Signals systems and transforms 4e by philips

DC Power Supplies, 10 Analogous Systems, 12Analog-to-Digital Converter, 14 Numerical Integration, 16 Picture in a Picture, 17 Compact Disks, 18 Sampling in Telephone Systems, 19 Data-Acq

SIGNALS, SYSTEMS, AND TRANSFORMS

F OURTH E DITION

CHARLES L PHILLIPS

Emeritus Auburn University Auburn, Alabama

JOHN M PARR

University of Evansville Evansville, Indiana

EVE A RISKIN

University of Washington Seattle, Washington

Upper Saddle River, NJ 07458

SIGNALS, SYSTEMS, AND TRANSFORMS

Phillips, Charles L

Signals, systems, and transforms / Charles L Phillips, John M Parr,

Eve A Riskin.—4th ed.

p cm.

Includes bibliographical references and index.

ISBN-13: 978-0-13-198923-8

ISBN-10: 0-13-198923-5

1 Signal processing–Mathematical models 2 Transformations

(Mathematics) 3 System analysis I Parr, John M II Riskin, Eve A.

(Eve Ann) III Title.

TK5102.9.P47 2008

621.382'2—dc22

2007021144

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Taylor, Justin, Jackson, Rebecca, and Alex

Michaela, Cadence, Miriam, and Connor Duncan,

Gary, Noah, and Aden

DC Power Supplies, 10 Analogous Systems, 12

Analog-to-Digital Converter, 14 Numerical Integration, 16 Picture in a Picture, 17 Compact Disks, 18 Sampling in Telephone Systems, 19 Data-Acquisition System, 21

Time Transformations, 24 Amplitude Transformations, 30

Even and Odd Signals, 32 Periodic Signals, 34

CONTENTS

2.3 Common Signals in Engineering 39

Unit Step Function, 45

Unit Impulse Function, 49

Relation to Physical Systems, 118

Frequency Response of Linear Systems, 243

Frequency Spectra of Signals, 252

Summary, 255

Energy Density Spectrum, 255

Power and Energy Transmission, 261

Contents xi

Summary 324

Problems 324

Transforms with Complex Poles, 370

Functions with Repeated Poles, 373

Bilateral Transform from Unilateral Tables, 384

Inverse Bilateral Laplace Transform, 386

8.3 Solution of State Equations 408

Laplace-Transform Solution, 409

Convolution Solution, 414

Infinite Series Solution, 415

Unit Step and Unit Impulse Functions, 447

Equivalent Operations, 449

Time Transformations, 451

Amplitude Transformations, 456

Even and Odd Signals, 459

Systems with Memory, 475

Contents xiii

Two Standard Forms, 523

Shorthand Notation for the DFT, 620

Frequency Resolution of the DFT, 621

Validity of the DFT, 622

Summary, 626

Decomposition-in-Time Fast Fourier Transform Algorithm, 627

Decomposition-in-Frequency Fast Fourier Transform, 632

Summary, 635

Contents xv

Calculation of Fourier Transforms, 635

Complex-Number Arithmetic, 724

Euler’s Relation, 727

Conversion Between Forms, 728

E Solution of Differential Equations 730

The basic structure and philosophy of the previous editions of Signals, Systems, and

Transforms are retained in the fourth edition New examples have been added and

some examples have been revised to demonstrate key concepts more clearly Thewording of passages throughout the text has been revised to ease reading and im-prove clarity In particular, we have revised the development of convolution and theDiscrete Fourier Transform Biographical information about selected pioneers inthe fields of signal and system analysis has been added in the appropriate chapters.References have been removed from the end of each chapter and are collected inAppendix I

Many end-of-chapter problems have been revised and numerous new lems are provided Several of these new problems illustrate real-world concepts indigital communications, filtering, and control theory The end-of-chapter problemshave been organized so that multiple similar problems are provided The answer to

prob-at least one of each set of similar problems is provided in Appendix H The intent is

to allow students to develop confidence by gaining immediate feedback about theirunderstanding of new material and concepts All MATLAB examples have beenupdated to ensure compatibility with the Student Version Release 14

