IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 61, NO. 11, NOVEMBER 2012
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Testing ADC Spectral Performance Without Dedicated Data Acquisition Jingbo Duan, Member, IEEE, Le Jin, Member, IEEE, and Degang Chen, Senior Member, IEEE
Abstract—Static linearity and spectral performance are the most important parameters that need to be measured in an analog-to-digital converter (ADC) test. Testing these two types of parameters contributes the most significant part of ADC test cost. The relationship between integral nonlinearity (INL) and spectral performance has been used to reduce test cost in many research works. This paper comprehensively investigates the relationship and presents a low-cost test method that measures ADCs’ spectral performance based on measured INL values. By making use of INL measurement results, the method eliminates dedicated hardware and test time for spectral performance measurement. It only needs a small amount of computation to obtain ADC’s spectral performance. The method is useful in application where spectral performance only needs to be measured at low frequency. Simulation and experimental results show that the proposed method achieves the same measurement accuracy as a standard fast Fourier transform method. Index Terms—Analog-to-digital converter (ADC), Fourier transform (FT), harmonic distortion, integral nonlinearity (INL), spectral performance, test.
I. I NTRODUCTION
A
NALOG-TO-DIGITAL converters (ADCs) have two main categories of specifications. One is defined based on the ADC’s transfer curve, including integral nonlinearity (INL), differential nonlinearity (DNL), offset, and gain error. These specifications are usually tested by a histogram method using either a sine wave or a triangular wave as the input signal. The other one is spectral performance, including total harmonic distortion (THD), spurious-free dynamic range (SFDR), signal-tonoise ratio (SNR), etc. These parameters are usually measured by the fast Fourier transform (FFT) method using a sine wave with very high spectral purity as the input signal [1], [2]. Different applications have emphasis on different specifications. For example, INL and DNL are very important in control loops and image processing, whereas THD and SFDR are very important in audio and communication systems [1]. Fig. 1 illustrates a typical ADC test procedure in industry. Among these steps, step 4 for the transfer curve linearity test Manuscript received February 7, 2012; revised April 14, 2012; accepted April 15, 2012. Date of publication July 10, 2012; date of current version October 10, 2012. The Associate Editor coordinating the review process for this paper was Dr. Dario Petri. J. Duan is with Broadcom Corp., San Jose, CA 95134 USA (e-mail:
[email protected]). L. Jin is with Maxim Integrated Products, Sunnyvale, CA 94086 USA (e-mail:
[email protected]). D. Chen is with the Department of Electrical and Computer Engineering, Iowa State University, Ames, IA 50011 USA (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIM.2012.2202949
Fig. 1. Typical ADC test flow.
and step 5 for the spectral performance test are the most timeconsuming processes and impose the most stringent hardware requirements, where they significantly drive up the ADC test cost. In recent years, many research works have been conducted in order to reduce the test cost. The ADC test cost consists of hardware cost and test time. Many research works have been published to reduce these two types of test cost. Several new methods are presented in [3]–[6] to reduce hardware cost by removing stimulus or reducing accuracy requirement of stimulus. To guarantee coherent sampling, accurate frequency control is another part of hardware cost. Methods are proposed in [8] and [9] to test spectral performance with a noncoherent input sine wave. These two methods reduce hardware cost by avoiding the use of accurate frequency synthesizers. Many research works on reducing test time can be found in the literature [10]–[16]. In [10], the measurement is completed in a time domain by comparing the output signal with a reference signal. By running in real time, the method reduces the test time by 4 times. The method in [11] and [12] measures INL by modeling the nonlinearity with Chebyshev polynomials. The measurement is based on the FFT test result and avoids a time-consuming code density test. In some applications, INL does not need to be measured very accurately. This method can be used to measure a rough INL value or trend with much smaller test time. To gain more accurate INL measurement while still reducing test time, a tradeoff was made in [13]. The authors combined spectral and histogram tests. A polynomial fitting method with sinewave input measures the low code frequency part of the INL. Histogram method with small triangular-wave input measures the high code frequency part of the INL. Other than polynomial fitting, Fourier series method is the other published approach to estimate INL from a frequency domain. In [14], INL is
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estimated from lower order Fourier coefficients with wobbling method. The authors in [15] investigate the theory of polynomial fitting and Fourier series. These two approaches are also compared with respect to efficiency and hardware requirement. A test flow for massive production test is presented in [16]. Again, INL is measured from a spectral analysis so that a timeconsuming histogram test is avoided. The method is limited to large population and stringent ADC specification requirement. A common feature of [11]–[16] is that the INL can be roughly measured from spectral performance measurement results. It is well known in the ADC test and design area that a relation exists between static INL and harmonic distortion. Many methods are presented to estimate INL from spectral analysis based on this relation. Many practitioners may also estimate harmonic distortion profile from the INL curve from experience and experiments [17], [18]. Therefore, the relation is worth to be comprehensively investigated with theory and experimental validation. The relation has been preliminarily investigated in the authors’ paper [19]. Based on the conference paper, this paper detailedly investigates the relationship between INL and harmonic distortion starting from the fundamental theory . It provides theory, error analysis, extensive simulation, and experiment validation. Based on the investigation, a low-cost ADC test method is proposed for test time reduction. The spectral performance is measured from INL test results so that step 5 in Fig. 1 can be eliminated. Since spectral performance test time is much smaller than INL test time, the test time reduction is not as much as the methods presented in [11]– [15]. However, the test time reduction is not accompanied by any accuracy degradation. Furthermore, when accurate spectral testing results are needed, both data acquisition time and computation time in standard test flow are significant. In addition, when INL is tested from a triangular wave, which cannot be used for a spectral test, the method eliminates all hardware cost in step 5 for spectral performance test. Only a small amount of computation is needed to measure harmonic power. From the proposed method, spectral performances are obtained almost for free. The proposed method is only useful in certain circumstances due to its limitations. First, it only measures pseudostatic spectral performance at one single low frequency. The dynamic effect or performance cannot be captured by this method. Second, it cannot detect a spurious signal at nonharmonic frequency. Given these limitations, the method is still useful. First, many ADCs’ spectral performances are only tested at one low input frequency. Second, commercial ADC products are usually well designed and would not have big spurs signal at nonharmonic frequencies. A production survey of 60 different commercial ADCs shows 27 of them are only tested at a single low frequency (input signal frequency is 10 times less than sampling frequency). Since both INL and spectral performance of these ADCs need to be measured, the proposed method can reduce the test time and hardware by computing spectral performance from measured INL values. The rest of this paper is organized as follows. In Section II, the standard spectral performance test is reviewed first. Then, the INL-test ADC model and the new method are introduced. Section III first describes how the measured INL values are
Fig. 2.
