1. Signals and Systems with MATLAB Computing and Simulink Modeling Fourth EditionSteven T. Karris Includes step-by-stepmn proceduresN 1 j2 -- -- -- - N X[ m ] = x [n ]e for designingn=0analog and digital filters Orchard Publications www.orchardpublications.com

2. Signals and Systemswith MATLAB Computingand Simulink Modeling Fourth EditionSteven T. KarrisOrchard Publicationswww.orchardpublications.com 3. Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth EditionCopyright 2008 Orchard Publications. All rights reserved. Printed in the United States of America. No part of thispublication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system,without the prior written permission of the publisher.Direct all inquiries to Orchard Publications, info@orchardpublications.comProduct and corporate names are trademarks or registered trademarks of the Microsoft Corporation and TheMathWorks Inc. They are used only for identification and explanation, without intent to infringe.Library of Congress Cataloging-in-Publication DataCatalog record is available from the Library of CongressLibrary of Congress Control Number: 2008927083ISBN13: 9781934404126ISBN10: 1934404128Copyright TX 5471562 4. PrefaceThis text contains a comprehensive discussion on continuous and discrete time signals andsystems with many MATLAB and several Simulink examples. It is written for junior andsenior electrical and computer engineering students, and for selfstudy by working professionals.The prerequisites are a basic course in differential and integral calculus, and basic electric circuittheory.This book can be used in a twoquarter, or one semester course. This author has taught thesubject material for many years and was able to cover all material in 16 weeks, with 2 lecturehours per week.To get the most out of this text, it is highly recommended that Appendix A is thoroughlyreviewed. This appendix serves as an introduction to MATLAB, and is intended for those whoare not familiar with it. The Student Edition of MATLAB is an inexpensive, and yet a verypowerful software package; it can be found in many college bookstores, or can be obtained directlyfromThe MathWorks Inc., 3 Apple Hill Drive, Natick, MA 017602098Phone: 508 6477000, Fax: 508 6477001http://www.mathworks.comemail: info@mathworks.comThe elementary signals are reviewed in Chapter 1, and several examples are given. The purpose ofthis chapter is to enable the reader to express any waveform in terms of the unit step function, andsubsequently the derivation of the Laplace transform of it. Chapters 2 through 4 are devoted toLaplace transformation and circuit analysis using this transform. Chapter 5 is an introduction tostatespace and contains many illustrative examples. Chapter 6 discusses the impulse response.Chapters 7 and 8 are devoted to Fourier series and transform respectively. Chapter 9 introducesdiscretetime signals and the Z transform. Considerable time was spent on Chapter 10 to presentthe Discrete Fourier transform and FFT with the simplest possible explanations. Chapter 11contains a thorough discussion to analog and digital filters analysis and design procedures. Asmentioned above, Appendix A is an introduction to MATLAB. Appendix B is an introduction toSimulink, Appendix C contains a review of complex numbers, and Appendix D is an introductionto matrix theory.New to the Second EditionThis is an extensive revision of the first edition. The most notable change is the inclusion of thesolutions to all exercises at the end of each chapter. It is in response to many readers whoexpressed a desire to obtain the solutions in order to check their solutions to those of the authorand thereby enhancing their knowledge. Another reason is that this text is written also for self 5. study by practicing engineers who need a review before taking more advanced courses such asdigital image processing.Another major change is the addition of a rather comprehensive summary at the end of eachchapter. Hopefully, this will be a valuable aid to instructors for preparation of view foils forpresenting the material to their class.New to the Third EditionThe most notable change is the inclusion of Simulink modeling examples. The pages where theyappear can be found in the Table of Contents section of this text. Another change is theimprovement of the plots generated by the latest revisions of the MATLAB Student Version,Release 14.The author wishes to express his gratitude to the staff of The MathWorks, the developers ofMATLAB and Simulink, especially to Ms. Courtney Esposito, for the encouragement andunlimited support they have provided me with during the production of this text.Our heartfelt thanks also to Ms. Sally Wright, P.E., of Renewable Energy Research LaboratoryUniversity of Massachusetts, Amherst, for bringing some errors on the previous editions to ourattention.New to the Fourth EditionThe most notable change is the inclusion of Appendix E on window functions. The plots weregenerated generated with the latest revisions of the MATLAB R2008a edition. Also, two end-of- chapter exercises were added in Chapter 10 to illustrate the use of the fft and ifft MATLABfunctionsThe author wishes to express his gratitude to the staff of The MathWorks, the developers ofMATLAB and Simulink, especially to The MathWorks Book Program Team, for theencouragement and unlimited support they have provided me with during the production of thisand all other texts by this publisher.Orchard Publicationswww.orchardpublications.cominfo@orchardpublications.com2 6. Table of Contents1 Elementary Signals 111.1Signals Described in Math Form .............................................................................111.2The Unit Step Function ..........................................................................................121.3The Unit Ramp Function ......................................................................................1101.4The Delta Function ............................................................................................... 111 1.4.1 The Sampling Property of the Delta Function ............................................112 1.4.2 The Sifting Property of the Delta Function ................................................1131.5Higher Order Delta Functions...............................................................................1141.6Summary ................................................................................................................1221.7Exercises.................................................................................................................1231.8Solutions to EndofChapter Exercises ................................................................124MATLAB ComputingPages 120, 121Simulink ModelingPage 1182The Laplace Transformation21 2.1 Definition of the Laplace Transformation...............................................................21 2.2 Properties and Theorems of the Laplace Transform ...............................................22 2.2.1 Linearity Property ........................................................................................23 2.2.2 Time Shifting Property .................................................................................23 2.2.3 Frequency Shifting Property ........................................................................24 2.2.4 Scaling Property ...........................................................................................24 2.2.5 Differentiation in Time Domain Property ...................................................24 2.2.6 Differentiation in Complex Frequency Domain Property ...........................26 2.2.7 Integration in Time Domain Property .........................................................26 2.2.8 Integration in Complex Frequency Domain Property .................................28 2.2.9 Time Periodicity Property ............................................................................28 2.2.10 Initial Value Theorem..................................................................................29 2.2.11 Final Value Theorem .................................................................................210 2.2.12 Convolution in Time Domain Property.....................................................211 2.2.13 Convolution in Complex Frequency Domain Property.............................212 2.3 The Laplace Transform of Common Functions of Time.......................................214 2.3.1 The Laplace Transform of the Unit Step Function u 0 ( t ) ..........................214 2.3.2 The Laplace Transform of the Ramp Function u 1 ( t ) ................................214 2.3.3 The Laplace Transform of t n u0 ( t ) ..............................................................215Signals and Systems with MATLAB Computing and Simulink Modeling, Third Edition iCopyright Orchard Publications 7. 