A New Method For Estimating Spectral Performance Of ADC .
A New Method for Estimating Spectral Performance of ADC from INL1Jingbo Duan1, Le Jin2, and Degang Chen1Department of Electrical and Computer EngineeringIowa State University, Ames, IA 500112National Semiconductor, Santa Clara, CA 95051AbstractLinearity test and spectral test are two main contributors ofADC test cost which includes data acquisition time andaccurate instrumentation. This paper presents a new methodfor estimating an ADC’s spectral performance from itstested INL data. The method does not require additionaldedicated test circuitry or data acquisition. The results fromINL test are used to compute harmonic distortions and otherspectral specifications of the ADC. Memory andcomputation requirements are very small comparing tothose in traditional spectral testing. When combined with aBIST approach for INL testing, the proposed method offersa very low cost BIST solution to ADC spectral testing. Bothsimulation and experimental results show that the proposedmethod can estimate THD and SFDR values accurately.1. IntroductionAs more mixed-signal functions are deeply embedded inSystem on Chip (SoC) applications and as customersdemand higher performance, accurate and cost-effectivetesting of ADCs becomes significantly more challenging. Inproduction test, ADC static linearity and spectralperformance are the two categories of specifications that aremost time consuming and impose most stringent hardwareand software test requirements. Static linearity, includingINL and DNL, is conventionally tested using the histogrammethod with either a sine wave or triangular wave input.Spectral performance, including SNR, THD, and SFDR, istested using the FFT method with a sine wave input havingvery high spectral purity [1].To reduce test time, methods have been introduced forestimating ADC static linearity from spectral testing results[2, 3]. However, due to the loss of “high-frequency” detailsof ADC’s transition levels, these methods are unacceptablein real applications in which transition levels matter. Forexample, in measurement instrumentation, automotivecontrol, and high resolution imaging, INL performance iscritical and must be measured accurately. This paper takesanother direction to achieve test time reduction by trying toestimate spectral performance based on INL test results.One might argue that saving spectral test time is not as big asaving as saving linearity test time. But when accuratelinearity test is mandated by the application, saving spectraltest time is the best one can hope for. When accuratespectral testing results are needed, both data acquisition timeand computation time in traditional methods are significant.Furthermore, the stringent spectral purity requirement on theinput sine wave generator is a major challenge, especiallyfor on-chip built-in self test. Removing this challenge is agiant step toward enabling built-in self test of deeplyembedded ADCs.The idea of estimating the spectral performance usinglinearity test data becomes more valuable in applications ofADC built-in self-test (BIST), where testing circuitry’s areais more concerned than test time. Significant research resultsof ADC BIST have been published in literature over the lasttwo decades. BIST schemes of SNR and other frequencyspecifications testing were presented in [4, 5]. Low costBIST schemes of testing static performances have beenpresented in [6, 7]. Recently, research results have beenpublished on reducing the accuracy requirement on linearitytesting signal and simplifying its generation circuitry, whichmakes it possible to realize ADC linearity test on chip [8][10]. Using the method developed in this paper, it takes verylittle additional resources to obtain the spectral performanceof an ADC based on BIST results of its linearity. Thismethod eliminates the need of accurate sine wavegeneration on chip for spectral testing, making ADC BISTone step easier to implement.In this paper, a method of estimating THD and SFDRbased on INL of an ADC is introduced. The methodcomputes THD and SFDR without requiring any additionalhardware or data acquisition. Only a small amount ofcomputation is required to estimate THD and SFDRaccurately. The rest of this paper is organized as following.In Section 2, the traditional method of testing THD andSFDR and its challenges are reviewed. In Section 3, the newmethod is described in detail. First a model of ADC test ispresented. Then how distortion is extracted from INL data ispresented. A way of efficiently compute harmonic power ispresent at the end. Error analysis is given in Section 4.Section 5 and Section 6 gives simulation results andmeasurement results respectively.2. Existing ChallengesIn traditional ADC spectral performance testing, a pure,sine wave with large amplitude is used as input signal of thePaper 23.3INTERNATIONAL TEST CONFERENCE978-1-4244-7207-9/10/ 26.00 2010 IEEE1
