Continuous Time Signals and Systems UVic ca

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Continuous Time Signals and Systems,Version 2013 09 11. Michael D Adams,Department of Electrical and Computer Engineering. University of Victoria Victoria BC Canada,Copyright c 2013 Michael D Adams. The author has taken care in the preparation of this book but makes no expressed or implied warranty of any kind and. assumes no responsibility for errors or omissions No liability is assumed for incidental or consequential damages in. connection with or arising out of the use of the information or programs contained herein. Copyright c 2013 Michael D Adams, Published by the University of Victoria Victoria BC Canada. Photography by Michael Adams, This book is licensed under a Creative Commons Attribution NonCommercial NoDerivs 3 0 Unported CC BY NC.
ND 3 0 License A copy of this license can be found in the section titled License on page xix of this book For a. simple explanation of the rights granted by this license see. http creativecommons org licenses by nc nd 3 0, MATLAB is a registered trademark of The MathWorks Inc. Image Processing Toolbox Optimization Toolbox Symbolic Math Toolbox Signal Processing Toolbox and Wavelet. Toolbox are registered trademarks of The MathWorks Inc. UNIX and X Window System are registered trademarks of The Open Group. Windows is a registered trademark of Microsoft Corporation. This book was typeset with LATEX, Library and Archives Canada Cataloguing in Publication. Adams Michael D 1969 author,Continuous time signals and systems Michael D. Includes index,ISBN 978 1 55058 495 0 pbk,ISBN 978 1 55058 506 3 PDF. 1 Signal theory Telecommunication Textbooks,2 System analysis Textbooks 3 MATLAB Textbooks.
TK5102 5 A33 2013 621 382 23 C2013 904334 9,To my students past present and future. License xix,Preface xxvii,Acknowledgments xxvii,About the Author xxix. 1 Introduction 1,1 1 Signals 1,1 1 1 Dimensionality of Signals 1. 1 1 2 Continuous Time and Discrete Time Signals 1, 1 1 3 Notation and Graphical Representation of Signals 2. 1 1 4 Examples of Signals 2,1 2 Systems 3,1 2 1 Classification of Systems 3.
1 2 2 Examples of Systems 4,1 3 Continuous Time Signals and Systems 5. 1 4 Why Study Signals and Systems 5,2 Continuous Time Signals and Systems 7. 2 1 Transformations of the Independent Variable 7,2 1 1 Time Reversal 7. 2 1 2 Time Scaling 7,2 1 3 Time Shifting 8,2 1 4 Combining Time Scaling and Time Shifting 8. 2 2 Transformations of the Dependent Variable 10,2 2 1 Amplitude Scaling 10.
2 2 2 Amplitude Shifting 10,2 2 3 Combining Amplitude Scaling and Shifting 12. 2 3 Signal Properties 12,2 3 1 Even and Odd Signals 12. 2 3 2 Periodic Signals 14,2 3 3 Support of Signals 16. 2 3 4 Signal Energy and Power 16,2 3 5 Examples 17. 2 4 Elementary Signals 18,2 4 1 Real Sinusoidal Signals 18.
2 4 2 Complex Exponential Signals 19, 2 4 3 Relationship Between Complex Exponential and Real Sinusoidal Signals 21. 2 4 4 Unit Step Function 21,2 4 5 Unit Rectangular Pulse 22. Version 2013 09 11 Copyright c 2013 Michael D Adams. vi CONTENTS,2 4 6 Unit Triangular Pulse 22,2 4 7 Cardinal Sine Function 24. 2 4 8 Unit Impulse Function 24, 2 5 Signal Representation Using Elementary Signals 27. 2 6 Continuous Time Systems 30,2 6 1 Block Diagram Representation 31.
2 6 2 Interconnection of Systems 31,2 7 Properties of Continuous Time Systems 31. 2 7 1 Memory 32,2 7 2 Causality 32,2 7 3 Invertibility 33. 2 7 4 Stability 33,2 7 5 Time Invariance 34,2 7 6 Linearity 35. 2 7 7 Examples 37,2 8 Problems 40, 3 Continuous Time Linear Time Invariant Systems 45. 3 1 Introduction 45,3 2 Continuous Time Convolution 45.
