Volterra Filtering for Integrating ADC Error

V olterra Filtering for Integrating ADC Error Correction,Based on an A Priori Error ModelPavol Mikulik and Ján ˇSaligaAbstract—Dynamic nonlinear effects contribute significantly toanalog-to-digital converters (ADCs)distortion.Volterra filtering can present an effective method for modeling and compensation of these phenomena.Considering an a priori error model of ADC allows finding an efficient inverse Volterra model for error correction.Method effectiveness is demonstrated by experimental measurements.Index Terms—ADC modeling,nonlinearity,volterra.I.I NTRODUCTIONDIGITIZATION chain performance improvement is still in the focus of scientific interest.The crucial part of digiti-zation systems is the analog-to-digital converter (ADC)or the system of a sample and hold circuit and an analog-to-digital con-verter (S/H-ADC)(in the following text,we use the term ADC for both ADC and S/H-ADC if not otherwise noted).A great effort has been devoted to the study of ADC error modeling and dependence on the particular ADC architecture [1].Typically,the ADC nonidealities are predominated by the dynamic non-linear behavior.The main motivation for study of ADC error models is to reduce ADC distortion.This paper is based on de-veloping an original idea initially published in [9].A.ADC Error Compensation Using Post-DistortionPost-distortion methods [2]–[6]are based on application of an inverse distortion to the digital output signal of the ADC.The simplest approach is to subtract the value of the integral non-linearityfunctionfrom each outputcode .In this case,theis considered a one-dimensional (1-D)function of outputcodeof the input signal [3].Error correction is thenachieved by subtracting the corresponding value of the phase plane from the output code,where the time slope can be esti-mated from the values of previous and subsequent samples.A similar method uses the current sample-previous sample phase plane [4].Manuscript received May 29,2001;revised May 7,2002.P.Mikulik is with the SCA Hygiene Products spol.s r.o.GemerskáHôrka,Slovak Republic (e-mail:pavol.mikulik@).J.ˇSaliga is with the Department of Electronics and Telecommunications,Technical University of Koˇs ice,Letná,Koˇs ice,Slovak Republic (e-mail:jan.saliga@tuke.sk).Digital Object Identifier 10.1109/TIM.2002.803513B.Volterra Filtering for ADC Error CompensationConceptually different methods utilize mathematical models of the dynamic nonlinear system,such as V olterra or Wiener models [5]–[7],[9].The V olterra model is an exact mathematical approach for description of causal time-invariant systems,where dynamic and nonlinear phenomena are present simultaneously [7].According to this model,the output signal of the nonlinear system can be expressed as a series of V olterra functionals,i.e.,by means of a series of multi-dimensional convolution integrals[5]–[10]is the input signal,is the output signal,andis the V olterra kernelofFig.1.Principle of error correction by means of an inversemodel.Fig.2.V olterra kernel identification by an adaptivemethod.Fig.3.Inverse volterra kernel identification by an adaptive method.coefficients were calculated on the base of the sequence of input signal values in the sampling instants,and the corresponding sequence of distorted output codes of the ADC,according to the least-mean-squares error optimization criterion.For the calibration signal,a sequence of low amplitude sinusoidal waveforms was used,superimposed on the dc level varying by steps smaller than the double of the sinewave amplitude,thus gradually mapping the whole ADC input range.Since sinewaves can be generated with high accuracy,problems concerning calibration signal selection and generation can be easily overcome.Moreover,the adaptive V olterra filter is not required since the (time invariant)V olterra filter coefficients are obtained directly.B.Volterra Filter Kernel CalculationFor clarification of V olterra filter kernels (i.e.,filter co-efficients)calculation,a brief review of time-invariant V olterra filtering theory is recalled.The discrete equivalent of the general formula (1)is[8]is the discrete input signal,is the filter outputsequence,and -thorderFig.4.Direct V olterra kernel calculation.V olterra kernel.The (2)is the mathematical model of a V olterrafilter of infinite order.In particular,the truncated model witha finiteorderis considered[8]and,according to a given optimization criterion.The least-mean-squares error optimization criterion can beconsideredfrom the finite sequence of deterministic input values and cor-responding outputcodes(13)whereis the vector of output code products obtained sim-ilarly as in (4),andbyapplying a test signal according to Fig.5(14)III.M ODIFICATION OF V OLTERRA S ERIES C ONSIDERING ANA-P RIORI ADC E RROR M ODEL The mathematical complexity of the V olterra series approach is the cost for its general ability of dynamic nonlinear system description.In many cases,this is a serious limitation of its practical application.However,knowledge of the structure of the system,where V olterra filtering will be applied,sometimes allows assuming symmetry properties of V olterra kernels,and on this basis,simplifying the V olterra kernels.In Section IV ,a modified V olterra filter expression is derived by exploiting the mathematical knowledge of the error in a given ADC architecture [1],[2].The method can be applied for ADC error correction,when the integralnonlinearityand the inputslope(15)Sinceistheis the sampling period.According to the conventional modified V olterra model [8]derived from (3),by using thesubstitution,the second-order V olterra series with memory ofsamples ,corresponding to the inverse nonlinearmodel(20)After further modification,this equation isobtained(a)(b)Fig.7.