LINEAR TIME-INVARIANT SYSTEMS. MARTIN SCHETZEN Northeastern University IEEE IEEE PRESS. A John Wiley 81 Sons, Inc., Publication

Transcription

1 LINEAR TIME-INVARIANT SYSTEMS MARTIN SCHETZEN Northeastern University IEEE IEEE PRESS A John Wiley 81 Sons, Inc., Publication

2 Copyright by The Institute of Electrical and Electronics Engineers. All rights reserved. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, , fax , or on the web at Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 11 1 River Street, Hoboken, NJ 07030, (201) , fax (201) , permreq@wiley.com. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services please contact our Customer Care Department within the U.S. at , outside the U.S. at or fax Wiley also publishes its books in a variety of electronic formats. Some content that appears in print, however, may not be available in electronic format. Library of Congress Cataloging-in-Publication Data is available. Schetzen, Martin. Linear time-invariant systems/ Martin Schetzen p. cm. Includes index. ISBN (cloth: alk. paper) Printed in the United States of America

3 IEEE Press 445 Hoes Lane, Piscataway, NJ IEEE Press Editorial Board Stamatios V Kartalopoulos, Editor in Chief M. Akay M. E. El-Hawary M. Padgett J. B. Anderson R. J. Herrick W. D. Reeve R. J. Baker D. Kirk S. Tewksbury J. E. Brewer R. Leonardi G. Zobrist M. S. Newman Kenneth Moore, Director of IEEE Press Catherine Faduska, Senior Acquisitions Editor John Griffin, Acquisitions Editor Tony VenGraitis, Project Editor Technical Reviewers Rik Pintelon, Yrije Universiteit Brussel Bing Sheu, Nassda Corporation

4 PREFACE This is a text on continuous-time systems. In our use, a system is defined as an object with inputs and outputs. An input is some excitation that results in a system response or output. System theory is the study of the relations between the system inputs (or stimuli) and the corresponding system outputs (or responses). Two major categories of system theory are analysis and synthesis. Analysis is the determination and study of the system input-output relation; synthesis is the determination of systems with a desired input-output relation. For analysis, signals are used as inputs to probe the system, and in synthesis the desired output is expressed as a desired operation on a class of input signals. Thus signals are an important topic in system theory. However, this is not a text on signal theory. In signal theory, the object being studied is a signal and systems are chosen to transform the signal into some desired waveform. In system theory, on the other hand, the object being studied is a system, and signals are chosen to be used either as probes of the system or as the medium used to express system input-output relation. Thus the discussion of signals in this text will be in terms of their application in system theory. System theory lies at the base of science because system theory is the theory of models, and a basic concern of science is the formation and study of models. In science, one performs experiments and observes some of the quantities involved. A model involving these quantities is then constructed such that the relation between the quantities in the model is the same as that observed. The model is then used to predict the results of other experiments. The model is considered valid as long as experimental results agree with the predictions obtained using the model. If they do not agree, the model is modified so that they do agree. The model is continually improved by comparing a large variety of experimental results with predictions obtained using the model and modifying the model so that they agree. One does ix