Standard test of ADC spectral performance.
processed for computing harmonic distortion. Then, an efficient way of calculating harmonic distortion power is described. Error analysis is given in Section IV. Simulation results and experimental results are presented in Section V and Section VI, respectively. Section VII concludes this paper. II. S PECTRAL T ESTING OF THE INL-T EST ADC M ODEL A. General Definition of THD and SFDR According to the fundamental signals and systems textbook [20], a periodical signal with a frequency of fi can be expressed by Fourier series as follows: x(t) = C0 +C1 cos(fi t+ϕ1 )+C2 cos(2fi t+ϕ2 )+· · ·
(1)
where C0 is the dc component. C1 and ϕ1 are the amplitude and the initial phase of the fundamental signal. C2 and ϕ2 are the amplitude and the initial phase of the second-order harmonic. Based on the Fourier series of periodic signal, the SFDR of the signal is defined as the ratio of the fundamental signal power to the largest harmonic power, whereas THD is defined as the ratio of the total harmonic power to the fundamental signal power, and they can be expressed as (2) and (3) H
THD = 10 log SFDR = 10 log
k=2
Ck2 (2)
C12 C12 . max (Ck2 )
(3)
In the test community, it is usually assumed that the input signal of a device under test (DUT) is a pure sine wave, as shown in Fig. 2. If the DUT is ideal, output signal x(t) is also a pure sine wave that has infinite THD and SFDR values. When the DUT is not ideal, distortion introduced by the DUT makes the output different from the input. Therefore, the output signal captures all harmonic distortion of the DUT. In addition, the THD and SFDR of the output signal calculated by (1)–(3) are those values of the DUT. B. IEEE Standard Testing of ADCs’ Spectral Performances Continuous-time signal cannot be acquired by real equipment pieces. Therefore, in DSP-based testing, the output signal of the DUT is sampled and recorded at discrete time. The harmonic distortion of the DUT can be obtained by applying discrete Fourier transform (DFT) to its output signal x(kTs ) when the input signal is a pure sine wave, and this can be expressed as follows: X(nfr ) = DFT (x(kTs ))
n = 1, 2 . . . M
(4)
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where Ts is the sampling time, M is the total number of samples, and fr is the frequency resolution. The relation between fr and Ts is fr = 1/(M · Ts ). Suppose that the frequency of the input sine wave fi = ni · fr , X(ni · fr ) is the coefficient of fundamental component, and X(i · ni · fr ) is the coefficient of the ith harmonic component, and assume that the largest spurious signal happens at one of the harmonic frequencies, then THD and SFDR are calculated by (5) and (6) H
THD = 10 log SFDR = 10 log
|X(i · ni · fr )|2
i=2
|X(ni · fr )|2 |X(ni · fr )|2 . max |X(i · ni · fr )|2
(5) Fig. 3. Spectral performance test of ADC.
(6)
In the DSP-based tests, both Nyquist criterion and coherent sampling conditions must be satisfied in the sampling process in order to avoid aliasing distortion and spectrum leakage, respectively. Coherent sampling conditions must guarantee that integer number of signal periods is sampled with different phases in different periods. This usually can be satisfied by choosing odd integer number of signal periods and M to be the power of 2. When the DUT is an ADC, the output signal x(kTs ) is not only discrete in time but also discrete in values. According to the IEEE Standard [1], a spectrally pure sine wave is converted into a digital code sequence by the ADC under test. Amplitude of the sine wave should be slightly smaller than the full scale range of the ADC. If the full scale range of the ADC is normalized into [0, 1], the input signal can be written as Vin (t) =
1 A + sin(2πfi · t + ϕi ) 2 2
(7)
in which A and fi are the peak-to-peak value and the frequency of the input sine wave, respectively. In most situations, A is set to be slightly smaller than the ADC’s full scale range. The frequency fi of the input sine wave must satisfy coherent sampling conditions with the sampling frequency of the ADC. After data acquisition, M digital codes are acquired and stored in total. The output signal is a set of discrete voltages interpreted from input sine wave and can be written as x(tk ) = C(tk ) · LSB
(8)
where C(tk ) is the kth output code, tk is the time when the kth point is sampled, and its value is tk = k · T s
k = 1, 2, . . . , M
(9)
where Ts is sampling period of the ADC. When k goes from 1 to M , x(tk ) is a sine wave. After data acquisition is finished, DFT is applied to the set of output voltages X(nfr ) = DFT (C(tk ) · LSB)
n = 1, 2 . . . M.
