VOLTERRA FILTERING FOR ADC ERROR CORRECTION. Abstract. 2. Volterra filtering for ADC error correction. Keywords. 1. Introduction

Transcription

1 Radoengneerng Volterra Flterng for ADC Error Correcton 39 Vol., No. 3, September P. MIKULIK, J. ŠALIGA VOLERRA FILERING FOR ADC ERROR CORRECION Pavol MIKULIK SCA Hygene Products spol. s r.o. 49 Gemerská Hôrka, Slovak Republc Ján ŠALIGA Dept. of Electroncs and Communcatons echncal Unversty of Košce, Letná 9/A, 4 Košce, Slovak Republc, Abstract Dynamc non-lnearty of analog-to-dgtal converters (ADC) contrbutes sgnfcantly to the dstorton of dgtzed sgnals. hs paper ntroduces a new effectve method for compensaton such a dstorton based on applcaton of Volterra flterng. Consderng an a-pror error model of ADC allows fndng an effcent nverse Volterra model for error correcton. Effcency of proposed method s demonstrated on expermental results. Keywords Analog-to-dgtal converters, Volterra flterng, dynamc non-lnearty correcton. Introducton Improvement of dgtzaton chan performance s constantly n the center of many scentfc studes. he crucal part of dgtzaton systems s the Analog-to-Dgtal Converter (ADC) or the system of a Sample and Hold Crcut and an Analog-to-Dgtal Converter (S/H-ADC). A bg effort has been devoted to study of ADC error modelng on dependence of the partcular ADC archtecture []. ypcally, the non-dealtes of an ADC are predomnated by ts dynamc non-lnear behavor. he man motvaton for study of ADC error models s to reduce the ADC dstorton. he man effort of many studes s developng a method, whch enables compensaton of ADC nonlnearty by post nverse dstorton of recorded dgtzed samples [-6]. he smplest approach s to subtract the value of ntegral non-lnearty functon INL(k) from each output code. In ths case, the INL s consdered as a one-dmensonal functon of output code k, and t descrbes the statc features of ADC,.e. the behavor of the ADC wth constant or slowly vared nput sgnal. For proper characterzaton of ADC behavor wth hgher frequency nput sgnal, a more complex model s ntroduced (phase plane approach), where a two-dmensonal INL s consdered, whch s the functon of output code k, and tme slope s of the nput sgnal [3]. Error correcton s then acheved by subtractng the correspondng value of the phase plane from the output code, where the tme slope can be estmated from values of prevous and subsequent samples. A smlar method uses current sample-prevous sample phase plane [4].. Volterra flterng for ADC error correcton Conceptually dfferent methods utlze mathematcal models of dynamc non-lnear system, such as Volterra or Wener models [5-7], []. he Volterra model s an exact mathematcal approach for descrpton of causal tmenvarant systems, where dynamc and non-lnear phenomena are present smultaneously [7]. Accordng to ths model, the output sgnal of the non-lnear system can be expressed as seres of Volterra functonals [5-]: dµ () j = j= j= = h... h ( µ,µ,...,µ ) x(t-µ j) where x(t) s the nput sgnal, y^(t) s the output sgnal, and h (µ, µ,, µ ) s the Volterra kernel of the -th order. Several studes have been devoted to Volterra kernel dentfcaton [5-7], [,]. In the conventonal methods, usually an adaptve Volterra flter s utlzed n a confguraton called dentfer of non-lnear systems (Fg. ) [5], [8-]. Once adapted, the flter provdes the Volterra kernels modelng a specfc non-lnear system under test. Input x(t) Dynamc non-lnear system Adaptve Volterra flter ŷ() t Fg. Volterra kernel dentfcaton by an adaptve method From the measured kernels, the nverse Volterra kernels can be derved analytcally n order to determne an -

