李·爱德华与瓦里亚雅：信号与系统结构与解释

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《信号与系统结构与解释》是由Edward A. Lee和Pravin Varaiya合著的一本经典教材，由Addison-Wesley出版社于2000年出版。本书专注于电信号与系统领域的深入探讨，旨在为电子工程与计算机科学专业的学生提供坚实的理论基础和实践理解。作者们是来自加州大学伯克利分校的学者，他们在书中结合实例，如音频信号、图像、视频信号等，来阐述信号的不同类型及其在实际通信系统中的应用。章节1以“信号与系统”开篇，首先介绍了信号的基本概念。作者区分了连续和离散信号，举例说明了音频信号（如声音波形）和图像（像素矩阵表示），强调了这些信号如何代表物理世界的各种属性。书中还讨论了家庭用电系统，展示了信号在日常生活中的应用。对于视频信号，作者指出其包含时间与空间信息的复合特性，而信号序列则进一步扩展了对时间维度的处理。接着，章节1.2探讨了系统，将系统定义为能够处理信号的功能实体。作者解释了系统如何作为函数来操作输入信号，并特别提到了电信通信系统，包括无线通信和卫星电话（如低轨道电话），以及数据加密技术在语音通信中的应用。这部分内容涵盖了信号的存储与检索，如音频文件的播放和下载，以及调制解调器在数据传输中的角色，如速率匹配和协议谈判。在第2章，“定义信号与系统”，作者深入探讨了函数的概念，通过赋值语句和图形方式来帮助读者理解系统的输入-输出关系。这部分内容对于建立数学模型和解析系统行为至关重要，同时也介绍了如何通过声明式编程方法来设计和分析系统。《信号与系统结构与解释》是一本理论与实践并重的教材，它以清晰的语言和丰富的示例，引导读者理解信号的多样性和系统的设计原理。无论是对初学者还是专业工程师，这本书都提供了深入学习信号与系统理论的坚实平台。通过阅读这本书，读者不仅能掌握基本概念，还能拓展到实际工程应用中的各种信号处理和控制系统设计技巧。

xiv

Preface

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 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 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,

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

dem, for example, faces 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 face reg-

ulatory constraints that are more limiting than their physical constraints. 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

differential equations, we frequently need discrete math, set theory, and mathematical logic. Indeed,

a major theme of this book is to illustrate that formal techniques can be used in a very wide range

of contexts. 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.

Again, Abelson and Sussman faced a similar conundrum.

“... we believe that the essential material to be addressed by a subject at this level is

not the syntax of particular programming language constructs, nor clever algorithms

for computing particular functions efﬁciently, nor even the mathematical analysis of

algorithms and the foundations of computing, but rather the techniques used to control

the intellectual complexity of large software systems.”

This 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

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.

Preface

“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 fram-work for dealing precisely with notions of ‘how to’.”

Indeed, a major theme in our book is the connection between imperative (computational) and declar-

ative (mathematical) descriptions of signals and systems. The laboratory component of the text, in

our view, is an essential part of a complete study. The web content, with its extensive applets il-

lustrating computational concepts, is an essential complement to the mathematics. Traditional elec-

trical engineering has emphasized the declarative view. Modern electrical engineering has much

closer ties to computer science, and has to complement this declarative view with an imperative

one.

Besides being a proper part of the intellectual discipline, an imperative view has another key advan-

tage. It enables a much closer connection with “real signals and systems,” which are often too messy

for a complete declarative treatment. While a declarative treatment can easily handle a sinusoidal

signal, an imperative treatment can easily handle a voice signal. One of our objectives in designing

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 text-

books by Steiglitz, A Digital Signal Processing Primer – with Applications to Digital Audio and

Computer Music (Addison-Wesley), and McClellan, Schafer and Yoder, DSP First – A Multimedia

Approach. Steiglitz leverages natural human interest in the very human ﬁeld of music to teach,

spectacularly gently, very sophisticated concepts in signals and systems. McClellan, Schafer and

Yoder beautifully integrate web-based content and laboratory exercises, complementing the tradi-

tional mathematical treatment with accessible and highly motivational 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.

Notes to Instructors

The course begins by describing signals as functions, focusing on characterizing the domain and the

range. Systems are also described as functions, but now the domain and range are sets of signals.

Characterizing these functions is the topic of this course. Sets and functions provide the unifying

notation and formalism.

