Signals and Systems

S. Palani

Signals and SystemsSecond Edition

S. PalaniProfessor (Retired)National Institute of TechnologyTiruchirappalli, India

ISBN 978-3-030-75741-0 ISBN 978-3-030-75742-7 (eBook)https://doi.org/10.1007/978-3-030-75742-7

Jointly published with ANE Books Pvt. Ltd.The print edition is not for sale in South Asia (India, Pakistan, Sri Lanka, Bangladesh, Nepal and Bhutan)and Africa. Customers from South Asia and Africa can please order the print book from: ANE BooksPvt. Ltd.ISBN of the Co-Publisher’s edition: 978-9-383-65627-1

1st edition: © Ane Books Pvt. Ltd. 20172nd edition: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whetherthe whole or part of the material is concerned, specifically the rights of reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformation storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoes not imply, even in the absence of a specific statement, that such names are exempt from the relevantprotective laws and regulations and therefore free for general use.The publishers, the authors, and the editors are safe to assume that the advice and information in this bookare believed to be true and accurate at the date of publication. Neither the publishers nor the authors orthe editors give a warranty, express or implied, with respect to the material contained herein or for anyerrors or omissions that may have been made. The publishers remain neutral with regard to jurisdictionalclaims in published maps and institutional affiliations.

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Preface to Second Edition

I have ventured to bring out the second edition of the book Signals and Systems in itsnew form due to the success and wide patronage extended to the previous edition andreprints by the members of the teaching faculty and student community. The presentedition as in the previous edition covers the undergraduate syllabus in Signals andSystems for the B.E. degree courses. A thorough revision of all the chapters in theprevious edition has been undertaken. Few errors noticed in the previous edition havebeen removed and appropriate corrections have been made. Signal representation isa vital topic to understand the importance of the theoretical concepts in Signalsand Systems. A large number of numerical problems have been included in Chap. 1which describes signal representation (both continuous and discrete time signals).Similarly, the classification of systems is well explained in Chap. 2 with graphicalillustration wherever possible. More number of numerical problems have been addedin Chap. 4 which describes Fourier Series Analysis. Further, the properties of FS arewell explained and applied in solving many FS problems by cutting short lengthyprocedures. Similarly, in Chap. 6, explanation is provided for the Fourier Transformmethod of Analysis and for the properties of FT which are frequently used to solvenumerical problems in an easier way. However, in FS and FT, conventional methodsof solving the numerical problems are also retained. In Chap. 8, numerical problemsusing LT properties have been solved. I hope that the readers of this book wouldappreciate the above attempts. Since every theoretical concept is explained by avariety of numerical examples which are presented in a graded manner, the book isvoluminous. I take this opportunity to thank Ane Books Pvt. Ltd and the publisherfor taking up this difficult job.

Pudukkottai, India S. Palani

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Preface to First Edition

The book SIGNALSANDSYSTEMS presents a comprehensive treatment of signalsand linear systems for the undergraduate level study. It is a rich subject withdiverse applications such as signal processing, control systems and communica-tion systems. It provides an integrated treatment of continuous-time and discrete-time forms of signals and systems. These two forms are treated side by side. Eventhough continuous-time and discrete-time theory havemanymathematical propertiescommon between them, the physical processes that are modelled by continuous-timesystems are very much different from the discrete-time systems counterpart.

I have written this book with the material I have collected during my long experi-ence of teaching signals and systems to the undergraduate level students in nationallevel reputed institutions. The book in the present form is written to meet the require-ments of undergraduate syllabus of Indian Universities in general and Anna Univer-sity in particular for B.E./B.Tech. degree courses. The organization of the chaptersis as follows.

Chapter 1 deals with the representation of signals and systems. It motivates thereader as to what signals and systems are and how they are related to other areassuch as communication systems, control systems and digital signal processing. Inthis chapter, various terminologies related to signals and systems are defined. Further,mathematical description, representation and classifications of signals and systemsare explained.

Chapter 2 presents a detailed descriptions of system classifications. Under broadercategory, systems are classified as continuous-time and discrete-time systems. Eachof them is further classified as linear and non-linear, time invariant and time varying,static and dynamic, causal and non-causal, stable and unstable and invertible andnon-invertible. Systems are identified accordingly.