C. INL-Tested ADC Model ADCs’ pseudostatic linearity performance is characterized by INL and DNL, where they are tested by a histogram method, and the input signal can be either sine or triangular wave [2]. An ADC can be modeled as a set of transition levels, denoted as {T (i), i = 1, 2 . . . 2n }. In Fig. 3, the vertical axis shows the relation between transition level T (i) and IN L(i) of the ADC. The horizontal axis represents time, and tk is the kth sampling time. Scales marked by × are the ideal transition levels I(i); those marked by gray line are linear transition levels L(i) obtained by the endpoint fit line of the real transition levels, and those marked by black line are the ADC’s real transition levels T (i). Relations among them are illustrated as follows: ⎧ ⎨ I(i) = i · LSB (11) L(i) = I(i) + Eos · LSB + 2in Eg · LSB ⎩ T (i) = L(i) + IN L(i) · LSB in which Eos is the offset, Eg is the gain error of the ADC, and IN L(i) is the INL error at transition level T (i). Based on (11), the ADC characteristics can be modeled by its INL. D. IEEE ADC Spectral Testing of the INL-Test ADC Model Based on the discussion of part C, we assume that the spectral performance of the ADC is determined by the INLtest model. This assumption is only valid at low frequency where dynamic effect does not appear. Under the assumption, THD and SFDR can be tested by mathematically applying the IEEE standard method to this model. The ADC under test is replaced by an INL-test model, which is characterized by a set of transition levels lying on the vertical axis in Fig. 3. At time tk , the input is Vin (tk ) and the output is C(tk ) = i. Then, the corresponding output voltage is I(i) = C(tk ) · LSB. From Fig. 3, the relationship between the input and the output is Vin (tk ) = C(tk ) · LSB + Eos · LSB +
C(tk ) Eg · LSB 2n
+IN L (C(tk )) · LSB + Q(tk ).
(12)
(10)
From the set of DFT coefficients, THD and SFDR are computed as in (5) and (6).
From (12), continuous input sine wave is interpreted into discrete transition voltages plus error and noise. In traditional sine wave testing, since only digital codes are analyzed to obtain
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Fig. 4. New ADC test flow when the new method is used.
spectral performance, (12) can be rewritten into the following form: Eg C(tk ) · 1 + n · LSB = Vin (tk ) − Q(tk ) 2 −IN L (C(tk )) · LSB − Eos · LSB.
(13)
All values of C(tk ) · LSB over the testing time 0 ≤ tk ≤ 1 represent the input signal, which is a single-tone sine wave. After Fourier transform (FT), (13) becomes 1 · {F T (Vin (tk )) F T (C(tk ) · LSB) = 1 + Eg /2n −F T (Q(tk ))−F T (Eos ·LSB)−F T (IN L (C(tk ))·LSB)} . (14) In this equation, F T (C(tk ) · LSB) is the FT of ADC output codes, which is the key in standard spectral performance test, as shown in (4). Equation (14) shows that F T (C(tk ) · LSB) consists of several components that are shown at the right side. F T (Vin (tk )) is the FT of input sine wave, which corresponds to X(f0 ) and X(fs − f0 ) in Section III. F T (Q(tk )) is the noise floor in the spectrum. F T (Eos · LSB) is the dc component from the ADC offset, which corresponds to X(0). The last term F T (IN L(C(tk )) · LSB) is the most important term in the new method because all harmonic distortion power is only carried by this term. It also contains a very small amount of noise. The DFT of INL data contains the same harmonic distortion power as the spectrum of a digital output code. Therefore, THD and SFDR can be computed from INL, which has been already tested after the first four steps are completed. There is no need to carry out step 5. E. New ADC Test Flow By computing harmonics from INL data generated from a transfer curve linearity test (see step 4 in Fig. 1), the step for spectral performance test can be removed from the flow. The new test flow is shown in Fig. 4. From the tested INL data, all harmonics can be obtained through a simple computation. This method is valid for all ADC structures since the ADC under test was treated as a black box in the test model.