2.3.4 The Laplace Transform of the Delta Function ( t ) ................................. 218 2.3.5 The Laplace Transform of the Delayed Delta Function ( t a ) .............. 218 2.3.6 The Laplace Transform of e at u 0 ( t ) .......................................................... 219 at 2.3.7 The Laplace Transform of t n e u 0 ( t ) ....................................................... 219 2.3.8 The Laplace Transform of sin t u 0 t ......................................................... 220 2.3.9 The Laplace Transform of cos t u 0 t ......................................................... 220 2.3.10 The Laplace Transform of e at sin t u 0 ( t ) ................................................. 221 2.3.11 The Laplace Transform of e at cos t u 0 ( t ) ................................................. 222 2.4 The Laplace Transform of Common Waveforms .................................................. 223 2.4.1 The Laplace Transform of a Pulse............................................................... 223 2.4.2 The Laplace Transform of a Linear Segment .............................................. 223 2.4.3 The Laplace Transform of a Triangular Waveform .................................... 224 2.4.4 The Laplace Transform of a Rectangular Periodic Waveform.................... 225 2.4.5 The Laplace Transform of a HalfRectified Sine Waveform ..................... 226 2.5 Using MATLAB for Finding the Laplace Transforms of Time Functions ............ 227 2.6 Summary ................................................................................................................ 228 2.7 Exercises................................................................................................................. 231 The Laplace Transform of a Sawtooth Periodic Waveform ............................... 232 The Laplace Transform of a FullRectified Sine Waveform.............................. 232 2.8 Solutions to EndofChapter Exercises................................................................. 2333The Inverse Laplace Transform31 3.1 The Inverse Laplace Transform Integral ..................................................................31 3.2 Partial Fraction Expansion........................................................................................31 3.2.1 Distinct Poles..................................................................................................32 3.2.2 Complex Poles ................................................................................................35 3.2.3 Multiple (Repeated) Poles..............................................................................38 3.3 Case where F(s) is Improper Rational Function.....................................................313 3.4 Alternate Method of Partial Fraction Expansion...................................................315 3.5 Summary .................................................................................................................319 3.6 Exercises..................................................................................................................321 3.7 Solutions to EndofChapter Exercises .................................................................322 MATLAB Computing Pages 33, 34, 35, 36, 38, 310, 312, 313, 314, 3224Circuit Analysis with Laplace Transforms 41 4.1 Circuit Transformation from Time to Complex Frequency.................................... 41 4.1.1 Resistive Network Transformation ............................................................... 41 4.1.2 Inductive Network Transformation .............................................................. 41 4.1.3 Capacitive Network Transformation ............................................................ 41ii Signals and Systems with MATLAB Computing and Simulink Modeling, Third EditionCopyright Orchard Publications 8. 4.2Complex Impedance Z(s).........................................................................................484.3Complex Admittance Y(s) .....................................................................................4114.4Transfer Functions .................................................................................................4134.5Using the Simulink Transfer Fcn Block.................................................................4174.6Summary.................................................................................................................4204.7Exercises .................................................................................................................4214.8Solutions to EndofChapter Exercises.................................................................424MATLAB ComputingPages 46, 48, 412, 416, 417, 418, 426, 427, 428, 429, 434Simulink ModelingPage 4175 State Variables and State Equations 515.1Expressing Differential Equations in State Equation Form................................... 515.2Solution of Single State Equations ........................................................................ 565.3The State Transition Matrix ................................................................................. 595.4Computation of the State Transition Matrix ...................................................... 511 5.4.1 Distinct Eigenvalues ................................................................................. 511 5.4.2 Multiple (Repeated) Eigenvalues ............................................................. 5155.5 Eigenvectors......................................................................................................... 5185.6 Circuit Analysis with State Variables.................................................................. 5225.7 Relationship between State Equations and Laplace Transform.......................... 5305.8 Summary .............................................................................................................. 5385.9 Exercises .............................................................................................................. 5415.10 Solutions to EndofChapter Exercises .............................................................. 543MATLAB ComputingPages 514, 515, 518, 526, 536, 548, 551Simulink ModelingPages 527, 537, 5456 The Impulse Response and Convolution61 6.1 The Impulse Response in Time Domain ................................................................ 61 6.2 Even and Odd Functions of Time .......................................................................... 64 6.3 Convolution ............................................................................................................ 67 6.4 Graphical Evaluation of the Convolution Integral................................................. 68 6.5 Circuit Analysis with the Convolution Integral ................................................... 618 6.6 Summary ............................................................................................................... 621 6.7 Exercises................................................................................................................ 623Signals and Systems with MATLAB Computing and Simulink Modeling, Third EditioniiiCopyright Orchard Publications 9. 