2762 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 57, NO. 12, DECEMBER 2008 ADC Testing With Verification Balázs Fodor and István Kollár, Fellow, IEEE Abstract—An important method for analog-to-digital-converter (ADC) testing is sine wave fitting. In this method, the device is excited with a sine wave, and another sine wave is fitted to the samples at the output of the ADC. The acquisition device can be analyzed by looking at the differences between the fitted signal and the samples. The fit is done using the least-squares (LS) method. If the samples of the error (the difference between the fitted signal and the samples) were random and independent of each other and of the signal, the LS fit would have very good properties. However, when the error is dominated by quantization error, particularly when a low bit number is used or the level of the measured noise is low, these conditions are not fulfilled. The estimation will be biased, and therefore, it must be corrected. The independence of the error samples is more or less true if the sine wave is noisy or dither is used. In these cases, the correction is not necessary. Therefore, it is reasonable to analyze the effect of the potentially unnecessary correction to noisy data, and it is desirable to determine the magnitude of the noise from the measurements. In this paper, these two questions are investigated. The variance of the corrected estimator is investigated, and a new noise estimation method is developed and analyzed. Index Terms—ADC test, analog-to-digital converter (ADC), effective number of bits (ENOB), elimination of samples, IEEE Standard 1241-2000, least-squares (LS) fit, noise estimation, sine wave fitting, sine wave test. I. I NTRODUCTION S INE FITTING may be the most important method in the testing of analog-to-digital converters (ADCs) under IEEE Standard 1241-2000. The essence of this method is the fitting of a sine wave to the samples that appear at the ADC output. The errors of the converter can be analyzed by looking at the difference between the fitted signal and the samples. Fitting is executed using the least-squares (LS) method. Error e is defined as the difference between observations y and model m. The observations are the samples, and the model is the test signal, whose parameters are unknown. Minimizing the sum of e 2 n , the LS fit is performed, i.e., min m  n (y n − m n ) 2 = min m  n e 2 n . (1) Manuscript received July 5, 2007; revised May 23, 2008. First published August 1, 2008; current version published November 12, 2008. This work was supported by the Hungarian Scientific Research Fund (OTKA) under Grant TS49743. The Associate Editor coordinating the review process for this paper was Dr. Richard Thorn. B. Fodor is with the Institute of Communications Technology, Braunschweig Technical University, 38106 Braunschweig, Germany (e-mail: balazs.fodor@ web.de). I. Kollár is with the Department of Measurement and Information Systems, Budapest University of Technology and Economics, 1521 Budapest, Hungary (e-mail: kollar@mit.bme.hu). Digital Object Identifier 10.1109/TIM.2008.928404 Fig. 1. Original sine wave, quantized samples, and quantization error (B =3 bits, dc =0). Fig. 2. PDF of the quantization error. (B =3 bits, and the dc component of the sine is nonzero.) The LS method has very good properties, particularly when the error is random and zero-mean Gaussian, and the samples are independent. However, when error e is dominated by quan- tization noise, neither of these is true. The main problem is that, even by ideal quantization, the quantization error strongly depends on the input signal (see Fig. 1). Therefore, the estimated parameters, particularly the esti- mated amplitude, will usually be biased. An easy-to-see exam- ple is the case when the dc level of the input signal is not zero. The peak in the probability density function (pdf) of the quan- tization error (the part that is related to the peaks of the sine wave) is no longer at the middle, and the mean value of the error is no longer zero (see Fig. 2). This causes an error in the amplitude estimation. However, the amplitude estimation can still be erroneous, even if the error distribution is symmetric to zero. 0018-9456/$25.00 © 2008 IEEE Authorized licensed use limited to: IEEE Xplore. Downloaded on December 20, 2008 at 04:27 from IEEE Xplore. Restrictions apply.