3 3 Properties of Convolution 53,3 3 1 Commutative Property 53. 3 3 2 Associative Property 55,3 3 3 Distributive Property 55. 3 4 Representation of Continuous Time Signals Using Impulses 56. 3 5 Continuous Time Unit Impulse Response and Convolution Integral Representation of LTI Systems 56. 3 6 Unit Step Response of LTI Systems 59, 3 7 Block Diagram Representation of Continuous Time LTI Systems 59. 3 8 Interconnection of Continuous Time LTI Systems 61. 3 9 Properties of Continuous Time LTI Systems 62,3 9 1 Memory 63. 3 9 2 Causality 63,3 9 3 Invertibility 64,3 9 4 Stability 67.
3 10 Response of Continuous Time LTI Systems to Complex Exponential Signals 70. 3 11 Problems 72,4 Continuous Time Fourier Series 77. 4 1 Introduction 77, 4 2 Definition of Continuous Time Fourier Series 77. 4 3 Determining the Fourier Series Representation of a Continuous Time Periodic Signal 79. 4 4 Convergence of Continuous Time Fourier Series 85. 4 5 Properties of Continuous Time Fourier Series 87. 4 5 1 Linearity 87,4 5 2 Time Shifting 90,4 5 3 Time Reversal 91. 4 6 Fourier Series and Frequency Spectra 91,4 7 Fourier Series and LTI Systems 94. 4 8 Filtering 95,4 9 Problems 99,4 10 MATLAB Problems 101.
Copyright c 2013 Michael D Adams Version 2013 09 11. CONTENTS vii,5 Continuous Time Fourier Transform 103. 5 1 Introduction 103, 5 2 Development of the Continuous Time Fourier Transform 103. 5 3 Definition of the Continuous Time Fourier Transform 107. 5 4 Convergence of the Continuous Time Fourier Transform 109. 5 5 Properties of the Continuous Time Fourier Transform 110. 5 5 1 Linearity 110,5 5 2 Time Domain Shifting 112. 5 5 3 Frequency Domain Shifting 112,5 5 4 Time and Frequency Domain Scaling 113. 5 5 5 Conjugation 114,5 5 6 Duality 115,5 5 7 Time Domain Convolution 116.
5 5 8 Frequency Domain Convolution 117,5 5 9 Time Domain Differentiation 118. 5 5 10 Frequency Domain Differentiation 119,5 5 11 Time Domain Integration 120. 5 5 12 Parseval s Relation 121, 5 6 Continuous Time Fourier Transform of Periodic Signals 122. 5 7 Fourier Transforms 125,5 8 Frequency Spectra of Signals 129. 5 9 Bandwidth of Signals 135,5 10 Frequency Response of LTI Systems 135.
5 11 Frequency Response and Differential Equation Representations of LTI Systems 136. 5 12 Energy Spectral Density 139,5 13 Power Spectral Density 140. 5 14 Filtering 141,5 15 Sampling and Interpolation 147. 5 15 1 Sampling 149, 5 15 2 Interpolation and Reconstruction of a Signal From Its Samples 153. 5 15 3 Sampling Theorem 154,5 16 Amplitude Modulation 155. 5 16 1 Modulation With a Complex Sinusoid 156,5 16 2 DSB SC Amplitude Modulation 158.
5 16 3 SSB SC Amplitude Modulation 159,5 17 Equalization 159. 5 18 Problems 163,5 19 MATLAB Problems 167,6 Laplace Transform 169. 6 1 Introduction 169,6 2 Motivation Behind the Laplace Transform 169. 6 3 Definition of the Laplace Transform 169, 6 4 Relationship Between Laplace Transform and Continuous Time Fourier Transform 170. 6 5 Laplace Transform Examples 171, 6 6 Region of Convergence for the Laplace Transform 174.
