信号与系统：陈启宗的全新视角

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"陈启宗的《信号与系统：全新视角》是他在纽约州石溪分校退休后免费提供的一本关于信号处理的书籍。" 陈启宗的《信号与系统：全新视角》是一本深入探讨信号处理及其在系统设计中应用的专业书籍。作者，陈启宗，是 IEEE 会士，并在石溪大学有着丰富的教学和研究经验。这本书旨在填补大学教育与工业界实际需求之间的差距，更注重教会读者如何学习而非单纯传授知识。书中的内容涵盖了信号与系统的基础知识，包括信号在系统设计中的作用以及用于描述这些系统的一系列数学方法。书中讨论的系统类型相对有限，但重点放在了对工业界广泛应用的主题上。此外，书中还提到了作者的其他著作，如《线性系统理论与设计》、《模拟和数字控制系统设计》、《数字信号处理：谱计算与滤波器设计》以及《信号与系统》的第三版，这些书籍进一步扩展了他在控制理论和数字信号处理领域的专业知识。信号处理是电子工程和计算机科学的重要分支，它涉及到对信号的获取、分析、变换和控制。在《信号与系统：全新视角》中，读者可以学习到如何使用不同数学工具，如傅里叶变换、拉普拉斯变换和状态空间模型，来理解和建模各种系统。这些工具对于理解通信、图像处理、音频编码、生物医学信号分析等领域的核心技术至关重要。系统控制方面，作者介绍了如何使用传递函数和状态空间方法来设计和分析控制系统，这些都是自动控制理论的基础。通过学习这些概念，工程师可以设计出能够稳定和优化系统性能的控制器。这本书的独特之处在于它的实用性和教育理念，强调理论与实践的结合，以及培养读者自我学习的能力。它不仅适合大学学生作为教材，也适用于工业界的工程师作为参考书，帮助他们在工作中更好地理解和应用信号处理技术。《信号与系统：全新视角》是一本全面而深入的教材，对于想要深入了解信号处理和系统控制的读者来说，它提供了宝贵的资源和指导。通过阅读这本书，读者将能够掌握信号分析的关键原理，以及如何将这些原理应用于实际的系统设计中。

6 CHAPTER 1. INTRODUCTION

variables, one for each spatial dimension. Thus a signal may have one or more independent

variables. The more independent variables in a signal, the more complicated the signal. In

this text, we study signals that have only one independent variable. We also assume the

independent variable to be time.

To transform an acoustic wave into an electrical signal requires a transducer. To obtain

the signal in Figure 1.4 requires a voltmeter, an ampliﬁer and a recorder. To obtain the

temperature in Figure 1.5(a) requires a temperature sensor. All transducers and sensors

are systems. There are other types of systems such as ampliﬁers, ﬁlters, and motors. The

computer program which generates the 90-day moving average in Figure 1.5(d) is also a

system.

1.2 Physics, mathematics, and engineering

Even though engineering is based on mathematics and physics, it is very diﬀerent from them.

Physics is concerned with discovering physical laws that govern natural phenomena. The

validity of a law is determined by whether it is consistent with what is observed and whether

it can predict what will happen in an experiment or in a natural phenomenon. The more

accurately it describes the physical world and the more widely it is applicable, the better

a physical law is. Newton’s laws of motion and universal gravitation (1687) can be used

to explain falling apples and planetary orbits. However they assume space and time to be

uniform and absolute, and cannot be used to explain phenomena that involve speeds close to

the speed of light. In these situations, Newton’s laws must be superseded by Einstein’s special

theory of relativity (1905). This theory hypothesizes that time is not absolute and depends

on relative speeds of observers. The theory also leads to the famous equation E = mc

, which

relates energy (E), mass (m), and the speed of light (c). Einstein also developed the general

theory of relativity (1915) which predicted the bending of light by the sun by an amount twice

of what was computed from Newton’s law. This prediction was conﬁrmed in 1919 during a

solar eclipse and established once and for all the paramount importance of Einstein’s theory

of relativity.

Although the eﬀects of magnets and static electric charges were recognized in ancient time,

it took more than eighteen centuries before human could generate and manipulate them.

Alessandro Volta discovered that electric charges could be generated and stored using two

dissimilar metals immersed in a salt solution. This led to the development of the battery in

the 1800s and a way of generating a continuous ﬂow of charges, called current, in a conducting

wire. Soon afterwards, it was discovered that current will induce a magnetic ﬁeld around a

wire. This made possible man-made magnetic ﬁelds. Michael Faraday demonstrated in 1821

that passing a current through a wire which is placed in a magnetic ﬁeld will cause the wire to

move, which led to the development of motors. Conversely, a moving wire in a magnetic ﬁeld

will induce a current in the wire, which led to the development of generators. These eﬀects

were uniﬁed by the physicist James Maxwell (1831-1879) using a set of four equations, called

Maxwell’s equations.

