信号与系统分析：变换方法与MATLAB应用

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"本书《Signals and Systems - 2ed 2011》是迈克尔·J·罗伯茨（Michael J. Roberts）教授撰写，他在田纳西大学电气与计算机工程系任教。这本书是第二版，专注于信号与系统的分析，主要采用变换方法和MATLAB工具进行讲解。" 在信号与系统领域，书中涵盖了基本概念和关键函数，包括： 1. **阶跃信号（Step Signal）**：阶跃信号通常表示为u(t)，当t >= 0时，其值为1，否则为0。它是分析线性时不变系统（LTI）的重要基础。 2. **单位冲激信号（Delta Function）**：δ(t)是单位冲激信号，也称为狄拉克δ函数，它在t = 0处的值为无穷大，但其积分等于1。它在理论计算中起着关键作用，可以视为“瞬时”的信号。 3. **符号函数（Signum Function）**：sgn(t)函数，当t > 0时，其值为1；t < 0时，值为-1；t = 0时，值为0。这个函数常用于描述信号的符号变化。 4. ** sinc 函数**：sinc(t)是正弦函数除以πt的函数，当t = 0时，sinc(t)的值为1，而其他情况下则根据t的值变化。 5. **三角波信号（Triangle Wave）**：tri(t)是一个周期性的三角形波形，其值在-1到1之间变化，具有平滑的上升和下降斜率，是滤波器设计和频率分析中的一个有用工具。 6. **矩形波信号（Rectangular Wave）**：rect(t)是具有固定幅度并在有限时间区间内为1的信号，通常用作脉冲或开关信号的模型。 7. **斜坡信号（Ramp Signal）**：ramp(t)是一个线性增长的信号，当t >= 0时，其值为t，否则为0。它在系统响应分析中很有用。 8. **离散时间信号**：书中也讨论了离散时间信号，如δ[n]，u[n]，sgn[n]和ramp[n]，这些都是连续时间信号在时间上采样后的版本，对于数字信号处理至关重要。 9. **傅里叶变换**：书中的变换方法可能包括傅里叶变换，它可以将信号从时域转换到频域，从而揭示信号的频率成分。 10. **MATLAB应用**：书中强调了使用MATLAB进行信号处理和系统分析，这是一个强大的计算工具，特别适合于信号与系统的模拟和实验。通过这些基本信号和变换方法，读者可以深入理解信号如何在系统中传输、变换和处理，这对于通信、控制理论、图像处理等领域的工程师和技术人员来说是至关重要的基础知识。

xii

MOTIVATION

I wrote the first edition because I love the mathematical beauty of signal and sys-

tem analysis. That has not changed. The motivation for the second edition is to

improve the book based on my own experience using the book in classes and also

by responding to constructive criticisms from students and colleagues.

AUDIENCE

This book is intended to cover a two-semester course sequence in the basics of

signal and system analysis during the junior or senior year. It can also be used (as

I have used it) as a book for a quick one-semester master’s-level review of trans-

form methods as applied to linear systems.

CHANGES FROM THE FIRST EDITION

Since writing the ﬁ rst edition I have used it, and my second book, Fundamentals of

Signals and Systems, in my classes. Also, in preparation for this second edition I have

used drafts of it in my classes, both to test the effects of various approaches to introduc-

ing new material and to detect and (I hope) correct most or all of the errors in the text

and exercise solutions. I have also had feedback from reviewers at various stages in the

process of preparing the second edition. Based on my experiences and the suggestions

of reviewers and students I have made the following changes from the ﬁ rst edition.

In looking at other well-received books in the signals and systems area, one ﬁ nds

that the notation is far from standardized. Each author has his/her preference and

each preference is convenient for some types of analysis but not for others. I have

tried to streamline the notation as much as possible, eliminating, where possible,

complicated and distracting subscripts. These were intended to make the material

precise and unambiguous, but in some cases, instead contributed to students’

fatigue and confusion in reading and studying the material in the book. Also, I

have changed the symbols for continuous-time harmonic functions so they will

not so easily be confused with discrete-time harmonic functions.