5 X PREFACE not say that the refined model represents reality; rather, one only claims that experiments proceed in accordance with the model. In this sense, science is not directly concerned with reality. The question of reality is addressed in philosophy, not science. However, there are areas of science and philosophy which do influence each other. Some of these are briefly mentioned in our discussion of system models. As an illustration, the electron is a model of an object that has not been observed directly. In an attempt to predict certain experiments, the electron was fist modeled as a negatively charged body with a given mass which moves about the nucleus of an atom in certain orbits. This model of the electron helped to predict the results of many experiments in which the atom is probed with certain inputs such as charged particles and the output is the observed scattered particles. This model also helped predict the results of experiments in which the atom is probed with electromagnetic fields and the output is the observed spectra of the radiation from the atom. However, to predict the results of other experiments, this model of the electron had to be modified. The model of the electron has been modified by giving it spin, a wavelength, and other properties. Does the electron really exist? Science does not address that question. Science just states that experiments proceed as if the electron exists. The modem development of engineering and science requires a deeper understanding of the basic concepts of system theory. Consequently, rather than an applications-oriented presentation, basic concepts and their system interpretation are emphasized in this text. The chapter problems are to help the reader gain a better understanding of the concepts presented. To study this text, the student need not know mathematics beyond basic calculus. Any additional required mathematical concepts are logically developed as needed in the text. Even so, all the mathematics used in the text is used with care. Mathematical rigor is not used; that is the province of the mathematician. However, mathematics is used with precision. For example, the impulse is not something with zero width and infinite height. The accurate development of the impulse presented also lends greater insight to its various applications discussed in the text. The careful discussion and application of mathematics results in the student having a better appreciation of the role of mathematics and a more sophisticated understanding of its application in science and engineering. Linear systems from a functional viewpoint is logically developed in this text. Each topic discussed lays the basis and motivation for the next topic. In order that the development be consistent with a systems orientation, many new results and also new derivations of classic results from a systems viewpoint are included in this text. Thus, many topics such as the Fourier and Laplace transforms and their inverse are not just stated. Rather, I have developed new methods to motivate and derive them from system concepts that had been developed previously. The text begins with a discussion of systems in general terms followed by a discussion and development of the various system classifications in order to motivate the approach taken in their analysis. The time-domain theory of continuous-time linear time-invariant (LTI) systems is then developed in some depth. This development leads naturally into a discussion of the system transfer function, gain, and phase shift. This lays the basis for a development of the Fourier transform and its

6 PREFACE Xi inverse together with its system theory interpretation and implications such as the relation between the real and imaginary parts of the system transfer function. The discussion of the Fourier transform and its inverse motivates the development of the bilateral Laplace transform and a full discussion of its system interpretation. One important class of systems which is analyzed is that of passive LTI systems. Although, as discussed in the text, there is no physical law that requires a system to be causal, it is shown that a passive LTI system must be causal. Constraints that the impedance function must satisfy are then obtained and interpreted. A new approach to the unilateral Laplace transform is presented by which the bilateral Laplace transform can be used in the transient analysis of LTI systems. The s-plane viewpoint is then used to discuss basic filter analysis and design techniques. The discussion of the s-plane viewpoint of systems concludes with the analysis of feedback systems and their stability, interconnected systems, and block diagram reduction. Because system theory is the theory of models and the construction of models is one of the main objectives of science, a discussion of the consistency of models and some of the paradoxes in LTI system theory that can arise due to improper modeling is given. The text concludes with an introductory discussion of the state-variable approach to system analysis and the types of problems for which this approach is advantageous. Thus the textual material lays a solid foundation for further study of system modeling, control theory, filter theory, discrete system theory, state-variable theory, and other subjects requiring a systems viewpoint. I thank Prof. John Proakis, who was department chairman during the years I spent writing this text, for his support and assistance. Also, my heartfelt thanks to my wife, Jeannine, for her encouragement and help in the tedious job of proofreading. Brookline, Massachusetts October MARTIN SCHETZEN

7 CONTENTS Preface General System Concepts 1.1 The System Viewed as a Mapping 1.2 System Analysis Concepts 1.3 Time-Invariant (TI) Systems 1.4 No-Memory Systems 1.5 Simple Systems with Memory 1.6 A Model of Echoing Problems ix Linear Time-Invariant (LTI) Systems 2.1 Linear Systems 2.2 Linear Time-Invariant (LTI) Systems 2.3 The Convolution Integral 2.4 The Unit-Impulse Sifting Property 2.5 Convolution Problems Properties of LTI Systems 3.1 Tandem Connection of LTI Systems V