2 4 Volterra Flterng for ADC Error Correcton Radoengneerng P. MIKULIK, J. ŠALIGA Vol., No. 3, September nverse model of the system [5]. hs allows the non-lnear system error to be corrected accordng to the nverse-model, by applyng post-dstorton. Inverse Volterra kernels can be also obtaned drectly by usng the confguraton presented n Fg.. Input x(t) Dynamc non-lnear system Adaptve Volterra flter ĝ() t Fg. Inverse Volterra kernel dentfcaton by an adaptve method However, the prncple of non-lnear system dentfcaton by means of adaptve Volterra flters requres a statonary nput sgnal [5], [8], []. hs causes the twofold problems of tryng to fnd a proper calbraton sgnal. he calbraton sgnal has to: () map the phase plane k, s and the memory plane (e.g. the state-prevous state k, k- plane) smultaneously, () be statonary over the adaptaton process n order to have nvarant condtons durng the adaptaton. Moreover, ths sgnal should be generated wth suffcent accuracy wth relaton to the tested ADC. Generaton such a sgnal establshes a serous problem.. Volterra kernel dentfcaton usng determnstc calbraton sgnal For reasons descrbed above, a dfferent approach was proposed by applcaton of a known determnstc calbraton sgnal on the nput of the tested ADC (Fg. 3) []. Generator of calbraton sgnal ADC under test Fg. 3 Drect Volterra kernel calculaton _ Calculaton of Volterra flter coefcents Volterra flter coeffcents were calculated on the base of the sequence of nput sgnal values n the samplng nstants, and the correspondng sequence of dstorted output codes of the ADC, accordng to the least-mean-squares error optmzaton crteron. As calbraton sgnal a sequence of low ampltude snusodal waveforms was used, supermposed on DC level varyng by steps smaller than the double of snewave ampltude, thus gradually mappng the whole ADC nput range. he problems concernng calbraton sgnal selecton and generaton can be easly overcome because of possblty to generate smply the snewave wth hgh accuracy. Moreover, the adaptve Volterra flter s not requred, snce the tme nvarant Volterra flter coeffcents are obtaned drectly.. Volterra flter kernels calculaton he dscrete equvalent of the general formula () s accordng to [8-]: = h... h m, m,..., m = m= m= m= j= x(n mj) where x( s the dscrete nput sgnal, y^( s the flter output sequence, and h s the element of th order Volterra kernel. he equaton () s the mathematcal model of a Volterra flter of nfnte order. In partcular, the truncated model wth a fnte order M, and a fnte memory of samples N s consdered: = h M N N... m, m,..., m = m= m= m= j = N h () x(n mj). (3) Organzng the Volterra kernel elements h and h, and the nput sample products j= x ( n mj) nto convenent block vectors H( and X(, respectvely, the (3) can be rewrtten as: y ( = H ( X( = X ( H( he desgn of an optmal Volterra flter requres the search for a set of Volterra kernels that mnmzes the error functon between the desred sgnal, and the flter output y^(, accordng to a gven optmzaton crteron. Meansquare error optmzaton crteron can be consdered: [ e ( n )] = E y ( n ) y ( n ) mn (4) E (5) he soluton leads to the equaton: * R H =, (6) R XY provdng that det R, (7) H * = R R XY (8) where: H * s the vector of the optmum Volterra kernels (optmal flter coeffcents), [ X( ] R = E (9) XY s the hgher-order mutual correlaton vector of the nput and the desred sgnals, and [ X( X ( ] R = E () s the hgher-order autocorrelaton matrx of nput sgnal [8], []. Although R, and R XY are statstcal characterstcs, prmarly defned on the base of the statstcal proper-

3 Radoengneerng Volterra Flterng for ADC Error Correcton 4 Vol., No. 3, September P. MIKULIK, J. ŠALIGA tes of the nput and the desred sgnals, ther value can be estmated from the fnte sequence of determnstc nput values and correspondng output codes: L R XY x( X( () L L n= n = L R X () ( X ( where X( s the vector of output code products obtaned smlarly as n (4), L s the length of the data sequence..3 Verfcaton of method effcency Method effcency s evaluated n terms of sgnal-tonose rato by applyng a test snewave sgnal accordng to Fg. 4. he sgnals from tested ADC before and after Volterra flterng are recorded. hen the deal sgnal s computed by four parameters best fttng method (sne-wave ampltude, DC offset, phase and normalzed sgnal frequency to samplng frequency) [6]. Subtractng the calculated deal sne-wave and recorded data the addtve nose s determned and consecutvely analyzed. Generator