Histogram Tests for Wideband Applications

Transcription

1 Histogram Tests for Wideband Applications Niclas Björsell 1 and Peter Händel 2 1 University of Gävle, ITB/Electronics, SE Gävle, Sweden niclas.bjorsell@hig.se, Phone: , Fax: Royal Institute of Technology, Signal Processing Lab, SE Stockholm, Sweden peter.handel@ee.kth.se Abstract Characterization and testing of analog-to-digital converters (ADCs) are important in many different aspects. Histogram test is a common method to characterize the linearity features of an ADC. Two commonly used stimuli signals are sine waves and Gaussian noise. This paper presents a metrological comparison between Gaussian and sine wave histogram tests for wideband applications; that is we evaluate the performance in characterization of the ADC and the usability of post-correction. A post-correction procedure involves characterization of the ADC non-linearity and then use of this information by processing the ADC output samples to remove the distortion. The results show that the Gaussian histogram test gives reasonable accuracy to measure nonlinearities. However, it does not result in a suitable model for post-correction in wideband applications. A single-tone sine wave histogram will be a better basis for post-correction. Best result can be obtained if the look-up table is trained with several single-tone sine waves in the frequency band. Keywords Analog to Digital Converters, ADC, Histogram, Test, Measurements I. INTRODUCTION Characterization and testing of ADCs are important in many different aspects. ADCs have for a long time been the major bottleneck in radio frequency (RF) instrumentation design and

2 telecommunication applications. An important effect is that ADC testing is a major factor of cost for manufacturers, and shortening of the test cycle implies large savings. Another is that a proper characterization of the ADC can be used for error compensation by using post-correction methods [1]-[2]. A histogram test is a common method to characterize the linearity features of an ADC. The ADC histogram test method is considered as an estimation problem where the task is to estimate an arbitrary transition level (T k ) based on the ADC output. Histogram tests give the number of occurrences of each code at the output of the converter. The resulting histogram is then compared with the probability density function (PDF) of the stimulus signal. Sine waves are commonly used as stimulus signals, due to the easiness with which they can be generated and their spectral purity. Another signal suitable for histogram tests is noise sequences. In [3], the advantages of using a Gaussian noise stimulus instead of sinusoid were discussed and some advantages were presented like: (i) The noise is as easy or easier to generate than a sine wave, (ii) if the generated noise obeys a Gaussian distribution then any additional noise, associated with the test ensemble just adds its variance to the one of the generated noise, assuming that it also possesses a Gaussian distribution, (iii) white noise is a wide band signal. In a single characterization run we can thus relate merit figures of the converter to a frequency band. The statistical efficiency of an unbiased estimator can be evaluated by comparing its variance to a corresponding theoretical minimum value, denoted Cramér-Rao lower bound (CRLB). An asset of histogram tests is that the estimator variance can be made arbitrary close to the CRLB [4]. The Gaussian histogram test (GHT) has been analyzed with respect to the estimator statistical performance [5]. It has been proven that the GHT provides asymptotically (as the number of data tends to infinity) efficient and unbiased estimates subject to a Gaussian stimulus. Further, the performance of GHT has been compared to the performance of sine wave histogram test (SHT) with respect to the estimator variance. The performance difference is mostly affected

3 by the signal to noise ratio (SNR) of the SHT stimulus. However, for reasonable SNR the SHT performance is superior to the performance of GHT for a given data length, in terms of estimator variance. On the other hand GHT, after compensation for gain error, does not depend on external noise. Moreover, if the ADC will be used for a wide band signal several sine wave tests are required. In this paper a metrological comparison between GHT and SHT is presented, primarily for wideband applications such as RF instrumentation and telecommunication. Measurements are performed on a commercial 12 bit, 210 MSps ADC, that is the Analog Devices AD9430. The purpose is to evaluate the performance in characterization of a state-of-the-art ADC and the usability for post-correction in both methods. A focus will be on the examination to what extent a single characterization by white noise can be useful as a characterization over a frequency band. II. APPROACH In [3], several advantages over alternative methods and a theoretical analysis of using Gaussian noise as stimuli are presented. In addition, the statistical efficiency of the GHT as a transition level estimator is analyzed in [5]. With that, a solid theoretical basis for using GHT is given. Moreover, in [6] a successful evaluation based on measurements are performed. However, the comparison is carried out on only two single-tone measurements. The focal point in our work is to compare the Gaussian stimuli with several single-tones that cover a wide frequency band. The different SHT frequencies will produce different result due to the dynamic errors of the device under test (DUT). Therefore it is interesting to characterize ADCs and their postcorrection in a frequency band rather than a single frequency. Our first approach is to use a set of sine waves in the frequency range from 120 to 200 MHz to cover the band, and secondly, to use a white Gaussian noises (WGN) as a one-shot broad band signal.

