We present an offline calibration method to correct the non-linearity due to DAC element mismatch in distributed feedback SigmaDelta-modulation A/D-converters. The improvement over previous methods is that not only the first feedback DAC is calibrated, but also the DACs that are coupled to later stages can be calibrated as well. This is needed in the case of Sigma Delta modulators with a low OSR, where the contribution of the second feedback DAC should not be neglected. The technique is based on a calibration measurement with a two-tone input signal

Babaie Fishani, Amir

De Bock, Maarten

Rombouts, Pieter

English

Ghent University Academic Bibliography

1Improved offline calibration for DAC mismatch inlow OSR Sigma Delta ADCs with distributedfeedbackMaarten De Bock, Amir Babaie-Fishani and Pieter RomboutsThis document is an author’s draft version submitted for publication to the 2014 IEEE Int. Symp. Circuits and Syst. (ISCAS2014).The actual version was published as [1].REFERENCES[1] M. De Bock, A. Babaie-Fishani, and P. Rombouts, “Improved offline calibration for DAC mismatch in low OSR Sigma Delta ADCs with distributedfeedback,” in Proc. IEEE Int. Symp. Circuits and Syst. (ISCAS), 2014, pp. 938–941.Improved offline calibration for DAC mismatch inlow OSR Σ∆ ADCs with distributed feedbackMaarten De Bock, Amir Babaie-Fishani, Pieter RomboutsGhent University (UGent), Dept. ELIS, Sint-Pietersnieuwstraat 41, 9000 Ghent, BelgiumEmail: maarten.debock,amir.babaiefishani,rombouts@elis.UGent.beAbstract—We present an offline calibration method to correctthe non-linearity due to DAC element mismatch in distributedfeedback SigmaDelta-modulation A/D-converters. The improve-ment over previous methods is that not only the first feedbackDAC is calibrated, but also the DACs that are coupled to laterstages can be calibrated as well. This is needed in the case ofSigma Delta modulators with a low OSR, where the contributionof the second feedback DAC should not be neglected. Thetechnique is based on a calibration measurement with a two-toneinput signal.I. INTRODUCTIONThe continuous evolution toward higher bandwidth hasincreased the interest for Σ∆-modulators with a low over-sampling ratio (OSR). In this case multibit quantization isalmost unavoidable. It is well known that the resolution isthen limited by the linearity of the feedback digital-to-analogconverter (DAC) [1]. This way, some kind of linearizationscheme is needed if linearity beyond the intrinsic devicematching is required. For this purpose, in the past, dynamicelement matching (DEM) techniques have been successfullyexploited [1]. With such techniques the DAC mismatch errorsare spectrally shaped and as a result greatly reduced withinthe low-frequency signal band as well. But DEM techniquesinherently become less efficient when the OSR is reduced anda larger fraction of the Nyquist band is occupied by the signalband. This is particularly true for the popular Data weightedaveraging (DWA)[2] which achieves only first order shapingof the DAC mismatch errors. Moreover, such DEM-techniquemust act upon the feedback DAC, which is inside the loop.This way all DEM-techniques put stress on the latency budgetand become problematic at high clock speeds.A solution for these problems is to compensate the DACmismatch errors in the digital domain. This can be done outsidethe modulator loop and as such doesn’t limit the modulatorspeed. The implementation of this is based on a look-up table(LUT) in which a digital estimation of the DAC mismatcherrors is stored. This LUT is then used to correct the digitaloutput of the modulator during normal operation. Obviously,the performance of such a LUT-based compensation schemedepends on the accuracy of the digital estimation of themismatch