Elsevier

Measurement

Volume 45, Issue 2, February 2012, Pages 148-154
Measurement

The Student’s t distribution to measure the word error rate in analog-to-digital converters

https://doi.org/10.1016/j.measurement.2011.03.002Get rights and content

Abstract

The paper aims to the Word Error Rate (WER) assessment in Analog to Digital Converters (ADCs) as specified in Annex A of the IEEE Std. 1241. A more detailed WER estimation, based on a new statistical approach, is proposed with respect to the approach described in the IEEE standard. In particular, Student’s t distribution has been introduced for a more accurate WER measurement considering n successive observations. The proposed method has been experimentally verified by means of WER measurements, where the testing results were compared to standard method. The final scope of this research is to give a contribution for the next version of such standard.

Introduction

There has been a rapid increase in the speed and accuracy of the data conversion systems. Apart from being a design challenge, testing of high-performance data converters has become a huge challenge for the engineers. For this reason, in order to provide a guide to engineers designing ADC test methods and systems, many technical standards have been developed in the years by IEEE [1], [2].

Beyond test accuracy, test speed is one of the most important topic especially in electronic industry where new products are outmoded quickly. According to this requirement, several research papers can be found proposing test methods to improve the standard ones in order to obtain higher speed with the same accuracy or better accuracy with the same speed [3], [4]. The main research contributions focus on dynamic testing in the time and frequency domains looking at fast assessment of Signal to Noise Ratio, Signal to Noise and Distortion, Effective Number of Bits, Total Harmonic Distortion, Spurious Free Dynamic Range, Integral and Differential Nonlinearities [3], [4], [5], [6]. All of the previous figures of merit concern with the analog circuitry of the ADCs. Less contributions can be found on the digital counterparts and the figures of merit that qualify them, like the Word Error Rate (WER).

Typical causes of word errors can be individuated in the metastability [1], [7], [8], [9] and timing jitter [1], [8], [9], [10] of comparators within the ADC. In particular, considering the metastability in comparators, a word error occurs when the input voltage to a pipeline stage is extremely close to the voltage threshold with which the comparator compares it, resulting in a classic metastability condition which can take an extraordinarily long time to settle [9].

The WER is defined by the IEEE Std. 1241 as the probability of receiving an erroneous code for an input, after correction is made for gain, offset, and nonlinearity errors, and a specified allowance is made for noise regardless the cause [1]. This definition should be better expressed taking into account that the number of metastability-induced word errors increases with sampling frequency [7], [8], while the jitter-induced errors depend on the signal amplitude and frequency [10].

From the manufacturers’ point of view, a comprehensive document which could be used as a guide of ADC standards and testing methods, and meet the requirements of different applications, is required [11]. In particular, the Ref. [7] puts in evidence that the Word or Bit Error Rate is not usually specified on most ADC data sheets. The ADC can give wrong indications if its BER is not sufficiently small. Excessive ADC Bit Error Rates in communications applications, for instance, can degrade the overall system Bit Error Rate. As it can be seen from [7], the WER measurement requires extensive characterization of the devices under a variety of input conditions and error limits, resulting in a complex and long procedure. Due to the typical WER values, given as parts per million, or parts per billion, a typical WER test session requires several days, thus making such assessment difficult to do within an industrial production process. Referring WER measurement in the industry scenario, papers [12], [13] assume only the contribution of comparator metastability, which is estimated by simulations starting from a simple model of the comparator. Instead, the application note [9] shows a WER measurement methodology and characterization results for a particular family of converters. A statistical approach is also discussed but it’s not for a general purpose use. Moreover, an evaluation of the uncertainty correlated with WER measurement is missing.

By taking in account the points of view of the different stakeholders the IEEE Instrumentation and Measurement Society TC-10 “Waveform generation, measurement and analysis” standardized the WER measurement for ADCs in the IEEE Std. 1241 [1].

In particular, the present paper focuses on the static test, the most used one, that is matter of concern for the industrial testing due to the long acquisition times required to achieve an adequate number of WER observations. The number of observations to be used for the WER measurement can be obtained according to the Annex A of the 1241 standard that discusses the presence of statistical errors associated with WER measurement. In this context the source of word errors is assumed to be purely random and both the number of observed word errors and the total number n of trial samples are supposed to be statistically significant. Even when users are not interested in the knowledge of the exact WER, but they would be satisfied with an upper limit the standard [1] suggests to acquire, at least, ten times the number of samples expected to be affected by a word error. With such assumptions the WER tests can be very time expensive due to the significant number of acquisitions needed to use the standard model. It’s important to note that only when the samples belong to normal population with known standard deviation a normal distribution may be assumed and the WER values can be obtained using the method suggested in the Annex A of the IEEE Std. 1241 [1].