6.8 Solutions to EndofChapter Exercises................................................................ 625 MATLAB Applications Pages 612, 615, 6307Fourier Series 71 7.1 Wave Analysis......................................................................................................... 71 7.2 Evaluation of the Coefficients................................................................................. 72 7.3 Symmetry in Trigonometric Fourier Series ............................................................. 767.3.1 Symmetry in Square Waveform..................................................................... 787.3.2 Symmetry in Square Waveform with Ordinate Axis Shifted ........................ 787.3.3 Symmetry in Sawtooth Waveform................................................................. 797.3.4 Symmetry in Triangular Waveform ............................................................... 797.3.5 Symmetry in Fundamental, Second, and Third Harmonics........................ 710 7.4 Trigonometric Form of Fourier Series for Common Waveforms.......................... 7107.4.1 Trigonometric Fourier Series for Square Waveform ................................... 7117.4.2 Trigonometric Fourier Series for Sawtooth Waveform............................... 7147.4.3 Trigonometric Fourier Series for Triangular Waveform ............................. 7167.4.4 Trigonometric Fourier Series for HalfWave Rectifier Waveform............. 7177.4.5 Trigonometric Fourier Series for FullWave Rectifier Waveform.............. 720 7.5 Gibbs Phenomenon ............................................................................................... 724 7.6 Alternate Forms of the Trigonometric Fourier Series .......................................... 724 7.7 Circuit Analysis with Trigonometric Fourier Series............................................. 728 7.8 The Exponential Form of the Fourier Series........................................................ 731 7.9 Symmetry in Exponential Fourier Series .............................................................. 7337.9.1 Even Functions ........................................................................................... 7337.9.2 Odd Functions ............................................................................................ 7347.9.3 Half-Wave Symmetry ................................................................................. 7347.9.4 No Symmetry .............................................................................................. 7347.9.5 Relation of C n to C n ................................................................................ 734 7.10 Line Spectra.......................................................................................................... 736 7.11 Computation of RMS Values from Fourier Series................................................ 741 7.12 Computation of Average Power from Fourier Series ........................................... 744 7.13 Evaluation of Fourier Coefficients Using Excel ................................................ 746 7.14 Evaluation of Fourier Coefficients Using MATLAB ........................................ 747 7.15 Summary ............................................................................................................... 750 7.16 Exercises ............................................................................................................... 753 7.17 Solutions to EndofChapter Exercises ............................................................... 755 MATLAB Computing Pages 738, 747iv Signals and Systems with MATLAB Computing and Simulink Modeling, Third EditionCopyright Orchard Publications 10. Simulink ModelingPage 7318 The Fourier Transform818.1Definition and Special Forms ................................................................................ 818.2Special Forms of the Fourier Transform ................................................................ 82 8.2.1 Real Time Functions.................................................................................. 83 8.2.2 Imaginary Time Functions ......................................................................... 868.3Properties and Theorems of the Fourier Transform .............................................. 89 8.3.1 Linearity...................................................................................................... 89 8.3.2 Symmetry.................................................................................................... 89 8.3.3 Time Scaling............................................................................................. 810 8.3.4 Time Shifting............................................................................................ 811 8.3.5 Frequency Shifting ................................................................................... 811 8.3.6 Time Differentiation ................................................................................ 812 8.3.7 Frequency Differentiation ........................................................................ 813 8.3.8 Time Integration ...................................................................................... 813 8.3.9 Conjugate Time and Frequency Functions.............................................. 813 8.3.10 Time Convolution .................................................................................... 814 8.3.11 Frequency Convolution............................................................................ 815 8.3.12 Area Under f ( t ) ........................................................................................ 815 8.3.13 Area Under F ( ) ...................................................................................... 815 8.3.14 Parsevals Theorem................................................................................... 8168.4Fourier Transform Pairs of Common Functions.................................................. 818 8.4.1 The Delta Function Pair .......................................................................... 818 8.4.2 The Constant Function Pair .................................................................... 818 8.4.3 The Cosine Function Pair ........................................................................ 819 8.4.4 The Sine Function Pair............................................................................. 820 8.4.5 The Signum Function Pair........................................................................ 820 8.4.6 The Unit Step Function Pair .................................................................... 822 j 0 t 8.4.7 The eu0 ( t ) Function Pair .................................................................... 824 8.4.8 The ( cos 0 t ) ( u 0 t ) Function Pair ............................................................... 824 8.4.9 The ( sin 0 t ) ( u 0 t ) Function Pair ............................................................... 825 8.5 Derivation of the Fourier Transform from the Laplace Transform .................... 825 8.6 Fourier Transforms of Common Waveforms ...................................................... 827 8.6.1 The Transform of f ( t ) = A [ u 0 ( t + T ) u 0 ( t T ) ] ....................................... 827 8.6.2 The Transform of f ( t ) = A [ u 0 ( t ) u 0 ( t 2T ) ] ........................................... 828 8.6.3 The Transform of f ( t ) = A [ u 0 ( t + T ) + u 0 ( t ) u 0 ( t T ) u 0 ( t 2T ) ] ........... 829Signals and Systems with MATLAB Computing and Simulink Modeling, Third EditionvCopyright Orchard Publications 11. 8.6.4 The Transform of f ( t ) = A cos 0 t [ u0 ( t + T ) u 0 ( t T ) ] .............................. 830 8.6.5 The Transform of a Periodic Time Function with Period T..................... 831 8.6.6 The Transform of the Periodic Time Function f ( t ) = A n = ( t nT ) .... 832 8.7Using MATLAB for Finding the Fourier Transform of Time Functions............ 833 8.8The System Function and Applications to Circuit Analysis............................... 