8 Vi CONTENTS 3.2 A Consequence of the Commutative Property 3.3 The Unit Impulse Revisited 3.4 Convolution Revisited 3.5 Causality 3.6 Stability 3.7 System Continuity 3.8 The Potential Integral Problems The Frequency Domain Viewpoint 4.1 The Characteristic Function of a Stable LTI System 4.2 Sinusoidal Response 4.3 Tandem-Connected LTI Systems 4.4 Continuous Frequency Representation of a Waveform Problems The Fourier Transform 5.1 The Fourier Transform 5.2 An Example of a Fourier Transform Calculation 5.3 Even and Odd Functions 5.4 An Example of an Inverse Fourier Transform Calculation 5.5 Some Properties of the Fourier Transform 5.6 An Application of the Convolution Property 5.7 An Application of the Time- and Frequency-Shift Properties 5.8 An Application of the Time-Differentiation Property 5.9 An Application of the Scaling Property 5.10 A Parseval Relation and Applications 5.11 Transfer Function Constraints Problems The Bilateral Laplace Transform 6.1 The Bilateral Laplace Transform 6.2 Some Properties of the RAC 6.3 Some Properties of the Bilateral Laplace Transform Problems

9 CONTENTS Vii 7 The Inverse Bilateral Laplace Transform The Inverse Laplace Transform The Linearity Property of the Inverse Laplace Transform The Partial Fraction Expansion Concluding Discussion and Summary 223 Problems Laplace Transform Analysis of the System Output The Laplace Transform of the System Output Causality and Stability in the s-domain Lumped Parameter Systems Passive Systems The Differential Equation View of Lumped Parameter Systems 252 Problems S-Plane View of Gain and Phase Shift Geometric View of Gain and Phase Shift The Pole-Zero Pair Minimum-Phase System Functions Bandpass System Functions Algebraic Determination of the System Function 291 Problems Interconnection of Systems 305 IO. 1 Basic LTI System Interconnections 10.2 Analysis of the Feedback System 10.3 The Routh-Hurwitz Criterion 10.4 System Block Diagrams 10.5 Model Consistency 10.6 The State-Space Approach Problems

10 Viii CONTENTS Appendix A A Primer on Complex Numbers and Complex Algebra 353 Appendix B Energy Distribution in Transient Functions 363 Index 367

11 CHAPTER 1 GENERAL SYSTEM CONCEPTS 1.1 THE SYSTEM VIEWED AS A MAPPING In scientific usage, a system is defined as an assemblage of interacting things. The interaction of these things is described in terms of certain parameters such as voltage, current, force, velocity, temperature, and chemical concentration. For example, the solar system is a system in which the things are the sun, planets, asteroids, and comets, which interact due to the forces that are associated with their gravitational fields. An electronic circuit is another example of a system. The things in an electronic circuit are the resistors, capacitors, inductors, transistors, and so on, that interact due to their currents and voltages. The solar system is said to be an autonomous system because there is no external input driving the system. On the other hand, an electronic circuit such as an audio amplifier is a nonautonomous system because there is an input-the input audio voltagethat drives the system to produce the amplified output audio voltage. An input of a nonautonomous system is an independent source of a parameter. A nonautonomous system output is an input-dependent parameter that is desired to be known. Figure is a general schematic representation of a nonautonomous system with several inputs and several outputs. For simplicity, we shall restrict our discussion in this text to nonautonomous single-input single-output (SISO) systems. For our development then, a system can be defined in a mathematical sense as a rule by which an input x is transformed into an output y. The rule can be expressed symbolically as Y = r [XI (1.I-1) 1