A companion web site at http://www.ee.washington.edu/class/SST_textbook/

textbook.html contains sample laboratories, lecture notes for Chapters 1–7 and

Chapters 9–12, and the MATLAB files listed in the textbook as well as several ditional MATLAB files It also contains a link to a second web site at

ad-http://www.ee.washington.edu/class/235dl/, which contains interactive versions of

the lecture notes for Chapters 1–7 Here, students and professors can find out solutions to all the examples in the lecture notes, as well as animated demon-strations of various concepts including transformations of continuous-timesignals, properties of continuous-time systems (including numerous examples ontime-invariance), convolution, sampling, and aliasing Additional examples for dis-crete-time material will be added as they are developed

worked-This book is intended to be used primarily as a text for junior-level students inengineering curricula and for self-study by practicing engineers It is assumed that

PREFACE

the reader has had some introduction to signal models, system models, and ential equations (as in, for example, circuits courses and courses in mathematics),and some laboratory work with physical systems.

differ-The authors have attempted to consistently differentiate between signal andsystem models and physical signals and systems Although a true understanding ofthis difference can be acquired only through experience, readers should understandthat there are usually significant differences in performance between physical sys-tems and their mathematical models

We have attempted to relate the mathematical results to physical systems thatare familiar to the readers (for example, the simple pendulum) or physical systemsthat students can visualize (for example, a picture in a picture for television) Thedescriptions of these physical systems, given in Chapter 1, are not complete in anysense of the word; these systems are introduced simply to illustrate practical appli-cations of the mathematical procedures presented

Generally, practicing engineers must, in some manner, validate their work Tointroduce the topic of validation, the results of examples are verified, using differentprocedures, where practical Many homework problems require verification of theresults Hence, students become familiar with the process of validating their ownwork

The software tool MATLAB is integrated into the text in two ways First,

in appropriate examples, MATLAB programs are provided that will verify thecomputations Then, in appropriate homework problems, the student is asked toverify the calculations using MATLAB This verification should not be difficultbecause MATLAB programs given in examples similar to the problems areapplicable Hence, another procedure for verification is given The MATLAB

programs given in the examples may be downloaded from http://www.ee.

washington.edu/class/SST_textbook/textbook.html Students can alter data

statements in these programs to apply them to the end-of-chapter problems.This should minimize programming errors Hence, another procedure for verifi-cation is given However, all references to MATLAB may be omitted, if theinstructor or reader so desires

Laplace transforms are covered in Chapter 7 and z-transforms are covered in

Chapter 11 At many universities, one or both transforms are introduced prior to thesignals and systems courses Chapters 7 and 11 are written such that the material can

be covered anywhere in the signals and systems course, or they can be omittedentirely, except for required references

The more advanced material has been placed toward the end of the chapterswherever possible Hence, this material may be omitted if desired For example,Sections 3.7, 3.8, 4.6, 5.5, 7.9, 10.7, 12.6, 12.7, and 12.8 could be omitted by instructorswithout loss of continuity in teaching Further, Chapters 8 and 13 can be skipped if

a professor does not wish to cover state-space material at the undergraduate level.The material of this book is organized into two principal areas: continuous-time signals and systems, and discrete-time signals and systems Some professorsprefer to cover first one of these topics, followed by the second Other professorsprefer to cover continuous-time material and discrete-time material simultaneously

Preface xix

The authors have taken the first approach, with the continuous-time material ered in Chapters 2–8, and the discrete-time material covered in Chapters 9–13 Thematerial on discrete-time concepts is essentially independent of the material oncontinuous-time concepts so that a professor or reader who desires to study the dis-crete-time material first could cover Chapters 9–11 and 13 before Chapters 2–8 Thematerial may also be arranged such that basic continuous-time material and dis-crete-time material are intermixed For example, Chapters 2 and 9 may be coveredsimultaneously and Chapters 3 and 10 may also be covered simultaneously

cov-In Chapter 1, we present a brief introduction to signals and systems, followed

by short descriptions of several physical continuous-time and discrete-time systems

In addition, some of the signals that appear in these systems are described Then avery brief introduction to MATLAB is given

In Chapter 2, we present general material basic to continuous-time signals andsystems; the same material for discrete-time signals and systems is presented inChapter 9 However, as stated above, Chapter 9 can be covered before Chapter 2 orsimultaneously with Chapter 2 Chapter 3 extends this basic material to continuous-time linear time-invariant systems, while Chapter 10 does the same for discrete-timelinear time-invariant systems