of calbraton sgnal ADC under test Volterra flter Fg. 4 Verfcaton of developed method effcency 3. Modfcaton of Volterra seres consderng an a-pror ADC Flterng gan evaluaton Mathematcal complexty of Volterra seres approach s a cost for ts general ablty of dynamc nonlnear system descrpton. In many cases, ths s a serous lmt of practcal applcaton. However, knowledge of structure of the system, where Volterra flterng has to be appled, sometmes allows assumng symmetry propertes of Volterra kernels, and on ths base smplfyng the Volterra kernels. In the followng secton, modfed Volterra flter expresson s derved by explotng the mathematcal knowledge of the error n a gven ADC archtecture [], []. he method can be appled for ADC error correcton, when the ntegral non-lnearty INL Q (k,s), consdered n terms of a two-dmensonal error functon of the output code k and the nput slope s presents a smooth surface and can be modeled wth a low-order polynomal. hs s partcularly fulflled for ntegratng ADCs (IADCs) [], [3], [4]. In typcal IADCs, INL Q error functon can be modeled []: INL Q ( k, s) = B Bk B k B s B ks B s... (3) Consderng order of flter M=, and a memory of samples N=3, accordng to the conventonal modfed Volterra model [8] based on equaton (3) we obtan: = s s s n ) s n ) s s k ( s k ( n ) s n ) s n ) n ) s n ) k ( n ) (4) where the subscrpted coeffcents s denote conventonal modfed Volterra kernels. Snce ING Q (k, s) k = k - k d [], where k d s the k- th deal code, thus k d k - ING Q (k, s). On ths bass, the mathematcal model of the Volterra flter s expressed n the followng form:! = kd ( = B ( B ) Bk ( (5) B s( B s( B s ( he slope s estmated by central dfference equaton [3]: ( [ n ) n ) ] S 5 s =, (6) where S s the samplng perod. Usng the substtuton n=m-, the second-order Volterra seres correspondng to the followng nverse nonlnear model G' n can be stated: m ) = g' g' m) g' m ) g' m ) g' g' g' m) m ) g' k ( m ) g' k ( m) m) m ) m ) m ) g' k ( m ) Substtutng (6) to (7), assumng from symmetry that: g =, g' ' g' g' the followng equaton s obtaned: g' 4 (7) =, g ' = g ' /= g, (8) m ) = g' g' m ) g' k ( m ) g' S g' s ( m ) S ' S s( m ) m ) s( m ) After further modfcaton, t s obtaned: = g g g s( g k ( g s( g s ( (9) () Comparng (5) and (), eqn. () provdes sought formula for the Volterra flter necessary for the correcton of an ADC accordng to the nverse-model prncple. Vector X( takes the followng form: [ ] ( =,, s(, k (, s(, s ( X () R, and R XY are calculated accordng to (), (). he

4 4 Volterra Flterng for ADC Error Correcton Radoengneerng P. MIKULIK, J. ŠALIGA Vol., No. 3, September vector of the optmal flter coeffcents s obtaned analogously as n (8): G [ g, g, g, g 3, g 4, g 5 ] = R R XY = () R R U - R4 D D C R3 he fnal flter expresson for corrected output calculaton s the followng: L7 y ( = G X( (3) Fg. 6b Appled dynamc non-lnear crcut 4. Expermental results For expermental verfcaton of the method, the followng confguraton was chosen: he non-deal ADC s substtuted by an analogue crcut modellng dynamc nonlneartes, and a subsequent "almost deal" ACD. (Fg. 5). he almost deal ADC was mplemented by usng a precse -bt ADC and consderng the upper 9 bts only. Input sgnal x(t) Dynamc, weakly non-lnear crcut Almost deal ADC Fg. 5 Realzed model of non-deal ADC hs method allows ncludng a controlled dstorton, and studyng the flterng effects on a physcal model wth known propertes. he measurement was realzed n confguraton accordng to Fg. 6. Both calbraton, and flter effcency verfcaton sgnals were generated wth precson drect dgtal synthess generator (HP Aglent). he dynamc non-lnear crcut was created usng L7 operatonal amplfer wth nonlnear elements n the feedback loop. For data acquston a Natonal Instruments A-MIO-6E- multfuncton I/O board was used. he needed testng and data processng software was developed partally n software developng envronment LabWndows/CVI (Natonal Instruments) and partally n Matlab. PC Flterng effectveness s demonstrated on snusodal test sgnal wth frequency 