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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Over the last 60 years, electronics has undergone important and rapid developments. This has generated a large range of theoretical and practical notions. This book presents a comprehensive treatise on the evolution of electronics and allows the reader to grasp both the fundamental concepts and the associated practical applications through examples and exercises. Following on from Volume 1, which studied elementary devices, their electrical models and basic functions, Volume 2 was devoted to linear and stationary systems in the continuous-time regime. This third volume deals with the properties of discrete-time and quantized level systems over two chapters. The first presents an analysis of sampled signals and systems, with applications on switched capacitors circuits, analog and digital phase locked loops, frequency synthesis and filters characterized by either finite or infinite impulse response. Most tools are useful to elucidate the properties of both analog and digital systems. The second chapter focuses on the properties of analog-to-digital and digital-to-analog converters. Various principles that are used to perform these conversions are described. Finally, a large section is devoted to sigma-delta converters. Throughout this whole chapter, the signal-to-noise ratio, which is a central issue in these quantized level systems, is analyzed and discussed. Both chapters are followed by useful exercises which illustrate the general principles addressed. The exercises further build on the material covered in the chapters, particularly that which may not have been covered in detail.
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Veröffentlichungsjahr: 2018
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Table of Contents
Cover
Preface
Introduction
1 Discrete-time Signals and Systems
1.1. Discrete-time signals
1.2. Discrete time–continuous time interface circuits
1.3. Phase-shift measurements; phase and frequency control; frequency synthesis
1.4. Sampled systems
1.5. Discrete-time state-space form
1.6. Exercises
2 Quantized Level Systems: Digital-to Analog and Analog-to-Digital Conversions
2.1. Quantization noise
2.2. Characteristics of converters
2.3. Digital-to-analog conversion
2.4. Analog-to-digital conversion
2.5. “Sigma-delta” conversions
2.6. Exercises
Bibliography
Index
End User License Agreement
List of Tables
1 Discrete-time Signals and Systems
Table 1.1. Continuous-time, sampled signals and their z-transforms
Table 1.2. Impulse response and z-transform of elementary analog filters
Guide
Cover
Table of Contents
Begin Reading
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Fundamentals of Electronics 3
Discrete-time Signals and Systems, and Quantized Level Systems
Pierre Muret
First published 2018 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd
27-37 St George’s Road
London SW19 4EU
UK
www.iste.co.uk
John Wiley & Sons, Inc.
111 River Street
Hoboken, NJ 07030
USA
www.wiley.com
© ISTE Ltd 2018
The rights of Pierre Muret to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.
Library of Congress Control Number: 2018930834
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-78630-183-3
Preface
Today, we can consider electronics to be a subject derived from the theoretical advances achieved during the 20th Century in areas comprising the modeling and conception of components, circuits, signals and systems, together with the tremendous development attained in integrated circuit technology. However, such development led to something of a knowledge diaspora that this work will attempt to contravene by collecting the general principles at the center of all electronic systems and components, together with the synthesis and analysis methods required to describe and understand these components and subcomponents. The work is divided into three volumes. Each volume follows one guiding principle from which various concepts flow. Accordingly, Volume 1 addresses the physics of semiconductor components and the consequences thereof, that is, the relations between component properties and electrical models. Volume 2 addresses continuous time systems, initially adopting a general approach in Chapter 1, followed by a review of the highly involved subject of quadripoles in Chapter 2. Volume 3 is devoted to discrete-time and/or quantized level systems. The former, also known as sampled systems, which can either be analog or digital, are studied in Chapter 1, while the latter, conversion systems, are addressed in Chapter 2. The chapter headings are indicated in the following general outline.
Each chapter is paired with exercises and detailed corrections, with two objectives. First, these exercises help illustrate the general principles addressed in the course, proposing new application layouts and showing how theory can be implemented to assess their properties. Second, the exercises act as extensions of the course, illustrating circuits that may have been described briefly, but whose properties have not been studied in detail. The first volume should be accessible to students with a scientific literacy corresponding to the first 2 years of university education, allowing them to acquire the level of understanding required for the third year of their electronics degree. The level of comprehension required for the following two volumes is that of students on a master’s degree program or enrolled in engineering school.
In summary, electronics, as presented in this book, is an engineering science that concerns the modeling of components and systems from their physical properties to their established function, allowing for the transformation of electrical signals and information processing. Here, the various items are summarized along with their properties to help readers follow the broader direction of their organization and thereby avoid fragmentation and overlap. The representation of signals is treated in a balanced manner, which means that the spectral aspect is given its proper place; to do otherwise would have been outmoded and against the grain of modern electronics, since now a wide range of problems are initially addressed according to criteria concerning frequency response, bandwidth and signal spectrum modification. This should by no means overshadow the application of electrokinetic laws, which remains a necessary first step since electronics remains fundamentally concerned with electric circuits. Concepts related to radio-frequency circuits are not given special treatment here, but can be found in several chapters. Since the summary of logical circuits involves digital electronics and industrial computing, the part treated here is limited to logical functions that may be useful in binary numbers computing and elementary sequencing. The author hopes that this work contributes to a broad foundation for the analysis, modeling and synthesis of most active and passive circuits in electronics, giving readers a good start to begin the development and simulation of integrated circuits.