Presented in Chapters 4, 5, and 6 are the Fourier series and the Fourier form for continuous-time signals and systems The Laplace transform is then devel-oped in Chapter 7 State variables for continuous-time systems are covered inChapter 8; this development utilizes the Laplace transform

trans-The z-transform is developed in Chapter 11, with the discrete-time Fouriertransform and the discrete Fourier transform presented in Chapter 12 However,Chapter 12 may be covered prior to Chapter 11 The development of the discrete-time Fourier transform and discrete Fourier transform in Chapter 12 assumes thatthe reader is familiar with the Fourier transform State variables for discrete-timesystems are given in Chapter 13 This material is independent of the state variablesfor continuous-time systems of Chapter 8

In Appendix A, we give some useful integrals and trigonometric identities In eral, the table of integrals is used in the book, rather than taking the longer approach

gen-of integration by parts Leibnitz’s rule for the differentiation gen-of an integral andL’Hôpital’s rule for indeterminate forms are given in Appendix B and are referenced inthe text where needed Appendix C covers the closed forms for certain geometric series;this material is useful in discrete-time signals and systems In Appendix D, we reviewcomplex numbers and introduce Euler’s relation, in Appendix E the solution of lineardifferential equations with constant coefficients, and in Appendix F partial-fractionexpansions Matrices are reviewed in Appendix G; this appendix is required for thestate-variable coverage of Chapters 8 and 13 As each matrix operation is defined,MATLAB statements that perform the operation are given Appendix H providessolutions to selected chapter problems so that students can check their work inde-pendently Appendix I lists the references for the entire text, arranged by chapter.This book may be covered in its entirety in two 3-semester-hour courses, or inquarter courses of approximately the equivalent of 6 semester hours With the omis-sion of appropriate material, the remaining parts of the book may be covered with

fewer credits For example, most of the material of Chapters 2, 3, 4, 5, 6, 8, 9, 10, 11and 12 has been covered in one 4-semester-hour course The students were alreadyfamiliar with some linear-system analysis and the Laplace transform.

We wish to acknowledge the many colleagues and students at Auburn sity, the University of Evansville, and the University of Washington who have con-tributed to the development of this book In particular, the first author wishes toexpress thanks to Professors Charles M Gross, Martial A Honnell, and Charles L.Rogers of Auburn University for many stimulating discussions on the topics in thisbook, and to Professor Roger Webb, director of the School of Electrical Engineer-ing at the Georgia Institute of Technology, for the opportunity to teach the signaland system courses at Georgia Tech The second author wishes to thank ProfessorsDick Blandford and William Thayer for their encouragement and support for thiseffort, and Professor David Mitchell for his enthusiastic discussions of the subjectmatter The third author wishes to thank the professors and many students inEE235 and EE341 at the University of Washington who contributed comments tothis book and interactive web site, in particular Professors Mari Ostendorf andMani Soma, Eddy Ferré, Wai Shan Lau, Bee Ngo, Sanaz Namdar, Jessica Tsao, andAnna Margolis We would like to thank the reviewers who provided invaluablecomments and suggestions They are Leslie M Collins, Duke University; WilliamEads, Colorado State University; Aleksandar Dogandzic, Iowa State University;and Bruce Eisenstein, Drexel University The interactive web site was developedunder a grant from the Fund for the Improvement of Postsecondary Education(FIPSE), U.S Department of Education

Univer-CHARLES L PHILLIPS Auburn University

University of Evansville

EVE A RISKIN University of Washington

engineer-ing These topics involve the modeling of physical signals by mathematical functions, the modeling of physical systems by mathematical equations, and the solutions of the

equations when excited by the functions

Engineers must model two distinct physical phenomena First, physical systems are modeled by mathematical equations For systems that contain no sampling (continuous-time, or analog, systems), we prefer to use ordinary differential equa-

tions with constant coefficients; a wealth of information is available for the analysisand the design of systems of this type Of course, the equation must accuratelymodel the physical systems An example of the model of a physical system is a linearelectric-circuit model of Figure 1.1:

where is the force applied to the mass M and is the resulting displacement

of the mass

A second physical phenomenon to be modeled is called signals Physical

sig-nals are modeled by mathematical functions One example of a physical signal is the

voltage that is applied to the speaker in a radio Another example is the ture at a designated point in a particular room This signal is a function of time be-cause the temperature varies with time We can express this temperature as

tempera-(1.3)

Consider again Newton’s second law Equation (1.2) is the model of a physical

physical systems, we apply mathematics to the models of systems and signals, not to

the physical systems and signals The usefulness of the results depends on the racy of the models

accu-In this book, we usually limit signals to having one independent variable We

choose this independent variable to be time, t, without loss of generality Signals are divided into two natural categories The first category to be considered is continuous-

time, or simply, continuous, signals A signal of this type is defined for all values of

time A continuous-time signal is also called an analog signal A continuous-time

signal is illustrated in Figure 1.2(a)

x(t)f(t)

x(t)f(t)

u(t)

temperature at a point = u(t),

x(t)f(t)

Figure 1.2 (a) Continuous-time signal; (b) discrete-time signal.

