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• M. D. Adams, Continuous-Time Signals and Systems (Version 2013-09-11), Uni-

versity of Victoria, Victoria, BC, Canada, Sept. 2013, xxx + 308 pages, ISBN

978-1-55058-495-0 (paperback), ISBN 978-1-55058-506-3 (PDF).

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Continuous-Time Signals and Systems

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Michael D. Adams

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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. AdamsPublished 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.

Adams.

Includes index.

ISBN 978-1-55058-495-0 (pbk.)

ISBN 978-1-55058-506-3 (PDF)

1. Signal theory (Telecommunication)Textbooks.

2. System analysisTextbooks. 3. MATLABTextbooks.

I. Title.

TK5102.5.A33 2013 621.38223 C2013-904334-9

http://creativecommons.org/licenses/by-nc-nd/3.0/

• To my students, past, present, and future

• v

Contents

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 . . . . .