Chapter 8 of the ﬁ rst edition on correlation functions and energy and power

spectral density has been omitted. Most junior-level signals and systems

courses do not cover this type of material, leaving it to be covered in courses on

probability and stochastic processes.

Several appendices from the printed ﬁ rst edition have been moved to the book’s

website, www.mhhe.com/roberts. This, and the omission of Chapter 8 from the

ﬁ rst edition, signiﬁ cantly reduce the size of the book, which, in the ﬁ rst edition,

was rather thick and heavy.

I have tried to “modularize” the book as much as possible, consistent with the

need for consecutive coverage of some topics. As a result the second edition

has 16 chapters instead of 12. The coverages of frequency response, ﬁ lters,

communication systems and state-space analysis are now in separate chapters.

■

PREFACE

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® MATLAB is a registered trademark of The MathWorks, Inc.

The ﬁ rst ten chapters are mostly presentation of new analysis techniques, theory

and mathematical basics. The last six chapters deal mostly with the application

of these techniques to some common types of practical signals and systems.

The second edition has more examples using MATLAB

than the ﬁ rst edition

and MATLAB examples are introduced earlier than before.

Instead of introducing all new signal functions in the chapters on signal

description I introduced some there, but held some derived functions until the

need for them arose naturally in later chapters.

In Chapter 4 on system properties and system description, the discussion of

mathematical models of systems has been lengthened.

In response to reviewers’ comments, I have presented continuous-time

convolution ﬁ rst, followed by discrete-time convolution. Even though

continuous-time convolution involves limit concepts and the continuous-time

impulse, and discrete-time convolution does not, the reviewers felt that the

students’ greater familiarity with continuous-time concepts would make this

order preferable.

More emphasis has been placed on the importance of the principle of

orthogonality in understanding the theoretical basis for the Fourier series, both in

continuous and discrete time.

The coverage of the bilateral Laplace and z transforms has been increased.

There is increased emphasis on the use of the discrete Fourier transform to

approximate other types of transforms and some common signal-processing

techniques using numerical methods.

Material on continuous-time angle modulation has been added.

The “comb” function used in the ﬁ rst edition, deﬁ ned by

comb( and combttn n nmN

)() [] [=− = −

=−∞

∞

∑

␦␦

]

m=−∞

∞

∑

in which a single impulse is represented by δ (t) in continuous time and by

δ [n] in discrete time, has been replaced by a “periodic impulse” function. The

periodic impulse is represented by δ

(t) in continuous time and by δ

[n] in

discrete time where T and N are their respective fundamental periods. They

are defined by

␦␦ ␦ ␦

ttnT nnmN() ( ) [ ] ( ).=− = −

=−∞

∞

=−∞

∞

∑∑

and

The continuous-time comb function is very elegant mathematically, but I have

found from my experience in my own classes that its simultaneous time-scaling

and impulse-strength scaling under the change of variable t → at confuses the

students. The periodic impulse function is characterized by having the spacing

between impulses (the fundamental period) be a subscript parameter instead of

being determined by a time-scaling. When the fundamental period is changed the

impulse strengths do not change at the same time, as they do in the comb function.

This effectively separates the time and impulse-strength scaling in continuous time

and should relieve some confusion among students who are already challenged by

■

Preface xiii

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the abstractions of various other concepts like convolution, sampling and inte-

gral transforms. Although simultaneous time and impulse-strength scaling do not

occur in the discrete-time form, I have also changed its notation to be analogous to

the new continuous-time periodic impulse.

OVERVIEW

The book begins with mathematical methods for describing signals and systems, in both

continuous and discrete time. I introduce the idea of a transform with the continuous-

time Fourier series, and from that base move to the Fourier transform as an extension

of the Fourier series to aperiodic signals. Then I do the same for discrete-time signals.

