信号与系统解析：李爱德华与瓦拉亚的经典教材

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"Structure and Interpretation of Signals and Systems" 是一本由Edward A. Lee和Pravin Varaiya合著的信号处理领域的经典教材，源自UC Berkeley。这本书深入探讨了信号与系统的关系，对于学习和理解信号处理有极大的帮助。在书中，作者首先介绍了“信号”的概念。信号是数据的一种表现形式，它可以是声音、图像、视频，甚至可以代表物理属性。例如，音频信号用于传输声音信息，图像信号则包含视觉图像的细节。书中特别提到了家用电力系统作为信号的一个示例，展示信号如何表示和传递能量变化的信息。此外，视频信号讨论了动态图像的处理，进一步扩展到颜色和光的理解，这些都是图像信号的重要组成部分。接着，书中阐述了“系统”的概念。系统可以被视为作用于输入信号并产生输出信号的函数或过程。它们可以是简单的单向转换，也可以是复杂的多阶段处理。书中以通信系统为例，解释了系统如何在发送者和接收者之间传输信息。书中还进一步探讨了无线通信和低地球轨道（LEO）卫星通信系统，这些都是现代通信技术的关键领域。在信号的类型上，书中区分了连续信号和离散信号，并讲解了采样理论。离散信号在时间上不连续，通常与数字系统相关，而采样是将连续信号转换为离散信号的过程，这对于数字信号处理至关重要。这个过程遵循奈奎斯特定理，以确保无损地恢复原始连续信号。本书不仅限于基本概念，还可能深入到滤波器设计、系统分析、信号频谱分析以及系统的时域和频域表示等多个高级主题。对于希望深入理解信号处理的读者来说，"Structure and Interpretation of Signals and Systems" 提供了一个全面且深入的学习框架。通过这本书，读者可以掌握信号处理的基础知识，为在相关领域如通信工程、电子工程、计算机科学等进行更深入研究打下坚实基础。

xiv

CONTENTS

C.8.2 In-lab section ................................. 591

C.8.3 Independent section ............................. 592

C.9 Plucked string instrument ............................... 595

C.9.1 Background .................................. 595

C.9.2 In-lab section ................................. 597

C.9.3 Independent section ............................. 598

C.10 Modulation and demodulation ............................ 601

C.10.1 Background .................................. 601

C.10.2 In-lab section ................................. 606

C.10.3 Independent section ............................. 607

C.11 Sampling and aliasing ................................. 610

C.11.1 In-lab section ................................. 610

Index 615

Preface

This book addresses a sea change in our discipline. We have all been aware for some time that

“electrical engineering” has lost touch with the “electrical.” Electricity provides the impetus, the

pressure, the potential, but not the body. How else could microelectromechanical systems (MEMS)

become so important in EE? Is this not mechanical engineering? Or signal processing. Is this not

mathematics? Or digital networking. Is this not computer science? How is it that control system

techniques are proﬁtably applied to aeronautical systems, structural mechanics, electrical systems,

and options pricing?

This change is recognized in our graduate programs. We have research programs in networking,

multimedia communications, and embedded control, and we teach courses in queuing networks,

wavelets, and hybrid systems. But our undergraduate curriculum resists change.

Like most engineering schools, Berkeley required an introductory course in analog circuits. This

requirement was abandoned since it did not meet current needs. Far too many schools still require

such a course. The requirement impedes the modernization of subsequent courses in signal pro-

cessing, communications, and control. Sensing that the electrical engineering curriculum fails to

prepare them for exciting opportunities, students move towards computer science and engineering.

Circuits used to be at the heart of electrical engineering. It is arguable that today it is the techniques

of analysis and design that emerged from circuit theory that are at the heart of the discipline. The

circuits themselves have become an area of specialization. It is an important area of specialization,

to be sure, but it is a specialization nonetheless.