12 2 GENERAL SYSTEM CONCEPTS Fig Schematic of a system. The output and the input usually are functions of an independent variable such as position or time. If they are only functions of time t, Eq. (1.1-1) is written in the form m = T [ml (1.1-2) and represented schematically as shown in Fig As an example, in the study of the time variation of the pupillary diameter, d(t), of a person's eye as a function of light intensity, i(t), the system considered is one with the output d(t) for the input i(t) so that This is a symbolic representation of the rule that governs the changes in the diameter of the eye's pupil due to changes in the light intensity impinging on the eye. As another example, to study changes in the intensity of light, i(t), from a light bulb due to changes in the voltage, u(t), impressed across its filament, the system is one in which the input is the filament voltage, which is a function of time, and the output is the light intensity, which also is a function of time, so that Although our main consideration in this text will be of nonautonomous systems with inputs and outputs that are functions of time, it should be noted that they need not be functions of time. For example, in electrostatics the potential field 4(p), which is a function of position, p, can be considered as the output of a system with the input being the charge density distribution p(p) and expressed as 4(P) = 1 MP)l (1.1-5) In this system, both the input charge density and the output potential field are functions only of position, p. Another example in which the input and output of a system are functions only of position is a system used to study the deflection of beams. In such a system, the input is the force on the beamf(p), which is a function of the position along the beam at which the force is applied, and the output is the Fig X f 4 SYSTEM "' Schematic of a SISO system.

13 1.1 THE SYSTEM VIEWED AS A MAPPING 3 beam deflection d(p), which also is a function of position along the beam. Thus the relation can be expressed as (1.1-6) In Eq. (1.1-2), T is called an operator because the output function y(t) can be viewed as being produced by an operation on the input function x(t). The statement that y(t) is the response of a system to the input x(t) means that there exists an operator 1 or, equivalently, a rule by which a given output time function y(t) is obtained from a given input time function x(t). Another equivalent way of thinking about a system is to consider the input x(t) being mapped into the output y(t). This viewpoint can be conceptualized as shown in Fig In the figure, the set of all the possible inputs is denoted by X and the set of all possible outputs is denoted by Y. As illustrated, the input time functions denoted by x1 and x, are both mapped into the same output time function denoted by y,. Note that there can be only one time function resulting from a rule being applied to a given input. Thus if the relation between the system input and output is y(t) = ~?(t), the rule is that the output value at any instant of time to, is equal to the square of the input value at the same time to. Clearly, for this system there is only one output for any given input. However, note that many different inputs can produce the same output. For example, consider the input to be a waveform that jumps back and forth between +2 and -2. Irrespective of the times at which the jumps take place, the output will be +4 so that for this system, there are an infinite number of input waveforms that produce the same output waveform. On the other hand, an example of a relation that cannot be modeled as a system is one in which y(t) = [x(t)]'/*. This cannot be modeled as a system because, in taking the square root of x(t), there is no general rule by which the correct sign of y(t) can be known because if x(to) = 4, is y(to) = +2 or is y(to) = -2? However, if the rule is that the output is the positive square root of the input, then there is only one possible output for any given input and the relation can be modeled as a system. In terms of the mapping concept, a system is said to be a many-to-one mapping because many different inputs can result in a particular output, but a given input cannot result in more than one output. Our discussion in this text will center on SISO systems with inputs and outputs that are functions of time. The inverse of a given system is one that undoes the mapping of the given system as shown in Fig Note that the system inverse is a system, so that its operator X Y Fig The mapping of a the operator 1.