834 8.9Summary .............................................................................................................. 842 8.10 Exercises............................................................................................................... 847 8.11 Solutions to EndofChapter Exercises .............................................................. 849 MATLAB Computing Pages 833, 834, 850, 854, 855, 856, 859, 8609DiscreteTime Systems and the Z Transform 91 9.1Definition and Special Forms of the Z Transform ............................................... 91 9.2Properties and Theorems of the Z Transform...................................................... 939.2.1 Linearity ..................................................................................................... 939.2.2 Shift of f [ n ]u 0 [ n ] in the DiscreteTime Domain ..................................... 939.2.3 Right Shift in the DiscreteTime Domain ................................................ 949.2.4 Left Shift in the DiscreteTime Domain................................................... 95 n 9.2.5 Multiplication by a in the DiscreteTime Domain................................. 96 naT9.2.6 Multiplication by e in the DiscreteTime Domain ........................... 969.2.7 Multiplication by n and n2 in the DiscreteTime Domain ..................... 969.2.8 Summation in the DiscreteTime Domain ............................................... 979.2.9 Convolution in the DiscreteTime Domain ............................................. 989.2.10 Convolution in the DiscreteFrequency Domain ..................................... 999.2.11 Initial Value Theorem ............................................................................... 999.2.12 Final Value Theorem............................................................................... 910 9.3The Z Transform of Common DiscreteTime Functions.................................. 9119.3.1 The Transform of the Geometric Sequence .............................................9119.3.2 The Transform of the DiscreteTime Unit Step Function ......................9149.3.3 The Transform of the DiscreteTime Exponential Sequence .................9169.3.4 The Transform of the DiscreteTime Cosine and Sine Functions ..........9169.3.5 The Transform of the DiscreteTime Unit Ramp Function....................918 9.4Computation of the Z Transform with Contour Integration .............................920 9.5Transformation Between s and zDomains .......................................................922 9.6The Inverse Z Transform ...................................................................................925vi Signals and Systems with MATLAB Computing and Simulink Modeling, Third EditionCopyright Orchard Publications 12. 9.6.1 Partial Fraction Expansion ..................................................................... 9259.6.2 The Inversion Integral............................................................................ 9329.6.3 Long Division of Polynomials ................................................................ 9369.7 The Transfer Function of DiscreteTime Systems ............................................ 9389.8 State Equations for DiscreteTime Systems ...................................................... 9459.9 Summary............................................................................................................. 9489.10Exercises ............................................................................................................. 9539.11Solutions to EndofChapter Exercises............................................................. 955MATLAB ComputingPages 935, 937, 938, 941, 942, 959, 961Simulink ModelingPage 944Excel PlotsPages 935, 94410The DFT and the FFT Algorithm 10110.1 The Discrete Fourier Transform (DFT) ............................................................10110.2 Even and Odd Properties of the DFT ................................................................10910.3 Common Properties and Theorems of the DFT ..............................................1010 10.3.1 Linearity ...............................................................................................1010 10.3.2 Time Shift ............................................................................................1011 10.3.3 Frequency Shift....................................................................................1012 10.3.4 Time Convolution ...............................................................................1012 10.3.5 Frequency Convolution .......................................................................101310.4 The Sampling Theorem ...................................................................................101310.5 Number of Operations Required to Compute the DFT ..................................101610.6 The Fast Fourier Transform (FFT) ..................................................................101710.7 Summary...........................................................................................................102810.8 Exercises ...........................................................................................................103110.9 Solutions to EndofChapter Exercises...........................................................1033MATLAB ComputingPages 105, 107, 1034Excel Analysis ToolPakPages 106, 10811Analog and Digital Filters11.1 Filter Types and Classifications......................................................................... 11111.2 Basic Analog Filters........................................................................................... 112Signals and Systems with MATLAB Computing and Simulink Modeling, Third Edition viiCopyright Orchard Publications 13. 11.2.1 RC LowPass Filter ............................................................................... 11211.2.2 RC HighPass Filter .............................................................................. 11411.2.3 RLC BandPass Filter.............................................................................11711.2.4 RLC BandElimination Filter ................................................................118 11.3 LowPass Analog Filter Prototypes ..................................................................111011.3.1 Butterworth Analog LowPass Filter Design .......................................111411.3.2 Chebyshev Type I Analog LowPass Filter Design..............................112511.3.3 Chebyshev Type II Analog LowPass Filter Design ............................113811.3.4 Elliptic Analog LowPass Filter Design ...............................................1139 11.4 HighPass, BandPass, and BandElimination Filter Design..........................1141 11.5 Digital Filters ....................................................................................................1151 11.6 Digital Filter Design with Simulink..................................................................117011.6.1 The Direct Form I Realization of a Digital Filter.................................117011.6.2 The Direct Form II Realization of a Digital Filter................................117111.6.3 The Series Form Realization of a Digital Filter ....................................117311.6.4 The Parallel Form Realization of a Digital Filter .................................117511.6.5 The Digital Filter Design Block............................................................1178 11.7 Summary ...........................................................................................................1187 11.8 Exercises ...........................................................................................................1191 11.9 Solutions to EndofChapter Exercises ...........................................................1197 MATLAB Computing Pages 113, 114, 116, 117, 119, 1115, 1119, 1123, 1124, 1131, 1135, 1136, 1137, 1138, 1140, 1141, 1142, 1143, 1145, 1146, 1148, 1150, 1155, 1156, 1157, 1160, 1162, 1164, 1167, 