6 7 Properties of the Laplace Transform 178,6 7 1 Linearity 179. 6 7 2 Time Domain Shifting 182,6 7 3 Laplace Domain Shifting 182. 6 7 4 Time Domain Laplace Domain Scaling 183,6 7 5 Conjugation 186. Version 2013 09 11 Copyright c 2013 Michael D Adams. viii CONTENTS,6 7 6 Time Domain Convolution 187,6 7 7 Time Domain Differentiation 188. 6 7 8 Laplace Domain Differentiation 189,6 7 9 Time Domain Integration 190.
6 7 10 Initial Value Theorem 191,6 7 11 Final Value Theorem 191. 6 8 More Laplace Transform Examples 193, 6 9 Determination of the Inverse Laplace Transform 197. 6 10 Characterizing LTI Systems Using the Laplace Transform 202. 6 11 System Function and System Properties 203,6 11 1 Causality 203. 6 11 2 Stability 204,6 11 3 Invertibility 206,6 12 LTI Systems and Differential Equations 207. 6 13 Interconnection of LTI Systems 211,6 14 Unilateral Laplace Transform 214.
6 15 Solving Differential Equations Using the Unilateral Laplace Transform 216. 6 16 Problems 223,6 17 MATLAB Problems 226,A Complex Analysis 227. A 1 Introduction 227,A 2 Complex Numbers 227,A 3 Representations of Complex Numbers 228. A 4 Arithmetic Operations 229,A 4 1 Conjugation 229. A 4 2 Addition 229,A 4 3 Multiplication 230,A 4 4 Division 231. A 4 5 Miscellany 232,A 5 Arithmetic Properties of Complex Numbers 232.
A 5 1 Commutative Property 232,A 5 2 Associative Property 232. A 5 3 Distributive Property 232,A 6 Roots of Complex Numbers 232. A 7 Euler s Relation and De Moivre s Theorem 233, A 8 Conversion Between Cartesian and Polar Form 234. A 9 Complex Functions 234,A 10 Circles Disks and Annuli 235. A 11 Limit 237,A 12 Continuity 237,A 13 Differentiability 237.
A 14 Analyticity 238,A 15 Zeros and Singularities 239. A 16 Quadratic Formula 241,A 17 Problems 242,A 18 MATLAB Problems 243. B Partial Fraction Expansions 245,B 1 Problems 249. Copyright c 2013 Michael D Adams Version 2013 09 11. CONTENTS ix, C Solution of Constant Coefficient Linear Differential Equations 251. C 1 Overview 251, C 2 Constant Coefficient Linear Differential Equations 251.
C 3 Solution of Homogeneous Equations 251, C 4 Particular Solution of Nonhomogeneous Equations 253. C 5 General Solution of Nonhomogeneous Equations 255. C 6 Problems 259,C 7 MATLAB Problems 259,D Miscellaneous Information 261. D 1 Integrals 261,D 2 Derivatives 261,D 3 Arithmetic and Geometric Series 262. E MATLAB 263,E 1 Introduction 263,E 2 Octave 263,E 3 Invoking MATLAB 263. E 3 1 UNIX 263,E 3 2 Microsoft Windows 264,E 4 Command Line Editor 264.
E 5 MATLAB Basics 264,E 5 1 Identifiers 265,E 5 2 Basic Functionality 265. E 6 Arrays 267,E 6 1 Arrays with Equally Spaced Elements 268. E 6 2 Array Subscripting 268,E 6 3 Other Array Functions 269. E 7 Scripts 269,E 8 Relational and Logical Operators 270. E 9 Operator Precedence 272,E 10 Control Flow 272,E 10 1 If Elseif Else 272.
