A finite-memory discrete-time convolution approach for the nonlinear dynamic modelling of S/H–ADC devices

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Abstract

A nonlinear dynamic “black box” model for sample-hold and analogue to digital conversion devices (S/H–ADCs) is proposed in this paper. It derives from the expansion of a Volterra-like series (previously introduced by authors for the general-purpose modelling of nonlinear dynamic systems) and it is based on a discrete convolution in the time domain, which describes the nonlinear dynamics of the system. The model carries out a functional description of the S/H–ADC nonlinear behaviour as the sum of two blocks which take separately into account the static and the purely dynamic contributions. Model characterization can be performed by means of a simple and reliable measurement procedure and the model analytical representation allows an easy implementation in the framework of commercial available CAD tools for circuit analysis and design. Preliminary experimental results, which validate the proposed approach, are presented in the paper together with some examples which show model capabilities in characterizing the device nonidealities under both DC and large-signal operating conditions.

Introduction

Characterization and modelling of S/H–ADC devices are of basic importance for the performance analysis of measurement instruments based on signal digital acquisition and processing. With the aim of identifying suitable procedures for the correction of the device nonidealities, a high accuracy in the prediction of device behaviour under large-signal operating conditions is needed. This can be obtained only if the modelling approach takes into account not only the nonlinearities associated with the S/H–ADC static characteristic, but also those which deal with device dynamics. In the last years, many efforts were made in order to improve the previous static-oriented S/H–ADC modelling techniques towards the characterization and prediction of dynamic effects, which are important sources of error not only in high-performance ADCs (flash converters), but also in more “general-purpose” architectures (e.g. those based on successive approximations or sigma–delta conversion). Some examples of such techniques can be found in [1], [2], [3], [4], [5], which follow very different approaches. In particular, models can be characterized starting from the extraction of circuit parameters (resistors and coupling capacitances in the voltage reference ladders or in the S/H switches), classical dynamic error parameters (droop and slew rates, feedthrough, aperture and settling times, and jitters in the S/Hs, gain, bandwidth, harmonic distortion in input amplifiers), information on the dynamic behaviour of comparators or other kinds of data such as technology-dependent parameters of active devices. All this information can be obtained by direct measurements and/or derived from manufacturers' data sheets. Interesting “white box” approaches have been recently proposed [6], which start from some aprioristic assumptions on the error mechanisms in order to reduce the space of needed parameters. Particular analytical procedures at the bit level (e.g. the Walsh transform [7]) have been applied, as well.

From a more general point of view, an S/H–ADC device can be approached as a nonlinear dynamic system to be characterized by means only of input–output data (“black box” behavioural approach). Traditionally, the behavioural modelling of nonlinear dynamic systems found an important theoretical basis in the Volterra series formulation [8], [9], [10], [11]: many examples are available in literature, which show the application of such an approach to a wide range of fields of interest [12], [13], [14], [15], [16] (telecommunications, microwave circuit design, image processing, robotics, physiology, physics and many others). Nevertheless, the intrinsic properties of the Volterra series formulation lead to well-documented general limitations on its reliable practical application. In fact, in presence of important nonlinearities associated with the input–output system behaviour, the number of terms, which must be taken into account in the series expansion in order to achieve an acceptable system characterization, quickly increases. Even for mildly nonlinear systems (this is the case of S/H–ADC devices), at least three or four Volterra series kernels must be identified to obtain models capable of predicting the system dynamic behaviour with a good accuracy. Strong efforts have been made in the last years for the identification of feasible measuring procedures for higher-order kernels, but the proposed solutions (e.g. [17], [18], [19], [20]) still seem to suffer from several limitations. More precisely, Volterra kernels are usually experimentally characterized taking into account their frequency multidimensional domain representation. The procedures devoted to this aim perform only a partial kernel extraction, excluding particular regions of the frequency domain and/or carrying out kernel evaluation over a grid of points whose resolution hardly leads to a sufficient information about system behaviour. In addition, the techniques for kernel characterization usually involve the exploitation of high-order statistics or the generation of complicate input test signals.

In the aim of overcoming such limitations, the authors previously introduced a modified Volterra series [21] based on the dynamic deviation of the system input signal with respect to its value assumed in the instant when the output is evaluated: this Volterra-like series does not introduce either approximations or additional hypotheses with respect to the conventional one, but it is characterized by definitely different convergence properties, which allow to truncate its expansion to the first-order term if some assumptions are made on the system memory time duration with respect to the minimum period of the input signal. The need for the identification of only one term of the series, without introducing relevant truncation errors in the model accuracy, is an important advantage with respect to the classical approach, which proposes the modified series-based method as a valid alternative to it when the hypotheses on the system behaviour are satisfied. The general-purpose approach to the modelling of nonlinear dynamic systems, based on the Volterra-like series, has been applied to the case of S/H–ADC devices, as described in the following.

In order to characterize the nonlinear dynamic effects, it has been previously shown that an S/H–ADC device with input signal sI(t) can be described through the “black box” functional model of Fig. 1 [22], [23]: the actual device is modelled as an ideal instrument, sampling and converting to digital a signal y(t) which is the result of the input/output relationship of a nonlinear dynamic system (nonlinear system with memory). The nonlinear block in Fig. 1 is then described by the cascade of a purely linear dynamic system with a virtually nonfinite memory and a nonlinear block associated with finite, relatively short-memory effects (Fig. 2).