These equations showed that the electric and magnetic ﬁelds, called

electromagnetic (EM) waves, could travel through space. The traveling speed was computed

from Maxwell’s equations as 3 × 10

meters per second, same as the speed of light. Thus

Maxwell argued that light is an EM wave. This was generally accepted only after EM waves

were experimentally generated and detected by Heinrich Hertz (1857-1894) and were shown

to have all of the properties of light such as reﬂection, refraction, and interference. All EM

waves propagate at the speed of light but can have diﬀerent frequencies or wavelengths. They

are classiﬁed according to their frequency ranges as radio waves, microwaves, infrared, visible

light, ultraviolet, x-rays, and so forth. Light as an EM wave certainly has the properties of

The original Maxwell’s formulation consisted of twenty equations which were to o complicated to be useful

for engineers. The twenty equations were condensed to the now celebrated four equations in 1883 by Oliver

Heaviside (1850-1925). Heaviside was a self-taught electrical engineer and was keenly interested in applying

mathematics into practical use. He also developed resistive operators to study electrical circuits. See Section

9.10.1 and References [C2, N1].

1.2. PHYSICS, MATHEMATICS, AND ENGINEERING 7

wave. It turns out that light also has the properties of particles and can be looked upon as a

stream of photons.

Physicists were also interested in the basic structure of matter. By the early 1900s, it

was recognized that all matter is built from atoms. Every atom has a nucleus consisting

of neutrons and positively charged protons and a number of negatively charged electrons

circling around the nucleus. Although electrons in the form of static electrical charges were

recognized in ancient times, their properties (the amount of charge and mass of each electron)

were exp erimentally measured only in 1897. Furthermore, it was experimentally veriﬁed

that the electron, just as light, has the dual properties of wave and particle. These atomic

phenomena were outside the reach of Newton’s laws and Einstein’s theory of relativity but

could be explained using quantum mechanics developed in 1926. By now it is known that

all matters are built from two types of particles: quarks and leptons. They interact with

each other through gravity, electromagnetic interactions, and strong and weak nuclear forces.

In summary, physics tries to develop physical laws to describe natural phenomena and to

uncover the basic structure of matter.

Mathematics is indispensable in physics. Even though physical laws were inspired by

concepts and measurements, mathematics is needed to provide the necessary tools to make

the concepts precise and to derive consequences and implications. For example, Einstein had

some ideas about the general theory of relativity in 1907, but it took him eight years to ﬁnd

the necessary mathematics to make it complete. Without Maxwell’s equations, EM theory

could not have been developed. Mathematics, however, has developed into its own subject

area. It started with the counting of, perhaps, cows and the measurement of farm lots. It

then developed into a discipline which has little to do with the real world. It now starts with

some basic entities, such as p oints and lines, which are abstract concepts (no physical point

has zero area and no physical line has zero width). One then selects a number of assumptions

or axioms, and then develops logical results. Note that diﬀerent axioms may lead to diﬀerent

mathematical branches such as Euclidean geometry, non-Euclidean geometry, and Riemann

geometry. Once a result is proved correct, the result will stand forever and can withstand any

challenge. For example, the Pythagorean theorem (the square of the hypotenuse of a right

triangle equals the sum of the squares of both sides) was ﬁrst proved about 5 B.C. and is

still valid today. In 1637, Pierre de Fermat claimed that no positive integer solutions exist

in a

+ b

= c

, for any integer n larger than 2. Even though many special cases such as

n = 3, 4, 6, 8, 9, . . . had been established, nobody was able to prove it for all integers n > 2

for over three hundred years. Thus the claim remained a conjecture. It was ﬁnally proven

as a theorem in 1994 (see Reference [S3]). Thus, the bottom line in mathematics is absolute

correctness, whereas the bottom line in physics is truthfulness to the physical world.

Engineering is a very broad discipline. Some involve in designing and constructing skyscrap-

ers and bridges; some in developing useful materials; some in designing and building electron-

ics devices and physical systems; and many others. Sending the two exploration rovers to

Mars (launched in Mid 2003 and arrived in early 2004) was also an engineering task. In

this ambitious national project, budgetary concerns were secondary. For most engineering

products, such as motors, CD players and cell phones, cost is critical. To be commercially

successful, such products must be reliable, small in size, high in performance and competitive

in price. Furthermore they may require a great deal of marketing. The initial product design

and development may be based on physics and mathematics. Once a working model is devel-

oped, the model must go through repetitive modiﬁcation, improvement, and testing. Physics

and mathematics usually play only marginal roles in this cycle. Engineering ingenuity and

creativity play more prominent roles.