Eliminating step 5 would be a significant reduction of ADC test cost. It not only reduces test time but also eliminates hardware cost for spectral performance test. Fig. 5 shows the test setup for traditional ADCs’ spectral performance test [1]. In this setup, two conditions must be satisfied to achieve valid testing. The first condition is that the sine wave must be pure enough so that its distortion is much lower than the resolution of the ADC under test. To obtain a pure-enough sine wave, a complicated sine wave generator is usually needed. In addition, a low- or bandpass filter, which is very area consuming, is sometimes used to purify the sine wave further [21]. The second condition is that input signal frequency must be well controlled to achieve coherent sampling. A precise frequency synthesizer is often used in traditional testing to generate a fractional frequency for an input signal [1]. In contrast, the new method eliminates the need of a precise generator, a high-order low- or bandpass filter, and an accurate frequency synthesizer, which significantly reduces the overall test cost. Nevertheless, the new method has some of the following limitations. First, the obtained THD and SFDR are pseudostatic because of the pseudostatic characteristics of INL test. This is equivalent to the FFT method when input sine-wave frequency is much lower than sampling frequency. However, a large number of commercial ADCs are only tested at one low input frequency. Therefore, this limitation does not reduce any value of the new method. Second, some ADCs are tested at multiple input frequencies for spectral performance, but this method does not support these types of applications. Third, this method cannot capture a spurious signal that is not at harmonic frequencies. There is one more concern, i.e., histogram-based INL test cannot identify nonmonotonicity of ADC . THD and SFDR may be tested to be good by this method although the ADC is nonmonotonic. However, this is not a problem, since after finishing four early test steps, the ADC is known to be monotonic and has no missing code and reasonably small INL. Otherwise, there is no need to test other performances. III. T ESTING THD AND SFDR F ROM INL In this section, a new method is presented to measure harmonic distortion power from existing INL test data, which needs no dedicated hardware setup or data acquisition, but only simple computation. At first, the process of extracting harmonic distortion power from INL data is described. Then, the computation process is discussed. Last, the method is briefly summarized. A. Obtaining Harmonics From Tested INL Data From (14), it has shown that only the INL term carries harmonic distortion information. However, direct DFT of INL does not give the right harmonics. To find out the correct way, the traditional method of obtaining harmonic distortion is considered at first, where a pure sine wave is applied to the ADC and output digital codes are acquired. Fig. 6 shows an example when a 4-bit ADC samples a period of sine wave. On the vertical axis, black scales are the ADC’s real transition levels whereas gray scales are the linear transition levels. The
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Fig. 5.
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Sine-wave test setup of ADC.
of constructing a new data series by sampling INL according to the output sinusoidal code space of the ADC. The curve at the left side is the original INL, which is sampled by ADC output codes of sine wave. The new INL sequence experienced by the sine wave is shown at the bottom of Fig. 7. The pattern of the new sequence consists of six repeats of original INL. However, the total number of elements of the new data series IN Lsin can be much smaller than the number of elements of original INL. From the spectrum of this new INL sequence, we can calculate the ADC’s harmonic distortion power and thus producing THD and SFDR. The difficulty of sinusoidal sampling is that ADC’s output code of sine wave is not available because only code density is recorded in histogram INL test. To overcome this, the sinusoidal digital codes are acquired by ideal quantization Fig. 6.
J ·k C(k) = 2n−1 · 1 + sin 2π M
Sampling of a sine wave.
difference between the ith real transition level and the linear transition level is IN L(i). Eight points are sampled from time t1 to time t8 , and the corresponding output codes are {9, 13, 13, 10, 6, 3, 3, 5}. Standard method applies DFT to these digital codes to measure harmonic performance. Alternatively, the new method measures them from INL data. Substitute C(tk ) in F T (IN L(C(tk ))) by above digital codes, we obtain a new data series {IN L(9), IN L(13), IN L(13), IN L(10), IN L(6), IN L(3), IN L(3), IN L(5)}. The elements of the data series are INL values, but sequence is determined by ADC’s output codes. We can say that this is the INL (or distortion) experienced by the input sine wave. According to the F T (IN L(C(tk ))) term in (14), the DFT of this data series gives ADC’s harmonic distortion. To obtain INL experienced by the sine wave, we need to resample the INL according to ADC output codes. Consider the tested INL data of the ADC as a data series IN Lorig IN Lorig (i)
i = 1, 2, 3, . . . 2n − 2
(15)
in which n is the resolution of the ADC. Use C(tk ) as index to read elements in data series IN Lorig , we can construct a new data series IN Lsin = IN Lorig (C(tk ))
k = 1, 2, 3, . . . M.
(17) in which n is the resolution of the ADC under test, and C(k) is the quantized digital code. Now, the value of C(k) can be used as the index to read the IN L from the original INL data and construct a new data set IN Lvsin . The frequency of the sine wave in (17) can be selected to be any value that makes computation convenient. Assume H is the number of harmonics will be calculated, and then the sine-wave frequency can be set to be around 1/(2H) so that the first H harmonics distribute within half sampling frequency. From this ideally quantized sine wave, another new sequence of INL is constructed as IN Lvsin = IN Lorig (C(k))
k = 1, 2, 3, . . . M.
(18)
IN Lvsin is slightly different from IN Lsin in (16), which may cause error in harmonic distortion computation. However, for reasonably good ADCs, this error is very small. Details of error analysis will be discussed in Section V. From the spectrum of IN Lvsin , the power of every harmonic component can be calculated, and then THD and SFDR can be expressed as follows: H
(16)
In (16), series IN Lsin is the distortion experienced by the input sine wave, C(tk ) is the ADC output code of the sine wave, and M is the total number of points in the sine wave test. M can be either larger or smaller than 2n − 2. Fig. 7 shows the process
k = 1, 2 . . . M
THD = SFDR =
Ph (i)
i=2
A2 /8 A2 /8 max Ph (i)
i=2:20
(19) (20)
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Fig. 7. Resample INL with sinusoidal code space.
where A is the full scale range of ADC, Ph (i) is the ith-order harmonic power. Because an ideal sine wave with amplitude equal to ADC’s full scale range is used to sample the INL, the signal power is theoretical full-scale sine-wave power, which is A2 /8.
in which Ei = e−j· M ·i·k1 . 2π
(24)
To avoid calculating the exponential value, Ei can be stored onchip and used for FT coefficient computation. The ith harmonic Xi can be calculated by a for loop
B. Improving Computation Efficiency Although the FT of INL can be easily computed by an onchip processor, the computation can be further simplified. To calculate THD and SFDR, harmonic distortion power is the only necessity. Instead of implementing FFT algorithm on-chip, the coefficient of the FT of INL is computed as X(k) =
1 M
M −1
x(n) · e−j· M ·n·k 2π
k = 0, 1, 2, . . . , M − 1.