I introduce the Laplace transform both as a generalization of the continuous-time Fou-

rier transform for unbounded signals and unstable systems and as a powerful tool in

system analysis because of its very close association with the eigenvalues and eigen-

functions of continuous-time linear systems. I take a similar path for discrete-time sys-

tems using the z transform. Then I address sampling, the relation between continuous

and discrete time. The rest of the book is devoted to applications in frequency-response

analysis, communication systems, feedback systems, analog and digital ﬁ lters and

state-space analysis. Throughout the book I present examples and introduce MATLAB

functions and operations to implement the methods presented. A chapter-by-chapter

summary follows.

CHAPTER SUMMARIES

CHAPTER 1

Chapter 1 is an introduction to the general concepts involved in signal and system

analysis without any mathematical rigor. It is intended to motivate the student by

demonstrating the ubiquity of signals and systems in everyday life and the impor-

tance of understanding them.

CHAPTER 2

Chapter 2 is an exploration of methods of mathematically describing continuous-time

signals of various kinds. It begins with familiar functions, sinusoids and exponentials

and then extends the range of signal-describing functions to include continuous-time

singularity functions (switching functions). Like most, if not all, signals and systems

textbooks, I deﬁ ne the unit step, the signum, the unit impulse and the unit ramp func-

tions. In addition to these I deﬁ ne a unit rectangle and a unit periodic impulse function.

The unit periodic impulse, along with convolution, provides an especially compact

way of mathematically describing arbitrary periodic signals.

After introducing the new continuous-time signal functions, I cover the common

types of signal tranformations, amplitude scaling, time shifting, time scaling, differen-

tiation and integration and apply them to the signal functions. Then I cover some char-

acteristics of signals that make them invariant to certain transformations, evenness,

oddness and periodicity, and some of the implications of these signal characteristics in

signal analysis. The last section is on signal energy and power.

CHAPTER 3

Chapter 3 follows a path similar to Chapter 2 except applied to discrete-time sig-

nals instead of continuous-time signals. I introduce the discrete-time sinusoid and

xiv Preface

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Preface xv

exponential and comment on the problems of determining the period of a discrete-

time sinsuoid. This is the student’s first exposure to some of the implications of

sampling. I define some discrete-time signal functions analogous to continuous-

time singularity functions. Then I explore amplitude scaling, time-shifting, time

scaling, differencing and accumulation for discrete-time signal functions, point-

ing out the unique implications and problems that occur, especially when time

scaling discrete-time functions. The chapter ends with definitions and discussion

of signal energy and power for discrete-time signals.

CHAPTER 4

This chapter addresses the mathematical decription of systems. First I cover the

most common forms of classiﬁ cation of systems, homogeneity, additivity, linearity,

time-invariance, causality, memory, static nonlinearity and invertibility. By exam-

ple I present various types of systems that have, or do not have, these properties and

how to prove various properties from the mathematical description of the system.

CHAPTER 5

This chapter introduces the concepts of impulse response and convolution as com-

ponents in the systematic analysis of the response of linear, time-invariant systems.

I present the mathematical properties of continuous-time convolution and a graphi-

cal method of understanding what the convolution integral says. I also show how

the properties of convolution can be used to combine subsystems that are connected

in cascade or parallel into one system and what the impulse response of the overall

system must be. Then I introduce the idea of a transfer function by ﬁ nding the re-

sponse of an LTI system to complex sinusoidal excitation. This section is followed

by an analogous coverage of discrete-time impulse response and convolution.

CHAPTER 6

This is the beginning of the student’s exposure to transform methods. I begin by graph-

ically introducing the concept that any continuous-time periodic signal with engineer-

ing usefulness can be expressed by a linear combination of continuous-time sinusoids,

real or complex. Then I formally derive the Fourier series using the concept of or-

thogonality to show where the signal description as a function of discrete harmonic

number (the harmonic function) comes from. I mention the Dirichlet conditions to let

the student know that the continuous-time Fourier series applies to all practical con-

tinuous-time signals, but not to all imaginable continuous-time signals.