Sec 1.1 Modeling 3

The second category for signals is discrete-time, or simply, discrete, signals A

discrete signal is defined at only certain instants of time For example, suppose that

signal processing (DSP).] Because a computer can operate only on numbers and not

on a continuum, the continuous signal must be converted into a sequence of

is available to the computer This sequence of

numbers is called a discrete-time signal Insofar as the computer is concerned, with n a noninteger does not exist (is not available) A discrete-time signal is illus-

trated in Figure 1.2(b)

We define a continuous-time system as one in which all signals are continuous time We define a discrete-time system as one in which all signals are discrete time.

Both continuous-time and discrete-time signals appear in some physical systems; we

call these systems hybrid systems, or data systems An example of a

sampled-data system is an automatic aircraft-landing system, in which the control functionsare implemented on a digital computer

The mathematical analysis of physical systems can be represented as inFigure 1.3 [1] We first develop mathematical models of the physical systems andsignals involved One procedure for finding the model of a physical system is to usethe laws of physics, as, for example, in (1.1) Once a model is developed, the equa-tions are solved for typical excitation functions This solution is compared with theresponse of the physical system with the same excitation If the two responses areapproximately equal, we can then use the model in analysis and design If not,

we must improve the model

Improving the mathematical model of a system usually involves making themodels more complex and is not a simple step Several iterations of the processillustrated in Figure 1.3 may be necessary before a model of adequate accuracyresults For some simple systems, the modeling may be completed in hours; for verycomplex systems, the modeling may take years An example of a complex model isthat of NASA’s shuttle; this model relates the position and attitude of the shuttle to

f(nT)f(nT), n = Á , - 2, - 1, 0, 1, 2, Á ,

f(t)f(t)

FIGURE 1.3 Mathematical solutions of physical problems.

Mathematical solution of equations Conceptional aspects

the engine thrust, the wind, the positions of the control surfaces (e.g., the rudder),and so on As an additional point, for complex models of this type, the equationscan be solved only by computer.

This book contains two main topics: (1) continuous-time signals and systemsand (2) discrete-time signals and systems Chapters 2 through 8 cover continuous-time signals and systems, while Chapters 9 through 13 cover discrete-time signalsand systems The material may be covered in the order of the chapters, in whichcontinuous-time topics and discrete-time topics are covered separately Alterna-tively, the basic material of the two topics may be intermixed, with Chapters 2 and 9covered simultaneously, followed by Chapters 3 and 10 covered simultaneously

In this section, we discuss several continuous-time physical systems The tions are simplified; references are given that contain more complete descriptions.The systems described in this and the next section are used in examples throughoutthe remainder of the book

descrip-We have already given the model of a rigid mass M in a frictionless environment,

[eq(1.2)]

that results from the force applied This model is a second-order linear differential

equation with constant coefficients.

Linearity is defined in Section 2.7 As we will see, an equation (or system) is

linear if the principle of superposition applies Otherwise, the equation is nonlinear.

Next we discuss several physical systems

Electric Circuits

In this section, we give models for some electric-circuit elements [2] We begin with

the model for resistance, given by

represented by the standard circuit symbol given in Figure 1.4 The dashed lines inthis figure indicate that the elements are parts of circuits For example, the resistancemust be a part of a circuit, or else v(t)is identically zero

(Æ)

i(t)v(t)

Rv(t),v(t) = Ri(t),

x(t)f(t)

2x(t)

dt2 ,

Sec 1.2 Continuous-Time Physical Systems 5

Figure 1.4 Electric-circuit elements (From

C L Phillips and R D Harbor, Feedback Control Systems, 3d ed., Prentice Hall,

Upper Saddle River, NJ, 1995.)