Engineering often involves tradeoﬀs or compromises between performance and cost, and

between conﬂicting speciﬁcations. Thus, there are often similar products with a wide range

in price. The concept of tradeoﬀs is less eminent in mathematics and physics; whereas it is

an unavoidable part of engineering.

For low-velocity phenomena, Newton’s laws are valid and can never be improved. Maxwell’s

equations describe electromagnetic phenomena and waves and have been used for over one

8 CHAPTER 1. INTRODUCTION

hundred years. Once all elementary particles are found and a Theory of Everything is devel-

oped, some people anticipate the death of (pure) physics and science (see Reference [L3]). An

engineering product, however, can always be improved. For example, after Faraday demon-

strated in 1821 that electrical energy could be converted into mechanical energy and vice

versa, the race to develop electromagnetic machines (generators and motors) began. Now,

there are various types and power ranges of motors. They are used to drive trains, to move

the rovers on Mars and to point their unidirectional antennas toward the earth. Vast numbers

of toys require motors. Motors are needed in every CD player to spin the disc and to position

the reading head. Currently, miniaturized motors on the order of millimeters or even microm-

eters in size are being developed. Another example is the ﬁeld of integrated circuits. Discrete

transistors were invented in 1947. It was discovered in 1959 that a number of transistors could

be fabricated as a single chip. A chip may contain hundreds of transistors in the 1960s, tens

of thousands in the 1970s, and hundreds of thousands in the 1980s. It may contain several

billion transistors as of 2007. Indeed, technology is open-ended and ﬂourishing.

1.3 Electrical and computer engineering

The ﬁeld of electrical engineering programs ﬁrst emerged in the US in the 1880s. It was

mainly concerned with the subject areas of communication (telegraph and telephone) and

power engineering. The development of practical electric lights in 1880 by Thomas Edison

required the generation and distribution of power. Alternating current (ac) and direct current

(dc) generators and motors were developed and underwent steady improvement. Vacuum

tubes ﬁrst appeared in the early 1900s. Because they could be used to amplify signals, long

distance telephony became possible. Vacuum tubes could also be used to generate sinusoidal

signals and to carry out modulation and demodulation. This led to radio broadcasting. The

design of associated devices such as transmitters and receivers spurred the creation of the

subject areas of circuit analysis and electronics. Because of the large number of electrical

engineers needed for the construction of infrastructure (power plants, transmission lines, and

telephone lines) and for the design, testing, and maintenance of devices, engineering colleges

taught mostly the aforementioned subject areas and the industrial practice of the time. This

mode of teaching remained unchanged until after World War II (1945).

During World War II, many physicists and mathematicians were called upon to participate

in the development of guided missiles, radars for detection and tracking of incoming airplanes,

computers for computing ballistic trajectories, and many other war-related projects. These

activities led to the new subject areas computer, control and systems, microwave technology,

telecommunications, and pulse technology. After the war, additional subject areas appeared

because of the advent of transistors, lasers, integrated circuits and microprocessors. Sub-

sequently many electrical engineering (EE) programs changed their name to electrical and

computer engineering (ECE). To show the diversity of ECE, we list some publications of

IEEE (Institute of Electrical and Electronics Engineering). For each letter, we list the num-

ber of journals starting with the letter and some journal titles

• A: (11) Antenna and Propagation, Automatic Control, Automation.

• B: (2) Biomedical Engineering, Broadcasting.

• C: (31) Circuits and Systems, Communication, Computers, Consumer Electronics.

• D: (6) Device and Material Reliability, Display Technology.

• E: (28) Electronic Devices, Energy Conversion, Engineering Management,

• F: (1) Fuzzy Systems.

• G: (3) Geoscience and Remote Sensing.

1.4. A COURSE ON SIGNALS AND SYSTEMS 9

• I: (17) Image Processing, Instrumentation and Measurement, Intelligent Systems, In-

ternet Computing.

• K: (1) Knowledge and Data Engineering.

• L: (3) Lightwave Technology.

• M: (14) Medical Imaging, Microwave, Mobile Computing, Microelectromechanical Sys-

tems, Multimedia.

• N: (8) Nanotechnology, Neural Networks.

• O: (2) Optoelectronics.

• P: (16) Parallel and Distributed Systems, Photonics Technology, Power Electronics,

Power Systems.

• Q: (1) Quantum Electronics.

• R: (4) Reliability, Robotics.

• S: (19) Sensors, Signal Processing, Software Engineering, Speech and Audio Processing.

• T: (1) Technology and Society.

• U: (1) Ultrasonics, Ferroelectrics and Frequency Control.

• V: (5) Vehicular Technology, Vary Large Scale Integration (VLSI) Systems, Visualiza-

tion and Computer Graphics.

• W: (1) Wireless Communications.

IEEE alone publishes 175 journals on various subjects. Indeed, the subject areas covered in

ECE programs are many and diversiﬁed.