errors. Therefore, the actual calibration, i.e. theprocess of determining the mismatch errors, is essential. Ina related work [3] a background calibration scheme for thiswas presented. But this increases the modulator complexityand also adds some blocks inside the loop, which might be aproblem in high speed modulators. An alternative is the offlinecalibration of [4]. It is based on a single measurement with aspectrally pure sine wave, from which the DAC mismatch er-Fig. 1. A CIFB Σ∆-modulator structure.rors are then estimated. This work was focused on modulatorswhere only one global feedback DAC needed to be calibrated.However, in a distributed feedback Σ∆-modulator, multiplefeedback DACs coupling to the successive integrator stages,are used (see fig. 1). Usually the first DAC will dominate theoverall distortion and by neglecting the errors of the otherDACs we can use the technique of [4]. However, as will beshown in section II, at low OSR also the contribution of thesubsequent DACs (DAC2, DAC3, ...) will become significantand limit the performance.In this paper, we present an improved offline calibrationmethod and compensation scheme that corrects the errorcontribution of these subsequent DACs as well. For this, westart with an exact analysis of the non-linearity of the overalldistributed feedback modulator due to DAC element mismatchin section II. The results of this analysis are then used insections III and IV where the improved calibration methodand compensation scheme are derived respectively. Section Vpresents the measurement results on an integrated prototypewhere the presented method is used for DAC mismatch com-pensation.II. EXACT ANALYSIS OF NON-LINEARITY DUE TOELEMENT MISMATCHFig. 1 shows a Σ∆-modulator with a distributed feedbacktopology (CIFB as in [1]). A quantizer is embedded in a controlloop with loop filter H(z). The loop filter H(z) is of Kth-order and consists of a cascade of K integrators, each witha gain ck. The digital output D of the multibit quantizer isfed back toward every integrator stage of the loop filter usingK feedback paths. In every feedback path the DAC convertsthe digital output d into an analog signal vk to which a gainak is applied. The additional feed-in paths of the input signalVin are added to create what is commonly known as a low-distortion topology [5]. A common analysis for this systemmodels the quantizer as an additive white noise contributionQ and all DACk in the feedback path are assumed to be ideal.The digital output D(z) of the complete system can then bewritten in the Z-domain as:D(z) =(H(z)1 +H(z)+ 1)︸ ︷︷ ︸STF(z)Vin(z) +11 +H(z)︸ ︷︷ ︸NTF(z)Q(z) (1)From eq. (1) it is readily seen that STF (z) = 1. Thequantization noise transfer function (NTF ) can in general bewritten as:NTF (z) =∏Kk=1(z − zk)PK(z)(2)Here, PK(z) is an Kth-order polynomial describing the poleplacement of the NTF . The zeros zi can be optimized byapplying local feedback around the integrator stages [6] andas such spreading them over the low pass signal band. Forthe remainder of this paper however, we will assume that allzeros zk = 1. This will ease the equations, but it should benoted that the analysis also holds for the general case wherethe NTF-zeros are spread over the signal bandwidth.In practice, all DACk will have some mismatch. We willassume that each DAC has a similar topology and consistsof N levels. To analyze the effect of this mismatch, first weintroduce the DAC-level selection signals xi, which are definedas:xi (n) ={+1, if the ith DAC-level is selected0, else(3)This way, there are N selection signals, and there is alwaysexactly one selection signal high at the same time. We cannow write each feedback signal vk asvk(n) =N∑i=1xi(n)·Ck,i (4)where Ck,i corresponds to the ith DAC-level of the