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Signals and Systems: A Fresh Look
Chi-Tsong Chen
Stony Brook University
i
Copyright c 2009 by Chi-Tsong Chen
Everyone is permitted to copy and distribute this book but changing it is not permitted.
Email: ctchen@ece.sunysb.edu or profctchen@gmail.com
Also by the same author:
ã Linear System Theory and Design, Oxford Univ. Press, [1st ed., (431 pp), 1970; 2nd
ed., (662 pp), 1980; 3rd ed., (332 pp), 1999].
ã Analog and Digital C

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Signals and Systems: A Fresh Look
Chi-Tsong ChenStony Brook University
i
Copyright c
2009 by Chi-Tsong ChenEveryone is permitted to copy and distribute this book but changing it is not permitted.Email: ctchen@ece.sunysb.edu or profctchen@gmail.comAlso by the same author:
ã
Linear System Theory and Design
, Oxford Univ. Press, [1st ed., (431 pp), 1970; 2nded., (662 pp), 1980; 3rd ed., (332 pp), 1999].
ã
Analog and Digital Control System Design: Transfer-Function, State-Space, and Alge-braic Methods
, Oxford Univ. Press, 1993.
ã
Digital Signal Processing: Spectral Computation and Filter Design
, Oxford Univ. Press,2001.
ã
Signals and Systems
, 3rd ed., Oxford Univ. Press, 2004.The following are widely recognized:
ã
There is a gap between what is taught at universities and what is used in industry.
ã
It is more important to teach how to learn than what to learn.These were the guiding principles in developing this book. It gives an overview of the subjectarea of signals and systems, discussing the role of signals in designing systems and variousmathematical descriptions for the small class of systems studied. It then focuses on topicswhich are most relevant and useful in practice. It also gives reasons for not stressing manyconventional topics. Its presentation strives to cultivate readers’ ability to think critically andto develop ideas logically.After teaching and writing for over forty years, I am on the verge of retirement and have de-cided to give this book away as a gift to the students in my class and any other interested read-ers. It, together with its solutions manual, can be downloaded from
www.ece.sunysb.edu/˜ctchen/
.If you ﬁnd the book useful, please spread the word. I also appreciate receiving any feedbackyou have.ii
Preface
Presently, there are over thirty texts on continuous-time (CT) and discrete-time (DT)signals and systems.
1
They typically cover CT and DT Fourier series; CT and DT Fouriertransforms; discrete Fourier transform (DFT); two- and one-sided Laplace and z-transforms;convolutions and diﬀerential (diﬀerence) equations; Fourier and Laplace analysis of signalsand systems; and some applications in communication, control, and ﬁlter design. About one-third of these texts also discuss state-space equations and their analytical solutions. Manytexts emphasize convolutions and Fourier analysis.Feedback from graduates that what they learned in university is not used in industryprompted me to ponder what to teach in signals and systems. Typical courses on signals andsystems are intended for sophomores and juniors, and aim to provide the necessary backgroundin signals and systems for follow-up courses in control, communication, microelectronic cir-cuits, ﬁlter design, and digital signal procession. A survey of texts reveals that the importanttopics needed in these follow-up courses are as follows:
ã
Signals
: Frequency spectra, which are used for discussing bandwidth, selecting samplingand carrier frequencies, and specifying systems, especially, ﬁlters to be designed.
ã
Systems
: Rational transfer functions, which are used in circuit analysis and in design of ﬁlters and control systems. In ﬁlter design, we ﬁnd transfer functions whose frequencyresponses meet speciﬁcations based on the frequency spectra of signals to be processed.
ã
Implementation
: State-space (ss) equations, which are most convenient for computercomputation, real-time processing, and op-amp circuit or specialized hardware imple-mentation.These topics will be the focus of this text. For signals, we develop frequency spectra
2
andtheir bandwidths and computer computation. We use the simplest real-world signals (soundsgenerated by a tuning fork and the middle C key of a piano) as examples. For systems, wedevelop four mathematical descriptions: convolutions, diﬀerential (diﬀerence) equations, ssequations, and transfer functions. The ﬁrst three are in the time domain and the last one isin the transform domain. We give reasons for downplaying the ﬁrst two and emphasizing thelast two descriptions. We discuss the role of signals in designing systems and the followingthree domains:
ã
Transform domain
, where most design methods are developed.
