110 IEEE TRANSACTIONS ON INSTRUMENTATION ANDMEASUREMENT, VOL. 54, NO. 1, FEBRUARY 2005 Uncertainty of ADC Random Noise Estimates Obtained With the IEEE 1057 Standard Test Francisco André Corrêa Alegria and António Manuel da Cruz Serra Abstract—In this paper, the uncertainty of the estimated stan- dard deviation of the random noise present in analog-to-digital converters (ADCs), using the test suggested in the IEEE 1057-94 Standard, is thoroughly analyzed. Expressions for the test result uncertainty and for determination of the test parameters are de- rived. Index Terms—Analog-to-digital converter (ADC) testing, random noise, uncertainty. I. INTRODUCTION T HE TRADITIONAL test method used to estimate analog-to-digital converters (ADCs) random noise is the one suggested in the IEEE Standard for Digitizing Waveform Recorders [1]. For low random noise standard deviation, a triangular stimulus signal is applied to the ADC input, and two sets of samples are acquired synchronously ( and ). The output codes obtained are then subtracted to eliminate possible ADC fixed errors, and the mean square error is calculated using mse (1) This error is then used to estimate the standard deviation of the random noise present in the ADC, , expressed in least signif- icant bit units (LSB), through mse mse (2) Actually, all the random additive noise present in the test setup is estimated. Special care should be taken to minimize the noise not due to the ADC itself, namely, the one introduced by the function generator and by improper connection cables. The knowledge of this kind of noise is important, for instance, in selection of the proper ADC for a given application. It is also important, when performing the standard static test [1] and the standard histogram test [1]–[7], for the determination of the un- certainty of the ADC transition voltages and code bin width es- timates. The estimation of the noise standard deviation obtained using (2) is biased, as referred in [1] and demonstrated in [8]. The Manuscript received July 9, 2003; revised June 5, 2004. This work was supported by the Portuguese National Research Project entitled “New error correction techniques for digital measurement instruments,” reference POCTI/ESE/46995/2002. The authors are with the Instituto Superior Técnico, Technical Uni- versity of Lisbon, 1049-001 Lisboa, Portugal (e-mail: falegria@lx.it.pt, acserra@ist.utl.pt). Digital Object Identifier 10.1109/TIM.2004.840226 value of the estimation error is very small, lower than 0.022 LSB, if a triangular amplitude greater than five ideal code bin widths is used [8]. The uncertainty of the noise standard deviation has, however, never been studied, as far as the authors are aware. We use the study presented in [8] and take it a step forward in order to an- alyze the probability distribution of the estimate of the random noise standard deviation and to obtain an expression for its un- certainty. An expression for the minimum number of samples required in order to limit the estimates uncertainty to a chosen level is also proposed. In Sections II–VII, we will analyze the probability distribu- tion of several random variables, starting with the ADC output codes, continuing with the square error, the mean square error, and finally, the estimated random noise standard deviation. Note that all voltage variables are normalized (divided) by the ideal ADC code bin width . II. ADC OUTPUT CODES As seen in [8], the output code of a sample has a discrete distribution with probability given by [8, eq. (11)] (3) where is the ADC number of bits, is the normalized tran- sition voltage , is the normalized input voltage and (4) The conditional mean, 2nd moment and variance of the output codes are, by definition [9] and (5) It was seen in [8] that the ADC output codes standard deviation, for small random noise standard deviation and when twice the amplitude of the triangular wave amplitude is an integer, is given by (6) 0018-9456/$20.00 © 2005 IEEE