fundamental, a 7th order lowpass filter will be needed.Building 7 poles at low frequency on chip consumes largearea and hence is not practical.To meet the second condition, the number of input signalperiods must be a coprime number of the total number ofsampling points. The relation between input sine wavefrequency and ADC sampling frequency is given by (4).f0 Fig.1. Spectrum of output signalADC under test [11, 12]. The frequency is set to satisfycoherent sampling condition, which usually leads to oddnumber of signal periods. M sampling points from the inputsignal are converted into digital binary codes by the ADC.DFT of these digital codes are computed, the magnitudespectrum of which looks like what is shown in Fig.1. Fromthe magnitude spectrum, THD and SFDR can be computedas following. H2 THD 10 log10 hmi2 Arms i 2 (2SFDR 10 log10 Armsmax ( hmi2 )i 2 H(1))(2)In above equations, H is the number of harmonics to becomputed, hmi is the magnitude of the component at the ithharmonic of DFT, Arms is the RMS value of input sine waveamplitude.Two conditions must be satisfied in traditional spectraltesting to achieve valid testing. The first condition is that thesine wave must be pure enough so that its distortion is muchlower than ADC resolution under test. The second conditionis input signal frequency must be well controlled to achievecoherent sampling. To make the sine wave pure enough, alow pass or band pass filter is usually put after the sine wavegenerator. Harmonics of the input sine wave must beattenuated to be much lower than ADC resolution. At boardlevel testing, a passive LC filter can be used to perform thisfunction [13]. LC filter shall not be implemented on chipsince it consumes large area. Even building active filter forthis purpose needs large area which can be shown by thefollowing example. Assume the input frequency is f0, theharmonic at frequency fh needs to be attenuated by R dB.The order of the Butterworth lowpass filter is given by thefollowing expression.r R20 log10 ( f h f 0 )(3)Assume the 2nd order harmonic of generated sine wave is50dB lower than fundamental. If we want to attenuate it by40dB so that the 2nd order harmonic is 90dB lower thanPaper 23.3PfsM(4)P is the number of input signal periods, M is the totalnumber of points will be sampled, and fs is the samplingfrequency of ADC. The value of M is usually a power of 2.P is usually chosen to be an odd integer to guarantee integernumber of periods is sampled and different phase of eachperiod is sampled. The coprime relation makes f0 be afractional frequency of fs. For example, when fs is 10M Hz,M is 8192, and P is 799, the input signal frequency is975.342K Hz. A frequency synthesizer is used in traditionaltesting to generate the fractional frequency [11]. Frequencysynthesizer design itself is a challenging task in currentSOC design. This block also consumes large area thus isunaffordable in ADC BIST. The high precision frequencyrequirement may be avoid by using window in DFT. But thewindowing will increase the computation complexity.As described above, generating a pure sine wave on chipat proper frequency with low cost is very challenging and isunpractical for ADC BIST. A new method proposed in thispaper avoids above challenges by estimating spectralperformance from INL data which has been acquired inlinearity test. INL can be tested on chip with low overheadby adopting SEIR method which only needs nonlineartriangular stimulus [8]. Estimating spectral performancefrom INL data does not need additional data acquisition andonly needs very small amount computation.3. New Method of Estimating THD and SFDRIn this section, a new method of estimating harmonicdistortion power and then THD and SFDR values ispresented. THD and SFDR are computed from existing INLdata without additional data acquisition. Digital circuitryneeded for this computation is available in SoC. In otherword, THD and SFDR can be estimated with almost noextra overhead. Another advantage of this approach is thatnoise in INL is much lower than normal ADC output codesbecause of average effect of histogram testing.The new method estimates THD and SFDR from testedINL data. Because of the static characteristic of INL,estimated THD and SFDR are pseudo static. This methodcannot capture spectral performance at high frequency orspurious not at harmonic frequencies.A new way of modeling ADC testing process is given atfirst. Secondly, how the harmonic distortion power isextracted from INL data and calculated is discussed. At last,efficient computation is presented.INTERNATIONAL TEST CONFERENCE2
