A survey on the dynamic characterization of A/D converters
Introduction
The A/D converter is the major component linking the analog world to the numerical one. This conversion reduces the quality of the analog signal which is converted in a numerical form. It is therefore important to measure the systematic error and defects introduced by the analog to digital conversion. In this paper, we present a theoretical and experimental approach for the dynamic characterization of the A/D converter.
In the first part of this paper is presented a theoretical background of the three major methods for the A/D converters characterization. First, the analysis of the ADC output signal is considered in time domain. The obtained parameters, which allow an ADC performance evaluation, are related to two standards: the JEDEC standard giving parameter definitions [1] and the IEEE-1057 Standard setting the measurement methods [2]. One of these parameters is the number of effective bits (neff). Indeed, an r-bit A/D converter is associated with a systematic error, the quantization noise, whose theoretical power Brms depends on r. When an A/D converter with the same resolution is tested, the measured quantization noise has a power Bq greater than Brms. So, an estimation of a number of effective bits associated with the measured quantization noise could be given by comparison with the theoretical one. In the time domain analysis, the Brms is deduced from the error signal, defined as the difference between the input and output signals of the ADC. In this case, in order to estimate the input signal, the output numerical signal is computed and fitted using a least-square method to obtain an estimation of its amplitude, offset, frequency and phase [2]. This method implies the resolution of a nonlinear equation system by iterative method, requiring initial values. These values generally correspond to the test setup with a perfect input signal generator. If generators were really perfect, any classical resolution algorithm (Newton, Gradient, Simplex) could converge to a correct result. But, in the case where the input frequency value is not precise, these algorithms would not converge. In order to eliminate this influence, we propose a method giving a direct estimation of the ADC conversion noise. It is based on the eigendecomposition value of the correlation matrix data [3]. This method will be presented in the third section of this paper.
Another analysis concerns statistics. Basically, a histogram is built, giving the number of occurrence codes H[i], depending on the code value (i). To be normalized, H[i] is then divided by the total number of samples N. The ADC transfer characteristic parameters, like differential nonlinearity DNL and integral nonlinearity INL, are directly deduced from the histogram. In this paper, three different characterization methods related to the statistical analysis are presented. Method 1 is the comparison between the effective histogram and the theoretical histogram obtained when considering a perfect ADC [4]. Method 2 and Method 3 are based on the cumulative histogram and compare the cumulative frequency occurrences of successive codes 5, 6.
For Method 1 and Method 2, the amplitude and offset of the input signal have to be known and for Method 3 a reference quantum qref is necessary. This assumption justifies a study of each method's robustness towards the estimation errors. Both simulations and experimental results showed a better behaviour for Method 2 and Method 3 [7].
Spectral analysis is then considered. From the power spectrum are deduced the signal-to-noise ratio, with or without harmonic distortion (SNDR or SNR), the harmonic distortion rate (THD), and the number of effective bits (neff) 8, 9. For the ADC input signal, all algorithms consider a full-scale and perfect sine wave.
In the first step, we show how parameters are deduced from the theoretical equations, when the hypothesis quoted earlier is fulfilled. Then we present eventual modifications by adding correction terms in the evaluation of SNDR, SNR, THD and neff, when, for instance, the input signal is not exactly a full-scale one.
A major concern in spectral analysis is the measurement of the harmonic distortion rate. Indeed, in the case of an input signal presenting harmonics, it becomes impossible to separate the ADC distortion from the signal generator one. To solve this problem, we developed a method called “DUAL-TONE”, using as input signal a sum of two sine waves. A theoretical study shows that the resulting intermodulation lines in the power spectrum are due to the single ADC nonlinearity. Experimental results confirm this assertion [10].
The “CanTest System” test bench, described in the last part of this article, proposes an automatic system dedicated to the dynamic characterization of A/D converters developed in our laboratory. The different elements included in this system are presented with their characteristics (sample frequency, bandwidth of the acquisition system, size of acquisition,…) [11]. In this section, several components characterized in the three analysis modes are also presented and the adequate analysis as a function of the A/D converter applications are discussed.
Section snippets
Theoretical background
Before describing the different types of analyses, the process of the analog-to-digital conversion is shown by the transfer function of the ADC. Fig. 1 shows the different elements of a transfer diagram for an ideal linear ADC. The analog input full-scale range of the A/D converter is divided into 2r equal quanta and each of these quanta is associated with an integer code value.
The quality of the conversion depends on the variation of the width of each quantum, the linearity of the transfer
Direct quantization noise Brms estimation
In the section dedicated to the time domain analysis, we have described two algorithms in order to compute the error signal. They are based on the knowledge input frequency: either its value is well know with high precision and the equation system is linear, either this condition is not fulfilled and the system is non linear. To have a comparison between the algorithms efficiency, we have studied the evolution of the estimation noise versus the difference between the real signal frequency fin
Robustness of the three different statistical analysis algorithms
The second section gives the theoretical background of the statistical analysis principle, assuming a perfect input sine wave or with a higher spectral purity than the ADC under test. In the real case, it will be very difficult to found a generator which fulfils this assumption. When the amplitude of the input signal doesn't sweep completely the ADC full-scale, or/and an additive noise appears during the acquisition, an edge effect occurs. It means that the measured histogram doesn't show
Concerning the drawbacks of the spectral analysis
In the experimental case, it is difficult, even impossible, to make an acquisition with a full-scale range amplitude. So, a correction has to be introduced in the measurement of SN̂DRdB, function of the ratio between the estimated input amplitude and the full-scale range of the converter, and function of the total harmonic distortion defined in Eq. (16).where  is an estimation of the input amplitude computed from the acquisition, and with PE
ADC dynamic evaluation system and experimental results
The CanTest system was developed at the IXL laboratory, dedicated to the dynamic characterization of A/D converters. The whole bench is organized as shown in Fig. 9. It allows data acquisition and parameter evaluation for a wide range of ADC, by different analytic methods. The main characteristics of CanTest are its adaptability to many external configurations, a very well developed user-interface, and the availability of a large number of data computation algorithms extracting specific
Conclusion
First, we have focused on the dynamic analysis in the time domain. We give a new approach based on eigenvalue decomposition (EVD) to determine the number of effective bits by a direct estimation of the quantization noise without knowing the error signal. Now, it would be interesting to include an algorithm to detect the number of harmonics contained in the acquisition vector in order to improve the measurement precision.
The aim of the statistical analysis part is to show that the different
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