Then I explore the properties of the Fourier series. I have tried to make the

Fourier series notation and properties as similar as possible and analogous to the

Fourier transform, which comes later. The harmonic function forms a “Fourier se-

ries pair” with the time function. In the first edition I used a notation for harmonic

function in which lowercase letters were used for time-domain quantities and up-

percase letters for their harmonic functions. This unfortunately caused some con-

fusion because continuous and discrete-time harmonic functions looked the same.

In this edition I have changed the harmonic function notation for continuous-time

signals to make it easily distinguishable. I also have a section on the convergence

of the Fourier series illustrating the Gibb’s phenomenon at function discontinui-

ties. I encourage students to use tables and properties to find harmonic functions

and this practice prepares them for a similar process in finding Fourier transforms

and later Laplace and z transforms.

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The next major section of Chapter 6 extends the Fourier series to the Fourier

transform. I introduce the concept by examining what happens to a continuous-time

Fourier series as the period of the signal approaches inﬁ nity and then deﬁ ne and de-

rive the continuous-time Fourier transform as a generalization of the continuous-time

Fourier series. Following that I cover all the important properties of the continuous-

time Fourier transform. I have taken an “ecumenical” approach to two different nota-

tional conventions that are commonly seen in books on signals and systems, control

systems, digital signal processing, communication systems and other applications of

Fourier methods such as image processing and Fourier optics: the use of either cyclic

frequency, f or radian frequency, ω. I use both and emphasize that the two are simply

related through a change of variable. I think this better prepares students for seeing

both forms in other books in their college and professional careers.

CHAPTER 7

This chapter introduces the discrete-time Fourier series (DTFS), the discrete Fou-

rier transform (DFT) and the discrete-time Fourier transform (DTFT), deriving

and deﬁ ning them in a manner analogous to Chapter 6. The DTFS and the DFT are

almost identical. I concentrate on the DFT because of its very wide use in digital

signal processing. I emphasize the important differences caused by the differences

between continuous and discrete time signals, especially the ﬁ nite summation

range of the DFT as opposed to the (generally) inﬁ nite summation range in the

CTFS. I also point out the importance of the fact that the DFT relates a ﬁ nite set

of numbers to another ﬁ nite set of numbers, making it amenable to direct numeri-

cal machine computation. I discuss the fast Fourier transform as a very efﬁ cient

algorithm for computing the DFT. As in Chapter 6, I use both cyclic and radian

frequency forms, emphasizing the relationships between them. I use F and Ω for

discrete-time frequencies to distinguish them from f and ω, which were used in

continuous time. Unfortunately, some authors reverse these symbols. My usage is

more consistent with the majority of signals and systems texts. This is another ex-

ample of the lack of standardization of notation in this area. The last major section

is a comparison of the four Fourier methods. I emphasize particularly the duality

between sampling in one domain and periodic repetition in the other domain.

CHAPTER 8

This chapter introduces the Laplace transform. I approach the Laplace transform

from two points of view, as a generalization of the Fourier transform to a larger

class of signals and as result that naturally follows from the excitation of a linear,

time-invariant system by a complex exponential signal. I begin by defining the

bilateral Laplace transform and discussing significance of the region of conver-

gence. Then I define the unilateral Laplace transform. I derive all the important

properties of the Laplace transform. I fully explore the method of partial-fraction

expansion for finding inverse transforms and then show examples of solving dif-

ferential equations with initial conditions using the unilateral form.

CHAPTER 9

This chapter introduces the z transform. The development parallels the develop-

ment of the Laplace transform except applied to discrete-time signals and sys-

tems. I initially define a bilateral transform and discuss the region of convergence.

Then I define a unilateral transform. I derive all the important properties and

xvi Preface

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