4 All SHT frequencies have been chosen according to the IEEE standard [7]. In this paper we will refer to signal frequencies in even megahertz [MHz]; though, the correct values can be found in Table 1. The sampling frequency has been Hz, or for short 210 MHz. This means that all frequencies are above Nyquist frequency and that under-sampling is used. Thereby one can expect to have more effects from dynamic errors compared to frequencies below Nyquist frequency, since the dynamic performance decreases with increased frequency according to the ADC datasheet. Table 1: The frequencies used are adjusted to obtain coherent sampling. Requested Frequency [MHz] Used Frequency [MHz] The ADC test-bed is composed of commercial state-of-the-art instruments and components designed for this test-bed. Even with a state-of the art signal generator (R&S SMU200A), additional signal conditioning is required. Filters are used to clean-up spurs and noise from the test signal. However, the filters attenuate the signals; therefore it has to be amplified to obtain sufficient drive level. For that purpose ultra low distortion ADC driver has been designed with a frequency range of MHz and 14dB gain. The amplifier ensures spectral purity even at

5 high output levels by >80dBm output 2 nd order intercept point (IP2), >49dBm IP3 and a noise level below 169dBm/Hz. A frame grabber (FG) acquires data from the ADC. The FG interface to the ADC under test and in real-time record the binary represented samples at a maximum speed of 350 MHz, width 16 bits, and depth 2 MSample (4 MByte). Samples are delivered from the ADC synchronously in binary format on low voltage differential signaling (LVDS) busses. After the acquisition process is completed the acquired data is uploaded to the PC over LAN for analyses. The digital outputs of the device under test are LVDS to provide for lowest possible internal device and printed circuit board (PCB) coupling. The test-bed is further described in [8]. A. Characterizing integral non-linearity One important performance parameter of an ADC is integral non-linearity (INL) [7], which can be described as the maximum difference between the ideal and the actual code transition levels after correction for gain and offset. A high demand on low distortion requires an ADC with low INL. The method to characterize ADC by using histogram tests is to a considerable extent the same for sine waves and Gaussian noise. The method is described in [7]. However, histogram methods can produce erroneous results if the device being tested has output codes that are swapped with other codes or exhibits other types of non-monotonic behavior. Such converters can produce seemingly good results; yet have large errors in the actual code transitions. To avoid these issues, converters should also be tested for dynamic performance to confirm that non-monotonic behavior is not significant.

6 B. Dynamic performance Commonly used measures of dynamic performance are often based on single-tone measurements; for example spurious free dynamic range (SFDR), total harmonic distortion (THD), and signal to noise and distortion ratio (SINAD). In addition there are measures such as intermodulation products, adjacent channel leakage ratio (ACLR), and missing tone that are characterized by multi-tone or wideband signals. Hence, the choice parameter for evaluation is not obvious; it must be done in accordance with the application. We have chosen THD to evaluate the ADC in the frequency domain. THD is the ratio of the root-mean-square value of the fundamental signal to the mean value of the root-sum-square of its harmonics. While evaluating the harmonic distortion the input signals are 0.5 db below fullscale. THD is a convenient figure of merit for non-linearity evaluations since non-linearity implies harmonic distortion in the output signal. Consequently, it will be a suitable parameter for evaluation to what extent we can use our knowledge of the non-linearity and thereby correct the signal to avoid harmonic distortion. C. Post-correction A post-correction procedure involves a characterization of the ADC non-linearity and then using this information by processing the ADC output samples to remove the distortion. Postcorrection can be performed in several ways [2]. One distinction between different methods is whether we shall use information about the input signal slope or not. Due to simplicity we have chosen look-up tables without slope information. Each table is trained by a specific signal according to the calibration method described in [1]. In this paper we have one table for each

7 frequency in Table 1 and one table for the noise stimulus. Moreover, we have trained one table with all frequencies which in some sense can be considered as a wideband alternative to the noise-trained table. New data series with the same frequencies as in Table 1 is then corrected with the different look-up tables and evaluated. The new data series for evaluation have lower amplitude (FS -0.5dB) compared to data series for training look-up tables (FS + over drive according to [7]). III. RESULTS The measurements are performed using an Analog Devices AD9430, 12 bit ADC at 210 MSps. The analogue input is in the range from 120 to 200 MHz. All frequencies are above Nyquist frequency, which means that under-sampling is used and thereby one can expect to find more influence from dynamic errors. A. Integral non-linearity Starting with the set of sine waves, single-tone measurements over all frequencies in the frequency band have been performed, and thereafter the INL was calculated. In Figure 1, the results are plotted in a three-dimensional diagram. One can see that the INL is slightly S-shaped over the transition levels. As expected the different frequencies produce different results, due to dynamic errors. The variations over frequency can be considered small but not negligible; this will be further discussed in the section of dynamic performance and post-correction. The maximum INL for each frequency is shown in Figure 2. The maximum INL is fairly constant over the full bandwidth floating around 2 LSB.

