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MIT 6 033 - Discrete-time Signals and Systems

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1 11 12009-09-29 13:11:30 UTC / rev b19331f50bbdDiscrete-time Signals and Systems2 22 2ii2009-09-29 13:11:30 UTC / rev b19331f50bbd3 33 32009-09-29 13:11:30 UTC / rev b19331f50bbdDiscrete-time Signals and SystemsAn Operator ApproachSanjoy Mahajan and Dennis FreemanMassachusetts Institute of Technology4 44 42009-09-29 13:11:30 UTC / rev b19331f50bbdTypeset in Palatino and Euler by the authors using ConTEXt and PDFTEXC Copyright 2009 Sanjoy Mahajan and Dennis FreemanSource revision: b19331f50bbd (2009-09-29 13:11:30 UTC)Discrete-time Signals and Systems by Sanjoy Mahajan and Dennis Freeman(authors) and ?? (publisher) is licensed under the . . . license.5 55 52009-09-29 13:11:30 UTC / rev b19331f50bbdBrief contentsPreface ix1 Difference equations 12 Difference equations and modularity 173 Block diagrams and operators: Two new representations 334 Modes 515 Repeated roots 63Bibliography 71Index 736 66 6vi2009-09-29 13:11:30 UTC / rev b19331f50bbd7 77 72009-09-29 13:11:30 UTC / rev b19331f50bbdContentsPreface ix1 Difference equations 11.1 Rabbits 21.2 Leaky tank 71.3 Fall of a fog droplet 111.4 Springs 142 Difference equations and modularity 172.1 Modularity: Making the input like the output 172.2 Endowment gift 212.3 Rabbits 253 Block diagrams and operators: Two new representations 333.1 Disadvantages of difference equations 343.2 Block diagrams to the rescue 353.3 The power of abstraction 403.4 Operations on whole signals 413.5 Feedback connections 453.6 Summary 494 Modes 514.1 Growth of the Fibonacci series 524.2 Taking out the big part from Fibonacci 554.3 Operator interpretation 574.4 General method: Partial fractions 595 Repeated roots 635.1 Leaky-tank background 645.2 Numerical computation 655.3 Analyzing the output signal 678 88 8viii2009-09-29 13:11:30 UTC / rev b19331f50bbd5.4 Deforming the system: The continuity argument 685.5 Higher-order cascades 70Bibliography 71Index 739 99 92009-09-29 13:11:30 UTC / rev b19331f50bbdPrefaceThis book aims to introduce you to a powerful tool for analyzing and de-signing systems – whether electronic, mechanical, or thermal.This book grew out of the ‘Signals and Systems’ course (numbered 6.003)that we have taught on and off to MIT’s Electrical Engineering and Com-puter Science students.The traditional signals-and-systems course – for example [16] – empha-sizes the analysis of continuous-time systems, in particular analog circuits.However, most engineers will not specialize in analog circuits. Rather, dig-ital technology offers such vast computing power that analogy circuits areoften designed through digital simulation.Digital simulation is an inherently discrete-time operation. Furthermore,almost all fundamental ideas of signals and systems can be taught usingdiscrete-time systems. Modularity and multiple representations , for ex-ample, aid the design of discrete-time (or continuous-time) systems. Simi-larly, the ideas for modes, poles, control, and feedback.Furthermore, by teaching the material in a context not limited to circuits,we emphasize the generality of these tools. Feedback and simulation aboundin the natural and engineered world, and we would like our students to beflexible and creative in understanding and designing these systems.Therefore, we begin our ‘Signals and Systems’ course with discrete-timesystems, and give our students this book. A fundamental difference frommost discussions of discrete-time systems is the approach using operators.Operators make it possible to avoid the confusing notion of ‘transform’. In-stead, the operator expression for a discrete-time system, and the system’simpulse response are two representations for the same system; they arethe coordinates of a point as seen from two different coordinate systems.Then a transformation of a system has an active meaning: for example,composing two systems to build a new system.10 1010 10x Preface2009-09-29 13:11:30 UTC / rev b19331f50bbdHow to use this bookAristotle was tutor to the young Alexander of Macedon (later, the Great).As ancient royalty knew, a skilled and knowledgeable tutor is the mosteffective teacher [3]. A skilled tutor makes few statements and asks manyquestions, for she knows that questioning, wondering, and discussing pro-mote long-lasting learning. Therefore, questions of two types are inter-spersed through the book:questions marked with a in the margin: These questions are what a tutormight ask you during a tutorial, and ask you to work out the next stepsin an analysis. They are answered in the subsequent text, where you cancheck your solutions and my analysis.numbered questions: These problems, marked with a shaded background,are what a tutor might ask you to take home after a tutorial. They givefurther practice with the tool or ask you to extend an example, use severaltools together, or resolve paradoxes.Try lots of questions of both types!Copyright licenseThis book is licensed under the . . . license.AcknowledgmentsWe gratefully thank the following individuals and organizations:For suggestions and discussions: . . .For the free software for typesetting: Donald Knuth (TEX); Han The Thanh(PDFTEX); Hans Hagen and Taco Hoekwater (ConTEXt); John Hobby (Meta-Post); Andy Hammerlindl, John Bowman, and Tom Prince (asymptote);Richard Stallman (emacs); the Linux and Debian projects.11 1111 112009-09-29 13:11:30 UTC / rev b19331f50bbd1Difference equations1.1 Rabbits 21.2 Leaky tank 71.3 Fall of a fog droplet 111.4 Springs 14The world is too rich and comlex for our minds to grasp it whole, for ourminds are but a small part of the richness of the world. To cope with thecomplexity, we reason hierarchically. We divide the world into small, com-prehensible pieces: systems. Systems are ubiquitious: a CPU, a memorychips, a motor, a web server, a jumbo jet, the solar system, the telephonesystem, or a circulatory system. Systems are a useful abstraction, chosenbecause their external interactions are weaker than their internal interac-tions. That properties makes independent analysis meaningful.Systems interact with other systems via forces, messages, or in general viainformation or signals. ‘Signals and systems’ is the study of systems andtheir interaction.This book studies only discrete-time systems, where time jumps ratherthan changes continuously. This restriction is not as severe as its seems.First, digital computers are, by design, discrete-time devices, so discrete-time signals and systems includes digital computers. Second,


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