n=0
(21) In which, x(n) is the value of L(n), X(k) is the kth coefficient of the Fourier series, and M is the total number of points used in estimation. The relation between input signal frequency and sampling frequency is set beforehand; thus, k1 is known, and X(k1 ) is the fundamental tone coefficient. The coefficient of the ith-order harmonic can be calculated by X(i · k1 ) =
M −1 2π 1 x(n) · e−j· M ·n·i·k1 M n=0
i = 2, 3, 4 . . . H. (22)
There is no need to calculate the fundamental component since it is not the power of input signal or part of distortion power. Only 19 coefficients need to be calculated for good estimation of THD and SFDR. The fundamental frequency is intentionally chosen and known so that k1 and M are always known. Rewrite (22) into X(i · k1 ) =
M −1 1 x(n) · (Ei )n M n=0
(23)
Xi = 0; for n = M : −1 : 1 Xi = Xi∗ Ei + x(n); end M multiplications and M summations are needed for each harmonic bin. To calculate THD, H multiplications, H summations, and 1 division are needed. To calculate an SFDR value, H comparisons and 1 division are needed. The total computation consists of H ∗ (M + 1) multiplications, H ∗ (M + 1) summations, H comparisons, 2 divisions, and H memory cells. C. Summary In summary, this method is based on the fact that harmonic distortion only comes from INL and the INL are INL values experienced by the sine wave. A concise guidance of how the method is used is shown as follows. 1) From tests that have been already done, the ADC is good enough to go to spectral performance test. 2) INL values are available from finished INL test. 3) Generate ideal sine-wave codes as (17) and use them as index to read INL and construct a new data set. 4) Calculate harmonics according to (23). 5) Calculate THD and SFDR by (19) and (20). Compared with the traditional test flow, cost saving of the proposed method includes data acquisition time and computation time. Assume that the number of samples is M . The traditional test flow needs M ∗ Ts data acquisition time and O(M log2 M ) operations for FFT, where Ts is the ADC’s clock
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period. The proposed method needs 0 data acquisition time and about 2H ∗ M operations for harmonic power. Moreover, to achieve the same measurement accuracy, the proposed flow needs less number for samples than the traditional test flow. IV. E RROR A NALYSIS Comparing (16) with (18), we can see that an approximation may cause estimation error in THD and the SFDR value. Ideal sine wave is used to resample INL instead of real sine-wave output codes so that F T (IN Lvsin ) will be slightly different from F T (IN Lsin ). It can be seen in (18) that C(k) is different from the actual output code of ADC C(tk ). For reasonably good ADC, C(tk ) is only several codes away from ideal value C(k) and the value of IN L changes very slowly. After sampling, the difference between IN Lsin and IN Lvsin will never be larger than the peak-to-peak value of INL, which we denote as ΔIN L. INL consists of three parts, including a part of input signal, distortion, and noise. Therefore, we can express INL as follows: IN Lsin (k) = h1 ej· M ·k·p + h2 ej· M ·k·2p + h3 ej· M ·k·3p + · · · 2π
2π
2π
+ higher order terms + noise
(25)
in which h1 is the coefficient of the fundamental component, h2 and h3 are the coefficients of the second and third harmonic components, and p is the number of periods of sine wave. Since IN Lsin is replaced by IN Lvsin , the coefficient of each harmonic will change ˜ 1 ej· 2π ˜ 2 ej· 2π ˜ 3 ej· 2π M ·k·p + h M ·k·2p + h M ·k·p + · · · IN Lvsin (k) = h + higher order term + noise
(26)
˜ 1 is the coefficient of the fundamental component in which h ˜ 3 are the coefficients of the second and third ˜ 2 and h and h harmonic components. Due to the difference between C(tk ) and C(k), the value of IN Lvsin (k) is actually equal to the value of IN Lsin (k ), in which k is very close to k. Equation (26) can be rewritten as IN Lvsin (k) = IN Lsin (k )
= h1 ej· M ·k ·p + h2 ej· M ·k ·2p + h3 ej· M ·k ·3p 2π
2π
2π
+ · · · + higher order term + noise.
(27)
Because k and k are very close to each other and the frequency of harmonic is low, we can expand each harmonic term ˜ i ej 2π ˜ i · j 2π · i · ej 2π ˜ i ej 2π M ·i·k = h M ·i·k + h M ·i·k (k − k ) h M 2π 2π ˜ · i · (k − k ) ej M ·i·k . (28) = hi 1 + j M Comparing (27) and (28), we have 2π ˜ · i · (k − k ) . hi = h i 1 + j M
(29)
Fig. 8. INL of ADC. (a) True INL of ADC. (b) First five periods of sinusoidally sampled INL.