8 INL (LSB) Transition levels (LSB) Frequency (MHz Figure 1: SHT characterizations for all nine frequencies separately. 3 max INL max INL (LSB) Sample Fequency (MHz) Figure 2: Maximum INL variation as a function of frequency. After the sequence of sine waves a Gaussian noise was used to characterize the same ADC. The purpose of this study is to investigate if a single GHT can be an alternative to performing several SHT tests. The data length of GHT is increased by a factor 10 and thereby has more competitive

9 performance than SHT according to [5]. The result from the measurement is presented in Figure 3. As can be seen the shape of the curve is close to the shape from the SHT that is plotted in Figure 1. A more illustrative comparison is the INL from GHT and a comprehensive INL for SHT. Figure 4 shows the corresponding INL from the look-up table trained by all frequencies. To some extent, the shape and amplitude from Figure 3 and Figure 4 are similar. However, the dynamic performance will show more deviation between the models, which will be further discussed in the next section. 1 INL INL (LSB) Transition levels (LSB) Figure 3: INL characterized for Gaussian stimuli signal.

10 1 INL INL (LSB) Transition levels (LSB) Figure 4: INL characterized by multiple sine waves. In order to evaluate the GHT there is a comparison to SHT performed in [6]. Although that comparison only covers two lower frequencies the results from [6] are in concordance with these results, although their deviation is somewhat larger. One reason for this might be that the noise in the current study is filtered with a 210 MHz low pass filter. B. Dynamic performance As was mentioned earlier the static non-linearity is not sufficient to characterize ADCs. Measurements in the frequency domain are required as well. Initially, we measured the SFDR for all frequencies from 120 MHz to 200 MHz. The purpose for this is twofold. First, it verifies the dynamic performance. Secondly, it is used to verify the measurement set-up. Testing dynamic performance of high-speed ADCs is regarded as difficult. By comparing to what extent the practical performance corresponds to data from the ADC datasheet the measurement set-up can be evaluated. When the measurement results represent the true performance of the ADC the

11 impact from the test-bed [8] can be neglected. In Figure 5, the measured SFDR is presented, and it corresponds to datasheet values. 80 SFDR SFDR (dbc) Frequency (MHz) Figure 5: The dynamic performance for all frequencies with reference to SFDR (solid line). As a reference, datasheet values (dotted line) are plotted. 75 THD THD (dbc) Frequency (MHz) Figure 6: The dynamic performance for all frequencies with reference to THD.

12 Total harmonic distortion is not given in the datasheets, but still is an interesting parameter to study. In Figure 6 the measured THD is presented. Below, THD will also be used to evaluate effectiveness of post-correction. C. Post-correction The first aim of this study has been to characterize non-linearities in ADCs. A second aim is to use the obtained information in order to correct for the distortion. By correcting for nonlinearities the harmonic distortion is supposed to be reduced. However, there is a problem by using the error model based on sine wave input signals. The correction is based on a look-up table that is trained for a single frequency only. This will result in inferior performance for signals with other frequencies. Figure 7 illustrates this phenomenon. A 130 MHz sine wave is corrected with a look-up table trained by frequencies from Table 1. Without any post-correction THD is 71.5 dbc. All models will improve the THD and as expected the best results is from a model trained on 130 MHz.

13 85 Signal frequency at 130 MHz 80 -THD (dbc) Training frequency (MHz) Figure 7: THD for a 130 MHz signal post-corrected by look-up tables trained by frequencies from 120 MHz to 200 MHz (solid line). THD without post-correction (dashed line) is printed as a reference. In order to study how much the THD depends on the training frequency. The outcomes of 81 post-corrected signals are compared in Figure 8. Each of the frequencies in Table 1 is used to train a look-up table. Thereafter, new signals from all frequencies are post corrected. On the x- axis is shown the difference between training frequency and signal frequency. On the y-axis is the improvement in THD. A mean value is added to illustrate a trend. However, the number of data is few and rather spread. Consequently, the mean value is just a trend and should not be considered as an exact value of improvement. As can be seen from the figure the compensation performance reduces by 4.5 db when using a model trained by a frequency other than the signal frequency. The model seems to be narrow banded. On the other hand, the distance from training frequency seems not to be of importance. A similar evaluation is performed in [9] with lower frequencies below Nyquist frequency, and with