1168, and 1197 through 11106 Simulink Modeling Pages 1171, 1174, 1177, 1178, 1180, 1182, 1183, 1184AIntroduction to MATLABA1A.1 MATLAB and Simulink ........................................................................... A1A.2 Command Window ......................................................................................... A1A.3 Roots of Polynomials ....................................................................................... A3A.4 Polynomial Construction from Known Roots ................................................. A4A.5 Evaluation of a Polynomial at Specified Values .............................................. A6A.6 Rational Polynomials ....................................................................................... A8A.7 Using MATLAB to Make Plots..................................................................... A10A.8 Subplots ......................................................................................................... A18A.9 Multiplication, Division, and Exponentiation .............................................. A18A.10Script and Function Files .............................................................................. A26A.11Display Formats ............................................................................................. A31viii Signals and Systems with MATLAB Computing and Simulink Modeling, Third EditionCopyright Orchard Publications 14. MATLAB ComputingPages A3 through A8, A10, A13, A14, A16, A17,A21, A22, A24, A27BIntroduction to SimulinkB1 B.1 Simulink and its Relation to MATLAB............................................................. B1 B.2 Simulink Demos ............................................................................................... B20 MATLAB Computing Page B4 Simulink Modeling Pages B7, B12, B14, B18CA Review of Complex Numbers C1 C.1 Definition of a Complex Number....................................................................... C1 C.2 Addition and Subtraction of Complex Numbers ............................................... C2 C.3 Multiplication of Complex Numbers.................................................................. C3 C.4 Division of Complex Numbers ........................................................................... C4 C.5 Exponential and Polar Forms of Complex Numbers.......................................... C4 MATLAB Computing Pages C6, C7, C8 Simulink Modeling Page C7D Matrices and Determinants D1D.1Matrix Definition.............................................................................................D1D.2Matrix Operations ...........................................................................................D2D.3Special Forms of Matrices................................................................................D6D.4Determinants .................................................................................................D10D.5Minors and Cofactors ....................................................................................D12D.6Cramers Rule ................................................................................................D17D.7Gaussian Elimination Method.......................................................................D19D.8The Adjoint of a Matrix ................................................................................D21D.9Singular and NonSingular Matrices ............................................................D21D.10 The Inverse of a Matrix .................................................................................D22D.11 Solution of Simultaneous Equations with Matrices ......................................D24D.12 Exercises.........................................................................................................D31Signals and Systems with MATLAB Computing and Simulink Modeling, Third EditionixCopyright Orchard Publications 15. MATLAB Computing Pages D3, D4, D5, D7, D8, D9, D10, D12, D19, D23, D27, D29 Simulink Modeling Page D3 Excel Spreadsheet Page D28E Window FunctionsE1E.1 Window Function Defined .................................................................................. E1E.2 Common Window Functions ............................................................................... E1E.2.1 Rectangular Window Function ................................................................. E2E.2.2 Triangular Window Function.................................................................... E5E.2.3 Hanning Window Function....................................................................... E7E.2.4 Hamming Window Function..................................................................... E9E.2.5 Blackman Window Function................................................................... E12E.2.6 Kaiser Family of Window Functions ....................................................... E14E.3 Other Window Functions .................................................................................. E15E.4 Fourier Series Method for Approximating an FIR Amplitude Response .......... E17ReferencesR1IndexIN1x Signals and Systems with MATLAB Computing and Simulink Modeling, Third Edition Copyright Orchard Publications 16. Chapter 1 Elementary SignalsT his chapter begins with a discussion of elementary signals that may be applied to electric networks. The unit step, unit ramp, and delta functions are then introduced. The sampling and sifting properties of the delta function are defined and derived. Several examples forexpressing a variety of waveforms in terms of these elementary signals are provided. Throughoutthis text, a left justified horizontal bar will denote the beginning of an example, and a right justi-fied horizontal bar will denote the end of the example. These bars will not be shown whenever anexample begins at the top of a page or at the bottom of a page. Also, when one example followsimmediately after a previous example, the right justified bar will be omitted.1.1 Signals Described in Math FormConsider the network of Figure 1.1 where the switch is closed at time t = 0 .R t = 0++ v out open terminals vSFigure 1.1. A switched network with open terminalsWe wish to describe v out in a math form for the time interval < t < + . To do this, it is conve-nient to divide the time interval into two parts, < t < 0 , and 0 < t < .For the time interval < t < 0 , the switch is open and therefore, the output voltage v out is zero.In other words, v out = 0 for < t < 0(1.1)For the time interval 0 < t < , the switch is closed. Then, the input voltage v S appears at theoutput, i.e., v out = v S for 0 < t < (1.2)Combining (1.1) and (1.2) into a single relationship, we obtain 0 < t < 0v out = (1.3) vS 0 < t < Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition11Copyright Orchard Publications 17. Chapter 1 Elementary SignalsWe can express (1.3) by the waveform shown in Figure 1.2.v out vS 0t Figure 1.2. Waveform for v out as defined in relation (1.3)The waveform of Figure 1.2 is an example of a discontinuous function. A function is said to be dis-continuous if it exhibits points of discontinuity, that is, the function jumps from one value toanother without taking on any intermediate values.1.2 The Unit Step Function u 0 ( t )A well known discontinuous function is the unit step function u 0 ( t ) * which is defined as 0 t0It is also represented by the waveform of Figure 1.3. u0 ( t )10 tFigure 1.3. Waveform for u 0 ( t )In the waveform of Figure 1.3, the unit step function u 0 ( t ) changes abruptly from 0 to 1 att = 0 . But if it changes at t = t 0 instead, it is denoted as u 0 ( t t 0 ) . In this case, its waveform anddefinition are as shown in Figure 1.4 and relation (1.5) respectively. 