6.4 Hz, ampltude.5v, and offset -V. Fgure 7a shows a detal from nose before flterng,.e. the dfference sgnal between best fttng snewave estmaton, and measured data on the output n the tme doman. Fgure 7b shows nose after flterng,.e. the dfference sgnal between nput, and fltrated output. Coeffcent Value ab. Correctng Volterra flter coeffcent calculated from results of measurement he man mprovements are () decreasng nose peak value almost tmes () a notable correcton of DC offset error Precse analyss of data before and after flterng ndcates that the flterng mproved sgnal-to-nose rato by 6.5dB. he exact ponts of sgnal mprovements can be found out from the spectrum after transformaton the expermental results nto frequency doman (Fg. 8). Sgnal generator Dynamc non-lnear crcut Hgh qualty multfuncton plug-n PC card NI A MIO-6E- Fg. 6a Confguraton of test setup he calbraton sgnal was created as a sequence of snusodal sgnals of frequency 5 Hz, and ampltude V supermposed on DC level that gradually vared from -4V to 4V wth step V. Actual values on the nput were estmated accordng to four-parametrc method of best-fttng sne-wave [5], [6]. Volterra flter coeffcents were calculated accordng to (), and are presented n ab.. Fg. 7 Detals of expermental results n tme doman: a) nose before flterng, b) nose after flterng

5 Radoengneerng Volterra Flterng for ADC Error Correcton 43 Vol., No. 3, September P. MIKULIK, J. ŠALIGA he method can be consdered as the next evoluton step of the a-pror approach to ADC error modellng [-], [], [4] toward to applcaton of an error correcton. he a- pror approach makes the analytcal dervaton of flter expresson easy by takng specfc characterstcs of the ADC error nto account. hs allows: () a more compact expresson to be used n model defnton () avodng to a complcate adaptve scheme n model determnaton (n each adaptaton step a non-lnear equaton system had to be solved), () easy computaton of the Volterra flter coeffcents n flter dentfcaton, (v) usng smpler sgnal n expermental calbraton. he theoretcal consderatons were verfed on real expermental stand. Volterra kernels of a non-lnear crcut, smulatng the behavor of an IADC, were measured, and consequently effcency of the developed method was evaluated by applyng the measured kernels for error correcton on test sgnal. A sgnfcant performance mprovement n terms of sgnal-to-nose rato was acheved. References Fg. 8 Detals of expermental results n frequency doman: power spectral denstes and power spectral dstrbuton functons of nose before and after flterng he man mprovements are () decreasng nose peak value almost tmes () a notable correcton of DC offset error Precse analyss of data before and after flterng ndcates that the flterng mproved sgnal-to-nose rato by 6.5 db. he exact ponts of sgnal mprovements can be found out from the spectrum after transformaton the expermental results nto frequency doman (Fg. 8). Expermental results transformed nto frequency doman are shown n Fg. 8. he addtve nose before and after flterng s expressed n form of power spectral densty functon and power spectral dstrbuton functon. Especally the power spectral dstrbuton functon very clearly ndcates the advance n sgnal after fltraton. he most evdent mprovements are: compensaton of DC offset by 5.5 db, decreasng dstorton caused by the second harmonc (.8 Hz) by 4. db. 5. Conclusons A new method based on Volterra flterng for nversemodel correcton of the ADC error has been proposed. he Volterra flter models effects of dynamc non-lnear error. [] ARPAIA, P., DAPONE, P., MICHAELI, L. A dynamc error model for ntegratng analog-to-dgtal converters. Measurement. 999, vol. 5, p [] MICHAELI, L. Fast Dynamc methods of the systematc error autocorrecton. In Proc. of the 5-th. Internatonal Symposum on Electrcal Measurng Instruments for Low and Medum Frequences. Venna: IMEKO C-4. 