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`COMPUTER-CONTROLLED
` SYSTEMS
`
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`Theory and Design
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`Theory and Design %
`
`_;
`5
`
`Karl J. Rstrfim
`Bjérn Wittenmark
`
`.
`
`g
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`Prentice-Hall, Inc., Englewood Cliffs, N.J. 07632
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`Library of Congress Cataloging in Publication Data
`Asmom; KARL J. (Karl Johan).
`(date)
`Computer controlled systems.
`Includes bibliographies and index.
`1». Automatic control——-Data processing.
`1. Wittenmark, E.
`II. Title.
`TJ2l3.A78 1984
`629.8’95
`83-17643
`ISBN 0-13-164319-3
`
`Editorial/production supervision
`and interior design: Karen Skrable
`Manufacturing buyer: Anthony Caruso
`
`©1984 by Prentice-Hall, Inc., Englewood Cliffs, N.J. 07632
`
`All rights reserved. No part of this book
`may be reproduced in any form or
`by any means without permission in writing
`from the publisher.
`
`Printed in the United States of America
`
`1098765432
`
`ISBN El-L3-1.E:iLI3]."l-3
`
`Prentice-Hall International, Inc., London
`Prentice-Hall of Australia Pty. Limited, Sydney
`Editora Prentice-Hall do Brasil, Ltda., Rio de Janeiro
`Prentice-Hall Canada Inc., Toronto
`Prentice-Hall of India Private Limited, New Delhi
`. Prentice-Hall of Japan, Inc., Tokyo
`Prentice-Hall of Southeast Asia Pte. Ltd., Singapore
`Whitehall Books Limited, Wellington, New Zealand
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`» To Introduce the Subject and to Give Some Historical
`
`Technology and TIleory.
`
` Background on the Development of Computer-Control
`
`1 .1 Introduction
`
`
`
`Digital computers are increasingly being used to implement control systems. It is
`therefore important to understand computer-controlled systems well. One can view
`computer-controlled systems as approximations of analog-control systems, but this
`is a poor approach because the full potential of computer control is not used. At best
`the results are only as good as those obtained with analog control. Alternatively, one
`‘can learn about computer-controlled systems, so that the full potential of computer
`control
`is used. The main goal of this book is to provide the required background.
`A computer-controlled system can be schematically described as in Fig. 1.]. The
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`|'—Compu't’er
`I
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`IIIII II III
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` yltl
` Process
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`Algorithm N D
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`_
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`Figure 1.1 Schematic diagram of a computer-controlled system.
`
`output from the process y(t) is a continuous-time signal. The output is converted into
`digital form by the analog-to-digital (A-D) converter. The A-D converter can be
`included in the computer or regarded as a separate unit, according to one’s preference.
`The conversion is done at the sampling times, t,,. The computer interprets the convert-
`ed signal, {y(t,,)}, as a sequence of numbers, processes the measurements using an
`algorithm‘ and gives a new sequence of numbers, {u(t,,)}. This sequence is converted to
`an analog signal by a digital—to-analog (D-A) converter. Notice that the system runs
`open loop in the interval between the A-D and the D-A conversion. The events are
`synchronized bv the real-time clock in the computer. The digital computer operates
`sequentially in time and each operation takes some time. The D-A converter must,
`however, produce a continuous-time signal. This is normally done by keeping the
`control signal constant between the conversions. The computer-controlled system
`contains both continuous-time signals and sampled, or discrete-time signals. Such
`systems have traditionally been called sampled-data systems, and this term will be used
`here as a synonym for computer-controlled systems.
`The mixture of different types of signals sometimes causes difliculties. In most
`cases it is. ‘however, sufficient to describe the behavior of the system at the sampling
`instants. The signals are then of interest only at discrete times. Such systems will be
`called discrete-time systems. Discrete-time systems deal with sequences of numbers,
`so a natural way to represent these systems is to use dilference equations.
`.
`The purpose of the book is to present the control theory that is relevant to the
`analysis and design of computer-controlled systems. This chapter provides some back-
`ground. A brief overview of the development of computer-control technology is given
`in Sec. 1.2. The need for a suitable theory is discussed in Sec. 1.3. Examples are used
`to demonstrate that computer-controlled systems cannot be fully understood by the
`theory of linear, time-invariant, continuous-time systems. An example shows not only
`that computer-controlled systems can be designed using continuous-timetheory and
`approximations, but also that substantial improvements can be obtained by other
`techniques that use the full potential of computer control. Sec. 1.4 gives some exam-
`ples of inherently sampled systems. The development of the theory of sampled-data
`systems is outlined in Sec. 1.5.