In the origins of our discipline, a signal was a voltage that varies over time, or an electromagnetic or

an acoustic waveform. Now it is likely to be a sequence of discrete messages sent over the internet

using TCP/IP. The state of a system used to be adequately captured by variables in a differential

equation. Now it is likely to be the registers and memory of a computer, or more abstractly, a

process continuation, or the state of a set of concurrent ﬁnite state machines. A system used to

be well-modeled by a linear time-invariant transfer function. Now it is likely to be described as

a computation in a Turing-complete computational engine. Despite these fundamental changes

in the medium with which we operate, the methodology remains robust and powerful. It is the

methodology, not the medium, that deﬁnes our ﬁeld. Our graduates are more likely to write software

than to push electrons, and yet we recognize them as electrical engineers.

Fundamental limits have also changed. Although we still face thermal noise and the speed of light,

we are likely to encounter other limits before we get to these, such as complexity, computability,

xvi

Preface

chaos, and, most commonly, limits imposed by other human constructions. A voiceband data mo-

dem, for example, uses the telephone network, which was designed to carry voice, and offers as

immutable limits such non-physical constraints as its 3 kHz bandwidth. DSL modems and wire-

less phones must meet regulatory rules that are more limiting than the physical constraints of their

transmission media. Computer-based audio systems face latency and jitter imposed by the operating

system.

The mathematical basis for the discipline has also shifted. Although we still use calculus and differ-

ential equations, we frequently need discrete math, set theory, and mathematical logic. Whereas the

mathematics of calculus and differential equations evolved to describe the physical world, the world

we face as system designers often has non-physical properties that are not such a good match to this

mathematics. Instead of abandoning formality, we need to broaden the mathematical base. Indeed,

the major goal of this book is to show the value of formal methods in a wide range of contexts.

Approach

Colleagues at other schools also recognize the need for engineering curriculum reform. But there

is less agreement on how to meet this need. We have been impressed by the approach of Abelson

and Sussman, in Structure and Interpretation of Computer Programs (MIT Press), who confronted

a similar transition in their discipline.

“Underlying our approach to this subject is our conviction that ‘computer science’ is

not a science and that its signiﬁcance has little to do with computers.”

Before Abelson and Sussman, computer programming was about getting computers to do your

bidding. In the preface to Structure and Interpretation of Computer Programs, they write

“First, we want to establish the idea that a computer language is not just a way of

getting a computer to perform operations but rather that it is a novel formal medium

for expressing ideas about methodology. Thus, programs must be written for people to

read, and only incidentally for machines to execute.

“... [The] essential material to be addressed by a subject at this level is ... the techniques

used to control the intellectual complexity of large software systems.”

Our book is about signals and systems, not about large software systems. But it takes a compu-

tational view of signals and systems. It focuses on the methods “used to control the intellectual

complexity,” rather than on the physical limitations imposed by the implementations of old. Appro-

priately, it puts emphasis on discrete-time modeling, which is pertinent to the implementations in

software and digital logic that are so common today. Continuous-time models describe the physical

world, with which our systems interact. But fewer of the systems we engineer operate directly in

this domain.

If imitation is the highest form of ﬂattery, then it should be obvious whom we are ﬂattering. Our

title is a blatant imitation of Structure and Interpretation of Computer Programs. The choice of title

Preface

xvii

reﬂects partly a vain hope that we might (improbably) have as much inﬂuence as they have. But

more realistically, it reﬂects a sympathy with their cause. Like us, they faced an identity crisis in

their discipline.

“The computer revolution is a revolution in the way we think and in the way we express

what we think. The essence of this change is what might best be called procedural epis-

temology—the study of the structure of knowledge from an imperative point of view,

as opposed to the more declarative point of view taken by classical mathematical sub-

jects. Mathematics provides a framework for dealing precisely with notions of ‘what

is.’ Computation provides a framework for dealing precisely with notions of ‘how to’.”

We develop two themes. The ﬁrst theme is the use of sets and functions as a universal language

to describe very diverse signals and systems. Different signals—voice, images, bit sequences—

are represented as functions with an appropriate domain and range. Systems are represented as

functions whose domain and range are themselves sets of signals. Thus, for example, a modem is

represented as a function that maps bit sequences into voice-like signals.

The second theme is that complex systems are constructed by connecting simpler subsystems in

standard ways—cascade, parallel, feedback, etc. The connections determine the description of

the interconnected system from the descriptions of component subsystems. The connections place

consistency requirements on the input and output signals of the systems being connected.

