A/D Conversion with Non-uniform Differential Quantization

Abstract

The result of measurement has an uncertain value-band depending on the dynamic behavior of the measured object and the measurement instrumentation. Digitalization as the main part of measurement consists of prefiltering, sampling, windowing, and quantization. In addition to this, the dynamics of the analogue-to-digital (A/D) conversion are also important. Here a trade-off between the number of references for generating the reference levels and the number of steps of the conversion is presented. On the one hand, the pure parallel techniques have considerable redundancy of the used reference levels for estimation of the current measurement value and its change; on the other hand, the successive-approximation techniques are relatively slower and they have no possibility of adaptation to the signal until the end of the conversion [1]. Serial-parallel A/D techniques have similar structures as differential pulse-code converters. Numerical value is attained with a successive approximation of the difference between the reference and the measured quantity, g _r and g respectively. Estimation of the difference between two levels of signal $\Updelta g$ is possible with at least two sampling pulses. The dynamic error is proportional to that difference. In the differential multi-step A/D techniques, the change of signal $\Updelta g$ in the time between two sampling points $\Updelta t$ is tracked by a suitably large step y of the reference quantity, $y \propto \Updelta g = g^{\prime}\left( t \right)\Updelta t$. Considering the presumption of dynamic error, a larger uncertainty can be added to a larger step. It is reasonable to use a finite set of possible step representatives ($y_{i} ,i = 1,2, \ldots$). They have to be selected in such a way that their quantization intervals $\Updelta_{i}$ ($\propto$ uncertainty intervals) have a minimum overlap. At the same time there should be no empty space between the quantization intervals. For effectiveness of differential tracking, the non-uniform quantization must fulfill three conditions: partitions into halves, increasing quantization uncertainty with difference, and low overlapping of the quantization intervals. The best trade-off between the number of decision levels and the settling time is with the pure exponential quantization rule: $y_{i} = a^{i - 1} \Updelta_{0} = 2^{i - 1} \Updelta_{0}$, where $\Updelta_{0} = {R \mathord{\left/ {\vphantom {R {2^{b} }}} \right. \kern-0pt} {2^{b} }}$ is the smallest quantization interval ($i = 1,2, \ldots ,b$, R—full scale of the measurement range, b—a number of bits). The fastest response is achievable with base 2. The principle that the larger the distance between two levels $\Updelta g$ is, the larger also the dynamic error of estimation is, shows the limitation of the information flow. In comparison with uniform quantization, the information H of the first step does not increase with an increase in the number of bits, b. It is limited to the constant value $H\left( {a = 2} \right)_{b \to \infty } = 3$. The quantity of information decreases when approaching the last step of conversion. The limited band of the measurement channel or more precisely the finite impulse response of the sampling device, which is not infinitely short, is the cause of the finite information capacity of the measurement channel. The number of bits required for the conversion decreases towards the end of conversion, and the sharp stop of the uniform constant information flow is smoothed. The differential tracking b-bit A/D conversion gives better results than the classical A/D conversion with the successive approximation procedure, due to b-times more available sampling points, and the adaptive property of the A/D procedure that every previous approximation step to signal becomes the center of observation with an exponential increase in resolution in the new step. Taking into consideration only the quantization noise contribution, the adaptive A/D conversion provides better results if the sampling ratio $s = {{f({\text{sampling}})} \mathord{\left/ {\vphantom {{f({\text{sampling}})} {f\left( {\text{signal}} \right)}}} \right. \kern-0pt} {f\left( {\text{signal}} \right)}}$ is high enough. Considering together the systematic and the random errors in the signal parameters estimation, shows the advantage of the adaptive A/D conversion at lower values of the sampling ratio, s [2].

Appendix A

After the second step of the A/D conversion with the exponential quantization and base $a = \;2$ the index i is reduced by 2 (9.21). The representatives with the index above $i = 5$ enable the following paths and their entropies:

$$ \begin{aligned} &y_{5} \to 2p_{3} H_{1} \hfill \\ &y_{6} \to 2\left( {p_{3} H_{1} + p_{4} H_{2} } \right) \hfill \\ &\vdots \hfill \\ &y_{i} \to 2\left( {p_{3} H_{1} + p_{4} H_{2} + \cdots + p_{i - 2} H_{i - 4} } \right) \hfill \\ \end{aligned} $$

(A1)

In an example with parameters $i = 8,\;a = 2\quad \Rightarrow \quad k_{ \hbox{max} } = 4$, the following information flow can be evaluated.