The ith harmonic power is calculated from the Fourier series coefficient at i · p. The estimation of the ith-order harmonic distortion power is
2 2π · i · (k − k ) . (30) ei = 10 log 1 + M Assume that the difference between k and k is less than 5 LSB and M takes 8192 as an example. The estimation errors of THD and SFDR values are smaller than 0.025 dB, which is negligible. From above analysis, it causes negligible error by using ideal quantized sine wave to sample INL instead of real ADC output codes. V. S IMULATION R ESULTS The method that measures THD and SFDR from INL data was validated in a MATLAB simulation. Both the standard FFT method and the new method are implemented in MATLAB. A 16-bit ADC is modeled as a set of randomly generated transition levels. Fig. 8(a) shows the true INL of the ADC, the maximum and minimum values of which are +1.9 LSB/ −1.3 LSB. To obtain spectral performance from the INL data, an ideal sine wave is mathematically quantized into a set of digital codes according to (17). In total, 655 periods and 215 digital codes are generated. According to these digital codes, the INL shown in Fig. 8(a) is sinusoidally sampled 215 times. In other words, the method uses these digital codes as index to read the INL values and construct another set of data. For computation convenience, 655 periods of sine wave are generated for 215 samples so that the first 20 harmonic bins are distributed within half sampling frequency. Fig. 8(b) shows the first five periods of the sinusoidally sampled INL. Fig. 9 compares the spectrum of ADC output codes in standard spectral testing and the spectrum of sinusoidally sampled INL data. The black curve is the spectrum of the ADC output voltage when the input is a sine wave. For good investigation of harmonic distortion, input noise in the standard FFT method is set to zero so that the SNR of this spectrum is 97.2 dB, which is very close to the theoretical SNR value of a 16-bit ADC. The gray curve is the spectrum of sinusoidally sampled INL data, the first five periods of which are shown in Fig. 8(b).
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Fig. 9. Spectrum of output code and sampled INL. TABLE I ACCURACY OF THE N EW M ETHOD W ITH D IFFERENT N UMBER OF S AMPLES (U NIT: D B)
Fundamental frequencies in two methods are set to be equal for convenient comparison. From the gray curve, we can see three parts, i.e., the small component at fundamental frequency, components at harmonic frequencies, and noise floor. It can be observed that the spectrum of sinusoidally sampled INL has the same harmonic bins as the spectrum of output signal. The zoomed-in plot around the third harmonic bin shows more details of this. From all these harmonic bins, THD power and maximum harmonic distortion power can be calculated. There is no real fundamental bin in the gray curve. When the INL is measured, all distortions within the full scale range of the ADC are excited. Therefore, the signal power is simply FS2/8, which is 1/8 or −9 dB according to the assumption in Section II. THD and SFDR values are computed based on THD power and maximum harmonic power. Another observation is that the noise floor of the INL spectrum is much lower than that of the digital output code spectrum, which is an advantage of INLbased measurement. In Fig. 9, the number of samples of the new method is chosen to be the same as the standard FFT method, which is 32 768. However, the new method can measure the spectral performance with much smaller number of samples due to the large number of samples in INL measurement. Above spectral performance measurements are based on true INL of the ADC and have shown good accuracy. In real world, only tested INL values are available for the new method. Due to limited testing resources such as stimulus generator and number of hits per code, tested INL may have a large amount of error. Fig. 10(a) shows the INL of the 16-bit ADC measured by histogram method using 32 hits per code. The linear input ramp used for testing contains additive noise with 0.5 LSB standard deviation. As a result, the measured INL is noisy compared with the true INL curve shown in Fig. 8(a). The measurement error can be as large as 0.4 LSB, as shown in Fig. 10(b), even the ramp is ideally linear.
Fig. 10.
(a) INL tested by histogram method. (b) INL measurement error.
Fig. 11. Comparison of traditional and new methods. (a) Estimation error distribution of THD. (b) Estimation error distribution of SFDR.
To investigate how the INL measurement error affects the new method, 500 different ADCs are randomly generated and measured by both the standard FFT method and the new
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TABLE II P ERFORMANCE OF F OUR ADC S
Fig. 12. (a) INL of ADC4. (b) Spectrum of ADC4 output code and INL.
method. The standard method measures the THD and SFDR with sine-wave input and 32 768 samples. The input sine wave contains noise with 0.5 LSB standard deviation. Therefore, the measured results of the 500 ADCs are different from their true performances. THD and SFDR values measured by a 32 768point standard FFT method with noise-free pure sine-wave input are considered as the true THD and SFDR. The measurement error of the standard method is obtained by subtracting the true THD and SFDR from its measured values. The new method computes the THD and SFDR from measured INL values. The INL values of these ADCs are measured by 32 hits per code histogram method with linear ramp input. The ramp contains noise with 0.5 LSB standard deviation, and thus, tested INL curves have similar error, as shown in Fig. 10(b). In the new method, 32 768 sine wave digital codes are mathematically generated first. Then, these digital codes are used as index to sinusoidally sample the measured INL, and a new set of data is constructed. THD and SFDR are then computed from
the new data set. Similarly, the measurement error of the new method is obtained by subtracting the true THD and SFDR from its measured values. Fig. 11 shows the distribution of 500 measurement errors of the new method. Measurement errors of the standard FFT method are also shown in Fig. 11 for comparison. The THD and SFDR estimation errors σ of the 32 768-point new method are 0.014 and 0.028 dB, respectively. The THD and SFDR measurement errors σ of the 32 768-point standard method are 0.055 and 0.089 dB, respectively. It can be seen that, with the same number of samples, the new method has higher accuracy than the standard method. The reason is that INL is usually tested from a large number of samples, which reduces noise effect. Benefit from this, the new method can use smaller number of samples to achieve same accuracy. Table I shows the accuracy of THD and SFDR measured by the new method with different number of samples. From the table, the accuracy of the new method with 2048 samples is same as that of the standard FFT method with 32 768 samples.