The model for inductance is given by

model for capacitance is given by

(1.6)

where C is the capacitance in farads The symbols for inductance and capacitance

are also given in Figure 1.4

For the ideal voltage source in Figure 1.4, the voltage at the terminals of the

that flows through the voltage source is determined by the circuit connected tothe source For the ideal current source, the current that flows through the current

that appears at the terminals of the current source is determined by the circuit nected to these terminals

con-Consider now a circuit that is an interconnection of the elements shown inFigure 1.4 The circuit equations are written using the models given in the figurealong with Kirchhoff’s voltage and current laws Kirchhoff’s voltage law may bestated as follows:

The algebraic sum of voltages around any closed loop in an electric circuit is zero.

Kirchhoff’s current law may be stated as follows:

The algebraic sum of currents into any junction in an electric circuit is zero.

v(t)i(t),

i(t)v(t)

t

- qv(t)dt,

Figure 1.5 Operational amplifier.

Operational Amplifier Circuits

A device called an operational amplifier (or op amp) [3] is commonly used in

cir-cuits for processing analog electrical signals We do not investigate the internalstructure of this amplifier, but instead present only its terminal characteristics

We denote an operational amplifier by the circuit symbol of Figure 1.5(a) Thecircles indicate amplifier terminals, and the dashed lines indicate connections exter-nal to the amplifier The signal-input terminals are labeled with a minus sign for the

inverting input and a plus sign for the noninverting input The power-supply

The op amp is normally shown as in Figure 1.5(b), with the power-supply terminals

voltage output is

The operational amplifier is designed and constructed such that the input

voltage if the amplifier is to operate in its linear range (not saturated)

105

vo(t)>vd(t)

i+(t)

i(t)

V

V

Sec 1.2 Continuous-Time Physical Systems 7

For this discussion, we assume that the amplifier is ideal, which is sufficientlyaccurate for most purposes The ideal op amp has zero input currents

Additionally, the ideal amplifier operates in its linear range with infinite

Because the op amp is a very high-gain device, feedback is usually added forstabilization The feedback is connected from the output terminal to the invertinginput terminal (the minus terminal) This connection results in negative, or stabiliz-ing, feedback and tends to prevent saturation of the op amp

An example of a practical op-amp circuit is given in Figure 1.6 In this circuit,

Figure 1.5(b) is assumed to be zero, the equation for the input loop in Figure 1.6 isgiven by

(1.7)

Using (1.7), we express this equation as

(1.8)

(1.8) is a linear algebraic equation.

A second practical op-amp circuit is given in Figure 1.7 We use the precedingprocedure to analyze this circuit Because the input loop is unchanged, (1.7) applies,

(1.9)

vi(t) - i(t)R - 1

ti(t)dt - vo(t) = 0

-Figure 1.6 Practical voltage amplifier.

Figure 1.7 Integrating amplifier.

Substitution of (1.7) into (1.9) yields

(1.10)

Thus, the equation describing this circuit is given by

(1.11)

This circuit is called an integrator or an integrating amplifier; the output voltage is

in-tegrator is a commonly used circuit in analog signal processing and is used in

sever-al examples in this book

If the positions of the resistance and the capacitance in Figure 1.7 are changed, the op-amp circuit of Figure 1.8 results We state without proof that theequation of this circuit is given by

inter-(1.12)

(The reader can show this by using the previous procedure.) This circuit is called a

differentiator, or a differentiating amplifier; the output voltage is the derivative of the

use in analog signal processing, because the derivative of a signal that changes

Sec 1.2 Continuous-Time Physical Systems 9

rapidly is large Hence, the differentiator amplifies any high-frequency noise inHowever, some practical applications require the use of a differentiator Forthese applications, some type of high-frequency filtering is usually required beforethe differentiation, to reduce high-frequency noise

Simple Pendulum

We now consider a differential-equation model of the simple pendulum, which is

pendulum bob is M, and the length of the (weightless) arm from the axis of rotation

to the center of the bob is L.

The force acting on the bob of the pendulum is then Mg, where g is the

gravi-tational acceleration, as shown in Figure 1.9 From physics we recall the equation ofmotion of the simple pendulum:

(1.13)

non-linear (Superposition does not apply.)

We have great difficulty in solving nonlinear differential equations; however,

Appendix D) by

(1.14)

is expressed in radians The error in this approximation is less than 10 percent for

decreases as becomes smaller We then express the model of the pendulum as,from (1.13) and (1.14),

(1.15)

d2u(t)

dt2+g

for small This model is a second-order linear differential equation with constant

coefficients.