We begin by describing systems using the notion of state, ﬁrst using automata theory and then

progressing to linear systems. 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 introduced 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 simple generalizations of the Fourier

series. Finally, the course closes by using these concepts to study sampling and aliasing, which

helps bridge the computational world with the physical world.

This text has evolved to support a course we now teach regularly at Berkeley to about 500 students

per year. An extensive accompanying web page is organized around Berkeley’s 15 week semester,

although we believe it can be adapted to other formats. The course organization at Berkeley is as

follows:

Week 1 – Signals as Functions – Chapters 1 and 2. The ﬁrst week motivates forthcoming material

by illustrating how signals can be modeled abstractly as functions on sets. The emphasis is on

characterizing the domain and the range, not on characterizing the function itself. The startup

sequence of a voiceband data modem is used as an illustration, with a supporting applet that plays

the very familiar sound of the startup handshake of V32.bis modem, and examines the waveform in

both the time and frequency domain. The domain and range of the following signal types is given:

sound, images, position in space, angles of a robot arm, binary sequences, word sequences, and

event sequences.

Week 2 – Systems as Functions – Chapters 1 and 2. The second week introduces systems as

functions that map functions (signals) into functions (signals). Again, it focuses not on how the

function is deﬁned, but rather on what is its domain and range. Block diagrams are deﬁned as

a visual syntax for composing functions. Applications considered are DTMF signaling, modems,

digital voice, and audio storage and retrieval. These all share the property that systems are required

to convert domains of functions. For example, to transmit a digital signal through the telephone

system, the digital signal has to be converted into a signal in the domain of the telephone system

xvii

xviii

Notes to Instructors

(i.e., a bandlimited audio signal).

Week 3 – State – Chapter 3. Week 3 is when the students start seriously the laboratory component

of the course. The ﬁrst lecture in this week is therefore devoted to the problem of relating declar-

ative and imperative descriptions of signals and systems. This sets the framework for making the

intellectual connection between the labs and the mathematics.

The purpose of this ﬁrst lab exercise is to explore arrays in Matlab and to use them to construct audio

signals. The lab is designed to help students become familiar with the fundamentals of Matlab, while

applying it to synthesis of sound. In particular, it introduces the vectorization feature of the Matlab

programming language. The lab consists of explorations with sinusoidal sounds with exponential

envelopes, relating musical notes with frequency, and introducing the use of discrete-time (sampled)

representations of continuous-time signals (sound).

Note that there is some potential confusion because Matlab uses the term “function” somewhat

more loosely than the course does when referring to mathematical functions. Any Matlab command

that takes arguments in parentheses is called a function. And most have a well-deﬁned domain and

range, and do, in fact, deﬁne a mapping from the domain to the range. These can be viewed formally

as a (mathematical) functions. Some, however, such as plot and sound are a bit harder to view

this way. The last exercise in the lab explores this relationship.

The rest of the lecture content in the third week is devoted to introducing the notion of state and

state machines. State machines are described by a function update that, given the current state and

input, returns the new state and output. In anticipation of composing state machines, the concept of

stuttering is introduced. This is a slightly difﬁcult concept to introduce at this time because it has

no utility until you compose state machines. But introducing it now means that we don’t have to

change the rules later when we compose machines.

Week 4 – Nondeterminism and Equivalence – Chapter 3. The fourth week deals with nonde-

terminism and equivalence in state machines. Equivalence is based on the notion of simulation, so

simulation relations and bisimulation are deﬁned for both deterministic and nondeterministic ma-

chines. These are used to explain that two state machines may be equivalent even if they have a

different number of states, and that one state machine may be an abstraction of another, in that it

has all input/output behaviors of the other (and then some).

During this week, students do a second set of lab exercises that explore the representation of images

in Matlab, relating the Matlab use of color maps with a formal functional model. It discusses the

ﬁle formats for images, and explores the compression that is possible with colormaps and with

more sophisticated techniques such as JPEG. The students construct a simple movie, reinforcing

the notions of sampling introduced in the previous lab. They also blur an image and create a simple

edge detection algorithm for the same image. This lab also reinforces the theme of the previous one

by asking students to deﬁne the domain and range of mathematical models of the relevant Matlab

functions. Moreover, it begins an exploration of the tradeoffs between vectorized functions and

lower-level programming constructs such as for loops. The edge detection algorithm is challenging

(and probably not practical) to design using only vectorized functions. As you can see, the content

of the labs lags the lecture by about one week so that students can use the lab to reinforce material

that they have already had some exposure to.

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