Outline
1) Volume 1: Electronic Components and Elementary Functions [MUR 17a].
i) Diodes and Applications
ii) Bipolar Transistors and Applications
iii) Field Effect Transistor and Applications
iv) Amplifiers, Comparators and Other Analog Circuits
2) Volume 2: Continuous-time Signals and Systems [MUR 17b].
i) Continuous-time Stationary Systems: General Properties, Feedback, Stability, Oscillators
ii) Continuous-time Linear and Stationary Systems: Two-port Networks, Filtering and Analog Filter Synthesis
3) Volume 3: Discrete-time Signals and Systems, and Quantized Level Systems.
i) Discrete-time Signals: Sampling, Filtering and Phase Control, Frequency control circuits
ii) Quantized Level Systems: Digital-to-analog and Analog-to-digital Conversions
Pierre MURETFebruary 2018
Introduction
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.
1Discrete-time Signals and Systems
1.1. Discrete-time signals
In the discretized time domain, where only specific moments are taken into consideration and identified, signals are represented by series of samples.
1.1.1. “Dirac comb” and series of samples
The “Dirac comb” distribution is the basic tool in the discrete-time domain.
1.1.1.1. Dirac comb in the phase space and in the time domain
Signals can be expressed by means of linear combinations of complex exponentials exp(jnθ) in the time and frequency domains (see Chapter 1 of Volume 2 [MUR 17b]), n being an integer and θ an angle proportional to the angular frequency-time product. In the case of a linear combination of 2N + 1 terms having the same amplitude, n varying from −N to +N, , a function IN(θ) is obtained which gives very sharp maxima (or lines) every time θ = 2kπ (Figure 1.1) with k integer, because it is the only case in which the images of exp(jnθ) in the complex plane are colinear and add up while their sum tends to cancel out when θ ≠ 2kπ:
Figure 1.1.Function IN(θ) with N = 10
The “Dirac comb” is a series of periodic Dirac impulses and can be defined from IN(θ) by taking the limit for N →∞.
Provided that and (see Chapter 1 and the Appendix in Volume 2 [MUR 17b]), or more generally , we define the “Dirac comb” distribution by means of a sum of Dirac impulses (Figure 1.2):
Figure 1.2.Dirac comb in the phase space
If θ = 2π f0t, where is a fixed frequency and t is the time variable, we obtain, by changing variable in the distribution Σδk, a Dirac comb in the time domain with a single line whenever f0t = k (k being an integer), that is t = k T0 (Figure 1.3).
Figure 1.3.Time “Dirac comb” distribution of time period T0
is dimensionless and has the dimension of a frequency, in order to preserve the dimension of the function to which the distribution is applied.
1.1.1.2. Frequency Dirac comb
Alternatively, if θ = 2π f T0, where is a fixed period and f is the frequency variable, we have a Dirac comb in the frequency domain with a single line every time f T0 = n (n being an integer) or f = n f0 (Figure 1.4).
has the dimension of time, and the presence of the factor can be verified by calculating the coefficient of the Fourier series of the frequency Dirac comb, which is periodic.
Figure 1.4.Frequency “Dirac comb” distribution of frequency period f0
1.1.1.3. Fourier series
These series of “Dirac comb” impulses Σδn do not directly represent temporal signals or their spectrum but rather virtual signals, since δn is not a regular function. These are operators associated with the “Dirac comb” distribution acting on a time or frequency function g, which can generally be denoted as a functional associated to Σδn: < TΣδn, g > (see Appendix in Volume 2 [MUR 17b]). We can also consider them as bases of the vector spaces of series or discrete functions upon which the continuous-time signal or the continuous-frequency spectrum is projected by computing either the distribution applied to the signal or to the spectrum, or the distribution applied to the product of the signal or spectrum by exp (±j2πft) when looking for the Fourier transform (FT) or its inverse.
The use of the “frequency Dirac comb” distribution corresponds to the decomposition of a periodic signal into Fourier series, already demonstrated in Chapter 1 of Volume 2 [MUR 17b] and derived from a description of the periodic time signals built on exponential function bases.