(a)Configuration of test setup.(b)Applied nonlinear circuit.to the inverse-model principle of Fig.2.Vector takes thefollowing form:4V toFig.8.(a)Noise before filtering.(b)Noise after filtering.filter models dynamic nonlinear error effects.The method can be considered as the natural evolution of the a priori approach to ADC error modeling[1],[2],[9],[12]toward a correction application.The a-priori approach makes easy the analytical derivation of the filter expression by taking into ac-count specific characteristics of the ADC error.This allows the following:1)in model definition,a more compact expression to be used;2)in model determination,a complicated adaptive scheme to be avoided(in each adaptation step a nonlinear equation system had to be solved);3)in filter identification, the V olterra filter coefficients to be computed very easily;and iv)in experimental calibration,a simpler signal to be used. The theoretical considerations were verified by a real exper-imental configuration.V olterra kernels of a nonlinear circuit, simulating the behavior of an Integrating ADC,were measured, and,consequently,method efficiency was evaluated by applying the measured kernels for error correction on the test signal.A significant performance improvement in terms of signal-to-noise ratio was achieved.Recently,we have been investigating possibilities to extend this method for other ADC architectures, such as sigma-delta ADCs.R EFERENCES[1]P.Arpaia,P.Daponte,and L.Michaeli,“A dynamic error model forintegrating analog-to-digital converters,”Measurement,vol.25,pp.255–264,1999.[2]L.Michaeli,“Fast dynamic methods of the systematic error autocorrec-tion,”in Proc.5th.Int.Symp.Electr.Meas.Instrum.Low Medium Freq., Vienna,Austria,April1992,IMEKO TC-4,pp.247–249.[3]T.A.Rebold and F.H.Irons,“A phase plane approach to the compen-sation of high speed analog-to-digital converters,”in Proc.IEEE Int.Symp.Circuits Syst.,Philadelphia,PA,May4–7,1987,pp.455–458.[4] F.H.Irons,D.M.Hummels,and S.P.Kennedy,“Improved compen-sation for analog-to-digital converters,”IEEE Trans.Circuits Syst.,vol.38,no.8,Aug.1991.[5]J.Tsimbinos and K.V.Lever,Applications of Higher Order Statisticsto Modeling,Identification and Cancellation of Nonlinear Distortion in High-Speed Samplers and Analogue-to-Digital Converters Using the Volterra and Wiener Models.New York:IEEE,1993.[6]J.Tsimbinos,“Identification and Compensation of Nonlinear Distor-tion,”Doctor of Philosophy,Inst.Telecommunications Res.School of Electron.Eng.,Univ.South Australia.[7] D.Mirri,G.Iuculano,F.Filicori,G.Pasini,and G.Vannini,“Mod-eling of nonideal dynamic characteristics in S/H-ADC devices,”in Proc.IMTC,Waltham,MA,Apr.22–26,1995,pp.27–32.[8] D.Kocur,“Algoritmy Adaptácie Adaptívnych V olterrových CíslicovýchFiltrov,”Habilitation(in Slovak),Technical University Koˇs ice,Koˇs ice, Slovak Republic,1994.[9]P.Arpaia and P.Mikulik,“Dynamic error correction of integratinganalog-to-digital converters by using volterra filtration,”in Proc.5th.Int.Workshop ADC Modeling Testing,Vienna,Austria,Sept.25–28, 2000,pp.39–44.[10] C.Evans,D.Rees,L.Jones,and M.Weiss,“Probing signals for mea-suring nonlinear volterra kernels,”in Proc.IMTC,Waltham,Mass,Apr.22–26,1995,pp.10–15.[11] D.Hummels,“Linearization of ADCs and DACs for all-digitalwide-bandwidth receivers,”in Proc.IWADC,4th IMEKO Workshop ADC Modeling Testing,Bordeaux,France,Sept.9–10,1999,pp.145–152.[12]P.Arpaia,P.Daponte,and L.Michaeli,“An a-priori approach to phase-plane modeling of SAR A/D converters,”IEEE Trans.Instrum.Meas., vol.47,no.4,pp.849–857,Aug.1998.Pavol Mikulik was born in Roˇzˇn ava,Slovak Republic,on August20,1961.He received the M.Sc.degree in electrical engineering in1985from the Depart-ment of Electronics and Telecommunications,Technical University of Koˇs ice, Koˇs ice,Slovak Republic.Since1996,he has been an external Ph.D.student with the same department in the field of ADC error correction.He is with SCA Hygiene Products s.r.o., GemerskáHôrka,Slovak Republic,as Electronic Support Manager.JánˇSaliga was born in1958.He received the Ing.degree in electrical engi-neering in1982and the Ph.D.degree in radioelectronics in1995from the Tech-nical University in Koˇs ice,Koˇs ice,Slovak Republic.Since1984,he has been Assistant Professor with the Department of Elec-tronics and Multimedial Telecommunication,Technical University in Koˇs ice, where he is now Associated Professor of electronic measurement,data acqui-sition systems,measurement in telecommunications,and computer interfacing. He was involved in many research and application,national and international projects.His present research activities are focused on ADC and DAC testing methodology,general measurement methods,virtual instrumentation,and dis-tributed measurement systems.。