Preface
The concepts of signals and systems are the corner-stone of a wide variety of areas, ranging from home-oriented consumer electronics and multimedia entertainment products to sophisticated communications, aeronautics and astronautics, and control. The ideas and approaches associated with these concepts have great e.ects on our life in one way or another. Although the signals and systems which arise across those .elds are naturally di.erent in their physical make-up and application, the principles and tools for analyzing signals and systems are the same and hence applicable to all of them. Therefore, an introductory course on signals and systems is fundamental and compulsory for an engineering under-graduate curriculum in any well-established tertiary (education) institutions. Such a course is commonly designed in one of the two forms below:
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a one-semester subject that intends to provide students with a rich set of concepts and tools for analyzing deterministic signals and an important class of systems known as linear time-invariant systems;
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a two-semester subject that expands on the one-semester course by adding more detailed treatment of signal processing and systems for speci.ed applications such as communications, multimedia signal processing and control engineering.
This book takes the .rst form and assumes that the students have a background in calculus and introductory physics.
Why another ¡°Signals and Systems¡±?
Given that there are many well-written textbooks on signals and systems available, our readers would be screaming when seeing this textbook: Why another ¡°Signals and Systems¡±?
The thought of writing such a textbook was stimulated in 2007 when the .rst author conducted a 2nd year course on signals and systems in Zhejiang University of Technology (ZJUT) in Hangzhou, which he just joined from Nanyang Technological University (NTU), Republic of Singapore. The course was designed as ¡°shuang yu ke¡± spelled in Chinese, meaning that the teaching materials such as the textbooks and slides are all in English and as to the class language, the lecturers can choose either Chinese, English or a mixture of the two¡ªa typical Chinese style! The textbooks adopted were those popularly used in the world but were found very di.cult to most of our students as their English is not good enough to deal with those textbooks of one thousand pages or so in length.1 This motivates him to write a new ¡°Signals and Systems¡±
1Here, we have those big masters of Signals and Systems to ¡°blame¡±¡ªwhy not writing their text-book in Chinese!
with a primary objective of providing a condensed version of ¡°Signals and Systems¡± in English, while keeping the important technical materials as much as possible.
Most of us agree that for a university study it is more important to teach students how to learn and analyze than what to be learned and analyzed. The .rst author still remembers what was said by Professor Y. Liu who taught him calculus in 1978 in Beijing Institute of Technology (BIT) (now, known as Bei Li Gong) that a good textbook, say of ten chapters, should be written in such a way that after the .rst four chapters taught by the lecturer, the rest can be studied by students themselves easily. What Prof. Liu really meant is that a textbook should be written to facilitate and reinforce self-study. This is another objective that this textbook is intended to achieve. Given the mathematical nature of this subject, rigidity should be sustained as much as possible, which is something very important to engineering students to learn. This is the third objective of this textbook.
How the book is structured ?
One of the reasons for the existing textbooks of signals and systems to have easily over eight hundred pages is due to the continuous-and discrete-time forms of signals and systems. The success of a signals and systems that achieves our primary object lies in how to provide a balanced and integrated treatment of the two forms in a pedagogical way that can help students see the fundamental similarities and di.erences between the two without too much repeating. With all of these in mind, this book is organized as follow.
. Chapter 1 is aimed to provide an overview of signals and systems. Compared with the one in most of the signals and systems, it is condensed with an emphasis on the periodical signals and linear time-invariant (LTI) systems.
. Chapter 2 deals with the time-domain approach to the LTI systems. Since the key to the development of this chapter is to exploit the properties of linearity and time-invariance as well as signal decompositions, the concepts of unit impulse re-sponse and convolution are developed in details for the discrete-time case, while the continuous-time counterparts are directly given. One of the remarkable points emphasized in this chapter is the equivalence between the LTI systems and con-volutions, which relates the physical interpretation and mathematical expression. Another important technical point in this chapter is the establishment of a con-clusion which states that any complete solution of a linear constant coe.cient di.erential/di.erence equation (LCCDE) can be characterized by the sum of the output of an LTI system excited by the force signal and a homogeneous solution of the LCCDE. This conclusion yields a very clear picture of the relationship be-tween an LCCDE and the systems that it can characterize. With the help of the
transforms to be discussed in Chapter 6, it also provides an easy way to .nd a particular solution and hence the set of complete solutions of the LCCDE.