14 4 GENERAL SYSTEM CONCEPTS Fig Schematic of a system inverse. must be a many-to-one mapping of the set of its inputs, Y, to the set of its outputs, X. This means that if a system inverse is to exist, the operator of the system must be a one-to-one mapping of the set of its inputs, X, to the set of its outputs, Y. As an example, if the system mapping were one as shown in Fig , there would be no rule for the system inverse to map its input y, because the correct output, x, or x2, could not be determined. We thus conclude that a system inverse exists ifand only if the system operator is a one-to-one mapping. Clearly, the inverse of a square-law device for which y(t) = 2(t) and x(t) is any waveform does not exist. However, if the set of inputs, X, is restricted to time functions that are never negative so that x(t) 3 0, then the mapping is one-to-one and a system inverse does exist relative to that restricted class of inputs. 1.2 SYSTEM ANALYSIS CONCEPTS System analysis is the determination of the rule 1 of a given system. If nothing is known a priori about the given system, then one could only perform a series of experiments on the system from which a list of various inputs and their corresponding outputs is made. The difficulty with this is manifold. First, it can be shown that it is theoretically impossible to make a listing of all possible inputs not because of human frailty but because there are more possible input time functions than it is theoretically possible to list. A mathematician would say that the set of possible inputs is not listable or, equivalently, not countable. To circumvent this problem, one might attempt to make a list of just a judicious selection of possible inputs and their corresponding outputs. However, such a listing would not characterize the mapping because the system output due to an input that is not on the list would not be known. Even if an input were close to one on the list (that is, they are approximately equal), it could not be concluded that the two corresponding outputs were close to each other. For example, it might be that the input, as opposed to the one on the list, caused a relay in the system to temporarily open and thereby result in an entirely different output. If it were known for the given system that inputs that are close to each other result in outputs that are close to each other (such systems are called continuous systems), then such a listing could be used to obtain the approximate system response to any input that is close to one on the list. We thus note that, without more knowledge of the system operator, 1, such a listing is relatively useless. Aside from these problems, a listing would not result in a comprehensive knowledge of the rule 1 because it would be an onerous, if not impossible, task to deduce the rule 1 just from examining a collection of waveforms. We thus conclude that to do any meaningful system analysis, some a priori knowledge of the system operator must be known. It is a common problem in

15 1.3 TIME-INVARIANT (TI) SYSTEMS 5 scientific studies that some a priori knowledge of an object is required in order to analyze the object. For example, in communication theory, a basic problem is the extraction of a signal from a noisy version of it. If nothing were known a priori about the signal or the noise, then nothing can be done because there would be no known characteristics of the signal or the noise that could be used to extract the signal from its noise environment. On the other hand, if everything about the signal were known, then there is no need to extract the signal because it is known. Thus, a basic problem in communication theory is to determine what is reasonable to know about the signal and the noise that would be useful in extracting the signal. In system theory too, we are faced with the problem of determining what is reasonable to know that would be useful for system analysis. For this, systems are classified based on certain characteristics that are reasonable to know. Some of the system classifications that have been found useful are presented and used in this text. 1.3 TIME-INVARIANT (TI) SYSTEMS One characteristic that has been found useful is time-invariance. A time-invariant (Tq system is one in which the rule 1 by which the system output is obtainedfiom the system input does not vary with time. For many mechanical and electrical systems it is reasonable to know whether the system is TI. For example, one can consider an electrical resistor as a system with the input being the current i(t) through the resistor and the output being the voltage e(t) across the resistor. The rule then is e(t) = r. i(t) in which r is the value of the resistance. This system is TI if r does not vary with time. To be realistic, the value r of any physical resistor will vary with time because physical objects do age and the resistance value, after some time that could be many years, will be significantly different. However, over any reasonable interval of time, the value of r can be considered to be constant. We similarly model many physical systems as being TI if their rules, 1, do not change over any interval of interest. Note that in making a model of a physical system, we are not trying to represent it exactly because the best model of any physical system is the physical system itself. Rather, the attempt is to make as simple a model as possible for which the difference between its output and that of the physical system being modeled is acceptably small for the class of inputs of interest. The model then serves as a basis not only for calculating responses of the physical system, but also to gain a deeper understanding of the physical system behavior. A difficulty with the above-given definition of a TI system is that it is not an operational definition. An operational definition is one that specifies an experimental procedure. Thus, an operational definition of a TI system is one which specifies an experimental procedure by which one can determine whether the system is TI. The only meaningful definitions in science are operational ones because, at its base, science is concerned with experimental procedures and results.