A comprehensive treatment of time domain analysis of continuous-time anddiscrete-time systems are given in Chapter 3. It develops convolution from the repre-sentation of an input signals as a superposition of impulses. To find the convolutionof two time signals, both analytical as well as graphical methods are explained.

Chapter 4 deals with the Fourier representation of continuous-time signals.Continuous time periodic signals are represented by trigonometric Fourier series,polar Fourier series and exponential Fourier series.

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In Chapter 5, discrete-time signals is represented by exponential Fourier seriesand their properties are derived. The Fourier spectra of discrete-time signal is alsodetermined in this chapter.

It is not possible to find Fourier series representation of non-periodic signals. InChapter 6, Fourier transform is introduced which can represent periodic as well asnon-periodic signals. In this chapter the Fourier transform for continuous-time signalis explained.

In Chapter 7, the representation of discrete-time signal using discrete time Fouriertransform is explained. Further, discrete Fourier transform and Fast Fourier Trans-form algorithm are also explained here. The Laplace transform is a very powerfultool in the analysis of continuous time signals and systems.

InChapter 8, the Laplace transformmethod is explained and its properties derived.The use of Laplace transform to solve differential. equation is described. Finallydifferent forms of structure realization of continuous-time systems are discussed.

Chapter 9 is devoted to the z-transform and its application to discrete time signalsand systems. The properties of z-transform and techniques for inversion are intro-duced in this chapter. The use of z-transform for solving difference equation isexplained. Different forms of structure realization of discrete-time system is alsoexplained in this chapter.

In Chapter 10, the sampling theorem is explained. The necessary condition toavoid aliasing is also explained here.

The notable features of this book includes the following:

1. The syllabus content of signals and systems for undergraduate level of most ofthe Indian Universities in general and Anna University in particular has beencovered.

2. The organization of the chapter are sequential in nature.3. Large number of numerical examples have been worked out.4. Chapter objectives and summary are given in each chapter.5. For the students to practice, short and long questions with answers are given at

the end of each chapter.

I take this opportunity to thank Shri. Sunil, Managing Director Ane Books India, forcoming forward to publish the book. I would like to express my sincere thanks toShri. R. Krishnamoorthi, sales manager Ane Books India who took the initiatives topublish the book in a short span of time. I would like to express my sincere thanks toMr. V. Ashok who has done a wonderful job to key the voluminous book like this ina very short time and beautifully too. My sincere thanks are also due to my colleague

Preface to First Edition ix

Mr. N. Sathurappan who gave some useful suggestions. I would also like to thank mywife Dr. S. Manimegalai, M.B.B.S., M.D., who was the source of inspiration whilepreparing this book.

S. Palani

Contents

1 Representation of Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Terminologies Related to Signals and Systems . . . . . . . . . . . . . . . 2

1.2.1 Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Continuous and Discrete Time Signals . . . . . . . . . . . . . . . . . . . . . 31.4 Basic Continuous Time Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4.1 Unit Impulse Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4.2 Unit Step Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4.3 Unit Ramp Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.4 Unit Parabolic Function . . . . . . . . . . . . . . . . . . . . . . . . . 101.4.5 Unit Rectangular Pulse (or Gate) Function . . . . . . . . . 101.4.6 Unit Area Triangular Function . . . . . . . . . . . . . . . . . . . . 111.4.7 Unit Signum Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4.8 Unit Sinc Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4.9 Sinusoidal Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4.10 Real Exponential Signal . . . . . . . . . . . . . . . . . . . . . . . . . 131.4.11 Complex Exponential Signal . . . . . . . . . . . . . . . . . . . . . 14

1.5 Basic Discrete Time Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.5.1 The Unit Impulse Sequence . . . . . . . . . . . . . . . . . . . . . . 151.5.2 The Basic Unit Step Sequence . . . . . . . . . . . . . . . . . . . . 151.5.3 The Basic Unit Ramp Sequence . . . . . . . . . . . . . . . . . . . 161.5.4 Unit Rectangular Sequence . . . . . . . . . . . . . . . . . . . . . . . 161.5.5 Sinusoidal Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.5.6 Discrete Time Real Exponential Sequence . . . . . . . . . . 19