Input sine waveVin (tk ) n(tk ) TC ( tk ) Eos LSB C (tk )Eg LSB Q(tk ) (5)2nIn this equation, tk is the testing time index, Vin(tk) is voltageof input sine wave at time tk, n(tk) is the input referred noiseincluding ADC noise and signal source noise, C(tk) is theoutput code at time tk, TC(tk) is the transition voltagecorresponding to output C(tk), Q(tk) is the quantization errorat time tk, Eos is the offset, and Eg is the gain error of theADC. Continuous input sine wave is represented by discretetransition voltages of the ADC plus error and noise. Thetransition voltage corresponding to output code C(tk) can beexpressed by equation (6)TC (tk ) C (tk ) LSB INLC (tk ) LSB(6)in which, INLC(tk) is the INL error of transition level TC(tk).Equation (7) can be obtained by substituting (6) into (5)and switching sides.C (tk ) LSB EgC (tk ) LSB2n Vin (tk ) n(tk ) Q (tk ) INLC (tk ) LSB Eos LSB(7)C(tk)·LSB is the output data of ADC. All values of C(tk)·LSBover the testing time 0 tk 1 represents the input signalwhich is a single tone sine wave.After Fourier transform, equation (7) becomes toFT ( C (tk ) LSB ) 1 FT V (t ) FT ( n(tk ) Q(tk ) ) 1 Eg 2n { ( in k ) (}) FT INLC (tk ) LSB FT ( Eos LSB )(8)In this equation, FT(C(tk)·LSB) is the Fourier transform ofADC output codes which is the key data in traditional FFTtesting. FT(C(tk)·LSB) consists of several components thatare shown at the right side of (8). FT (Vin (tk ) ) is the Fouriertransform of input sine wave, FT(n(tk)-Q(tk)) is the noisefloor, FT ( INLC (t ) LSB ) is the harmonic distortion causedkby nonideality of ADC, and FT ( Eos LSB ) is the part DCcomponent from ADC offset. Fig.1 shows a typicalspectrum of a digitized sine wave contains all componentsin equation (8). Signal power, harmonic distortion power,and noise power can be computed from the spectrum andeventually SNR, THD, and SFDR can be computed.It can be observed from equation (8) that all harmonicdistortion power is carried by FT ( INLC ( t ) ) term which alsokPaper 23.32ADC output codeWhen a sine wave is converted into digital codes by anADC, transfer characteristic of the ADC can be representedby equation (5).Original INL3.1. Model of ADC testing10-1-2Sinusoidally sampled INLData indexFig.2. Sinusoidal sampling of INLcontains a small amount of noise and a small part of inputsignal. Spectrum of INL data contains the same harmonicdistortion power as the spectrum of digital output datashown in Fig.1. To achieve the purpose of computing THDand SFDR value, we only need to do Fourier transform ofINL instead of output codes. All harmonics distortion powercan be calculated from spectrum of INL.3.2. Extracting distortion power from INLIn traditional spectral testing, a pure sine wave is appliedto ADC. Distortion information carried by output code is thedistortion experienced by the sine wave, which means thatthe distortion term FT ( INLC ( t ) ) in equation (8) is the INLkexperienced by the sine wave. To obtain INL correspondingto the input sine wave, we can sinusoidally sample INL bythe ADC output codes of the sine wave. Regard the originalINL data of the ADC as a series INLorig which has 2n-2elements as shown in (9).INLorig ( i )i 1, 2,3, 2n 2(9)in which n is the resolution of ADC. Sampling processconstructs a new series based on output codes and INLorigINLsin INLorig ( C ( tk ) )k 1, 2,3, M(10)In (10), series INLsin is the distortion experienced by sinewave, C(tk) is the ADC output code of sine wave, M is thetotal number of points in sine wave test. The value of M canbe either larger or smaller than 2n-2. Fig.2 shows the processof constructing a new data sequence by sinusoidallysampling INL according to output codes of ADC. The curveat the left side is the original INL, which is sampled byADC output codes of sine wave. The new INL sequenceexperienced by the sine wave is shown at the bottom ofFig.2. The pattern of the new sequence consists of 6 repeatsof original INL. But the total number of points of the newINTERNATIONAL TEST CONFERENCE3
sequence can much smaller than the number of points oforiginal INL. From spectrum of this new INL sequence, wecan calculate the spectral performance as following.H P (i )hTHD i 2H P (i )hTHD i 2(11)P0 Ph (1)SFDR (16)A2 8A2 8max Ph ( i )(17)i 2 20SFDR P0 Ph (1)(12)max Ph ( i )i 2 20In above two equations, Ph(i) is the ith harmonic power inthe power spectrum of the new INL sequence INLsin. Thepower spectrum corresponds to the term FT ( INLC (t ) LSB )where A is the full scale range of ADC. Equation (16) and(17) are computations carried out in the new method.Because there is no input signal component in spectrum ofINL data, the signal power is theoretical full scale sine wavepower. Every harmonic power can be calculated from thespectrum of INL data.kin (8). This term also contains a part of input signal powerPh(1). P0 is the signal power corresponding to FT (Vin (tk ) )term in (8). The numerator of (11) is the total power of firstH order harmonics, and the denominator is the signal power.The difficulty of sinusoidal sampling is that ADC’s outputcode of sine wave is not available because only code densityis recorded in histogram testing. To overcome this, weacquire the sinusoidal digital codes by virtually testing asine wave. Assume a sine wave Xin has frequency of f0 andamplitude of 1. An ideal ADC with the same full scale rangeconverts this sine wave into digital codes which can besimply calculated by N C (k ) (1 sin ( 2π