re ot egligible. Tis sttemet ws lso cofirmed experimetlly. Te proposed metod of ADC o-lierity correctio usig polyomil pproximtio of te itegrl o-lierity INL( d its iverse fuctio gives mostly good results but ot lwys. It cocers e.g. digitizers wit oticeble ysteresis wic is prticulrly commo for sigls wit frequecy er te mximum iput frequecy of digitizers. Geerlly, iverse fuctio used for post-correctio of INL is frequecy depedet. For tis reso it cot be directly used for wide-bd sigls. However, tis issue lso cocers oter postcorrectio metods, e.g. loo-up tble. 6. REFERENCES [] IEEE 4- Stdrd for Alog to Digitl Coverters, Te Istitute of Electricl d Electroics Egieers,. [] L. Miceli, P. Miclo, J. Slig, Fst Gestig of ADC Usig Uified Error Model, Te 7t IMEKO world cogress, pp , Dubrovi, Croti,. [] A.C. Serr, M. F. d Silv, P. Rmos, R. C. Mrtis, L. Miceli, J. Slig, Combied Spectrl d Histogrm Alysis for Fst ADC Testig, IEEE Trsctios o Istrumettio d Mesuremet, (54 4 (5. [4] N. Björsell, P. Hädel, Acievble ADC Performce by Postcorrectio Utilizig Dymic Modelig of te Itegrl Nolierity, EURASIP Jourl o Advces i Sigl Processig, vol. 8. [5] V. Hsz, M. Komre, J. Roztocil, D. Slepic, P. Suce, System for Testig Middle-Resolutio Digitizers Usig Test Sigl up to MHz, IMTC/6 Proceedigs of te t IEEE Istrumettio d Mesuremet Tecology Coferece. [6] C.L. Moteiro, P. Arpi, A.C. Serr, A compreesive pse-spectrum pproc to metrologicl crcteriztio of ysteretic ADCs, IEEE Trsctios o Istrumettio d Mesuremet, (5 4 ( pp [7] P. Suce, V. Hsz, D. Slepic, ADC Nolierity Correctio Bsed o INL( Approximtios. IEEE Itertiol Worsop IDAACS 9. Rede, Itly 9, pp [8] P. Suce, V. Hsz, D. Slepic, Evlutio of ADC No-lierity Correctio Bsed o INL( Approximtio, BEC t Bieil Bltic Electroics Coferece [CD- ROM]. Tlli, Estoi, pp. -4. Vldimír Hsz, Hed of te Dept. of Mesuremet of te Czec Tecicl Uiversity i Prgue, member of te IMEKO TC-4 Mesuremet of Electricl Qutities, member of te IEEE Istrumettio & Mesuremet Society. He is iterested i mesuremet of dymic prmeters of AD modules icludig EMC. Dvid Slepič, received te PD degree i Electricl Egieerig i 5 from te Czec Tecicl Uiversity i Prgue. He is curretly doig reserc o ig-resolutio ADC testig. Te reserc ivolves te developmet of metodologies for log d digitl sigl processig d te desig of DAQ systems. Petr Sucáe received te PD degree i Electricl Egieerig i from te Czec Tecicl Uiversity i Prgue. Te mi re of is reserc cocers te AD coverter modellig, results preseted ere followes from is PD tesis. Curretly, e wors for privte compy i Czec Aerospce idustry. 6