E 10 2 Switch 273,E 10 3 For 274,E 10 4 While 274,E 10 5 Break and Continue 275. E 11 Functions 276,E 12 Graphing 278,E 13 Printing 281. E 14 Symbolic Math Toolbox 282,E 14 1 Symbolic Objects 282. E 14 2 Creating Symbolic Objects 282,E 14 3 Manipulating Symbolic Objects 283. E 14 4 Plotting Symbolic Expressions 284,E 15 Signal Processing 284.
E 15 1 Frequency Responses 284,E 15 2 Impulse and Step Responses 286. E 15 3 Filter Design 286,E 16 Miscellany 287,E 17 Problems 289. Version 2013 09 11 Copyright c 2013 Michael D Adams. x CONTENTS,F Review Problems 293, Copyright c 2013 Michael D Adams Version 2013 09 11. List of Tables,4 1 Fourier Series Properties 91,5 1 Fourier Transform Properties 123. 5 2 Fourier Transform Pairs 125,6 1 Laplace Transform Properties 192.
6 2 Laplace Transform Pairs 193,6 3 Unilateral Laplace Transform Properties 216. 6 4 Unilateral Laplace Transform Pairs 217,C 1 Forms for particular solution 254. E 1 Keys for Command Line Editing 264,E 2 Predefined Variables 265. E 3 Operators 265,E 4 Elementary Math Functions 266. E 5 Other Math Related Functions 266,E 6 Exponential and Logarithmic Functions 266.
E 7 Trigonometric Functions 266,E 8 Radix Conversion Functions 267. E 9 Array Size Functions 268,E 10 Special Matrix Vector Functions 269. E 11 Basic Array Manipulation Functions 269,E 12 Relational Operators 271. E 13 Logical Operators 271,E 14 Relational and Logical Functions 271. E 15 Operator Precedence 272,E 16 Special Predefined Function Variables 277.
E 17 Basic Plotting Functions 278,E 18 Other Graphing Functions Commands 278. E 19 Line Styles 279,E 20 Line Colors 279,E 21 Marker Styles 279. E 22 Graph Annotation Functions 279,E 23 Special Symbols for Annotation Text 280. E 24 Functions related to signal processing 284,E 25 Miscellaneous Functions Commands 288. Version 2013 09 11 Copyright c 2013 Michael D Adams. xii LIST OF TABLES, Copyright c 2013 Michael D Adams Version 2013 09 11.
List of Figures, 1 1 Graphical representations of a continuous time and b discrete time signals 2. 1 2 Segment of digitized human speech 2,1 3 A monochromatic image 3. 1 4 System with one or more inputs and one or more outputs 3. 1 5 A simple RC network 4,1 6 Communication system 4. 1 7 Feedback control system 4,2 1 Example of time reversal 7. 2 2 Example of time scaling 8,2 3 Example of time shifting 9.
2 4 Two different interpretations of combined shifting and scaling transformation a Original signal. Results obtained by shifting followed by scaling b intermediate result and c final result Results. obtained by scaling followed by shifting d intermediate result and e final result 10. 2 5 Example of amplitude scaling 11,2 6 Example of amplitude shifting 12. 2 7 Example of an even signal 13,2 8 Example of an odd signal 13. 2 9 Example of a periodic signal 15, 2 10 Examples of a left sided b right sided c finite duration and d two sided signals 16. 2 11 Real sinusoidal signal 19, 2 12 Real exponential signal for a 0 b 0 and c 0 20. 2 13 Complex sinusoidal signal a Real and b imaginary parts 20. 2 14 Real part of a general complex exponential for a 0 b 0 and c 0 21. 2 15 Unit step function 22,2 16 Unit rectangular pulse 22.
2 17 Using the unit rectangular pulse to extract one period of a periodic waveform 23. 2 18 Unit triangular pulse 23,2 19 Unit impulse function 25. Continuous time signals and systems Michael D Adams Includes index ISBN 978 1 55058 495 0 pbk ISBN 978 1 55058 506 3 PDF 1 Signal theory Telecommunication Textbooks 2 System analysis Textbooks 3 MATLAB Textbooks I Title TK5102 5 A33 2013 621 382 23 C2013 904334 9

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