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Chapter 3 is the biggest chapter, dealing with four signal representations, namely, Fourier series (FS), discrete-time FS (DTFS), Fourier transform (FT), and discrete-time FT (DTFT). The most important concept in this chapter is signal decom-position. Along this line, the four signal transforms can be uni.ed as a linear combination of basis signals. This uni.cation allows us to condense the text sig-ni.cantly as one can just focus on the development and the properties of FT and DTFT. Great e.ort has been made to explain why a signal should be transformed, which is one of the di.cult points for most of the 2nd year students to understand given that they usually do not have a relevant background.
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Equipped with Chapters 1¨C3, a top student should be able to study on ones own the rest of this book, including Chapter 4 that studies the class of LTI systems using Fourier analysis techniques developed in Chapter 3. The key points of the chapter are the frequency response and the equivalence between convolution in time-domain and multiplication in frequency-domain and the development of the concepts in both continuous-and discrete-time domains are rotated to avoid un-necessary repetition.
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Chapter 5 mainly focuses on discrete processing of continuous-time signals. The development .ows mainly from the techniques derived in Chapters 2¨C4. Starting from sampling a signal that is continuous in time-domain, the relationship between the spectrum of the continuous-time signal and that of the discrete-time signal obtained is established, based on which the famous sampling theorem and hence the ideal reconstruction can be derived. Unlike most of the existing textbooks, this relation is derived directly using a simple mathematical procedure rather than the one that is obtained with the help of impulse train. Note that the DTFT of a discrete-time signal is a continuous function in frequency. Sampling (in frequency) of such a function leads to a new transform-discrete Fourier transform (DFT) that is popularly used in many applications of digital signal processing, including linear .ltering, correlation analysis, and spectral analysis. More profound discussions on the DFT, however, are beyond the scope of this book as the relevant topics are parts of the core contents for the subjects on digital signal processing.
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Chapter 6 deals with transform-domain approaches to signals and systems. The Laplace transform and z-transform are studied in a parallel way, while the appli-cations of the two transforms to LTI systems and .nding the complete solution of an LCCDE are given in a uni.ed manner, which once again allows us to condense the textbook. As two of the new features of this book, the problems of .nding inverse systems of a given LTI system & de-convolutions and decomposing system
responses are carefully treated. The former is rarely found in most of the existing textbooks, while the latter is intended organized to help students have an easy understanding of the unilateral transforms.
. The advantage of the uni.ed/integrated treatment of the continuous-and discrete-time forms is particularly demonstrated in Chapter 7 in study of block-diagram representations and structures of the LTI systems in transform-domain. As a special class of system structures, the state-space realizations of LTI systems are introduced.
. Following the traditional pedagogy, the tutorial questions given in each chapter are mainly designed for helping students to enhance the understanding of the concepts in an individual chapter only. This, however, would make most of the students problem-result-oriented and hence loses the real purpose of learning. To overcome this, Chapter 8 is devoted to providing a set of comprehensive exercises, each of which usually involves mathematical deviations and a more sophisticated applica-tion of the concepts and approaches provided in the entire text book rather than an individual chapter.
. Appendices are used to make the book concise and self-contained, and also sustain rigidity of the development.
Acknowledgements
It is our great pleasure to thank a number of people who helped us in writing this book with insightful suggestions and constructive feedback and those who did nothing directly to this book but we just feel good in doing so.