This derivation illustrates both a linear model (1.15) and a nonlinear model(1.13) and one procedure for linearizing a nonlinear model Models (1.13) and (1.15)have unusual characteristics because friction has been ignored Energy given to thesystem by displacing the bob and releasing it cannot be dissipated Hence, we expectthe bob to remain in motion for all time once it has been set in motion Note thatthese comments relate to a model of a pendulum, not to the physical device If wewant to model a physical pendulum more accurately, we must, as a minimum,include a term in (1.13) and (1.15) for friction

DC Power Supplies

Power supplies that convert an ac voltage (sinusoidal voltage) into a dc voltage(constant voltage) [3] are required in almost all electronic equipment Shown inFigure 1.10 are voltages that appear in certain dc power supplies in which the acvoltage is converted to a nonnegative voltage

The voltage in Figure 1.10(a) is called a half-wave rectified signal This signal is

generated from a sinusoidal signal by replacing the negative half cycles of the

the period of the waveform (the time of one cycle)

The signal in Figure 1.10(b) is called a full-wave rectified signal This signal is

generated from a sinusoidal signal by the amplitude reversal of each negative half

one-half that of the sinusoid and, hence, one-half that of the half-wave rectified signal.Usually, these waveforms are generated by the use of diodes The circuit symbolfor a diode is given in Figure 1.11(a) An ideal diode has the voltage–current charac-teristic shown by the heavy line in Figure 1.11(b) The diode allows current to flow

T0

T0u

Figure 1.10 Rectified signals: (a) half wave; (b) full wave.

Sec 1.2 Continuous-Time Physical Systems 11

Figure 1.11 (a) Diode; (b) ideal diode characteristic.

unimpeded in the direction of the arrowhead in its symbol and blocks current flow

in the opposite direction Hence, the ideal diode is a short circuit for current flow in

non-linear device; therefore, many circuits that contain diodes are nonnon-linear circuits.One circuit for a power supply is given in Figure 1.12(a) The power-supply

volt-age across the load is unidirectional; however, this voltvolt-age is not constant

A practical dc power supply is illustrated in Figure 1.13 The inductor–capacitor

(LC) circuit forms a low-pass filter and is added to the circuit to filter out the

A circuit that uses four diodes to generate a full-wave rectified signal is given inFigure 1.14 The diodes A and D conduct when the source voltage is positive, and thediodes B and C conduct when the source voltage is negative However, the current

Figure 1.12 Half-wave rectifier.

Figure 1.13 Practical dc power supply.

Figure 1.14 Full-wave rectifier.

voltage across the load is a full-wave rectified signal, as shown in Figure 1.10(b) As

in the half-wave rectified case in Figure 1.13, a filter is usually added to make theload voltage approximately constant

Analogous Systems

We introduce analogous systems with two examples The model of a rigid mass M in

a frictionless environment is given in (1.2):

(1.16)

that results from the applied force We represent this system with Figure 1.15(a)

an inductance The loop equation is given by

Sec 1.2 Continuous-Time Physical Systems 13

Figure 1.15 Analogous systems.

Figure 1.16 LC circuit.

We see that the model for the mass in (1.16) and for the circuit in (1.18) are of the

same mathematical form; these two systems are called analogous systems We

de-fine analogous systems as systems that are modeled by equations of the same ematical form

math-As a second example, consider the LC circuit in Figure 1.16, which is excited

by initial conditions The loop equation for this circuit is given by

fluidic systems, and so on Suppose that we know the characteristics of the LC

cir-cuit; we then know the characteristics of the simple pendulum We can transferour knowledge of the characteristics of circuits to the understanding of other types

d2u(t)

dt2+g



L C

i (t)

of physical systems This process can be generalized further; for example, we studythe characteristics of a second-order linear differential equation with constantcoefficients, with the knowledge that many different physical systems have thesecharacteristics.