1.6 Basic Operations on Continuous Time Signals . . . . . . . . . . . . . . . 191.6.1 Addition of CT Signals . . . . . . . . . . . . . . . . . . . . . . . . . . 201.6.2 Multiplications of CT Signals . . . . . . . . . . . . . . . . . . . . 211.6.3 Amplitude Scaling of CT Signals . . . . . . . . . . . . . . . . . 211.6.4 Time Scaling of CT Signals . . . . . . . . . . . . . . . . . . . . . . 221.6.5 Time Shifting of CT Signals . . . . . . . . . . . . . . . . . . . . . . 221.6.6 Signal Reflection or Folding . . . . . . . . . . . . . . . . . . . . . . 24

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1.6.7 Inverted CT Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.6.8 Multiple Transformation . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.7 Basic Operations on Discrete Time Signals . . . . . . . . . . . . . . . . . . 771.7.1 Addition of Discrete Time Sequence . . . . . . . . . . . . . . . 771.7.2 Multiplication of DT Signals . . . . . . . . . . . . . . . . . . . . . 771.7.3 Amplitude Scaling of DT Signal . . . . . . . . . . . . . . . . . . 791.7.4 Time Scaling of DT Signal . . . . . . . . . . . . . . . . . . . . . . . 791.7.5 Time Shifting of DT Signal . . . . . . . . . . . . . . . . . . . . . . 801.7.6 Multiple Transformation . . . . . . . . . . . . . . . . . . . . . . . . . 81

1.8 Classification of Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1031.8.1 Deterministic and Non-deterministic

Continuous Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1031.8.2 Periodic and Non-periodic Continuous Signals . . . . . . 1031.8.3 Fundamental Period of Two Periodic Signals . . . . . . . 1051.8.4 Odd and Even Functions of Continuous Time

Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1211.8.5 Energy and Power of Continuous Time Signals . . . . . 143

1.9 Classification of Discrete Time Signals . . . . . . . . . . . . . . . . . . . . . 1621.9.1 Periodic and Non-Periodic DT Signals . . . . . . . . . . . . . 1621.9.2 Odd and Even DT Signals . . . . . . . . . . . . . . . . . . . . . . . . 1701.9.3 Energy and Power of DT Signals . . . . . . . . . . . . . . . . . . 178

2 Continuous and Discrete Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . 1972.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1972.2 Linear Time Invariant Continuous (LTIC) Time System . . . . . . . 1982.3 Linear Time Invariant Discrete (LTID) Time System . . . . . . . . . 1992.4 Properties (Classification) of Continuous Time System . . . . . . . 199

2.4.1 Linear and Non-linear Systems . . . . . . . . . . . . . . . . . . . 2002.4.2 Time Invariant and Time Varying Systems . . . . . . . . . . 2122.4.3 Static and Dynamic Systems (Memoryless

and System with Memory) . . . . . . . . . . . . . . . . . . . . . . . 2162.4.4 Causal and Non-causal Systems . . . . . . . . . . . . . . . . . . 2202.4.5 Stable and Unstable Systems . . . . . . . . . . . . . . . . . . . . . 2252.4.6 Invertibility and Inverse System . . . . . . . . . . . . . . . . . . . 230

2.5 Discrete Time System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2412.6 Properties of Discrete Time System . . . . . . . . . . . . . . . . . . . . . . . . 241

2.6.1 Linear and Non-linear Systems . . . . . . . . . . . . . . . . . . . 2422.6.2 Time Invariant and Time Varying DT Systems . . . . . . 2462.6.3 Causal and Non-causal DT Systems . . . . . . . . . . . . . . . 2482.6.4 Stable and Unstable Systems . . . . . . . . . . . . . . . . . . . . . 2512.6.5 Static and Dynamic Systems . . . . . . . . . . . . . . . . . . . . . 2542.6.6 Invertible and Inverse Discrete Time Systems . . . . . . . 256