99, p [3] REBOLD,. A., IRONS, F. H. A phase plane approach to the compensaton of hgh speed analog-to-dgtal converters. Proceedngs of the IEEE Internatonal Symposum on Crcuts and Systems. Phladelpha: IEEE, 987, p [4] IRONS, F. H., HUMMELS, D. M., KENNEDY, S. P. Improved compensaton for analog-to-dgtal converters. IEEE ransactons on Crcuts and Systems. 99, vol. 38, no. 8. [5] SIMBINOS, J., LEVER, K. V. Applcatons of hgher order statstcs to modelng, dentfcaton and cancellaton of nonlnear dstorton n hgh-speed samplers and analogue-todgtal converters usng the Volterra and Wener models. IEEE, 993. [6] SIMBINOS, J. Identfcaton and compensaton of nonlnear dstorton, A thess submtted n accordance wth the requrements for the degree of Doctor of Phlosophy, Insttute for elecommuncatons Research School of Electronc Engneerng, Unversty of South Australa. [7] MIRRI, D., IUCULANO, G., FILICORI, F., PASINI, G., VANNI- NI, G. Modelng of non-deal dynamc characterstcs n S/H- ADC devces. Proceedngs of IMC 95. Waltham. 995, p [8] KOCUR, D. Algortmy adaptáce adaptívnych volterrových číslcových fltrov (n Slovak language). hess for habltaton. echncal Unversty Košce. 994, p. 9-. [9] KOCUR, D. Návrh časovo nvarantných nelneárnych Volterrových číslcových fltrov(n Slovak language), New trends n

6 44 Volterra Flterng for ADC Error Correcton Radoengneerng P. MIKULIK, J. ŠALIGA Vol., No. 3, September sgnal processng, part., Lptovský Mkuláš: ČSVS at VVŠ. 99, p [] KOCUR D. Adaptve Volterra flters: a tutoral revew. Dgtal sgnal processng and communcatons, Proceedngs of workshop, Oradea, Romana, 995, p. -3 [] ARPAIA, P., MIKULIK, P. Dynamc error correcton of ntegratng analog-to-dgtal converters by usng Volterra fltraton. Proceedngs of 5th. Internatonal Workshop on ADC Modellng and estng. Venna: U Venna., p [] EVANS, C., REES, D., JONES, L., WEISS, M. Probng sgnals for measurng nonlnear Volterra kernels, Proceedngs of IMC 95. Waltham. 995, p. -5. [3] MICHAELI, L: Bystroje zmerenje pogresnostej analogovo cfrovych pereobrazovatelej (n Russa. Izmertelnaja echnka. 993, no., p. 7-. [4] ARPAIA, P., DAPONE, P., MICHAELI, L. An a-pror approach to phase-plane modellng of SAR A/D converters. IEEE ransactons on Instrumentaton and Measurement. 998, vol. 47, no. 4, p [5] MICHAELI, L., ŠALIGA, J., SEDLÁK, V. An approach to dagnostc of the AD converter embedded on AMEL mcrocontrollers. Proceedngs of 4 th Workshop on ADC Modelng and estng. Bordeaux. 999, p [6] IEEE Std , Standard for Dgtzng Waveform Recorders". About authors... Pavol MIKULIK was born n Rožňava, Slovak Republc, on August, 96. He receved the Ing. (M. Sc.) degree n Electrcal Engneerng n 985 on the Dept. of Electroncs and elecommuncatons, on echncal Unversty of Košce. Snce 996 he s an external PhD student on the same department n the feld of ADC error correcton. He works n SCA Hygene Products s. r. o. Slovaka as Electronc Support Leader. Ján ŠALIGA receved Ing. degree n Electrcal Engneerng n 98 and hs PhD n Radoelectroncs n 995 from the echncal Unversty n Košce. Snce 984 he has been assstant professor at echncal Unversty n Košce, Dept. of Electroncs and Multmeda elecommuncaton. Hs research actvtes cover ADC and DAC testng methodology, vrtual nstrumentaton and dstrbuted measurement systems. RADIOENGINEERING REVIEWERS September, Volume, Number 3 I. BALÁŽ, Slovak Unversty of echnology P. GALAJDA, echncal Unversty, Košce I. CHROMÝ, Brno Unversty of echnology, Brno M. KLÍMA, Czech echncal Unversty, Prague D. KOCUR, echncal Unversty, Košce Z. KOLKA, Brno Unversty of echnology, Brno J. KOLOUCH, Brno Unversty of echnology, Brno M. LAIPER, Czech echncal Unversty, Prague M. MACHO, Mltary Research Insttute, Brno S. MARCHEVSKÝ, echncal Unversty, Košce L. MICHAELI, echncal Unversty, Košce J. POSPÍŠIL, Brno Unversty of echnology, Brno L. POUŠEK, Czech echncal Unversty, Prague A. PROKEŠ, Brno Unversty of echnology, Brno I. PROVAZNÍK, Brno Unversty of echnology, Brno Z. RAIDA, Brno Unversty of echnology, Brno J. ROZMAN, Brno Unversty of echnology, Brno V. ŘÍČNÝ, Brno Unversty of echnology, Brno Z. SMÉKAL, Brno Unversty of echnology, Brno R. ŠÁRA, Czech echncal Unversty, Prague V. ŠEBESA, Brno Unversty of echnology, Brno