16 6 GENERAL SYSTEM CONCEPTS To develop an operational definition of time invariance, we first note that if a system is TI, then for any value of to we have YO - to) = - to)] (1.3-1) This equation states that if the rule does not change with time, then the response to x(t) shifted by to seconds must be the output y(t) also shifted by the same amount, to. It is clear that if Eq. (1.3-1) is satisfied for a given system no matter what the input x(t) or value of to used, then the system is TI. We thus can state an operational definition of time invariance: A system is time-invariant (TI) ifeq. (1.3-1) is satisfied for any input, x(t), and any time shift, to. To illustrate this operational definition, consider the resistor network shown in Fig As shown, the resistance of each resistor varies with time. The system input is x(t), which is an applied voltage, and the system output y(t) is the voltage across the resistor rb(t). The relation between the system output y(t) and its input x(t) is (1.3-2) This is the rule by which the system response is obtained from its input. Clearly, this system is TI if the resistor values do not vary with time. However, is it possible for the resistor values to vary with time and yet the system be TI? We shall use the operational definition of TI to answer this question. For this, we must show that the system is TI if the system response to x(t - to) is equal to y(t - to) for any x(t) and for any value of to. This can be viewed schematically as shown in Fig The box that is labeled (t - to) represents an ideal delay system for which the output is its input delayed by to seconds. The output z(t) of the top block diagram is the system response to the input x(t - to), while the output of the bottom block diagram is y(t - to). To show that the system is TI, we must show that z(t) = y(t - to) for any input x(t) and for any delay to. For our illustrative example, we have from Eq. (1.3-2) that the system response to the input x(t - to) is (1.3-3) Fig A resistor network.

17 = 1.3 TIME-INVARIANT (TI) SYSTEMS 7 Fig Illustrating TI determination. On the other hand, (1.3-4) To understand the difference between these two equations, note that Eq. (1.3-3) is obtained by only shifting the input x(t) by to seconds whereas Eq. (1.3-4) is obtained by shifting the input x(t) and the system rule 1 by to seconds in accordance with Fig The system is TI if z(t) = y(t - to) for any input x(t) and for any value of to. From Eqs. (1.3-3) and (1.3-4) we observe that this is true for any x(t) only if (1.3-5) for any value of to. Our analysis is now made easier by considering the reciprocal of each side of Eq. (1.3-5): r,(t> Yb(t) + ru(t - to) f l (1.3-6) rb(t - Equation (1.3-6) is satisfied for any value of to only if Yb(t) = (1.3-7) in which c is a constant. To see this, assume that the value of the ratio in Eq. (1.3-7) at the times t = t, and t = t2 differ. If this were so, then Eq. (1.3-6) would not be satisfied for t = t, and to = t, - t2. We thus note that the system of our example is TI only if Eq. (1.3-7) is satisfied. If Eq. (1.3-7) is satisfied, then, from Eq. (1.3-2), the relation between the system input and output can be expressed as The fact that the system is TI is easily seen because it is clear from Eq. (1.3-8) that the rule by which y(t) is obtained from x(t) does not vary with time. Note that a system can be TI even though the elements of which the system is composed vary