Gang Li is grateful to his former NTU colleagues, particularly, G.A. Bi, C.R. Wan,
Z.P. Lin, Y.C. Lim, L.H. Xie, and W.J. Cai for insightful suggestions, constructive dis-cussions, and more importantly, the friendship. M.Y. Fu of the University of Newcastle (Australia), S. Chen of the University of Southampton (UK), J. Wu of Zhejiang Univer-sity, and Y.Q. Shi of the New Jersey Institute of Technology (USA) have shown their continuous support to Gang Li¡¯s work in ZJUT, which is very much appreciated. He would like to take this opportunity to thank the most important person to his academic career¡ªMichel Gevers of the Universit¡äe Catholique de Louvain (UCL) (Belgium), who was his Ph.D supervisor and taught him so much, particularly on how to write. This book comes true partially owing to the adventure they had together in writing their co-authored book ¡°Parametrizations in Control, Estimation and Filtering Problems: Accuracy Aspects¡± published by Springer-Verlag (London limited, 1993). Un Grand Mercy, Mig !
Liping Chang is grateful to her Ph.D supervisor, academician Z.Q. Lin, SIOM of Chinese Academy of Science, Shanghai, who taught her how to explore and solve scienti.c problems during her graduate period.
Sheng Li wants to say ¡®thank you for bring me to this field¡¯ to his Ph.D supervisor, Rodrigo C. de Lamare (Reader), University of York, U.K.. In addition, he is grateful to his ¡®boss¡¯ during his post-doctorial work, Martin Haardt (Professor), TU-Ilmenau, Germany. Both of these nice guys taught him many things and helped him to improve himself.
All of us would like to express our gratitude to many colleagues in the College of Information Engineering of ZJUT. Particularly, X.X. He, Y.L. Qin and S.Q. Guo are acknowledged for their full supports in various ways. It is really our pleasure to work with them and to have achieved something meaningful together. The contribu-tion of other members in the teaching team for the subject ¡°Signals and Systems¡± is acknowledged. It was due to a great collective e.ort such that this subject was horned as a National Model Course in English by the Chinese Ministry of Education and a Provincial Course of Excellency by the Education Board of Zhejiang province. Par-ticularly, Tao Wu¡¯s contribution and the support from the Academic Affairs Of.ce of ZJUT as well as the two projects are very much appreciated. It is also our pleasure to acknowledge the authors of many existing textbooks on signals and systems pub-lished in English and Chinese since their works have influenced our book in one way or another.
We thank many students who made contributions to this book with suggestions and questions that helped us over years to re.ne and rethink the organization and presentation of the materials in the book. Particularly, we would like to thank Chaogeng Huang, who has just obtained his Ph.D under the guidance of the 1st author, and Huang Bai for the excellent job done in terms of research and administration, and two very talent undergraduate students (Year 2007) Yue Wang and Dehui Yang who helped us with simulations, .gures and tutorial problems. Special thanks go to our teaching assistants Zhihui Zhu, Tao Hong, Qiuwei Li, Shuang Li, and Liang Zhu for their help in preparing tutorial solutions and some of the computer experiments. We are grateful to our students for their active participation and intensive interaction with us, which makes this book even more student-oriented.
We would like to take this opportunity to thank Professor Xianyi Gong, a Fellow of the Chinese Academy of Engineering, of Zhejiang university for sharing with us his insightful views on signal processing (SP) as well as many other issues. It is our pleasure to say that his understanding and conception of SP as well as the enthusiasm to education have great influence on us!
Lastly, we would like to thank the support from our families, which is the driving force for us to write this book. It may be felt unusual by those not close to Gang Li that the .rst one among his family members to mention is his brother-in-law Aiguo Li who has done everything possible for him and his family. Liping Chang wishes to thank her family members, particularly her husband Jia Li and their four-month old son for their deep love. Sheng Li has the pleasure to say ¡°I love you much more than myself ¡± to his darling wife Xiaolei Zhang and his parents.
Gang Li (ieligang@zjut.edu.cn) Liping Chang (clping@zjut.edu.cn) Sheng Li (shengli@zjut.edu.cn) July 01, 2014 at Xiao He Shan, Hangzhou