ã
Time domain
, where all processing is carried out.
ã
Frequency domain
, where design speciﬁcations are given.We also discuss the relationship between ss equations (an internal description) and transferfunctions (an external description).Because of our familiarity of CT physical phenomena and examples, this text studiesﬁrst the CT case and then the DT case with one exception. The exception is to use DTsystems with ﬁnite memory to introduce some system concepts because simple numericalexamples can be easily developed whereas there is no CT counterpart. This text stressesbasic concepts and ideas and downplays analytical calculation because all computation inthis text can be carried out numerically using MATLAB. We start from scratch and takenothing for granted. For example, we discuss time and its representation by a real numberline. We give the reason of deﬁning frequency using a spinning wheel rather than usingsin
ωt
or cos
ωt
. We also give the reason that we cannot deﬁne the frequency of DT sinusoidsdirectly and must deﬁne it using CT sinusoids. We make distinction between amplitudes andmagnitudes. Even though mathematics is essential in engineering, what is more important,
1
See the references at the end of this book.
2
The frequency spectrum is not deﬁned or not stressed in most texts.
iii
in our view, is its methodology (critical thinking and logical development) than its varioustopics and calculational methods. Thus we skip many conventional topics and discuss, at amore thorough level, only those needed in this course. It is hoped that by so doing, the readermay gain the ability and habit of critical thinking and logical development.In the table of contents, we box those sections and subsections which are unique in thistext. They discuss some basic issues and questions in signals and systems which are notdiscussed in other texts. We discuss some of them below:1. Even though all texts on signals and systems claim to study linear time-invariant (LTI)systems, they actually study only a very small subset of such systems which have the“lumpedness” property. What is true for LTI lumped systems may not be true forgeneral TLI systems. Thus, it is important to know the limited applicability of whatwe study.2. Even though most texts start with diﬀerential equations and convolutions, this textuses a simple RLC circuit to demonstrate that the state-space (ss) description is eas-ier to develop than the aforementioned descriptions. Moreover, once an ss equation isdeveloped, we can discuss directly (without discussing its analytical solution) its com-puter computation, real-time processing, and op-amp circuit implementation. Thus ssequations should be an important part of a text on signals and systems.3. We introduce the concept of coprimeness (no common roots) for rational transfer func-tions. Without it, the poles and zeros deﬁned in many texts are not necessarily correct.The concept is also needed in discussing whether or not a system has redundant com-ponents.4. We discuss the relationship between the Fourier series and Fourier transform which isdual to the sampling theorem. We give reasons for stressing only the Fourier transformin signal analysis and for skipping Fourier analysis of systems.5. This text discusses model reduction which is widely used in practice and yet not dis-cussed in other texts. The discussion shows the roles of a system’s frequency responseand a signal’s frequency spectrum. It explains why the same transfer functions can beused to design seismometers and accelerometers.A great deal of thought was put into the selection of the topics discussed in this text.
3
Itis hoped that the rationale presented is convincing and compelling and that this new text willbecome a standard in teaching signals and systems, just as my book
Linear system theory and design
has been a standard in teaching linear systems since 1970.In addition to electrical and computer engineering programs, this text is suitable formechanical, bioengineering, and any program which involves analysis and design of systems.This text contains more material than can be covered in one semester. When teaching aone-semester course on signals and systems at Stony Brook, I skip Chapter 5, Chapter 7 afterSection 7.4, and Chapter 11, and cover the rest. The topics in Chapter 3 are discussed wherethey are needed. Clearly other arrangements are also possible.Many people helped me in writing this book. Ms. Jiseon Kim plotted all the ﬁgures inthe text except those generated by MATLAB. Mr. Anthony Oliver performed many op-ampcircuit experiments for me. Dr. Michael Gilberti scrutinized the entire book, picked up manyerrors, and made several valuable suggestions. I consulted Professors Amen Zemanian andJohn Murray whenever I had any questions and doubts. I thank them all.C. T. ChenDecember, 2009
3
This text is diﬀerent from Reference [C8] in structure and emphasis. It compares four mathematicaldescriptions, and discusses three domains and the role of signals in system design. Thus it is not a minorrevision of [C8]; it is a new text.
iv

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