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`1.2 ComputerTechnology
`
`The idea of using digital computers as components in control systems emerged around
`1950. Applications in: missile and aircraft control were investigated first. Studies
`showed that there was no potential for using the general-purpose digital computers
`that were available at that time. The computers were too big, they consumed too much
`power, and they were not sufficiently reliable. For this reason special-purpose com-
`puters—digital differential analyzers (DDA)—were developed for the early aerospace
`applications.
`The major developments in computer control occurred in the process industries.
`The progress of these developments is illustrated in Fig. 1.2, which shows the growth
`of computers used for process control over a period of 25 years.
`
`100 M
`
`10M
`1”
`
`700*
`10 k
`
`1k
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`100
`10
`
`E
`g_
`‘E,
`3
`O_
`§
`5
`§
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`
`
`1960
`
`1970
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`1980
`
`' 1990
`
`Figure 1.2 Growth of computers used
`for industrial process control. For
`comparison the total number of
`computers is also given. The picture is
`compiled from several sources:
`Control Engineering, A. D. Little,
`Frost and Sullivan, and Diebold.
`(Redrawn from data published in
`‘Control Engineering, © 1980,
`Technical Publishing C0,, with
`permission.
`
`The idea of using digital computers for process control emerged in the mid-
`fifties. Serious work started in March 1956 when the aerospace company Thomson
`Ramo Woolridge (TRW) contacted Texaco to set up a feasibility study. After pre-
`liminary discussions it was decided to investigate a polymerization unit at the Port
`Arthur, Texas, refinery. A group of engineers from TRW and Texaco made a thorough
`feasibility study, which required about 30 people-years. A computer-controlled system
`for the polymerization unit was designed based on the RW-300 computer. The control
`system went on-line March 12, 1959. The system controlled 26 flows, 72 temperatures,
`3 pressures, and 3 compositions. The essential functions were to minimize the reactor
`pressure, to determine an optimal distribution among the feeds of 5 reactors, to con-
`trol the hot-water inflow based on measurement of catalyst activity, and to determine
`the optimal recirculation.
`The pioneering work done by TRW was noticed by many computer manufac-
`turers, who saw a large potential market for their products. Many different feasibility
`studies were initiated and vigorous development was started. The results of these
`efforts are reflected in the growth shown in Fig. 1.2.
`To discuss the dramatic developments, it is useful to introduce four periods.
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`Sec. 1.2
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`Computer Technology
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`=.,
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`Pioneering period
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`2 1955
`
`Direct-digital-control period c: 1962
`Minicomputer period
`z
`Microcomputer period
`2 1972
`
`It is difficult to give precise dates, because the development was highly diversi-
`V
`fied. There was a wide difference between different application areas and different
`industries; there was also considerable overlap.i The dates given refer to the first
`appearance of new ideas.
`A
`
`The Pioneering Period
`
`The work done by TRW and Texaco evoked substantial interest at process industries,
`among computer manufacturers, and in research organizations. The industries saw a
`potential tool for increased automation, the computer industries saw new markets,
`and universities saw a new research field. Many feasibility studies were initiated by the
`computer manufacturers because they were eager to learn the new technology and
`were very interested in knowing what a proper process—contro1 computer should look
`like. Feasibility studies continued throughout the sixties.
`'
`I
`The computer systemsithat were used were slow, expensive, and unreliable. The
`earlier systems used vacuum tubes. Typical data for a computer around 1958 were an
`addition time of 1 ms, a multiplication time of 20 ms, and a Mean Time Between
`Failures (MTBF) for a central processing unit of 50-100 h. To make full use of the
`expensive computers, it was necessary to have them perform many tasks. Because the
`computers were so unreliable, they controlled the process by printing instructions to
`the process operator or by changing the set points of analog regulators. These super-
`visory modes of operation were referred to as operator guide and set-point control.
`The major tasks of the computer were to find the optimal operating conditions,
`to perform scheduling and production planning, and to give reports about production
`and raw-material consumption. The problem of finding the best operating conditions
`was viewed as astatic optimization problem. Mathematical models of the processes
`were necessary in order to perform the optimization. The models used—which were
`quite _complicated—were derived from physical models and from regression analysis
`of process data. Attempts were alsomade to carry out on-line‘ optimization.
`'1
`Progress was often hampered by lack of process knowledge. It also become
`clear that it was not sufficient to View the problems simply as static optimization
`problems; dynamic models were needed. A significant proportion of the. effort in
`many of the feasibility studies was devoted to modeling, which was quite time con-
`suming because there was a lack of good modeling methodology. This stimulated‘
`research into system-identification methods.
