Table Of ContentMATHEMATICS
for
CIRCUITS
FILTERS
and
MATHEMATICS
for
CIRCUITS
and FILTERS
edited by
Wai-Kai Chen
Boca Raton London New York
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ISBN-13: 978-0-8493-0052-3 (hbk)
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Library of Congress Cataloging-in-Publication Data
Mathematics for circuits and filters I Wai-Kai Chen, editor.
p. em.
Includes bibliographical references and index.
ISBN 0-8493-0052-5 (alk. paper)
l. Electric circuits, Linear-Mathematical models. 2. Electric filters-Mathematical
models. 3. Electric engineering-Mathematics. I. Chen, Wai-Kai, 1936-
TK454. M3295 1999
621.3-dc21 99-043798
CIP
Library of Congress Card Number 99-043798
Visit the Taylor & Francis Web site at
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DOI: 10.1201/9781315214023
Preface
The purpose of M athematics for Circuits and Filters is to provide in a single volume a comprehensive
reference work covering the broad spectrum of mathematics and symbols that underlie numerous
applications i n e lectrical c ircuits and filters. It i s written a nd d eveloped for practicing e lectrical e ngineers
in industry, government, and academia. Over the years, the mathematical fundamentals of electrical
circuits and filters have evolved to include a wide range of topics and a broad range of practice. To
encompass s uch a w ide range o f k nowledge, the b ook f ocuses on t he k ey c oncepts, m odels, and e quations
that enable the electrical engineer to analyze, design, and predict the behavior of l arge-scale circuits,
devices, filters, and systems.
Wai-Kai Chen
Editor-in-Chief
Contributors
John R. Deller, Jr. Jelena Kovacevic Krishnaiyan Thulasiraman
Michigan S tate U niversity AT&T Bell Laboratories University o f O klahoma
East Lansing, M ichigan Murray H ill, N ew J ersey Norman, O klahoma
Igor Djokovic Michael K. Sain P. P. Vaidyanathan
California I nstitute o f T echnology Universtiy o f N otre Dame California Institute o f T echnology
Pasadena, California Notre D ame, I ndiana Pasadena, California
W. Kenneth Jenkins Cheryl B. Schrader
University o f I llinois University o f T exas
Urbana, I llinois San A ntonio, T exas
Contents
1 Linear O perators a nd M atrices Cheryl B. Schrader and M ichael K. Sain 1
1.1 Introduction . . . . . . . 1
1.2 Vector Spaces Over Fields . . . . . . . . 2
1.3 Linear O perators a nd M atrix R epresentations 4
1.4 Matrix O perations . . . . . . 6
1.5 Determinant, Inverse, and R ank 8
1.6 Basis Transformations 11
1.7 Characteristics: Eigenvalues, Eigenvectors, and S ingular V alues 15
1.8 On Linear Systems . . . . . . . . . . . . . . . . 17
2 Bilinear O perators a nd M atrices Michael K . Sain and C heryl B. Schrader 19
2.1 Introduction 19
2.2 Algebras 20
2.3 Bilinear O perators 21
2.4 Tensor Product 22
2.5 Basis Tensors 23
2.6 Multiple Products 26
2.7 Determinants 27
2.8 Skew S ymmetric P roducts 28
2.9 Solving Linear Equations 3 I
2.10 Symmetric P roducts 33
2.11 Summary 35
3 The L aplace Transform John R. Deller, ]r. 37
3.1 Introduction 37
3.2 Motivational E xample 38
3.3 Formal Developments 44
3.4 Laplace Transform A nalysis of L inear S ystems 68
3.5 Conclusions and F urther R eading . . . . . 78
3.6 Appendix A : The Dirac Delta (Impulse) Function 79
3.7 Appendix B : Relationships among t he L aplace, Fourier, and z -Transforms 80
4 Fourier S eries, Fourier T ransforms and t he D FT W. Kenneth Jenkins 83
4.1 Introduction .............. -. . . . . . . . . . . 83
4.2 Fourier S eries Representation of C ontinuous T ime Periodic S ignals 85
4.3 The Classical Fourier T ransform for Continuous T ime S ignals 89
4.4 The D iscrete T ime Fourier T ransform 93
4.5 The Discrete F ourier T ransform 97
4.6 Family T ree of F ourier T ransforms 102
4. 7 Selected Applications of Fourier Methods 104
4.8 Summary . . . . . . . . . . . . . . 110
5 z-Transform ]elena Kovacevic 113
5.1 Introduction . . . . . . . . 113
5.2 Definition o f t he z -Transform 114
5.3 Inverse z-Transform 117