The first block of such a cascade is simply characterized, in the time domain, by its pulse response h(t) (or, alternatively, in the frequency domain by the transfer function H(f)), while the latter can be conveniently modelled by the truncation to the first-order integral term of the modified Volterra series, previously proposed for the characterization of nonlinear dynamic systems under the hypothesis of “short” duration of their nonlinear dynamic effects. More precisely, following the general-purpose approach in Ref. [21], the output signal y of the nonlinear system in Fig. 2 can be further described as the sum of the outputs of a memoryless nonlinear block (i.e. the zero-order term of the series) and a purely dynamic nonlinear one (the first-order term in the series expansion), if the duration of its memory can be considered “short” with respect to the typical minimum period of the input signal s(t). This hypothesis represents the only important constraint required on the behaviour of a nonlinear dynamic system in order to successfully apply to it the approach proposed in Ref. [21], and it can be considered satisfied in the case of an S/H–ADC device. In fact, the ideal dynamic behaviour for this family of systems is characterized by purely linear effects (which are taken into account by the first block of the cascade in Fig. 2), usually introduced by the input signal conditioning circuits (amplifiers, attenuators, filters) and the sample/hold process, while the nonlinear dynamic effects represent an error source due to the presence of active devices in the system, which are usually characterized by fast dynamics and can be associated with a short memory.

The final S/H–ADC representation is, therefore, shown in Fig. 3. A nonlinear model, suitable for the characterization of S/H–ADC dynamic nonidealities, based on a frequency domain-oriented approach for the identification of the nonlinear blocks, was previously proposed by authors, starting from this functional description [23]. Although the previous model represented a correct application of the Volterra-like method [21] for the modelling of nonlinear dynamic systems with short-memory effects, its formulation in the frequency domain introduced some problems to be dealt with regarding the experimental techniques devoted to model parameter extraction. In particular, the characterization of the model proposed in Ref. [23] is possible by investigating the device frequency response in a wide bandwidth, definitely outside the device nominal operation region, with evident problems of feasibility and reliability for the overall experimental procedure. In this paper, an accurate time domain-oriented version of the S/H–ADC model, functionally described in Fig. 3, will be presented, whose formulation allows to easily characterize the model parameters by means of conventional measurement procedures in the time domain and reliable numerical algorithms.

Section snippets

The S/H–ADC discrete-time convolution model

The purely linear dynamic block at the input of the functional structure in Fig. 2 can be simply characterized by means of the conventional convolution integral of its pulse response h with respect to the signal sI, extended over a virtually nonfinite memory interval:s(t)=+h(τ)sI(t−τ)dτ

According to the modified Volterra series-based approach, the output of the cascaded nonlinear block in Fig. 2 can be described as the sum of two contributions represented, respectively, by the output y(S)(t)

Model characterization

The experimental extraction of behavioural models for nonlinear dynamic systems, such as those based on the classical Volterra series, is strongly limited by the difficulties that are related to the identification of all kernels which are needed in order to achieve a good prediction accuracy. In many cases, only a part of these kernels can be practically characterized, and the obtained models suffer from a strong approximation and reduced performances. The proposed approach overcomes such a

Experimental results

The proposed time domain discrete convolution model has been extracted for an S/H–ADC device implemented in the framework of SPICE circuit analysis CAD tool. The device has been realized through a sampling switch based on a Schottky diode (MBD701) bridge circuit [24], separated from the input and output by two Op Amps (OPA640). DC and both small- and large-signal analyses performed on this SPICE-based device have allowed to obtain the empirical data needed for the model characterization.

Conclusions

A finite-memory discrete-time convolution approach has been proposed for the nonlinear dynamic modelling of S/H–ADC devices. The model takes separately into account the static and the purely dynamic nonlinearities, described by means of a modified Volterra method. It can be characterized directly in the time domain, starting from experimental data obtained through conventional measurement procedures, by applying reliable numerical algorithms. The model has been fully extracted for a SPICE-based

Domenico Mirri (M'91) was born in Italy in 1936. He received his MS in Electronic Engineering from the University of Bologna, Bologna, Italy, in 1963. Currently, he is a Full Professor of Electronic Measurements at the University of Bologna. His current research interests are in the areas of digital measurement instruments, devices metrological characterization and biomedical measurements.

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    Domenico Mirri (M'91) was born in Italy in 1936. He received his MS in Electronic Engineering from the University of Bologna, Bologna, Italy, in 1963. Currently, he is a Full Professor of Electronic Measurements at the University of Bologna. His current research interests are in the areas of digital measurement instruments, devices metrological characterization and biomedical measurements.

    Gaetano Pasini (M'97) was born in Italy in 1964. He received his MS in Electronic Engineering from the University of Bologna, Bologna, Italy, in 1991. Currently, he is with the University of Bologna where he is Associate Professor in Electrical Measurement. His research activity is mainly oriented to digital signal processing in electronic instruments, power measurements and characterization of nonlinear systems with memory.

    Pier Andrea Traverso received his Laurea degree in Electronic Engineering and his PhD degree in Electronic and Computer Science Engineering from the University of Bologna, Italy, in 1996 and 2000, respectively. Presently, he is with the Department of Electronics, Computer Science and Systems, University of Bologna, as a Research Associate. His main research interests are in the fields of nonlinear dynamic system characterization and modelling, microwave electron device modelling and digital measurement instruments.

    Fabio Filicori (M'98) was born in Italy in 1949. He received his degree in Electronic Engineering from the University of Bologna, Bologna, Italy, in 1964. Presently, he is a Full Professor of Applied Electronics at the University of Bologna. His current research interests are in the areas of nonlinear circuit analysis and design, electronic devices modeling, digital measurement instruments and power electronics.

    Gaetano Iuculano (M'66) received his degree in Electronic Engineering from the University of Bologna, Italy. At present, he is a Professor of Electrical Measurements and Metrology at the Department of Electronic Engineering of the University of Florence. He has experience in calibration applications and planning experiments, in reliability analysis and life testing for electronic devices and systems, and considerable expertise in practical statistical analysis for electrical engineering. He has authored and coauthored more than 100 technical papers in his current research interests.

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