`A lot of experience was gained during the feasibility studies. It became clear that
`process control puts special demands on computers. The need to respond quickly to
`demands from the process led to development of the interrupt feature, which is a
`special hardware device that allows an external event to interrupt the computer in its
`current work so that it can respond to more urgent process tasks. Many sensors that
`
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`.y.;\lr..’I\.lT-5nt'1';;;E-
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`were needed were not available. There were also several difliculties in trying to intro-
`duce a new technology into old industries.
`.
`The progress made was closely monitored at conferences and meetings and in
`journals. A series of articles describing the use of computers in process control was
`published in the journal Control Engineering. By March 1961 thirty—seven systems had
`been installed. A year later the number of systems had grown to 159. The applications
`involved control of steel mills and chemical industries and generation of electric power.
`The development progressed at different rates in different industries. Feasibility studies
`continued through the sixties and the seventies.
`
`Direct Digital Control
`
`‘
`
`The early installations of control computers operated in supervisory mode, either as
`operator guide or as set-point control. The ordinary analog-control equipment was
`needed in both cases. A drastic departure from this approach was made by Imperial
`Chemical Industries (ICI) in England in 1962. A complete analog instrumentation for
`process control was replaced by one computer, a Ferranti Argus. The computer
`measured 224 variables and controlled 129 valves directly. This was the beginning of
`a new era in process control: Analog technology was simply replaced by digital
`technology; the function of the system was the same. The name Direct Digital Control
`(DDC) was coined to emphasize that the computer controlled the process directly.
`In 1962 a typical process-control computer could add two numbers in 100 ,us and
`multiply them in 1 ms. The MTBF was around 1000 h.
`Cost was the major argument for changing the technology. The cost of analog
`technology increased linearly with the number of control loops; the initial cost of a
`digital computer was large, but the cost of adding an additional loop was small. The
`digital system was thus cheaper for large systems. Another advantage was that the
`operator communication could be changed drastically; an operator communication
`panel could replace a large wall of analoginstruments. The panel used in the ICI
`system was very simple—a digital display and a few buttons.
`Flexibility was another advantage of the DDC systems. Analog systems were
`changed by rewiring; computer-controlled systems were changed by reprogramming.
`Digital technology also offered other advantages. It was easy to have interaction
`among several control loops. The parameters of a control loop could be made functions
`of operating conditions. The programming was simplified by introducing special
`DDC languages. A user of such a language did not need to know anything about
`programming, but simply introduced inputs, outputs, regulator types, scale factors,
`and regulator parameters into tables. To the user the systems thus looked like a con-
`nection of ordinary regulators. A drawback of the systems is that it was difficult to
`do unconventional control strategies. This certainly hampered development of control
`for many years.
`»
`DDC was a major change of direction in the development of computer-con-
`trolled systems. Interest was focused on the basic control functions instead of the
`supervisory functions of the earlier systems. Considerable progress was made in the
`years 1963—65. Specifications for DDC systems were worked out jointly between
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`Sec. 1.2
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`‘users and vendors. Problems related to choice of sampling period and control algo-
`rithms, as well as the key problem of reliability, were discussed extensively. The con-
`cept DDC was quickly accepted in spite of the fact that DDC systems often turned
`out to be more expensive than the corresponding analog systems.
`
`The Minicomputer Period
`
`There was substantial development of digital computer technology in the sixties. The
`requirements on a process—control computer were neatly matched with progress in
`integrated circuit technology. The computers became smaller, faster, more reliable,
`and cheaper. The term minicomputer was coined for the new computers that emerged.
`It was possible to design eflicient process—control systems by using minicomputers.
`The development of minicomputer technology combined with the increasing
`knowledge gained about process control with computers during the pioneering and
`DDC periods caused a rapid increase in applications of computer control. Special
`process—control computers were announced by several manufacturers. A typical
`process computer of the period had a word length of 16 bits. The primary memory was
`8—l24k words. A disc drive was commonly used as a secondary memory. The CDC
`1700 was a typical computer of this period, with an addition time of 2 ,us and a mul-
`tiplication time of 7 ,us. The MTBF for a central processing unit was about 20,000 h.
`An important factor in the rapid increase of computer control in this period
`was that digital computer control now came in a smaller “unit.” It was thus possible
`to use computer control for smaller projects and for smaller problems. Becauseof
`minicomputers, the number of process computers grew from about 5000 in 1970 to
`about 50,000 in 1975.