1u0 ( t t0 )t 0t0Figure 1.4. Waveform for u 0 ( t t 0 )* In some books, the unit step function is denoted as u ( t ) , that is, without the subscript 0. In this text, however, wewill reserve the u ( t ) designation for any input when we will discuss state variables in Chapter 5.12Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright Orchard Publications 18. The Unit Step Function0t < t0u0 ( t t0 ) = (1.5)1t > t0If the unit step function changes abruptly from 0 to 1 at t = t 0 , it is denoted as u 0 ( t + t 0 ) . Inthis case, its waveform and definition are as shown in Figure 1.5 and relation (1.6) respectively.u0 ( t + t0 )1 t0 0tFigure 1.5. Waveform for u 0 ( t + t 0 ) 0 t < t0 u0 ( t + t0 ) = (1.6) 1 t > t0Example 1.1Consider the network of Figure 1.6, where the switch is closed at time t = T .R t = T++v out open terminals vS Figure 1.6. Network for Example 1.1Express the output voltage v out as a function of the unit step function, and sketch the appropriatewaveform.Solution:For this example, the output voltage v out = 0 for t < T , and v out = v S for t > T . Therefore,v out = v S u 0 ( t T )(1.7)and the waveform is shown in Figure 1.7.Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition13Copyright Orchard Publications 19. Chapter 1 Elementary SignalsvS u0 ( t T )v out t 0TFigure 1.7. Waveform for Example 1.1Other forms of the unit step function are shown in Figure 1.8.ttt 00 0(a) (b) (c)AA A A u0 ( t )A u0 ( t T ) A u0 ( t + T )Au 0 ( t ) Au 0 ( t + T )Au 0 ( t T ) AA A t t 0t 0 (d)0 (e) (f)tt0 t 0 (g)0(h)(i) A AA A u0 ( t ) A u0 ( t + T )A u0 ( t T ) Figure 1.8. Other forms of the unit step functionUnit step functions can be used to represent other timevarying functions such as the rectangularpulse shown in Figure 1.9.u0 ( t )11ttt 0 100(c) (a)(b)u0 ( t 1 )Figure 1.9. A rectangular pulse expressed as the sum of two unit step functions14 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth EditionCopyright Orchard Publications 20. The Unit Step FunctionThus, the pulse of Figure 1.9(a) is the sum of the unit step functions of Figures 1.9(b) and 1.9(c)and it is represented as u 0 ( t ) u 0 ( t 1 ) .The unit step function offers a convenient method of describing the sudden application of a volt-age or current source. For example, a constant voltage source of 24 V applied at t = 0 , can bedenoted as 24u 0 ( t ) V . Likewise, a sinusoidal voltage source v ( t ) = V m cos t V that is applied toa circuit at t = t0 , can be described as v ( t ) = ( V m cos t )u 0 ( t t 0 ) V . Also, if the excitation in acircuit is a rectangular, or triangular, or sawtooth, or any other recurring pulse, it can be repre-sented as a sum (difference) of unit step functions.Example 1.2Express the square waveform of Figure 1.10 as a sum of unit step functions. The vertical dottedlines indicate the discontinuities at T, 2T, 3T , and so on. v(t) A T 2T 3Tt 0A Figure 1.10. Square waveform for Example 1.2Solution:Line segmenthas height A , starts at t = 0 , and terminates at t = T . Then, as in Example 1.1, thissegment is expressed as v1 ( t ) = A [ u0 ( t ) u0 ( t T ) ] (1.8)Line segment has height A , starts at t = T and terminates at t = 2T . This segment isexpressed asv 2 ( t ) = A [ u 0 ( t T ) u 0 ( t 2T ) ](1.9)Line segmenthas height A , starts at t = 2T and terminates at t = 3T . This segment is expressedas v 3 ( t ) = A [ u 0 ( t 2T ) u 0 ( t 3T ) ](1.10)Line segmenthas height A , starts at t = 3T , and terminates at t = 4T . It is expressed as v 4 ( t ) = A [ u 0 ( t 3T ) u 0 ( t 4T ) ](1.11)Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition15Copyright Orchard Publications 21. Chapter 1 Elementary SignalsThus, the square waveform of Figure 1.10 can be expressed as the summation of (1.8) through(1.11), that is,v ( t ) = v1 ( t ) + v2 ( t ) + v3 ( t ) + v4 ( t ) = A [ u 0 ( t ) u 0 ( t T ) ] A [ u 0 ( t T ) u 0 ( t 2T ) ] (1.12) +A [ u 0 ( t 2T ) u 0 ( t 3T ) ] A [ u 0 ( t 3T ) u 0 ( t 4T ) ]Combining like terms, we obtain v ( t ) = A [ u 0 ( t ) 2u 0 ( t T ) + 2u 0 ( t 2T ) 2u 0 ( t 3T ) + ] (1.13)Example 1.3Express the symmetric rectangular pulse of Figure 1.11 as a sum of unit step functions.i(t)A t T 20 T2Figure 1.11. Symmetric rectangular pulse for Example 1.3Solution:This pulse has height A , starts at t = T 2 , and terminates at t = T 2 . Therefore, with refer-ence to Figures 1.5 and 1.8 (b), we obtaini ( t ) = Au 0 t + -- Au 0 t -- = A u 0 t + -- u 0 t -- T TTT - 2 - - 2 - (1.14) 22Example 1.4Express the symmetric triangular waveform of Figure 1.12 as a sum of unit step functions.v(t) 1tT 2 0T2Figure 1.12. Symmetric triangular waveform for Example 1.4Solution:16 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth EditionCopyright Orchard Publications 22. The Unit Step FunctionWe first derive the equations for the linear segmentsand shown in Figure 1.13. 2 v( t) 2 -- t + 1-1 -- t + 1- T T tT 2 0 T2Figure 1.13. Equations for the linear segments of Figure 1.12For line segment,v 1 ( t ) = -- t + 1 u 0 t + -- u 0 ( t )2 TT - 2 - (1.15)and for line segment, v 2 ( t ) = -- t + 1 u 0 ( t ) u 0 t T 2----(1.16) T 2Combining (1.15) and (1.16), we obtainv ( t ) = v1 ( t ) + v2 ( t ) (1.17) = -- t + 1 u 0 t + T u 0 ( t ) + -- t + 1 u 0 ( t ) u 0 t T 2-2 --- ---- T 2 T 2Example 1.5Express the waveform of Figure 1.14 as a sum of unit step functions. v( t)321 t 0 1 2 3Figure 1.14. Waveform for Example 1.5Solution:Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition17Copyright Orchard Publications 23. Chapter 1 Elementary SignalsAs in the previous example, we first find the equations of the linear segments linear segmentsandshown in Figure 1.15. v(t) 3 22t + 1 1 t+3t 0123 Figure 1.15. Equations for the linear segments of Figure 1.14Following the same procedure as in the previous examples, we obtain v ( t ) = ( 2t + 1 ) [ u 0 ( t ) u 0 ( t 1 ) ] + 3 [ u 0 ( t 1 ) u 0 ( t 2 ) ] + ( t + 3 ) [ u0 ( t 2 ) u0 ( t 3 ) ]Multiplying the values in parentheses by the values in the brackets, we obtain v ( t ) = ( 2t + 1 )u 0 ( t ) ( 2t + 1 )u 0 ( t 1 ) + 3u 0 ( t 1 ) 3u 0 ( t 2 ) + ( t + 3 )u 0 ( t 2 ) ( t + 3 )u 0 ( t 3 )v ( t ) = ( 2t + 1 )u 0 ( t ) + [ ( 2t + 1 ) + 3 ]u 0 ( t 1 ) + [ 3 + ( t + 3 ) ]u 0 ( t 2 ) ( t + 3 )u 0 ( t 3 )and combining terms inside the brackets, we obtain v ( t ) = ( 2t + 1 )u 0 ( t ) 2 ( t 1 )u 0 ( t 1 ) t u 0 ( t 2 ) + ( t 3 )u 0 ( t 3 ) (1.18)Two other functions of interest are the unit ramp function, and the unit impulse or delta function.We will introduce them with the examples that follow.Example 1.6In the network of Figure 1.16 i S is a constant current source and the switch is closed at timet = 0 . Express the capacitor voltage v C ( t ) as a function of the unit step.18Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright Orchard Publications 24. The Unit Step Function R t = 0+vC ( t )iSC Figure 1.16. Network for Example 1.6Solution:The current through the capacitor is i C ( t ) = i S = cons tan t , and the capacitor voltage v C ( t ) ist1 *v C ( t ) = ---C- i C ( ) d (1.19)where is a dummy variable.Since the switch closes at t = 0 , we can express the current i C ( t ) as iC ( t ) = iS u0 ( t ) (1.20)and assuming that v C ( t ) = 0 for t < 0 , we can write (1.19) asiS 0 1t ---- u0 ( ) diStv C ( t ) = ---C- i S u 0 ( ) d = C+ --- C - 0 u 0 ( ) d (1.21) 0or iS v C ( t ) = ---- tu 0 ( t )- (1.22)CTherefore, we see that when a capacitor is charged with a constant current, the voltage across it isa linear function and forms a ramp with slope i S C as shown in Figure 1.17.vC ( t ) slope = i S C t 0Figure 1.17. Voltage across a capacitor when charged with a constant current source* Since the initial condition for the capacitor voltage was not specified, we express this integral with at the lower limit ofintegration so that any non-zero value prior to t < 0 would be included in the integration.Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition19Copyright Orchard Publications 25. Chapter 1 Elementary Signals1.3 The Unit Ramp Function u 1 ( t )The unit ramp function, denoted as u 1 ( t ) , is defined astu1 ( t ) = u 0 ( ) d (1.23)where is a dummy variable.We can evaluate the integral of (1.23) by considering the area under the unit step function u 0 ( t )from to t as shown in Figure 1.18.Area = 1 = = t1tFigure 1.18. Area under the unit step function from to tTherefore, we define u 1 ( t ) as0t 0 .Therefore, we conclude that if f ( t ) is of exponential order, L { f ( t ) } exists ifRe { s } = > 0 (2.9)where Re { s } denotes the real part of the complex variable s .Evaluation of the integral of (2.4) involves contour integration in the complex plane, and thus, itwill not be attempted in this chapter. We will see in the next chapter that many Laplace trans-forms can be inverted with the use of a few standard pairs, and thus there is no need to use (2.4)to obtain the Inverse Laplace transform.In our subsequent discussion, we will denote transformation from the time domain to the com-plex frequency domain, and vice versa, asf(t) F(s)(2.10)2.2 Properties and Theorems of the Laplace TransformThe most common properties and theorems of the Laplace transform are presented in Subsec-tions 2.2.1 through 2.2.13 below.0 t* A function f ( t ) is said to be of exponential order if f ( t ) < kefor all t 0 .22Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright Orchard Publications 44. Properties and Theorems of the Laplace Transform2.2.1 Linearity PropertyThe linearity property states that iff 1 ( t ), f 2 ( t ), , f n ( t )have Laplace transforms F 1 ( s ), F 2 ( s ), , F n ( s )respectively, andc 1 , c 2 , , c nare arbitrary constants, then, c1 f1 ( t ) + c2 f2 ( t ) + + cn fn ( t ) c 1 F1 ( s ) + c2 F2 ( s ) + + cn Fn ( s ) (2.11)Proof: L { c1 f1 ( t ) + c2 f2 ( t ) + + cn fn ( t ) } =t 0 [ c 1 f 1 ( t ) + c 2 f 2 ( t ) + + c n f n ( t ) ] dt st st st= c1 t 0f1 ( t ) e dt + c 2 t 0 f2 ( t ) edt + + c n t 0 fn ( t ) edt= c1 F1 ( s ) + c2 F2 ( s ) + + cn Fn ( s )Note 1:It is desirable to multiply f ( t ) by the unit step function u 0 ( t ) to eliminate any unwanted nonzero values of f ( t ) for t < 0 .2.2.2 Time Shifting PropertyThe time shifting property states that a right shift in the time domain by a units, corresponds to asmultiplication by ein the complex frequency domain. Thus, as f ( t a )u 0 ( t a ) e F(s)(2.12)Proof:a st st L { f ( t a )u 0 ( t a ) } =00e dt + a f( t a )edt(2.13)Now, we let t a = ; then, t = + a and dt = d . With these substitutions and with a 0 ,the second integral on the right side of (2.13) is expressed as s ( + a ) as s as0 f()ed = e 0 f ( ) e d = eF(s)Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition23Copyright Orchard Publications 45. Chapter 2 The Laplace Transformation2.2.3 Frequency Shifting PropertyThe frequency shifting property states that if we multiply a time domain function f ( t ) by an expo- atnential function e where a is an arbitrary positive constant, this multiplication will produce ashift of the s variable in the complex frequency domain by a units. Thus, at ef(t) F(s + a ) (2.14)Proof: at at st ( s + a )tL {e f( t) } = 0ef(t)edt = 0 f ( t ) edt = F ( s + a )Note 2:A change of scale is represented by multiplication of the time variable t by a positive scaling fac-tor a . Thus, the function f ( t ) after scaling the time axis, becomes f ( at ) .2.2.4 Scaling PropertyLet a be an arbitrary positive constant; then, the scaling property states that f ( at ) -- F - 1 s - - (2.15)a a Proof: st L { f ( at ) } = 0 f ( at ) edtand letting t = a , we obtaind - = --d ( ) = -- F - s ( a ) 1( s a ) 1s L { f ( at ) } = 0f( )e-a a - 0 f ( ) e-a a -Note 3:Generally, the initial value of f ( t ) is taken at t = 0 to include any discontinuity that may bepresent at t = 0 . If it is known that no such discontinuity exists at t = 0 , we simply interpretf ( 0 ) as f ( 0 ) .2.2.5 Differentiation in Time Domain PropertyThe differentiation in time domain property states that differentiation in the time domain corre-sponds to multiplication by s in the complex frequency domain, minus the initial value of f ( t ) att = 0 . Thus,24 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth EditionCopyright Orchard Publications 46. Properties and Theorems of the Laplace Transformd- f( t ) = ---- f ( t ) sF ( s ) f ( 0 )(2.16) dtProof: st L {f (t)} =0 f( t ) edtUsing integration by parts where v du= uv u dv (2.17) st stwe let du = f( t ) and v = e. Then, u = f ( t ) , dv = se, and thus st st st a L { f( t ) } = f ( t )e 0 +s 0 f(t)edt = limaf ( t )e 0 + sF ( s ) sa = lim [ ef ( a ) f ( 0 ) ] + sF ( s ) = 0 f ( 0 ) + sF ( s ) aThe time differentiation property can be extended to show that d2------- f ( t ) s 2 F ( s ) sf ( 0 ) f( 0 )- (2.18)2dtd3 ------- f ( t ) s 3 F ( s ) s 2 f ( 0 ) sf( 0 ) f( 0 ) -(2.19) 3 dtand in generalnd ------- f ( t ) s n F ( s ) s n 1 f ( 0 ) s n 2 f( 0 ) f - n1(0 ) (2.20) n dtTo prove (2.18), we let dg ( t ) = f( t ) = ---- f ( t ) -dtand as we found above,L { g( t ) } = sL { g ( t ) } g ( 0 )Then, L { f( t ) } = sL { f( t ) } f( 0 ) = s [ sL [ f ( t ) ] f ( 0 ) ] f( 0 )= s 2 F ( s ) sf ( 0 ) f( 0 )Relations (2.19) and (2.20) can be proved by similar procedures.Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition25Copyright Orchard Publications 47. Chapter 2 The Laplace TransformationWe must remember that the terms f ( 0 ), f( 0 ), f( 0 ) , and so on, represent the initial condi-tions. Therefore, when all initial conditions are zero, and we differentiate a time function f ( t ) ntimes, this corresponds to F ( s ) multiplied by s to the nth power.2.2.6 Differentiation in Complex Frequency Domain PropertyThis property states that differentiation in complex frequency domain and multiplication by minusone, corresponds to multiplication of f ( t ) by t in the time domain. In other words, d tf ( t ) ---- F ( s ) - (2.21)dsProof: st L { f( t)} = F( s) =0 f ( t ) edtDifferentiating with respect to s and applying Leibnitzs rule* for differentiation under the integral,we obtain dd st st st st---- F ( s ) = ----ds - ds-0 f( t)edt = 0 se f ( t )dt = 0 t ef ( t )dt = 0 [ tf ( t ) ] edt = L [ tf ( t ) ]In general, nn nd t f ( t ) ( 1 ) ------- F ( s ) - n (2.22) dsThe proof for n 2 follows by taking the second and higherorder derivatives of F ( s ) withrespect to s .2.2.7 Integration in Time Domain PropertyThis property states that integration in time domain corresponds to F ( s ) divided by s plus the ini-tial value of f ( t ) at t = 0 , also divided by s . That is, b* This rule states that if a function of a parameter is defined by the equation F ( ) = a f ( x, ) dxwhere f is some known function of integration x and the parameter , a and b are constants independent of x and , and the partial derivativedF b ( x, )f exists and it is continuous, then ------ =d - a ----------------- dx . ( )26 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth EditionCopyright Orchard Publications 48. Properties and Theorems of the Laplace Transform t F(s) f (0 ) f ( ) d ---------- + ------------ s s - (2.23)Proof:We begin by expressing the integral on the left side of (2.23) as two integrals, that is, t 0 t f ( ) d = f ( ) d + 0 f ( ) d(2.24)The first integral on the right side of (2.24), represents a constant value since neither the upper,nor the lower limits of integration are functions of time, and this constant is an initial conditiondenoted as f ( 0 ) . We will find the Laplace transform of this constant, the transform of the sec-ond integral on the right side of (2.24), and will prove (2.23) by the linearity property. Thus, st st ste 0 f ( 0 ) e 0 e L {f (0 )} = dt = f ( 0 ) dt = f ( 0 ) ------- -s 0 (2.25)f(0 )f(0 )= f ( 0 ) 0 ------------ = ----------- - - ssThis is the value of the first integral in (2.24). Next, we will show that tF(s) 0 f ( ) d ---------- sWe lettg(t) =0 f ( ) dthen, g ( t ) = f ( )and0 g( 0) = 0 f ( ) d = 0Now, L { g ( t ) } = G ( s ) = sL { g ( t ) } g ( 0 ) = G ( s ) 0sL { g ( t ) } = G ( s ) G(s) L { g ( t ) } = -----------sSignals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 27Copyright Orchard Publications 49. Chapter 2 The Laplace Transformation t F(s) L 0f ( ) d = ---------- s (2.26)and the proof of (2.23) follows from (2.25) and (2.26).2.2.8 Integration in Complex Frequency Domain PropertyThis property states that integration in complex frequency domain with respect to s corresponds todivision