18 8 GENERAL SYSTEM CONCEPTS with time. To be a TI system just means that the rule by which the output is obtained from the input does not vary with time. A circuit in which one or more elements vary with time is called a time-varying circuit. However, as we observe from our example, the circuit can be a time-varying circuit while the system defined from the circuit is time-invariant. Care must be taken not to confuse circuit theory and system theory. Note hrther that if the system output were defined as the current i(t) through the resistors in Fig instead of the voltage y(t), then the relation between the input x(t) and output i(t) would be (1.3-9) Such a system would be TI only if r,(t) + rb(t) = constant. Thus the system with the output y(t) can be time-invariant, while the system with the output i(t) is timevarying. We thus observe that whether the defined system is time-invariant depends on what is defined as the system input and the system output. This is so because it is only the rule by which inputs are mapped into outputs that determines whether the system is TI, and the rule depends on what is called the input and what is called the output. We shall be concerned almost exclusively with time-invariant systems in this text. 1.4 NO-MEMORY SYSTEMS Another characteristic that is reasonable to know is whether a given system is a nomemory system. A no-memory system is defined as one for which the output value at any time to, y(to), depends only on the input value ut the same time, x(to). Thus a square-law device in which y(t) = x2(t) is a time-invariant no-memory system. On the other hand, the system with the output y(t) = 2(t - 3), which is a square-law device with a 3-second delay, is time-invariant system with memory. It is not a nomemory system because the output at any given time, to, depends on the input at the time t = to - 3, which is 3 seconds before to and not at the time to. We shall discuss only time-invariant no-memory systems in this section. Because the rule 1 does not vary with time for such systems, the output amplitude at any time is a given function only of the input amplitude at the same time so that we can express the system input-output relation in the form (1.4-1) in which x is the input amplitude and y is the output amplitude. Equation (1.4-1) is called the transfer characteristic of the no-memory system. The function f in Eq. (1.4-1) is a rule by which an amplitude x is mapped into an amplitude y. As such it must be, as discussed in Section 1.1, a many-to-one mapping. The procedure for determining the response y(t) of a no-memory system simply is to determine, at each time instant, the output amplitude from the input amplitude at that instant in accor-

19 1.4 NO-MEMORY SYSTEMS 9 dance with Eq. (1.4-1). There are several types of no-memory systems of importance which are discussed below. Also included in this discussion is the definition of some notation and basic functions of importance for our subsequent discussions. 1.4A The Ideal Amplifier The ideal amplifier is a no-memory system for which the output is K times its input. The constant K is called the gain of the ideal amplifier. The relation between the output and the input thus is and the transfer characteristic of the ideal amplifier is y=kk (1.4-3) The graph of this relation is simply a straight line with a slope of K as shown in Fig In system representations, the ideal amplifier is represented by either of the block diagrams shown in Fig The Half-Wave Rectifier The half-wave rectifier is a no-memory system for which the transfer characteristic is (1.4-4) A graph of this relation is shown in Fig Equation (1.4-4) can be written more compactly in the form y = Eocu(x) (1.4-5) Fig Ideal amplifier transfer characteristic.

20 10 GENERAL SYSTEM CONCEPTS X ( t 4 Y O (4 (b) Fig Block diagrams of an ideal amplifier. in which ifx<o ifx>o (1.4-6) is called the unit step function. We shall find the function u(x) to be very useful in our study of systems. As an illustration of the half-wave rectifier operation, we determine the output y(t) when the input x(t) is a sinusoid. The sinusoid is a waveform given by Eq. (1.4-7) and illustrated in Fig x(t) = A sin(wt) (1.4-7) This is a periodic waveform with a fundamental period equal to T. A periodic waveform is one that repeats itself so that some time shift of the waveform results in the same waveform. That is, there are values of z for which x(t + z) = x(t) (1.4-8) Note that a periodic waveform must extend from t = -00 to t = 00 because otherwise no shift ofx(t) will result in the same time function. The positive values of z for which Eq. (1.4-8) is satisfied are called periods of x(t); the smallest positive value of z for which Eq. (1.4-8) is satisfied is called the fundamental period of the waveform x(t). In Fig , the fundamental period of the sinusoid is T, while 2T, 3T,... are simply periods of the sinusoid. The value of T for the sinusoid can be determined by substituting Eq. (1.4-7) into Eq. (1.4-8) to obtain A sin[w(t + z)] = A sin(ot) (1.4-9) Y slope = K 0 X Fig Half-wave rectifier characteristic. Some texts define u(0) = 1; others define u(0) = 0. I ve defined u(0) = 1/2 not to be different but rather for consistency in system theory. A reason for my choice will be given in Section 2.4.