f 0 k ) ) 2 k 1, 2 M(13)in which, N is the number of transition level of ADC, C(k) isthe output code, and M is the total number of samples thatwill be used for spectral performance estimation. Now thevalue of C(k) can be used as the index to read the value ofINL from the original INL data and construct a new data setINLvsin. Constructing a new data set from INL according tosine wave does not change distortion power. Frequency ofthe sine wave in (13) can be selected to be any value thatmakes computation convenient. Assume H is the number ofharmonics will be calculated. In order to let the first Hharmonics distribute within half sampling frequency, thesine wave frequency can be set asf0 12H(14)The value of f0 should be slightly adjusted to achievecoherent sampling. From these ideal digital codes for sinewave, another new sequence of INL is constructed asINLvsin INLorig ( C ( k ) )k 1, 2,3, M(15)From the spectrum of INLvsin, we can calculate THD andSFDR as following.Paper 23.33.3. Reducing computation requirementThough the Fourier transform of INL can be easilycomputed by on chip processer, the computation can befurther simplified. To calculate THD and SFDR, we onlyneed distortion power when full scale input signal is applied.Instead of implementing FFT algorithm on chip, discretetime Fourier series (DFS) of INL is computed as (18).X (k ) 1MM 1 x (n) e j 2π n kMk 0,1, 2,., M 1 (18)n 0In which, x(n) is the value of L(n), X(k) is the kth coefficientof the Fourier series, M is the total number of points used inTHD and SFDR estimation. The coefficient of thefundamental component is given by (19)X ( k1 ) 1MM 1 x ( n) e j 2π n k1M(19)n 0The relation between input signal frequency and samplingfrequency is set beforehand, thus value of k1 is known.Coefficient of ith order harmonic can be calculated by (20)X ( i k1 ) 1MM 1 x ( n) e j 2π n i k1Mi 2,3, 4.H(20)n 0There is no need to calculate fundamental component sinceit is not the power of input signal or part of distortion power.Only 19 coefficients need to be calculated for goodestimation of THD and SFDR value. The frequency of inputsine wave is selected by tester so that value of k1 and totalnumber of points M are always known. Rewrite (20) in to(21)1 M 1n(21)X ( i k1 ) x ( n ) ( Ei )M n 0In which,Ei eINTERNATIONAL TEST CONFERENCE j 2π i k1M(22)4
Instead of creating a look up table for exponential term, onlythe exponential value Ei needs to be stored on chip and usedfor DFS coefficient computation. In (21), there are n timesof multiplication. When n is large, it can be expressed inbinary form to reduce computation further.12n b0 b1 2 b3 4 b12 2INLvsin ( k ) INLsin ( k ′ ) h1e( Ei ) ( Ei ) ( Eib02 b14 b2) (E )i ( Ei4096 b12)2π k ' pM h2 ej 2π k ' 2 pM h3ej 2π k ' 3 pM . higher order term noise(27)(23)Because k and k’ are very close to each other and frequencyof harmonic is low, we can expand each harmonic term.(24)h i eThe exponential term in (21) can be rewritten asnj Values of different power Ei can be stored in memory. Thenumber of multiplications in exponential value computationis reduced to 13.j2π i kM h i ej2π i k 'M2π 2π j M i k ' h i j i e( k k ') M 2π2π j i k ' h i 1 j i ( k k ') e MM (28)Comparing (27) and (28), we have4. Error analysis2π i ( k k ') hi h i 1 jM 4.1. Approximations in equation derivationComparing (16) and (17) with equation (11) and (12),we can see 2 approximations may cause estimation error inTHD and SFDR value. The first approximation is usingideal sine wave to sample INL instead of real output codes,so that FT(INLvsin) will be slightly different from FT(INLsin).It can be seen from equation (13) that C(k) is different fromthe actual output code of ADC C(tk). For reasonably goodADC, C(k) is only several codes away from C(tk) and thevalue of INL changes very slowly. The difference betweenINLsin and INLsin will never be larger than the peak-to-peakvalue of INL which we denote as INL. INL consists ofthree parts including a part of input signal, distortion, andnoise. So we can express INL as following.INLsin ( k ) h1ej 2π k pM h2 ej 2π k 2 pM h3 ej 2π k 3 pM . higher order terms noise(25)in which h1 is the coefficient of fundamental component, h2and h3 are coefficients of 2nd and 3rd harmonic components,and p is the number of periods of sine wave. From (11) and(12) to (16) and (17), INLsin is replaced by INLvsin, so thecoefficient of each harmonic will change.2π2π2πj k pj k 2 pj k pINLvsin ( k ) h 1e M h 2 e M h 3e M . higher order term noise Paper 23.3The ith harmonic power is calculated from the Fourier seriescoefficient at i·p. The estimation of the ith order harmonicdistortion power is2 2π ei 10 log 1 i ( k k ') M (30)Assume difference be
coherent sampling. To make the sine wave pure enough, a low pass or band pass filter is usually put after the sine wave generator. Harmonics of the input sine wave must be attenuated to be much lower than ADC resolution. At board level testing, a pas
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