The hard work is to characterize the functional descriptions of signals and systems in ways that

facilitate their analysis and design. Our approach is to develop and relate both the imperative (com-

putational) and declarative (mathematical) descriptions of signals and systems. There is a need for

both views. The declarative view is suited to analysis; the imperative view is suited to implementa-

tion and design.

Introductory courses like the analog circuits course favor analysis and focus almost exclusively

on the declarative view. A modern view of signals and systems has much closer ties to computer

science, and has to complement this declarative view with an imperative one. The mathematical

treatment in the book presents the declarative view. The laboratory component of the text, and the

web content, with its extensive applets illustrating computational concepts, bring out the imperative

view. Both views are essential to a complete study.

An imperative view has another key pedagogical and intellectual advantage. It enables a much

closer connection with “real” signals and systems, which are often too messy for a complete declar-

ative treatment. While a declarative treatment can easily handle a sinusoidal signal, an imperative

treatment can easily handle a voice signal. An important feature of the approach in this text and the

course on which it is based is to illustrate concepts with real signals and systems at every step.

This is quite hard to do in a textbook. The print medium biases authors towards the declarative

view simply because its static nature is better suited to the declarative view than to the imperative

view. Our solution to this problem has been heavily inﬂuenced by the excellent and innovative

textbooks by Steiglitz, A Digital Signal Processing Primer—with Applications to Digital Audio

and Computer Music (Addison-Wesley, 1996), and McClellan, Schafer and Yoder, DSP First—A

Multimedia Approach (Prentice-Hall, 1998). Steiglitz leverages natural human interest in music

xviii

Preface

to teach very sophisticated concepts in signals and systems. McClellan, Schafer and Yoder nicely

integrate web-based content and laboratory exercises, complementing the traditional mathematical

treatment with accessible and motivating manipulation of real signals. If you are familiar with these

books, you will see their inﬂuence all over the laboratory exercises and the web content.

Content

A summary of each chapter is given in “Notes to Instructors.” Here we focus on the principal topics.

We begin by describing signals as functions, focusing on characterizing the domain and the range

for familiar signals that humans perceive such as sound, images, video, trajectories of vehicles, as

well as signals typically used by machines to store or manipulate information, such as sequences of

words or bits.

Systems, too, are described as functions, but now the domain and the range are themselves sets of

signals. The telephone handset converts voice into an analog electrical signal and the line card in the

telephone central ofﬁce converts the latter into a stream of bits. Systems can be connected to form

a more complex system, and the function describing these more complex systems is a composition

of functions describing the component systems.

Characterizing concretely the functions that describe signals and systems is the content of the book.

We begin to characterize systems using the notion of state, the state transition function and the

output function, all in the context of ﬁnite state machines. State machines are composed in various

ways (cascade, parallel, feedback). A telephone answering system is constructed step by step to

illustrate all the concepts. Applications to games and feedback control illustrate the power of the

state machine model.

Linear time-invariant systems are inﬁnite state machines with linear state transition and output func-

tions. The input-output behavior of these systems is now fully characterized by the zero-input re-

sponse and the zero-state impulse response.

Frequency domain concepts are introduced as a complementary toolset, different from that of state

machines, and much more powerful when applicable. Frequency decomposition of signals is moti-

vated using psychoacoustics, and gradually developed until all four Fourier transforms (the Fourier

series, the Fourier transform, the discrete-time Fourier transform, and the DFT) have been described.

We linger on the ﬁrst of these, the Fourier series, since it is conceptually the easiest, and then quickly

present the others as generalizations of the Fourier series.

Linear time-invariant systems have the fundamental property that complex exponentials are eigen-

functions. Consequently, they are also characterized by their frequency response—the main reason

that frequency domain methods are important in the analysis of ﬁlters and feedback control. Many

examples and exercises illustrate each concept. The discussion of frequency domain methods closes

by studying sampling and aliasing.

Chapters 5 and 6 form the bridge between state machine concepts and frequency-domain concepts.

Chapter 5 describes time-based systems as state machines with an inﬁnite state space. Chapter6

describes hybrid systems—systems whose descriptions require both state machines and time-based

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