kthfeedback DAC. The DAC non-linearity manifests itself inthe sense that the actual DAC levels Ck,i deviate from theirnominal values Cnm,k,i by a mismatch error εk,i. If we assumeall DACs to be equal, we can assume Cnm,k,i = Cnm,i. EachDAC output vk can now be written asvk(n) =N∑i=1xi(n)·(Cnm,i + εk,i) = vnm(n) +N∑i=1xi(n)·εk,i︸ ︷︷ ︸eDAC,k(n)(5)where we have introduced eDAC,k, the error signal of the ithDAC. Every feedback signal can thus be written as a nominalfeedback signal vnm and an error contribution eDAC,k.Fig. 2 then shows the model of the CIFB-structure wherethe mismatch signals eDAC,k are added to the system. Eachfeedback DACk is modeled as an ideal DAC with output vnmwith added mismatch contribution eDAC,k. If we now analyzethe digital output D of this modulator model, we find:D(z) = Vin(z) + NTF (z)Q(z)−K∑k=1ETF k(z)EDAC,k(z)(6)Fig. 2. The CIFB Σ∆-modulator model where the DACs are replaced byan ideal DAC and additional error signals eDAC,k .0 0.2 0.4 0.6 0.8 1−200−160−120−80−400frequency (f/fBW)(a)outputPSD(dB)0 0.2 0.4 0.6 0.8 1−200−160−120−80−400frequency (f/fBW)(b)Fig. 3. The output PSD of Σ∆-modulator with only mismatch in DAC2 (a)without and (b) with the proposed compensation method (eq. (19).Here, we have introduced the error transfer function (ETF )for each error signal eDAC,k, which denotes the contributionof that error signal to the output D. Each ETF k can be writtenas:ETF k(z) = NTF (z)akK∏j=kcjz − 1 (7)If we look at the contribution of the mismatch of the first DAC(eDAC,1) and using (2), we find:ETF 1(z) = NTF (z)a1(z − 1)KK∏j=1cj =a1PK(z)K∏j=1cj (8)We can write the contribution of subsequent DAC errors(eDAC,2, eDAC,3, ... eDAC,K) in terms of ETF 1(z):ETF k(z) =aka1(z − 1)k−1∏k−1j=1 cjETF 1(z) (9)To show that the contribution of the subsequent DACs canindeed be significant, fig. 3(a) shows the base-band outputpower spectral density (PSD) of a 3rd-order Σ∆-modulator(the NTF is designed according to [6] with ||H||∞ = 2.4and OSR=16) where only DAC2 has mismatch. The modu-lator uses 32 unit elements, each with a normally distributedrandom mismatch with σ = 1%. The linearity of this two-tone measurement is limited by the 3rd-order intermodulationproduct (IM3) at −71 dB.III. ESTIMATION OF MISMATCH ERRORSThe estimation of the mismatch errors occurs in an off-lineprocedure prior to the normal operation. For this we apply asignal s(t) to the Σ∆ ADC. Based on (6), we can write theoutput d(n) of the modulator as :d(n) = s(n) + ntf (n) ? q(n)−N∑k=1etfk (n) ? eDAC,k(n)(10)We will use a digital decimation low-pass filter L(z) (with cor-responding impulse response l(n)) to remove the quantizationnoise contribution:dLP(n) =sLP(n) + enoise,LP(n)−K∑k=1l(n) ? etfk (n) ? eDAC,k(n) (11)Here, enoise,LP contains the electrical noise as well as theresidual quantization noise that may still be present. Applyingthe definition of the DAC error signals eDAC,k of eq. (5) wefind:dLP(n) =sLP(n) + enoise,LP(n)−K∑k=1N∑i=1l(n) ? etf k(n) ? xi(n)·εk,i (12)Now, we introduce the filtered selection signals xEL,k,i =l(n) ? etf k(n) ? xi(n) and the residue signal r(n), which isequal to:r(n) = dLP(n)− sLP(n) (13)If the applied signal s(t) is known, then also it’s digitizedversion sLP(n) is known. E.g. if s(t) is a combination ofspectrally pure sine waves, sLP(n) can easily be estimatedusing standard curve fitting algorithms such as IEEE-STD-1057 [7], [8]. The residue signal r(n) is then evaluated as theremaining curve fit error after elimination of the different sinewaves and hence can be considered to be known. We now find:r(n) = enoise,LP(n)−K∑k=1N∑i=1xEL,k,i(n)·εk,i (14)From this equation, we observe that we can obtain an esti-mation ε̂i,j of the different DAC errors by performing a leastmean square minimization (LMS):E(r(n) +K∑k=1N∑i=1xEL,k,i · ε̂k,i)2= E{e2noise,LP}︸ ︷︷ ︸noise variance+ E(K∑k=1N∑i=1xEL,k,i · (ε̂k,i − εk,i))2︸ ︷︷ ︸calibration error variance(15)Observing the calibration error variance, we find that the LMSoptimization corresponds to a correct estimation of εi,j . Inpractice, we will use a finite measurement interval. As a result,(15) must be approximated as the LMS over a finite data set8Fig. 4. The calibration setup with a two-tone test signal s(t) .of T data points. This way, we will find the DAC errors εk,iby minimizingT∑n=1(r(n) +K∑k=1N∑i=1xEL,k,i(n) · ε̂k,i)2(16)with regard to ε̂k,i. This can be expanded into∂∂ε̂j,mT∑n=1(r(n) +K∑k=1N∑i=1xEL,k,i(n) · ε̂k,i)2= 0,∀j,m(17)with j = 1..K and m = 1..N . This yields a system of K×Nequations (N mismatch errors for K feedback DACs) for theK ×N unknown DAC errors ε̂k,i:T∑n=1r(n)xEL,j,m(n) =T∑n=1K∑k=1N∑i=1xEL,j,m(n)xEL,k,i(n) · ε̂k,i ,∀j,m (18)All coefficients in this system can easily be evaluated andsolving this system is trivial. Fig. 4 then shows the calibrationmeasurement setup. As shown in eq. (9), the contribution of thesubsequent DACs (DAC2, DAC3, ...DACK) is differentiated.It is however important that the applied calibration signals(t) results in a non-linear contribution in the output that isvery sensitive to the contribution of the error signals of thesubsequent DACs (eDAC,2, eDAC,3, ... eDAC,K). Therefore, thecalibration setup must be a two-tone test with frequencies atthe edge of the signal band where the differentiation factor(z − 1) is at its maximum. Eliminating the two sine wavesfrom the output signal is trivial using IEEE-STD-1057, and theremaining residue of the double sine fit is r(n) as in eq. (13).IV. LUT-BASED COMPENSATIONA. General compensation schemeAfter estimating the DAC mismatch errors ε̂k,i in an offlinecalibration cycle, they are permanently stored in a look-uptable (LUT). Fig. 5 then shows the modulator during normaloperation. The corrected output dcal(n) is then calculated as:dcal(n) = d(n) +N∑i=1K∑k=1etfk (n) ? xi(n)·ε̂k,i (19)Fig. 3(b) shows the simulated output spectrum of the sameΣ∆-modulator as in fig. 3(a), but this time using the proposedcompensation method (eq. (19). All distortion tones are nowreduced to beneath the quantization noise floor.Fig. 5. The Σ∆-modulator during normal operation with DAC mismatchcompensation.B. DiscussionThe filters ETF k(z) are of Kth-order (the order of themodulator) and the coefficients of PK are in general noninteger and as such require considerable calculation effort. Thisis not a problem during the one time offline calibration cyclewhere the mismatch errors are determined, but applying thesefilters to the selection signals during normal operation mightadd considerable effort. The compensation scheme of fig. 5can be simplified by reducing the complexity of the ETF k(z).Eq. (8) can be approximated to 1 for signals at DC (z ≈ 1):limz→1ETF1(z) = 1 (20)Using this, also the higher order ETF k(z) complexity can befurther reduced, as (9) can now be approximated as:ETF k(z) =aka1(z − 1)k−1∏k−1j=1 cj(21)The differentiation factor (z − 1)k−1 is easily calculated andthe gain factor ak/(a1∏k−1j=1 cj) in general can be expected tobe close to integer.As expected, we find that the contribution of each eDAC,kis (k − 1)-times differentiated. For high OSR, z − 1 ≈ 0 forsignals within the signal band and the contribution of all butthe first DAC can be neglected when looking at the non-linearity of the overall modulator. However, when targetinghigh accuracy Σ∆-modulators with low OSRs, the assumptionz − 1 ≈ 0 does not hold anymore, and the contribution of theother DACs (in practice DAC2) can not be neglected. Themismatch errors in the subsequent DACs are also increasedas subsequent integrator stages are in general