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3 Time Domain Analysis of Continuous and Discrete TimeSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2713.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2713.2 Time Response of Continuous Time System . . . . . . . . . . . . . . . . . 2723.3 The Unit Impulse Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2733.4 Unit Impulse Response and the Convolution Integral . . . . . . . . . 2733.5 Step by Step Procedure to Solve Convolution . . . . . . . . . . . . . . . . 2753.6 Properties of Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

3.6.1 The Commutative Property . . . . . . . . . . . . . . . . . . . . . . . 2763.6.2 The Distributive Property . . . . . . . . . . . . . . . . . . . . . . . . 2773.6.3 The Associative Property . . . . . . . . . . . . . . . . . . . . . . . . 2783.6.4 The Shift Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2793.6.5 The Width Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

3.7 Analytical Method of Convolution Operation . . . . . . . . . . . . . . . . 2803.7.1 Convolution Operation of Non-causal Signals . . . . . . . 288

3.8 Causality of an Linear Time Invariant Continuous TimeSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

3.9 Stability of a Linear Time Invariant System . . . . . . . . . . . . . . . . . 3243.10 Step Response from Impulse Response . . . . . . . . . . . . . . . . . . . . . 3313.11 Representation of Discrete Time Signals in Terms

of Impulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3363.12 The Discrete Time Unit Impulse Response . . . . . . . . . . . . . . . . . . 3383.13 The Convolution Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3393.14 Properties of Convolution Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

3.14.1 Distributive Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3423.14.2 Associative Property of Convolution . . . . . . . . . . . . . . . 3423.14.3 Commutative Property of Convolution . . . . . . . . . . . . . 3443.14.4 Shifting Property of Convolution . . . . . . . . . . . . . . . . . . 3453.14.5 The Width Property of Convolution . . . . . . . . . . . . . . . 3453.14.6 Convolution with an Impulse . . . . . . . . . . . . . . . . . . . . . 3453.14.7 Convolution with Delayed Impulse . . . . . . . . . . . . . . . . 3463.14.8 Convolution with Unit Step . . . . . . . . . . . . . . . . . . . . . . 3463.14.9 Convolution with Delayed Step . . . . . . . . . . . . . . . . . . . 3473.14.10 System Causality from Convolution . . . . . . . . . . . . . . . 3473.14.11 BIBO Stability from Convolution . . . . . . . . . . . . . . . . . 3483.14.12 Step Response in Terms of Impulse Response

of a LTDT System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3493.15 Response Using Convolution Sum . . . . . . . . . . . . . . . . . . . . . . . . . 351

3.15.1 Analytical Method Using Convolution Sum . . . . . . . . 3523.15.2 Convolution Sum of Two Sequences

by Multiplication Method . . . . . . . . . . . . . . . . . . . . . . . . 3813.15.3 Convolution Sum by Tabulation Method . . . . . . . . . . . 3863.15.4 Convolution Sum of Two Sequences byMatrix

Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3903.16 Convolution Sum by Graphical Method . . . . . . . . . . . . . . . . . . . . . 393

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3.17 Deconvolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4053.18 Step Response of the System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4093.19 Stability from Impulse Response . . . . . . . . . . . . . . . . . . . . . . . . . . 4103.20 System Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414

4 Fourier Series Analysis of Continuous Time Signals . . . . . . . . . . . . . 4294.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4294.2 Periodic Signal Representation by Fourier Series . . . . . . . . . . . . . 4314.3 Different Forms of Fourier Series Representation . . . . . . . . . . . . 431

4.3.1 Trigonometric Fourier Series . . . . . . . . . . . . . . . . . . . . . 4314.3.2 Complex Exponential Fourier Series . . . . . . . . . . . . . . . 4334.3.3 Polar or Harmonic Form Fourier Series . . . . . . . . . . . . 434

4.4 Properties of Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4454.4.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4454.4.2 Time Shifting Property . . . . . . . . . . . . . . . . . . . . . . . . . . 4464.4.3 Time Reversal Property . . . . . . . . . . . . . . . . . . . . . . . . . . 4474.4.4 Time Scaling Property . . . . . . . . . . . . . . . . . . . . . . . . . . . 4474.4.5 Multiplication Property . . . . . . . . . . . . . . . . . . . . . . . . . . 4484.4.6 Conjugation Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4494.4.7 Differentiation Property . . . . . . . . . . . . . . . . . . . . . . . . . 4494.4.8 Integration Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4504.4.9 Parseval’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450

4.5 Existence of Fourier Series—the Dirichlet Conditions . . . . . . . . 4704.6 Convergence of Continuous Time Fourier Series . . . . . . . . . . . . . 4714.7 Fourier Series Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

5 Fourier Series Analysis of Discrete Time Signals . . . . . . . . . . . . . . . . 5095.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5095.2 Periodicity of Discrete Time Signal . . . . . . . . . . . . . . . . . . . . . . . . 5095.3 DT Signal Representation by Fourier Series . . . . . . . . . . . . . . . . . 5105.4 Fourier Spectra of x[n] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5125.5 Properties of Discrete Time Fourier Series . . . . . . . . . . . . . . . . . . 512

5.5.1 Linearity Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5125.5.2 Time Shifting Property . . . . . . . . . . . . . . . . . . . . . . . . . . 5135.5.3 Time Reversal Property . . . . . . . . . . . . . . . . . . . . . . . . . . 5135.5.4 Multiplication Property . . . . . . . . . . . . . . . . . . . . . . . . . . 5145.5.5 Conjugation Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5155.5.6 Difference Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5165.5.7 Parseval’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516

6 Fourier Transform Analysis of Continuous Time Signals . . . . . . . . . 5376.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5376.2 Representation of Aperiodic Signal by Fourier

Integral—The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 5386.3 Convergence of Fourier Transforms—The Dirichlet

Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541

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6.4 Fourier Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5426.5 Connection Between the Fourier Transform and Laplace

Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5426.6 Properties of Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 556

6.6.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5566.6.2 Time Shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5576.6.3 Conjugation and Conjugation Symmetry . . . . . . . . . . . 5576.6.4 Differentiation in Time . . . . . . . . . . . . . . . . . . . . . . . . . . 5586.6.5 Differentiation in Frequency . . . . . . . . . . . . . . . . . . . . . . 5596.6.6 Time Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5596.6.7 Time Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5606.6.8 Frequency Shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5616.6.9 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5626.6.10 The Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5626.6.11 Parseval’s Theorem (Relation) . . . . . . . . . . . . . . . . . . . . 563

6.7 Fourier Transform of Periodic Signal . . . . . . . . . . . . . . . . . . . . . . . 5656.7.1 Fourier Transform Using Differentiation

and Integration Properties . . . . . . . . . . . . . . . . . . . . . . . . 566

7 Fourier Transform Analysis of Discrete Time Signalsand Systems—DTFT, DFT and FFT . . . . . . . . . . . . . . . . . . . . . . . . . . . 6517.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6517.2 Representation of Discrete Time Aperiodic Signals . . . . . . . . . . . 6527.3 Connection Between the Fourier Transform

and the z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6557.4 Properties of Discrete Time Fourier Transform . . . . . . . . . . . . . . 659

7.4.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6597.4.2 Time Shifting Property . . . . . . . . . . . . . . . . . . . . . . . . . . 6597.4.3 Frequency Shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6607.4.4 Time Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6617.4.5 Time Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6627.4.6 Multiplication by n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6627.4.7 Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6637.4.8 Time Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6647.4.9 Parseval’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6647.4.10 Modulation Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665

7.5 Inverse Discrete Time Fourier Transform (IDTFT) . . . . . . . . . . . 6767.6 LTI System Characterized by Difference Equation . . . . . . . . . . . 6807.7 Discrete Fourier Transform (DFT) . . . . . . . . . . . . . . . . . . . . . . . . . 687

7.7.1 The Discrete Fourier Transform Pairs . . . . . . . . . . . . . . 6887.7.2 Four Point, Six Point and Eight Point Twiddle

Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6907.7.3 Zero Padding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697

7.8 Properties of DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7027.8.1 Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702

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7.8.2 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7027.8.3 Complex Conjugate Symmetry . . . . . . . . . . . . . . . . . . . 7037.8.4 Circular Time Shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . 7037.8.5 Circular Frequency Shifting . . . . . . . . . . . . . . . . . . . . . . 7037.8.6 Circular Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7047.8.7 Multiplication of Two DFTs . . . . . . . . . . . . . . . . . . . . . . 7047.8.8 Parseval’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704

7.9 Circular Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7057.9.1 Circular Convolution—Circle Method . . . . . . . . . . . . . 7057.9.2 Circular Convolution-Matrix Multiplication

Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7067.9.3 Circular Convolution-DFT-IDFT Method . . . . . . . . . . 709

7.10 Fast Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7137.10.1 FFT Algorithm-Decimation in Time . . . . . . . . . . . . . . . 7137.10.2 FFT Algorithm-Decimation in Frequency . . . . . . . . . . 722

8 The Laplace Transform Method for the Analysisof Continuous Time Signals and Systems . . . . . . . . . . . . . . . . . . . . . . . 7378.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7378.2 Definition and Derivations of the LT . . . . . . . . . . . . . . . . . . . . . . . 738

8.2.1 LT of Causal and Non-causal Systems . . . . . . . . . . . . . 7398.3 The Existence of LT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7418.4 The Region of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741

8.4.1 Properties of ROCs for LT . . . . . . . . . . . . . . . . . . . . . . . 7448.5 The Unilateral Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . 7528.6 Properties of Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 753

8.6.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7538.6.2 Time Shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7538.6.3 Frequency Shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7548.6.4 Time Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7548.6.5 Frequency Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7558.6.6 Time Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7568.6.7 Time Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7578.6.8 Time Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7588.6.9 Complex Frequency Differentiation . . . . . . . . . . . . . . . 7598.6.10 Complex Frequency Shifting . . . . . . . . . . . . . . . . . . . . . 7598.6.11 Conjugation Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7608.6.12 Initial Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 7608.6.13 Final Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761

8.7 Laplace Transform of Periodic Signal . . . . . . . . . . . . . . . . . . . . . . 7858.8 Inverse Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 788

8.8.1 Graphical Method of Determining the Residues . . . . . 7888.9 Solving Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803

8.9.1 Solving Differential Equation without InitialConditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803

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8.9.2 Solving Differential Equation with the InitialConditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 810

8.9.3 Zero Input and Zero State Response . . . . . . . . . . . . . . . 8148.9.4 Natural and Forced Response Using LT . . . . . . . . . . . . 818

8.10 Time Convolution Property of the Laplace Transform . . . . . . . . . 8208.11 Network Analysis Using Laplace Transform . . . . . . . . . . . . . . . . 824

8.11.1 Mathematical Description of R-L-C- Elements . . . . . . 8248.11.2 Transfer Function and Pole-Zero Location . . . . . . . . . . 826

8.12 Connection Between Laplace Transform and FourierTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855

8.13 Causality of Continuous Time Invariant System . . . . . . . . . . . . . . 8568.14 Stability of Linear Time Invariant Continuous System . . . . . . . . 8578.15 The Bilateral Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 858

8.15.1 Representation of Causal and Anti-causalSignals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859

8.15.2 ROC of Bilateral Laplace Transform . . . . . . . . . . . . . . . 8608.16 System Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 870

8.16.1 Direct Form-I Realization . . . . . . . . . . . . . . . . . . . . . . . . 8718.16.2 Direct Form-II Realization . . . . . . . . . . . . . . . . . . . . . . . 8778.16.3 Cascade Form Realization . . . . . . . . . . . . . . . . . . . . . . . 8828.16.4 Parallel Structure Realization . . . . . . . . . . . . . . . . . . . . . 8848.16.5 Transposed Realization . . . . . . . . . . . . . . . . . . . . . . . . . . 890

9 The z-Transform Analysis of Discrete Time Signalsand Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9219.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9219.2 The z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9229.3 Existence of the z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9249.4 Connection Between Laplace Transform, z-Transform

and Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9249.5 The Region of Convergence (ROC) . . . . . . . . . . . . . . . . . . . . . . . . 9269.6 Properties of the ROC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9299.7 Properties of z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 937