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Fig. 13. (a) INL of ADC4. (b) Spectrum of ADC5 output code and INL. TABLE III E STIMATION R ESULTS OF THE N EW M ETHOD (U NIT: D B)
TABLE IV SFDR E STIMATION R ESULTS OF THE N EW M ETHOD (U NIT: D B)
VI. E XPERIMENTAL R ESULTS The method has been also validated by experimental results. Based on measured INL values, the new method measures THD and SFDR values with only simple computations. The measured accuracy is comparable with that of the standard FFT method, which needs high-quality sine wave as input and 32 768-point FFT computation. Nine different SAR ADCs are tested by both the standard FFT method and the new method. The resolution of ADC1
to ADC4 is 16 bits, and the resolution of ADC5 to ADC9 is 12 bits. Performances of these ADCs are listed in Table II, in which, SNR value varies from 71.9 to 92.23 dB and INL varies from 0.59 to 2 LSB. ADCs with various performances can provide good validation of the proposed method. THD and SFDR values of all ADCs are measured by the 32 768-sample standard FFT method first. Since true performances of these ADCs can never be known, the new method will be compared with the standard FFT method. Figs. 12 and 13 show test results
DUAN et al.: TESTING ADC SPECTRAL PERFORMANCE WITHOUT DEDICATED DATA ACQUISITION
of ADC1 and ADC5, including linearity and spectral testing. INL values of ADC1 to ADC4 are tested by 128 hits per code histogram method. INL values of ADC1 to ADC4 are tested by 80 hits per code histogram method. In the spectrum plots, black curves are spectrums of the ADCs’ output signal obtained from the 32 768-point standard FFT method. The orange curves are spectrums of sinusoidally sampled INL. Full spectrums of sinusoidally sampled INL values are obtained only for plotting and comparison. To measure THD and SFDR values, only FT coefficients of the first 20 harmonic components are computed in the new method. Zoomed-in plots of the region around maximum harmonic frequency show that the spectrum of sinusoidally sampled INL has the same harmonic bins as the spectrum of ADC output codes. It is worth to notice that, in Fig. 12, the INL curve has several very steep jumps, which means that the ADC has high-order harmonic distortions. The new method has accurate measurement in this case because it can capture every order harmonic distortion based on INL. Tables 3 and 4 show the measurement results of the new method. In the new method, 16-bit ADCs are measured with 1024 samples and 12-bit ADCs are measured with 8192 samples. These samples are sinusoidally sampled INL data. The second rows list THD and SFDR values of these ADCs measured by the 32 768-point standard FFT method. The third rows list THD and SFDR values of these ADCs measured by the new method. The fourth row is the direct subtraction of decibel values measured by the new method and the standard FFT method. All differences are very small, which means that the new method can achieve same accuracy as the standard FFT method while using much smaller number of samples. It can be noticed that the THD value difference of ADC1 is larger than others in the fourth row. The reason is that ADC1 has smaller harmonic distortion than other ADCs. Meanwhile, it has the same noise level as others. Therefore, noise has more effects in harmonic distortion measurement no matter what method is used. Measurement errors of harmonic distortion power of all ADCs are very small that the accuracy becomes noise limited. On the whole, Tables 3 and 4 show that the new method gives a good-enough estimation of THD and SFDR. VII. C ONCLUSION The relationship between INL and harmonic distortion has been investigated by theory, simulations, and experiments. A low-cost test method has been presented to measure ADC THD and SFDR values from ADC’s INL test results. This method eliminates both hardware and data acquisition time cost for spectral performance test of ADC. In the circumstance of only low-frequency test is needed and INL has been already tested, this method requires only simple computations to measure THD and SFDR values. The new method avoids another round of data acquisition for spectral performance measurement and additional dedicated hardware. Both simulation and measurement results show that the accuracy of the method is comparable with that of the standard FFT method, which needs a high-quality signal generator and FFT computation with a large number of points.
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R EFERENCES [1] IEEE Standard for Terminology and Test Methods for Analog-to-Digital Converters, IEEE Std. 1241, 2010. [2] J. Doernberg, H.