Considering that the WER measurements can be obtained only in very small numbers in a reasonable time, in the evaluation of the WER and its uncertainty interval, the use of the Gaussian model (as required by IEEE Std. 1241) it’s not possible: to this aim other statistical techniques [14], which can better harmonize and integrate what is stated in [1] and in [15], based on Student’s t and chi-square distribution, fixing the number of degrees of freedom and using the t table to determine the confidence level, are proposed in the paper.

If the distribution of the WER measurement cannot be considered normal, the uncertainty and the relative confidence intervals could be evaluated using the bootstrap method [16] that allows to define confidence intervals without the need of introducing hypotheses about the distribution of the measured parameters. This interesting technique will be object of a future development of this research work.

In the next section, after a brief recall of the theoretical bases, the proposed approach for the WER estimation is presented. In order to validate the proposal, several experiments have been carried out on an actual devices. The test setup and achieved results are reported and discussed in Section 3, before providing the conclusions.

Section snippets

WER estimation for n successive observations

It is well known that a generic normally distributed measure M = N(m,u) with expected value m and standard uncertainty u, can be expressed in a normalized form by using the M-mu=N(0,1), with expected value zero and unitary standard uncertainty. It is also known that in the presence of a number ν of normal random variables given in the reduced form N1(0, 1), …, Nν(0, 1), mutually independent and independent from M, the sum of squares of such variables, that is χν2=i=1nNi2(0,1), is distributed like

Test setup and results

The experimental validation of the proposed method has been carried out by using the test setup shown in Fig. 1. The Tektronix Arbitrary Waveform Generator AWG420 has been used to provide a sinusoidal signal to the Tektronix oscilloscope TDS 7704B. The records acquired by the oscilloscope have been then processed by a PC to compute the WER. Since the WER is small (usually measured in parts per million or parts per billion), a lot of samples must be collected to test for it [1]. Before starting

Conclusions

The aim of this paper is to introduce a new method to measure the Word Error Rate for improving the Annex A of the IEEE Std. 1241 [1]. This is made according with GUM [15] and its supplements [18] carried out by BIPM. It has been quantified the uncertainty level with relative confidence level in the case of n successive observations using Student’s t distribution, with the final aim to give a contribution for a new draft of such standard.

The proposed method has been applied to the WER values

Acknowledgments

The research work described in the paper has been supported by Italian Ministry of Education, University and Research in the framework of the program “Research Projects of National Interest (PRIN)”, Project n. 2008S9J8XE “Metrology of the A/D and D/A conversion: standardization of the figures of merit, error estimation and correction, uncertainty evaluation”.

References (18)

  • H. Kobayashi et al.

    ADC standard and testing in Japanese industry

    Computer Standard and Interface

    (2001)
  • IEEE Std. 1241, IEEE Standard for terminology an test methods for analog-to-digital converters, IEEE, Piscataway, NJ,...
  • IEEE Std. 1057, IEEE Standard for digitizing waveform recorders, IEEE, Piscataway, NJ, USA,...
  • A.C. Serra et al.

    Combined spectral and histogram analysis for fast ADC testing

    IEEE Transactions on Instrumentation and Measurement

    (2005)
  • R. Holcer et al.

    DNL ADC testing by the exponential shaped voltage

    IEEE Transactions on Instrumentation and Measurement

    (2003)
  • S. Rapuano et al.

    ADC parameters and characteristics

    IEEE Instrumentation and Measurement Magazine

    (2005)
  • E. Balestrieri et al.

    Experimental comparison of different standards for dynamic characterization of ADCs

    IEEE Transactions on Instrumentation and Measurement

    (2009)
  • Analog Device, Tutorial MT-011, Find Those Elusive ADC Sparkle Codes and Metastable States, 2009....
  • G. Chiorboli, B. De Salvo, G. Franco, C. Morandi, Some thoughts on the word error rate measurement of A/D converters,...
There are more references available in the full text version of this article.

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