`'
`'
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`Microcomputers
`
`The minicomputer was still a fairly large system. Even as performance continued to
`increase and prices to decrease, the price of a minicomputer mainframe in 1975 was
`still about $10,000. This meant that a small system rarely cost less than $100,000.
`Computer control was still out of reach for a large number of control problems. But
`with the development of the microcomputer in 1972, the price of a card computer
`with the performance of a 1975 minicomputer dropped to $500 in 1980. Another
`consequence was that digital computing power in 1980 came in quanta as small as $50.
`This meant, of course, that computer control could now be considered as an alterna-
`tive, no matter how small the application.
`Since there are even more drastic developments in microelectronics to come with
`the very large scale integration (VLSI) technology in the eighties, it is a safe guess
`that there will be a large increase in computer-control applications then. Micro-
`computers have already made an impact on control equipment: Microcomputers are
`replacing analog hardware even as single-loop controllers; small DDC systems have
`been made using microcomputers; operator communication has been vastly improved
`in these systems with the introduction of color video-graphics displays; hierarchical
`control systems with a large number of microprocessors have been constructed; and
`special-purpose regulators based on microcomputers have been designed.
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`The Future
`
`Based on the dramatic developments in the past it is tempting to speculate about the
`future. There are four areas that are important for the development of computer
`process control.
`_
`
`Process knowledge.
`Measurement technology.
`
`Computer technology.
`Control theory.
`
`Knowledge about process control and process dynamics is increasing slowly but
`steadily. The possibilities of learning about process characteristics are increasing
`substantially with the installation of process-control systems because it is then easy
`to collect data, perform experiments, and analyze the results. Progress in system
`identification and data analysis has also provided valuable information.
`Progress in measurement technology is hard to predict. Many things can be done
`using existing techniques. The possibility of combining outputs of several different
`sensors with mathematical models is interesting. It is also possible to obtain. automatic
`calibration with a computer. The advent of new sensors will,Vhowever, always offer
`new possibilities.
`Spectacular developments are expected in computer technology with the intro-
`duction of the VLSI. The ratio of price to performance will drop substantially. The
`microcomputers of the late eighties are expected to have computing power greater
`than the large mainframes of the late seventies. Substantial improvements are also
`expected in display techniques and in communications.
`Programming has so far been one of the bottlenecks. There were only marginal
`improvements in productivity in programming from 1950 to 1970. At the end of the
`seventies many computer-controlled systems were still programmed in assembler
`code. In the computer-control field, it has been customary to overcome some of the
`programming problems by providing table-driven software. A user of a DDC system
`is thus provided with a so-called DDC package that allows the user to generate a DDC
`system simply by filling in a table, so very little effort is needed to generate a system.
`The widespread use of packages hampers development, however, because it is very
`easy to use DDC but it is a major effort to do something else. So only the well-
`proven methods are tried.
`Control theory made substantial progress in the period 1955-70. Very little of
`this theory has, however, made its way into existing computer-control systems, even
`though feasibility studies have indicated that significant improvements can be made.
`One reason for this is the cost of programming. As already mentioned, it requires
`little eifort to use a package provided by a vendor. It is, however, a major effort to
`try to do something else. Several signs show that this situation can be expected to
`change. Personal computers with interactive languages like BASIC are starting to be
`used for process control. With an interactive language it is very easy to try new things.
`To a large extent this is also done by those who use real-time BASIC. It is, however,
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`Sec. 1.2
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`Computer Technology
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`unfortunately verydiflicult to write safe systems in BASIC. This will change as better
`interactive systems become available.
`Thus there are many signs that point to interesting developments in the field of
`computer-controlled systems. A good way to be prepared is to learn the theory pre-
`sented in this book!
`
`1.3 Computer-Control Theory
`
`A schematic diagram of a computer-controlled system is shown in Fig. 1.1. The
`system contains essentially five parts: the process, the A-D and D-A converters, the
`control algorithm, and the clock. Its operation is controlled by the clock. The times
`when the measured signals are converted to digital form are called the sampling
`instants; the time between successive samplings is called the sampling period and is
`denoted by h. Periodic sampling is normally used but there are, of course, many other
`possibilities. For example, it is possible to sample when the output signals have
`changed by a certain amount. It is also possible to use different sampling periods for
`dilferent loops in a system. This is called multirate sampling.
`The only difference between a computer-controlled system and an ordinary
`analog-feedback system is that the control law is implemented using a digital computer,
`so the class of control laws that can be used conveniently is greatly increased. For
`“example, it is easy to use nonlinear calculations, to incorporate logic and to perform
`substantial calculations in the controller. Tables can be used to store data in order to
`
`accumulate knowledge about the properties of the system.