We now describe a physical sampler and some discrete-time physical systems Inmany applications, we wish to apply a continuous-time signal to a discrete-time sys-tem This operation requires the sampling of the continuous-time signal; we considerfirst an analog-to-digital converter, which is one type of physical sampler Thisdevice is used extensively in the application of continuous-time physical signals todigital computers, either for processing or for data storage

Analog-to-Digital Converter

We begin with a description of a digital-to-analog converter (D/A or DAC), since this device is usually a part of an analog-to-digital converter (A/D or ADC) We as- sume that the D/A receives a binary number every T seconds, usually from a digital

computer The D/A converts the binary number to a constant voltage equal to thevalue of that number and outputs this voltage until the next binary number appears

at the D/A input The D/A is represented in block diagram as in Figure 1.17(a), and atypical response is depicted in Figure 1.17(b) We do not investigate the internal oper-ation of the D/A

Next we describe a comparator, which is also a part of an A/D A comparator

with a reference voltage vr(t).If vi(t)is greater than vr(t),the comparator outputs

Binary numbers

Sec 1.3 Samplers and Discrete-Time Physical Systems 15

Figure 1.18 Comparator.

logic 1; for example, logic 1 is approximately 5 V for TTL (transistor-to-transistor

for TTL The comparator is normally shown with the signal ground of Figure 1.18omitted; however, all voltages are defined relative to the signal ground

Several different circuits are used to implement analog-to-digital converters,with each circuit having different characteristics We now describe the internal

operation of a particular circuit The counter-ramp A/D is depicted in Figure 1.19(a), with the device signals illustrated in Figure 1.19(b) [4] The n-bit counter

begins the count at value zero when the start-of-conversion (SOC) pulse arrives

vr(t),

vi(t)

Figure 1.19 Counter-ramp analog-to-digital converter (From C L Phillips and H T Nagle, Digital

Control Systems Analysis and Design, 3d ed., Prentice Hall, Upper Saddle River, NJ, 1995.)

v o

v i  v r

v i

(b) 0

SOC EOC

from the controlling device (usually, a digital computer) The count increases by

one with the arrival of each clock pulse The n-bit D/A converts the count to a

voltage

halts the clock input through an AND gate The end-of-conversion (EOC) pulsethen signals to the controlling device that the conversion is complete At this time,the controlling device reads the counter output, which is a binary number that is ap-

converter are discussed in Ref 4

Numerical Integration

In Section 1.2, we considered the integration of a voltage signal by

operational-amplifier circuits We now consider numerical integration, in which we use a digital

computer to integrate a physical signal

Integra-tion by a digital computer requires use of a numerical algorithm In general, ical algorithms are based on approximating a signal that has an unknown integralwith a signal that has a known integral Hence, all numerical integration algorithmsare approximate in nature

numer-We illustrate numerical integration with Euler’s rule, which is depicted in

the rectangular areas shown In this figure, the step size H (the width of each tangle) is called the numerical-integration increment The implementation of this al-

analog-to-digital converter

Let

(1.21)y(t) =

Lt 0x(t)dt

x(nH),

x(t)

x(t)x(t)

Sec 1.3 Samplers and Discrete-Time Physical Systems 17

in (1.21),

(1.22)

Ignoring the approximations involved, we express this equation as

(1.23)

Equa-tion (1.23) is called a first-order linear difference equaEqua-tion with constant coefficients Usually, the factor H that multiplies the independent variable n in (1.23) is omitted,

resulting in the equation

(1.24)

We can consider the numerical integrator to be a system with the input x[n] and put y[n] and the difference-equation model (1.24) A system described by a differ- ence equation is called a discrete-time system.

out-Many algorithms are available for numerical integration [5] Most of these gorithms have difference equations of the type (1.23) Others are more complex andcannot be expressed as a single difference equation Euler’s rule is seldom used inpractice, because faster or more accurate algorithms are available Euler’s rule ispresented here because of its simplicity

al-Picture in a al-Picture

We now consider a television system that produces a picture in a picture [6] This

system is used in television to show two frames simultaneously, where a smaller ture is superimposed on a larger picture Consider Figure 1.21, where a TV picture

pic-is depicted as having six lines (The actual number of lines pic-is greater than 500.) pose that the picture is to be reduced in size by a factor of three and inserted intothe upper right corner of a second picture

Sup-First, the lines of the picture are digitized (sampled) In Figure 1.21, each lineproduces six samples (the actual number can be more than 2000), which are called

picture elements (pixels) Both the number of lines and the number of samples per

line must be reduced by a factor of three to reduce the size of the picture Assumethat the samples retained for the reduced picture are the four circled in Figure 1.21.(In practical cases, the total number of pixels retained may be greater than 100,000.)

y[n] - y[n - 1] = Hx[n - 1]

t = nH.x(t)

y(nH)

y(nH) = y[(n - 1)H] + Hx[(n - 1)H]

=L(n - 1)H 0

x(t)dt +

LnH (n - 1)Hx(t)dt

LnH 0x(t)dt

Figure 1.21 Television picture within a picture.