Prior to World War II, there were some master’s degree programs in electrical engineering,

but the doctorate programs were very limited. Most faculty members did not hold a Ph.D.

degree and their research and publications were minimal. During the war, electrical engineers

discovered that they lacked the mathematical and research training needed to explore new

ﬁelds. This motivated the overhaul of electrical engineering education after the war. Now,

most engineering colleges have Ph.D. programs and every faculty member is required to have

a Ph.D.. Moreover, a faculty member must, in addition to teaching, carry out research and

publish. Otherwise he or she will be denied tenure and will be asked to leave. This leads to

the syndrome of “publish or perish”.

Prior to World War II, almost every faculty member had some practical experience and

taught mostly practical design. Since World War II, the majority of faculty members have

been fresh doctorates with limited practical experience. Thus, they tend to teach and stress

theory. In recent years, there has been an outcry that the gap between what universities

teach and what industries practice is widening. How to narrow the gap between theory and

practice is a challenge in ECE education.

1.4 A course on signals and systems

The courses oﬀered in EE and ECE programs evolve constantly. For example, (passive RLC)

Network Syntheses, which was a popular course in the 1950s, no longer appears in present day

ECE curricula. Courses on automatic control and sampled or discrete-time control systems

ﬁrst app eared in the 1950s; linear systems courses in the 1960s and digital signal processing

courses in the 1970s. Because the mathematics used in the preceding courses is basically the

same, it was natural and more eﬃcient to develop a single course that provides a common

10 CHAPTER 1. INTRODUCTION

background for follow-up courses in control, communication, ﬁlter design, electronics, and

digital signal processing. This contributed to the advent of the course signals and systems

in the 1980s. Currently, such a course is oﬀered in every ECE program. The ﬁrst book on

the subject area was Signals and Systems, by A. V. Oppenheim and A. S. Willsky, published

by Prentice-Hall in 1983 (see Reference [O1]). Since then, over thirty b ooks on the subject

area have been published in the U.S.. See the references at the end of this text. Most

bo oks follow the same outline as the aforementioned book: they introduce the Fourier series,

Fourier transform, two- and one-sided Laplace and z-transforms, and their applications to

signal analysis, system analysis, communication and feedback systems. Some of them also

introduce state-space equations. Those books are developed mainly for those interested in

communication, control, and digital signal processing.

This new text takes a fresh look on the subject area. Most signals are naturally generated;

whereas systems are to be designed and built to process signals. Thus we discuss the role

of signals in designing systems. The small class of systems studied can be described by four

types of equations. Although they are mathematically equivalent, we show that only one

type is suitable for design and only one type is suitable for implementation and real-time

processing. We use operational ampliﬁers and simple RLC circuits as examples because they

are the simplest possible physical systems available. This text also discusses model reduction,

thus it is also useful to those interested in microelectronics and sensor design.

1.5 Confession of the author

This section describes the author’s personal evolution regarding teaching. I received my Ph.D.

in 1966 and immediately joined the EE department at the State University of New York at

Stony Brook. I have been teaching there ever since. My practical experience has been limited

to a couple of summer jobs in industry.

As a consequence of my Ph.D. training, I became fascinated by mathematics for its abso-

lute correctness and rigorous development — no ambiguity and no approximation. Thus in

the ﬁrst half of my teaching career, I basically taught mathematics. My research was in linear

systems and focused on developing design methods using transfer functions and state-space

equations. Because my design methods of pole placement and model matching are simpler

both in concept and in computation, and more general than the existing design methods,

I thought that these methods would be introduced in texts in control and be adopted in

industry. When this expectation was not met, I was puzzled.

The steam engine developed in the late 18th century required three control systems: one

to regulate the water level in the boiler, one to regulate the pressure and one to control

the speed of the shaft using a centrifugal ﬂyball. During the industrial revolution, textile

manufacturing became machine based and required various control schemes. In mid 19th

century, position control was used to steer big ships. The use of thermostats to achieved

automatic temperature control began in the late 19th century. In developing the airplane in

the early 20th century, the Wright brothers, after the mathematical calculation failed them,

had to build a wind tunnel to test their design and ﬁnally led to a successful ﬂight. Airplanes

which required many control systems played an important role in World War I. In conclusion,

control systems, albeit designed using empirical or trial-and-error methods, had been widely

and successfully employed even before the advent of control and systems courses in engineering

curricula in the 1930s.

The analysis and design methods discussed in most control texts, however, were all de-

veloped after 1932

. See Subsection 9.10.1. They are applicable only to systems that are

describable by simple mathematical equations. Most, if not all, practical control systems,

however, cannot be so described. Furthermore, most practical systems have some physical

constraints and must be reliable, small in size, and competitive in price. These issues are

not discussed in most control and systems texts. Thus textbook design methods may not be

The only exception is the Routh test which was developed in 1877.

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