`
`Is There a Need for a Theory for Computer-Controlled Systems ?
`
`A good theory should make it possible to understand how a system like the one in
`Fig. 1.1 works and how it should be designed. It seems clear that a sampled system
`would behave as a continuous-time system if the sampling period were sufficiently
`small. This is certainly true under very reasonable assumptions. Is there then any need
`for a special theory for computer-controlled systems?
`Some examples will be used to show that the system in Fig. 1.1 cannot be fully
`understood in terms of the theory of time-invariant, linear systems even if the process
`to be controlled is a linear, time-invariant, continuous-time system.
`
`Example 1.1--Time dependence
`Suppose. that we want to implement a compensator that is simply a first-order lag.
`Such agcompensator can be implemented using A-D conversion, a digital computer,
`and D-A conversion. The first-order differential equation is approximated by a first-
`order difference equation. The step response of such a system is shown in Fig. 1.3.
`The figure clearly ‘shows that the sampled system is not time invariant because the
`response depends on the time when the step occurs. If the input is delayed, then the
`output is delayed by the same amount only if the delay is a multiple of the sampling
`period.
`
`The phenomenon illustrated in Fig. 1.3 depends on the fact that the system is
`controlled by a clock (compare with Fig. 1.1). The response of the system to an exter-
`
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`1.4 inherently Sampled Systems
`
`Sampled models are natural descriptions for many phenomena. The theory of sampled-
`data systems, therefore, has many applications outside the field of computer control.
`A few examples follow.
`‘
`
`Discrete-Time Systems as Models for Computer Algorithms
`
`Algorithms in computers can be described as discrete-time systems. This will be
`illustrated with an iterative algorithm and a real-time application.
`
`Example 1.5—I.te1-ative solution
`Iterative algorithms are examples of inherently sampled systems. Assume that the
`solution to an equation of the form
`
`x—f(x)=0
`is desired. One way to find the solution is to guess an initial value and then use Picard’s
`algorithm, i.e., to use the iterative scheme
`
`x(k + 1) = f[X(k)]
`
`where x(k) is the kth iteration. The numerical algorithm can thus be interpreted as a
`discrete-time system in which the time represents the number of iterations.
`Specifically, assume that f(x) = 3 — 4/7. In. this case it is easy to show that the
`solution is x = (7 — ,\/T13/2 x 1.697. The sequence of numbers shown in Fig. 1.7 is
`obtained if one starts with the initial guess x(0) = 0.
`
`xlk)
`
`0
`
`5
`
`10 k
`
`b)
`
`3 2 1 0
`
`
`
`a)
`
`Figure 1.7 Two graphic illustrations of the iterative scheme in Example 1.5.
`
`Example 1.6—Control algorithm
`
`A simple computer algorithm for a proportional and integral (PI) controller follows:
`
`uc:=adin(in1)
`y :=adin(in2)
`e:=uc — y
`u :=k*(e + i)
`dout(u)
`i:=i + h*e/ti
`
`{read reference value}
`{read process value}
`
`{output control signal}
`
`The program is executed every sampling period by a scheduling program, as illustrated
`in Fig. 1.8. The computer code is equivalent to the following difference equations:
`
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`Figure2.8 Frequencyfolding.
`
`The sampled spectrum is then obtained by adding the contributions with proper
`phase from all sheets.
`’
`
`Prefiltering
`
`A practical difficulty is that real signals do not have Fourier transforms that vanish
`outside a given frequency band. The high-frequency components may appear to be
`low-frequency components due to aliasing. The problem is particularly serious if
`there are periodic high-frequency components. To_ avoid the alias problem-, it is neces-
`sary to filter the analog signals before sampling. This may be done in many different
`ways.
`
`Practically‘ all analog sensors have some kind of filter; but the filter is seldom
`chosen for a particular control problem. It is therefore often possible to modify the
`filter so that the signals obtained do not have frequencies_above the Nyquist frequency.
`Sometimes the simplest solution is to introduce an analog filter in fron of the
`sampler. A standard analog circuit for a second-order filter with a transfer f nction
`' 2
`
`G,(s) =
`
`(2.13)
`
`is shown in Fig. 2.9.
`Higher-order filters are obtained by cascading first- and second-order systems.
`Examples of filters are given in Table 2.1.
`The Bessel filter has a linear phase curve, which means that the shape of the
`signal is not distorted much. The Bessel filters are therefore common in high—per-
`formance systems.
`
`28
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`Sampling of Continuous-Time Signals
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`Chap. 2
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