Now let the digitized full picture in Figure 1.21 represent a different picture;the four pixels of the reduced picture then replace the four pixels in the upperright corner of the full picture The inserted picture is outlined by the dashedlines

The generation of a picture in a picture is more complex than as describedhere, but we do see the necessity to reduce the number of samples in a given line.Note that the information content of the signal is reduced, since information (sam-ples) is discarded We can investigate the reduction of information only after math-ematical models are developed for the process of converting a continuous-timesignal to a discrete-time signal by sampling This reduction in the number of sam-

ples as indicated in Figure 1.21, called time scaling, is discussed in Section 9.2 The

effects of sampling on the information content of a signal are discussed in Chapters 5and 6

Compact Disks

We next discuss compact disks (CDs) These disks store large amounts of data in

a sampled form We initially consider the audio compact disk (CD) [7] The audio

CD is a good example of a practical system in which the emphasis in sampling is

on maintaining the quality of the audio signal A continuous-time signal to bestored on a CD is first passed through an analog antialiasing bandpass filter with abandwidth of 5 to 20,000 Hz This is because humans typically can hear frequen-cies only up to about 20 kHz Any frequency component above 20 kHz would beinaudible to most people and therefore can be removed from the signal withoutnoticeable degradation in the quality of the music The filtered signal is then sam-

The data format used in storage of the samples includes correcting bits and is not discussed here

error-The audio CD stores data for up to 74 minutes of playing time For stereomusic, two channels (signals) must be stored The disk stores 650 megabytes of data,with the data stored on a continuous track that spirals outward So that data from

Sec 1.3 Samplers and Discrete-Time Physical Systems 19

the CD may be read at a constant rate, the angular velocity of the motor drivingthe disk must decrease in time as the radius of the track increases in order to main-tain a constant linear velocity The speed of the motor varies from 200 rpm down to

50 rpm

As a comparison, computer hard disks store data in circular tracks Thesedisks rotate at a constant speed, which is commonly 3600 rpm; hence, the data ratevaries according to the radius of the track being read High-speed CD-ROM drivesuse constant angular velocities as high as 12,000 rpm

Because the data stored on the CD contain error-correcting bits, the CD playermust process these data to reproduce the original samples This processing is rathercomplex and is not discussed here

The audio CD player contains three servos (closed-loop control systems) Oneservo controls the speed of the motor that rotates the disk, such that the data areread off the disk at a rate of 44,100 samples per second The second servo points thelaser beam at the required position on the CD, and the third servo keeps the laserbeam focused on this position

The digital video disc (DVD) is a popular medium for viewing movies andtelevision programs It stores sampled video and audio and allows for high-quality

playback Over two hours of video can be stored on one DVD The DVD uses video

compression to fit this much data on one disk (We will discuss the mathematics

used in some video-compression algorithms in Chapter 12.)

Sampling in Telephone Systems

In this section, we consider the sampling of telephone signals [8] The emphasis inthese sampling systems is to reduce the number of samples required, even thoughthe quality of the audio is degraded Telephone signals are usually sampled at

This sampling allows the transmission of a number of telephone nals simultaneously over a single pair of wires (or in a single communications chan-nel), as described next

sig-A telephone signal is passed through an analog antialiasing filter with a band of 200 to 3200 Hz, before sampling Frequencies of less than 200 Hz are atten-uated by this filter, to reduce the 60-Hz noise coupled into telephones circuits from

pass-ac power systems The 3200-Hz cutoff frequency of the filter ensures that no

fil-ter severely reduces the quality of the audio in telephone conversations; thisreduction in quality is evident in telephone conversations

The sampling of a telephone signal is illustrated in Figure 1.22 The pulses are

of constant width; the amplitude of each pulse is equal to the value of the telephone

signal at that instant This process is called pulse-amplitude modulation; the

infor-mation is carried in the amplitudes of the pulses

Several pulse-amplitude-modulated signals can be transmitted simultaneouslyover a single channel, as illustrated in Figure 1.23 In this figure, the numeral 1 denotesthe samples of the first telephone signal, 2 denotes the samples of the second one, and

so on The process depicted in Figure 1.23 is called time-division multiplexing because

fs>2 = 4000 Hz