As a matter of fact, by taking the FT of a periodic signal yT0(t) = yT0(t + kT0) in the time domain, then reducing the interval to the period T0 and finally by performing a summation over all periods, we find:
expression of the product of the Dirac comb by the integral between brackets, to be evaluated for the only line frequencies . It is therefore possible to place the result of the integral calculated at
f
=
n f
0
inside the summation by replacing
f
by
n f
0
and write:
The inverse FT again yields the sum of the complex Fourier series, calculated by the distribution associated with the frequency Dirac comb applied to exp(j2πft):
Figure 1.5.Time “Dirac comb” distribution of period Te
In the case of the time Dirac comb, a (virtual) periodic signal of period Te (Figure 1.5), the coefficients cn are obtained by means of , which is a result independent of n. Consequently, the spectrum of the time Dirac comb (Figure 1.6) is also a Dirac comb, but in the frequency domain:
Figure 1.6.Spectrum of the time Dirac comb (frequency Dirac comb)
1.1.1.4. Sampled (or discrete time) signal and periodicity of the spectrum
The time Dirac comb corresponds to a base upon which a continuous time signal y(t) can be projected to obtain a discrete signal or sampled regularly with the period Te = 1 / fe in the time domain. To obtain any sample y(kTe), we calculate the distribution associated with a Dirac impulse δ (t − kTe) applied to y(t):
To obtain the FT of the signal being sampled, denoted by , the time Dirac comb distribution applied to y(t) exp(−j2π f t) has to be calculated:
that is to say, the FT of the regular product . It is thus the convolution product of the spectrum Y(f) of y(t) by the FT of the time Dirac comb, which is the frequency Dirac comb , namely:
that is to say, the spectrum Y(f) of the continuous-time signal is duplicated periodically around the frequencies
nf
e,
which are multiples of the sampling frequency,
and is multiplied by f
e
.
The spectrum of a sampled signal is therefore periodic (frequency domain, Figure 1.7):
Figure 1.7.Spectrum of a signal sampled at frequency fe
Conversely, it is possible to calculate the inverse FT of a periodic spectrum by the same technique already applied to periodic signals in the time domain:
where the integral in brackets only has to be evaluated for the sole moments where the samples are collected because the time Dirac comb only contains impulses at times tk.
Therefore, we can put the result of the integral calculated at tk = k Te inside the summation because it then becomes dependent of the index k when t is replaced by tk:
which is a series of samples of value yk with .
Conclusion: the time signal whose FT is periodic in the frequency domain is a sampled signal (or discrete) in the time domain.
1.1.2. Sampling (or Shannon’s) theorem, anti-aliasing filtering and restitution of the continuous-time signal using the Shannon interpolation formula
If the spectrum Y(f) of the continuous time signal y(t) has a bounded support smaller than [−fe/2, +fe/2], namely equal to 0 outside of an interval narrower than [−fe/2, +fe/2], can be replaced by feY(f) between the bounds of the integration interval [−fe/2, +fe/2] in the previous computation of yk with an upper bound fmax < fe/2:
Nonetheless, in the case of a bounded spectrum in the interval [−fmax, +fmax], itself included in the interval [−fe/2, +fe/2], (Figure 1.8), can be replaced by , the inverse FT of Y(f) evaluated at times kTe or still the amplitude yk of the samples. We can deduce the sampling (or Shannon’s) theorem:
If a signal y(t), whose spectrum Y(f) is 0 outside [−fmax fmax], is sampled with a sampling rate fe twice larger than the bound fmax, the samples calculated by the inverse transform of the spectrum of the sampled signal coincide with the values of the continuous-time signal y(t) evaluated at sampling times tk = kTe.
In the frequency domain, this leads us to state that:
The spectrum of the original signal is preserved after sampling in the interval [−fe/2, +fe/2] provided that it is bounded by a frequency fmax smaller than fe/2, corresponding to the condition:fe > 2 fmax.
Figure 1.8.Spectrum of a sampled signal which follows the Shannon theorem
Conversely, if Shannon’s theorem is not followed, the spectrum , which is a sum of all the spectra of the continuous-time signal shifted by nfe, exhibits in the interval [−fe/2, +fe/2] at least part of the spectrum Y(f ± fe) usually referred to as the aliased part (Figure 1.9):
Figure 1.9.Spectrum of a sampled signal for which the sampling theorem has not been followed
An anti-aliasing low-pass filter is then necessary. In practice, it still is, because a real signal is limited in time and can therefore be considered as being the product of a rectangular window from −T0 to + T0 by an unlimited signal. Its spectrum then being the convolution with (FT of the rectangular window), which is unlimited, is also unlimited. Since the ideal low-pass filtering with a cutoff frequency fe/2 proves impossible to be rigorously implemented, we approach it with a high-order analog filter having a linear phase-shift with frequency of the Bessel type (constant group delay). The signal y(t) should normally be filtered before sampling.
In all cases, the spectrum of the sampled signal is written by performing the FT:
