This third volume covers signals and systems dealing with variables or quantities that are discrete or quantized. This leads to distinguishing two chapters: the first concerning the discrete-time case and the second that of discrete (or quantized) levels. The electronic circuits and applications implemented are of analog, digital or mixed nature, and some make use of both types of discretization. Similar to the previous volume, it is fundamental to explain the signals and their properties in detail as well as the basic circuits that transform these signals before considering the functions performed by more complex arrangements, which we will refer to as systems.

The first chapter begins with the study of discrete-time signals, obtained by sampling continuous-time signals, first by means of ideal sampling, then by actual sampling or by using interpolation. The use of the Fourier transform is essential and allows us to demonstrate, on the one hand, equivalences between a discrete variable in one domain and the periodic nature of the quantity depending on the dual variable in the other domain and, on the other hand, the fundamental theorem which determines the possibility to preserve (or not) all the information contained within a signal when shifting from continuous-time to discrete-time domains, called the sampling theorem or the Shannon theorem. Basic analog circuits are described. The other transforms, relevant in cases where discretization can be applied in both time and frequency domains, are also indicated since these are the ones that are used in practice. Next follows the study of the measurement of the time delay and of the phase shift between periodic signals in circuits comprising basic analog and logic functions, which are now widely used. Since this measurement is only achieved once per period, the measured time and phase shifts become discrete quantities. However, in many cases, the approximation which consists of only considering the continuous-time domain, obtained by interpolation and assuming that stationarity is preserved, makes it possible to detail the operation of the analog phase-locked loop (PLL) and the correction strategies of this loop system. This approximation is also a means to establish a relation between phase and frequency, which proves very useful for the applications subsequently addressed. The PLL has undergone overly significant development since the 1970s, because it has allowed transformations of signals and their properties, which were very difficult or impossible to achieve without it, namely in areas such as instrumentation, computer sciences and communications (wireless broadcasting, wireline transmission, etc.) destined for conveying information. The main functions, grouped under the term “frequency synthesis,” are described. Digital PLLs are also covered in detail.

The last part of this chapter is dedicated to samples systems analysed with the z - transform (ZT), just as with the Laplace transform in the case of continuous-time systems. The properties of the ZT are carefully presented in order to provide all the tools that will be used in the end of the first chapter and in the next chapter, including the new meaning for the plane of the complex z variable. The study of switched-capacitor circuits is then discussed in a didactic manner, because it is based on the principles of electrostatics which are simple but nonetheless not necessarily familiar to the readers when applied to capacitor networks. These circuits have experienced major developments because of the possibility to integrate them naturally within CMOS technology and they constitute the basic building blocks for analog sampled filters, and modern digital-to-analog conversion (DAC) or analog-to-digital conversion (ADC). The first chapter logically proceeds through the study of two types of sampled filters (with infinite impulse response [IIR] and with finite impulse response [FIR]) and their properties, as well as approximations useful to recover second order transfer functions in the frequency domain. The notion of transmittance in the plane of the z variable is developed for all the basic functions useful for building these filters. The synthesis methods of these filters are briefly described in order to introduce the readers to the use of numerical functions available in MATLAB® or SciLab software. On the one hand, it should be noted that FIR filters allow one to access properties inaccessible to IIR and analog filters and, on the other hand, that all processing and analyses based on the ZT can be applied without the need for specifying if the technology being used for the implementation is either analog or digital. In the first case, these are switched-capacitor circuits that were previously studied and that are used, while the basic principle of the numerical functions necessary for the second case is described to conclude this chapter. Finally, we proceed with showing the power of state variable analysis in the discrete-time domain and in the plane of the z variable for sampled systems. In effect, it provides direct access to the mathematical modeling of these systems characterized by their fundamental parameters, namely transmittance poles in the plane of the z variable. Provided that the computation of successive samples is performed by computerized means, it enables, in addition, the avoidance of all the approximations previously employed. This model paves the way for the exact computation of the sampled time response in the case of nonlinear systems and/or undergoing frequency variations strong enough so that certain parameters of their transfer characteristics depend thereupon, which is the case in PLLs.

In the second chapter, we consider the principles and implementations of systems dealing with quantized signals, as is the case for ADCs and DACs. The digital quantity is a number encoded onto n bits in the binary system. The quantization of a signal induces some degradation, the first of which being quantization noise, which is presented and analyzed. The other imperfections, which can be likened to errors disturbing the original signal after its conversion, are then connected to the electrical characteristics of these converters. DAC is detailed through the various principles that can be implemented, on the one hand, with resistor ladder networks, historically the first ones to have been used, and, on the other hand, with switched-capacitor circuits, well-adapted to CMOS technology. The reverse conversion is then presented along with its different possible principles, which all have in common the development of an approximation of the analog quantity in digital form, then reconverted and compared to the original quantity, increasingly more precise during the successive stages of the conversion. Looped systems are therefore the main subject. In general, the complexity of systems increases if it is desirable to reduce the conversion time, and the quality of the analog comparator (or analog comparators) determines a very significant part of the accuracy of the conversion.

Finally, “sigma-delta” or “delta-sigma” conversions are addressed, which are the most recent in the field. In its basic principle, the “delta-sigma” conversion is easily understood if deduced from that using a ramp voltage and a count. However, when it is desirable to increase the performance of this type of converter, one is confronted, on the one hand, with a significant sophistication of the modeling, especially in optimizing the signal-to-noise ratio and, on the other hand, with the stability problems of the loop because it is necessary to increase the order of the filter(s) beyond two. In order to solve these problems, a large number of concepts presented in the first chapter are utilized. The core of this type of converter is formed by the modulator, a closed system that processes signals with a significantly much lower number of bits than the initial or final number, desired or imposed, but at a much faster rate than that of the input or the output. The operations carried out by the modulator yield a loss of resolution that will be recovered later by the decimator filter, and a displacement of the noise spectrum toward higher frequencies where it will be more easily filtered. The first-order modulator is examined in the first place and then followed by a generalization to higher-order modulators, which makes it possible to establish the transfer functions for the signal and for the quantization noise. Several types of stable modulator are examined. The role and the way to build the decimator filter are discussed and, based on this analysis, a scaling of the different frequencies to be used can be proposed. Finally, the principle and the implementation of the digital-to-analog “delta-sigma” converter are presented. Although in theory it is deduced from the ADC by swapping digital and analog functions, it is preferable to describe it by proceeding to digital resolution and rate conversions before the final DAC and the associated filtering, which conforms to the practical implementation.

This entire volume thus presents discrete-time and quantized-level signals and systems, the transformations of these signals into continuous-time or continuous-level signals as well as reverse transformations, analog, digital or mixed circuits, effective to achieve these operations and the models capable of calculating, predicting and scaling the responses of these systems. Corrected exercises are provided in order to address specific cases not fully detailed in the course, in order to illustrate it, to complete it and to show the methods adapted to solve the presented problems.