9.7.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9379.7.2 Time Shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9389.7.3 Time Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9399.7.4 Multiplication by n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9399.7.5 Multiplication by an Exponential . . . . . . . . . . . . . . . . . . 9409.7.6 Time Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9419.7.7 Convolution Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 9419.7.8 Initial Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 9429.7.9 Final Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943

9.8 Inverse z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9639.8.1 Partial Fraction Method . . . . . . . . . . . . . . . . . . . . . . . . . . 963

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9.8.2 Inverse z-Transform Using Power SeriesExpansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975

9.8.3 Inverse z-Transform Using ContourIntegration or the Method of Residue . . . . . . . . . . . . . . 981

9.9 The System Function of DT Systems . . . . . . . . . . . . . . . . . . . . . . . 9839.10 Causality of DT Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9839.11 Stability of DT System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9849.12 Causality and Stability of DT System . . . . . . . . . . . . . . . . . . . . . . 9849.13 z-Transform Solution of Linear Difference Equations . . . . . . . . . 998

9.13.1 Right Shift (Delay) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9989.13.2 Left Shift (Advance) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 999

9.14 Zero Input and Zero State Response . . . . . . . . . . . . . . . . . . . . . . . . 10159.15 Natural and Forced Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10189.16 Difference Equation from System Function . . . . . . . . . . . . . . . . . 10209.17 Discrete Time System Realization . . . . . . . . . . . . . . . . . . . . . . . . . 1024

9.17.1 Direct Form-I Realization . . . . . . . . . . . . . . . . . . . . . . . . 10259.17.2 Direct Form-II Realization . . . . . . . . . . . . . . . . . . . . . . . 10269.17.3 Cascade Form Realization . . . . . . . . . . . . . . . . . . . . . . . 10299.17.4 Parallel Form Realization . . . . . . . . . . . . . . . . . . . . . . . . 10309.17.5 The Transposed Form Realization . . . . . . . . . . . . . . . . . 1032

10 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105710.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105710.2 The Sampling Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105810.3 The Sampling Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105810.4 Signal Recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106110.5 Aliasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061

10.5.1 Sampling Rate ωs Higher than 2ωm . . . . . . . . . . . . . . . 106210.5.2 Anti-aliasing Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1062

10.6 Sampling with Zero-Order Hold . . . . . . . . . . . . . . . . . . . . . . . . . . . 106310.7 Application of Sampling Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 106410.8 Sampling of Band-Pass Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067

Appendix A: Mathematical Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077

About the Author

Dr. S. Palani obtained his B.E. degree in ElectricalEngineering in the year 1966 from the University ofMadras, M.Tech. in Control Systems Engineering fromIndian Institute of Technology Kharagpur in 1968, andPh.D. in Control Systems Engineering from the Univer-sity of Madras in 1982. He has a wide teaching expe-rience of over four decades. He started his teachingcareer in the year 1968 at the erstwhile RegionalEngineering College (now National Institute of Tech-nology), Tiruchirapalli in the department of EEE andoccupied various positions. As Professor and Head,he took the initiative to start the Instrumentation andControl Engineering Department. After a meritoriousservice of over three decades in REC, Tiruchirapalli,he joined Sudharsan Engineering College, Pudukkottaias the founder Principal. He established various depart-ments with massive infrastructure. Since 2006, he is theDean and Professor of the ECE department in the samecollege.

He has published more than a hundred researchpapers in reputed national and international journalsand conferences and has won many cash awards. Underhis guidance, seventeen research scholars were awardedPh.D. He has carried out several research projects worthabout several lakhs rupees funded by the Governmentof India and AICTE. As the theme leader of the Indo -UK,RECProject on energy, he has visitedmanyUniver-sities and industries in the United Kingdom. He is theauthor of the books titled Control Systems Engineering,Signals and Systems, Digital Signal Processing, LinearSystem Analysis, and Automatic Control Systems. Hisareas of research include the design of Controllers

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xx About the Author

for Dynamic systems, Digital Signal Processing, andImage Processing. Dr. S. Palani is the reviewer of tech-nical papers of reputed International Journals. He haschaired/organized many International conferences andWorkshops.