scaled down toreduce power consumption. As a result, these DACs will bemade with smaller devices which will increase their mismatch.V. EXPERIMENTAL RESULTSThe proposed calibration procedure and compensationscheme are tested on an integrated switched-capacitor Σ∆-modulator prototype with 3rd-order noise shaping and a signalbandwidth of 1.25 MHz for an OSR of 12. This is not a state-of-the art Σ∆-modulator in terms of signal bandwidth, but it ishere only used to illustrate the proposed calibration procedure.The quantizer has 9 quantization levels. The mismatch errors ofthe first two DACs (DAC1 and DAC2) are determined during acalibration cycle with a two-tone signal at frequencies 975 kHzand 1025 kHz and at −10 dBFS amplitude using the calibrationsetup as shown in fig. 4. The ETFk(z) are approximatedas in equations (20) and (21). Fig. 6(a) shows the measuredcompensated output spectrum when applying a two-tone signalat frequencies 1000 kHz and 1050 kHz with −9.5 dBFS when0.8 0.9 1 1.1 1.2−120−100−80−60−40−200frequency (MHz)(a)outputPSD(dB)0.8 0.9 1 1.1 1.2−120−100−80−60−40−200frequency (MHz)(b)Fig. 6. The measured output PSD for a two-tone test on a distributed feedbackΣ∆-modulator with 1.25 MHz bandwidth (OSR=12) when the mismatcherrors of (a) only DAC1 are compensated and (b) both DAC1 and DAC2are compensated.only the mismatch errors of DAC1 are compensated. We seethat the IM3 is at −75 dB. When the mismatch errors of DAC2are also compensated, the IM3 is further reduced to beneaththe noise floor as shown in fig. 6(b).VI. CONCLUSIONAn improved offline calibration method for DAC mismatcherrors in low oversampling multibit SigmaDelta ADCs hasbeen presented. It allows to calibrate every DAC in a dis-tributed feedback structure. The technique uses a single two-tone test and calculates the DAC mismatch errors from theresulting digital output signal. The mismatch errors are thenstored in a LUT from which the compensated digital outputsignal is calculated during normal operation.ACKNOWLEDGMENTThis work was supported by the Special Research Fund(BOF) of Ghent University.REFERENCES[1] R. Schreier and G. Temes, Understanding Delta-Sigma Data Converters.Press/Wiley Interscience, 2005.[2] R. T. Baird and T. S. Fiez, “Linearity enhancement of multibit DeltaSigma and D/A converters using data weighted averaging,” IEEE Trans.Circuits Syst.-II, vol. 42, no. 12, pp. 753–762, Dec. 1995.[3] P. Witte and M. Ortmanns, “Background dac error estimation using apseudo random noise based correlation technique for sigma-delta analog-to-digital converters,” Circuits and Systems I: Regular Papers, IEEETransactions on, vol. 57, no. 7, pp. 1500 –1512, july 2010.[4] M. De Bock, X. Xing, L. Weyten, G. Gielen, and P. Rombouts,“Calibration of DAC Mismatch Errors in Σ∆ ADCs Based on a Sine-Wave Measurement,” Circuits and Systems II: Express Briefs, IEEETransactions on, vol. 60, no. 9, pp. 567–571, 2013.[5] J. Silva, U. Moon, J. Steensgaard, and G. Temes, “Wideband low-distortion delta-sigma ADC topology,” Electron. Lett., vol. 37, no. 12,pp. 737–738, Jun. 7 2001.[6] R. Schreier, “An empirical study of high-order single-bit delta-sigmamodulators,” IEEE Trans. Circuits Syst.-II, vol. 40, no. 8, pp. 461–466,Aug. 1993.[7] P. Handel, “Properties of the IEEE-STD-1057 four-parameter sine wavefit algorithm,” IEEE Trans. Instrum. Meas., vol. 49, no. 6, pp. 1189–1193,Dec 2000.[8] “IEEE Standard for Digitizing Waveform Recorders,” IEEE Std. 1057,1994.

Improved offline calibration for DAC mismatch in low OSR Sigma Delta ADCs with distributed feedback

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Improved offline calibration for DAC mismatch in low OSR Sigma Delta ADCs with distributed feedback

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