-S. Lee, and D. A. Hodges, “Full-speed testing of A/D converters,” IEEE J. Solid-State Circuits, vol. SSC-19, no. 6, pp. 820– 827, Dec 1984. [3] K. Arabi and B. Kaminska, “Efficient and accurate testing of analog-todigital converters using oscillation-test method,” in Proc. ED&TC, Paris, France, Mar. 1997, pp. 348–352. [4] M. Flores, M. Negreiros, L. Carro, and A. Susin, “INL and DNL estimation based on noise for ADC test,” IEEE Trans. Instrum. Meas., vol. 53, no. 5, pp. 1391–1395, Oct. 2004. [5] L. Jin, K. Parthasarathy, T. Kuyel, D. Chen, and R. Geiger, “Accurate testing of analog-to-digital converters using low linearity signals with stimulus error identification and removal,” IEEE Trans. Instrum Meas., vol. 54, no. 3, pp. 1188–1199, Jun. 2005. [6] F. Alegria, P. Arpaia, A. M. da Cruz Serra, and P. Daponte, “Performance analysis of an ADC histogram test using small triangular waves,” IEEE Trans. Instrum. Meas., vol. 51, no. 4, pp. 723–729, Aug. 2002. [7] S. Vora and L. Satish, “ADC static nonlinearity estimation using linearity property of sinewave,” IEEE Trans. Instrum. Meas., vol. 60, no. 4, pp. 1083–1290, Apr. 2011. [8] P. Carbone, E. Nunzi, and D. Petri, “Windows for ADC dynamic testing via frequency-domain analysis,” IEEE Trans. Instrum. Meas., vol. 50, no. 6, pp. 1571–1576, Dec. 2001. [9] Z. Yu, D. Chen, and R. Geiger, “A computationally efficient method for accurate spectral testing without requiring coherent sampling,” Proc. ITC, pp. 1398–1407, 2004. [10] H. Mattes, S. Sattler, and C. Dworski, “Controlled sine wave fitting for ADC test,” in Proc. IEEE ITC, 2004, pp. 963–971. [11] F. Adamo, F. Attivissimo, N. Giaquinto, and M. Savino, “FFT test of A/D converters to determine the integral nonlinearity,” IEEE Trans. Instrum. Meas., vol. 51, no. 5, pp. 1050–1054, Oct. 2002. [12] F. Adamo, F. Attivissimo, N. Giaquinto, and I. Kale, “Frequency domain analysis for dynamic nonlinearity measurement in A/D converters,” IEEE Trans. Instrum. Meas., vol. 56, no. 3, pp. 760–769, Jun. 2007. [13] A. C. Serra, M. F. da Silva, P. M. Ramos, R. C. Martins, L. Michaeli, and J. Saliga, “Combined spectral and histogram analysis for fast ADC testing,” IEEE Trans. Instrum. Meas., vol. 54, no. 4, pp. 1617–1623, Aug. 2005. [14] N. Csizmadia and A. J. E. M. Janssen, “Estimating the integral nonlinearity of AD-converters via the frequency domain,” in Proc. IEEE Int. Test Conf., 1999, pp. 757–761. [15] V. Kerzérho, S. Bernard, P. Cauvet, and J. M. Janik, “A first step for an INL spectral-based BIST: The memory optimization,” J. Electron. Testing—Theory Appl., vol. 22, no. 4–6, pp. 351–357, Dec. 2006. [16] S. Bernard, M. Comte, F. Azais, Y. Bertrand, and M. Renovell, “A new methodology for ADC test flow optimization,” in Proc. Int. Test Conf., Sep. 2003, vol. 1, pp. 201–209. [17] A. Buchwald, Specifying & Testing ADCs, Feb. 2010, IEEE ISSCC Tutorial. [18] F. Azaıuml;s, S. Bernard, Y. Bertrand, M. Comte, and M. Renovell, “Correlation between static and dynamic parameters of A-to-D converters: In the view of a unique test procedure,” J. Electron. Testing— Theory Appl., vol. 20, no. 4, pp. 375–387, Aug. 2004. [19] J. Duan, L. Jin, and D. Chen, “A new method for estimating spectral performance of ADC from INL,” in Proc. IEEE Int. Test Conf., Austin, TX, May 2010, pp. 1–10. [20] A. V. Oppenheim, A. S. Willsky, and I. T. Young, Signals and Systems. Englewood Cliffs, NJ: Prentice-Hall, 1983. [21] B. Brannon and R. Reeder, “Understanding High Speed ADC Testing and Evaluation,” Analog Devices, 2006, Application Note AN-835.
Jingbo Duan (S’11–M’12) received the B.S. degree in electrical engineering from Chongqing University of Posts and Telecommunications, Chongqing, China, in 2004; the M.E. degree in electronic and communications from Tsinghua University, Beijing, China, in 2007; and the Ph.D. degree in electrical engineering from Iowa State University, Ames, in 2011. From 2007 to 2008, he was with Conexant System Inc., Beijing, as an Analog and Mixed Signal Circuit Design Engineer. In 2012, he joined Broadcom Corp., San Jose, CA, as a Staff Scientist. His research interests are in the fields of analog and mixed-signal circuit design.
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Le Jin (S’02–M’06) received the B.S. degree in electrical engineering from Shanghai Jiaotong University, Shanghai, China, in 2001 and the Ph.D. degree in electrical engineering from Iowa State University, Ames, in 2006. From 2006 to 2011, he was with the Data Conversion Systems Group, National Semiconductor Corporation, Santa Clara, CA. Since 2011, he has been with Maxim Integrated Products, Sunnyvale, CA. Dr. Jin is a member of Tau Beta Pi.
Degang Chen (S’90–M’92–SM’02) received the B.S. degree in instrumentation and automation from Tsinghua University, Beijing, China, in 1984 and the M.S. and Ph.D. degrees in electrical and computer engineering from the University of California, Santa Barbara, in 1988 and 1992, respectively. From 1984 to 1986, he was with Beijing Institute of Control Engineering. From March to August 1992, he was the John R. Pierce Instructor of Electrical Engineering at California Institute of Technology, Pasadena. Then, he joined Iowa State University, Ames, where he is currently a Professor with the Department of Electrical and Computer Engineering. He was with Boeing Company in summer 1999 and was with Dallas Semiconductor–Maxim in summer 2001. His current interests include analog- and mixed-signal VLSI integrated-circuit design and testing, particularly low-cost high-accuracy test and built-in self-test of analogand mixed-signal and RF circuits, as well as self-calibration and adaptive reconfiguration/repair strategies for enhancing the performance and yield of such circuits. Dr. Chen was the A. D. Welliver Faculty Fellow with Boeing Company in 1999. He was the recipient of the Best Paper Award at the 1990 IEEE Conference on Decision and Control; the Best Transaction Paper Award from the ASME Journal of Dynamic Systems, Measurement, and Control in 1995; and an SRC technology invention reward in 2005.