14 SFDR instead of THD as figure of merit. Despite these dissimilarities, both studies point out frequency dependency in the dynamic performance by using this method. 16 Improvment i THD Improvment in THD (db) MHz from training frequency (MHz) Figure 8: The difference (x-axis) between the training frequency and the actual signal frequency affects the improvement in THD (y-axis). A trend curve (solid line) is used to illustrate the dependency. The implication of the above results is that a broadband model is needed to build a look-up table. Two different methods are used and compared verses each other and the previously described single-tones models. First, an additional look-up table was trained with all the frequencies and in that way make a broadband model. In Figure 9, the obtained THD improvement is compare with the results from the single-tone models. Since their performance is frequency dependent there will be an interval for each frequency. The bars in Figure 9 are intervals where the upper value is the THD improvement from the model trained by the same frequency, and the lowest values is from the least fitting model. We can see that the broadband model always appears in the upper half of the interval and often close to the top value.

15 18 Improvment in THD Improvment in THD (db) Frequency (MHz) Figure 9: Post-correction based on look-up table trained with all frequencies. The vertical bars are the improvement interval for SHT trained by single frequencies. Secondly, a look-up table was trained by the noise signal. The evaluation follows the same principle as with multiple single tones and the result is presented in Figure 10. One may note that the performance is inferior compared with the performance of the previous model. In fact, in most cases it does not even reach the performance of the least fitting single tone model.

16 18 Improvment in THD Improvment in THD (db) Frequency (MHz) Figure 10: Post-correction based on look-up table trained with noise. The vertical bars are the improvement interval for SHT trained by single frequencies. IV. CONCLUSIONS Wideband signals are commonly used in many applications, for example in telecommunication and instrumentation industry. Hence, there is a need of characterizing ADC for wideband signals. By using sine wave histogram tests several single-tone measurements are required. However, Gaussian histogram tests theoretically seem to give a useful characterization for just one measurement, even though the data length has to be somewhat larger to reach the same performance. The focal point in this paper has been to examine to what extent a single characterization by white Gaussian noise can be useful as a characterization over a frequency band as an alternative to several single-tone characterizations. The objective was to evaluate the different stimuli so that the test procedure can be as efficient as possible to find an appropriate characterization for post-

17 correction based on look-up tables. The evaluation is based on measurements on a commercially available high-speed ADC. The results show that the Gaussian histogram test gives reasonable accuracy to characterize the integral non-linearity. However, when results from the GHT was stored in a lock-up table and used for post correction the improvement in THD was in the range of 1-3 db, which is lower than the improvement from almost all look-up tables based on single-tone characterization in the studied frequency range ( MHz). An additional look-up table was trained by several frequencies in the frequency range and resulted in an improvement of db after post correction; a result inferior to GHT as well as all single-tone characterization and thus the preferred stimulus to reduce distortion (THD) in wideband application. An interesting continuation on this paper would be to use dynamic models for characterization and post-correction. ACKNOWLEDGMENT This work has been supported in part by The Knowledge Foundation (KKS), Ericsson AB, Note AB, Syntronic AB, Racomna AB, Rohde & Schwarz GmbH and the Graduate School of Telecommunications (GST). REFERENCES [1] H. Lundin, M. Skoglund, and P. Händel, "A Criterion for Optimizing Bit-Reduced Post- Correction of AD Converters," IEEE Transactions on Instrumentation and Measurements, vol. 53, pp , 2004.

18 [2] E. Balestrieri, P. Daponte, and S. Rapuano, "A state of the art on ADC error compensation methods," Instrumentation And Measurement Technology Conference, Como, Italy, [3] R. C. Martins and A. M. Cruz Serra, "The use of a noise stimulus in ADC characterization," IEEE Int. Conf. on Electronics, Circuits and Systems, [4] N. Björsell and P. Händel, "A Statistical Evaluation of ADC Histogram Tests with Arbitrary Stimuli Signal," ADDA05, Limerick, [5] A. Moshitta, P. Carbone, and D. Petri, "Statistical Performance of Gaussian ADC Histogram Test," Int Workshop on ADC modelling and testing, [6] R. C. Martins and A. M. Cruz Serra, "Automated ADC Characterization Using the Histogram Test Stimulated by Gaussian Noise," IEEE trans. on Instrumentation and Measurement, vol. 48, pp , [7] IEEE, "Std IEEE Standard for Terminology and Test Methods for Analog-to- Digital Converters," [8] N. Björsell, O. Andersen, and P. Händel, "High dynamic range test-bed for characterization of analogue-to-digital converters up to 500MSPS," 10th Workshop on ADC modelling and testing, Gdynia, Poland, [9] P. Händel, M. Skoglund, and M. Pettersson, "A calibration scheme for imperfect quantizers," IEEE Transactions on Instrumentation and Measurement, vol. 49, pp , 2000.