of a time function f ( t ) by the variable t , provided that the limit lim f ( t ) exists. Thus,--------t0t f( t) -------- t s F ( s ) ds (2.27)Proof: stF(s) = 0 f ( t ) e dtIntegrating both sides from s to , we obtain stsF ( s ) ds = s 0 f ( t ) e dt dsNext, we interchange the order of integration, i.e., stsF ( s ) ds = 0 seds f ( t ) dtand performing the inner integration on the right side integral with respect to s , we obtain f ( t )1 st f(t) st s F ( s ) ds = 0 -- e - ts f ( t ) dt = 0 -------- e tdt = L -------- t 2.2.9 Time Periodicity PropertyThe time periodicity property states that a periodic function of time with period T corresponds to T st sTthe integral 0 f ( t ) edt divided by ( 1 e ) in the complex frequency domain. Thus, if we letf ( t ) be a periodic function with period T , that is, f ( t ) = f ( t + nT ) , for n = 1, 2, 3, we obtainthe transform pair28Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright Orchard Publications 50. Properties and Theorems of the Laplace Transform T st 0 f ( t ) edt f ( t + nT ) ----------------------------- sT- (2.28)1eProof:The Laplace transform of a periodic function can be expressed asT2T 3T st st st stL {f(t)} =0 f( t)edt =0f(t )edt + T f(t)e dt + 2T f ( t ) edt + In the first integral of the right side, we let t = , in the second t = + T , in the thirdt = + 2T , and so on. The areas under each period of f ( t ) are equal, and thus the upper andlower limits of integration are the same for each integral. Then,T TT s s ( + T ) s ( + 2T ) L {f(t)} =0 f( )ed + 0 f( + T)e d + 0 f ( + 2T ) ed + (2.29)Since the function is periodic, i.e., f ( ) = f ( + T ) = f ( + 2T ) = = f ( + nT ) , we can write(2.29) asT sT 2sT sL {f()} = (1 + e+e + ) 0 f ( ) e d(2.30)By application of the binomial theorem, that is,2 31 1 + a + a + a + = ----------- (2.31) 1awe find that expression (2.30) reduces toT s 0 f ( ) e dL { f ( ) } = ---------------------------------- sT1e2.2.10 Initial Value TheoremThe initial value theorem states that the initial value f ( 0 ) of the time function f ( t ) can be foundfrom its Laplace transform multiplied by s and letting s .That is,lim f ( t ) = lim sF ( s ) = f ( 0 )(2.32) t0 sSignals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition29Copyright Orchard Publications 51. Chapter 2 The Laplace TransformationProof:From the time domain differentiation property, d---- f ( t ) sF ( s ) f ( 0 ) -dtord d---- f ( t ) e st dt 0 L ---- f ( t ) = sF ( s ) f ( 0 ) = - - dt dtTaking the limit of both sides by letting s , we obtainT d st ----- f ( t ) e lim [ sF ( s ) f ( 0 ) ] = limlim dt ss T dt 0Interchanging the limiting process, we obtain T d stT dt lim [ sF ( s ) f ( 0 ) ] = lim ---- f ( t ) - lim e dt ss 0and since st lim e= 0 sthe above expression reduces to lim [ sF ( s ) f ( 0 ) ] = 0sor lim sF ( s ) = f ( 0 ) s2.2.11 Final Value TheoremThe final value theorem states that the final value f ( ) of the time function f ( t ) can be foundfrom its Laplace transform multiplied by s , then, letting s 0 . That is, lim f ( t ) = lim sF ( s ) = f ( ) (2.33)ts0Proof:From the time domain differentiation property, d---- f ( t ) sF ( s ) f ( 0 ) -dtor210Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth EditionCopyright Orchard Publications 52. Properties and Theorems of the Laplace Transformd d---- f ( t ) e st dt 0 L ---- f ( t ) = sF ( s ) f ( 0 ) = - - dt dtTaking the limit of both sides by letting s 0 , we obtainTd st ---- f ( t ) e lim [ sF ( s ) f ( 0 ) ] = lim lim - dts0 s0T dt 0and by interchanging the limiting process, the expression above is written asTd st ----- f ( t ) lim [ sF ( s ) f ( 0 ) ] = limlim e dt s0 T dt s00Also, since st lim e= 1 s0it reduces toT T d f ( t ) lim [ sF ( s ) f ( 0 ) ] = lim---- f ( t ) dt = lim -= lim [ f ( T ) f ( ) ] = f ( ) f ( 0 ) s0 T dtTT0 00Therefore,lim sF ( s ) = f ( )s02.2.12 Convolution in Time Domain PropertyConvolution* in the time domain corresponds to multiplication in the complex frequency domain,that is, f 1 ( t )*f 2 ( t ) F 1 ( s )F 2 ( s )(2.34)* Convolution is the process of overlapping two time functions f 1 ( t ) and f 2 ( t ) . The convolution integral indicates the amount of overlap of one function as it is shifted over another function The convolution of two time functions f1 ( t ) and f2 ( t ) is denoted as f 1 ( t )*f 2 ( t ) , and by definition, f 1 ( t )*f 2 ( t ) = f1 ( )f2 ( t ) d where is a dummy variable. Convolution is discussed in detail in Chapter 6.Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 211Copyright Orchard Publications 53. Chapter 2 The Laplace TransformationProof: st L { f 1 ( t )*f 2 ( t ) } = L f 1 ( )f 2 ( t ) d =0 0 f1 ( )f2 ( t ) d edt(2.35) st= 0 f 1 ( ) 0 f 2 ( t ) e dt dWe let t = ; then, t = + , and dt = d . Then, by substitution into (2.35), s ( + ) s sL { f 1 ( t )*f 2 ( t ) } = 0 f1 ( )0f2 ( ) ed d =0f 1 ( )ed 0 f 2 ( ) e d = F 1 ( s )F 2 ( s )2.2.13 Convolution in Complex Frequency Domain PropertyConvolution in the complex frequency domain divided by 1 2j , corresponds to multiplication inthe time domain. That is,1f 1 ( t )f 2 ( t ) ------- F 1 ( s )*F 2 ( s ) -(2.36) 2jProof: st L { f 1 ( t )f 2 ( t ) } =0 f1 ( t )f2 ( t ) edt(2.37)and recalling that the Inverse Laplace transform from (2.2) is + j1t f 1 ( t ) = ------- 2j - j F 1 ( )e dby substitution into (2.37), we obtain + j + j 1-t st 1- ( s )tL { f 1 ( t )f 2 ( t ) } = 0 -------2j jF 1 ( )e d f 2 ( t ) edt = ------- 2j j F1 ( ) 0 f 2 ( t ) edt dWe observe that the bracketed integral is F 2 ( s ) ; therefore, + j1 1L { f 1 ( t )f 2 ( t ) } = ------- 2j - j F1 ( )F2 ( s )d= ------- F 1 ( s )*F 2 ( s ) 2j -For easy reference, the Laplace transform pairs and theorems are summarized in Table 2.1.212Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth EditionCopyright Orchard Publications 54. Properties and Theorems of the Laplace Transform TABLE 2.1 Summary of Laplace Transform Properties and Theorems Property/TheoremTime Domain Complex Frequency Domain1Linearityc1 f1 ( t ) + c2 f2 ( t ) c1 F1 ( s ) + c2 F2 ( s )+ + cn fn ( t )+ + cn Fn ( s )2Time Shiftingf ( t a )u 0 ( t a ) aseF(s)3Frequency Shifting asF( s + a)ef(t)4Time Scaling f ( at )1 - -- F s - -a a5Time Differentiationd- sF ( s ) f ( 0 ) ---- f ( t ) See also (2.18) through (2.20) dt6Frequency Differentiationtf ( t ) d ---- F ( s ) - See also (2.22)ds7Time Integration t f ( ) dF( s) + f ( 0 )---------- ------------s s-8Frequency Integrationf(t)--------t s F ( s ) ds9Time Periodicity f ( t + nT )T st0 f ( t ) edt------------------------------ sT1e10 Initial Value Theorem lim f ( t ) lim sF ( s ) = f ( 0 ) t0 s11 Final Value Theorem lim f ( t ) lim sF ( s ) = f ( )t s012 Time Convolution f 1 ( t )*f 2 ( t )F 1 ( s )F 2 ( s )13 Frequency Convolutionf 1 ( t )f 2 ( t ) 1-------- F ( s )*F 2 ( s )2j 1Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition213Copyright Orchard Publications 55. Chapter 2 The Laplace Transformation2.3 The Laplace Transform of Common Functions of TimeIn this section, we will derive the Laplace transform of common functions of time. They are pre-sented in Subsections 2.3.1 through 2.3.11 below.2.3.1 The Laplace Transform of the Unit Step Function u 0 ( t )We begin with the definition of the Laplace transform, that is, stL { f( t)} = F( s) = 0 f ( t ) edtor st e = 0 -- = -- st1 0 1 e 1L { u0 ( t ) } = dt = -------- ---s s s 0Thus, we have obtained the transform pair1u 0 ( t ) -- - (2.38)sfor Re { s } = > 0 .*2.3.2 The Laplace Transform of the Ramp Function u 1 ( t )We apply the definition stL { f( t)} = F( s) = 0 f ( t ) edtor stL { u1 ( t ) } = L { t } = 0 t edtWe will perform integration by parts by recalling that u dv= uv v du(2.39)We let stu = t and dv = ethen, ste du = 1 and v = ---------- - s* This condition was established in relation (2.9), Page 22.214Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth EditionCopyright Orchard Publications 56. The Laplace Transform of Common Functions of TimeBy substitution into (2.39), st st st st t e e - dt = ------------ e - t e - ------- L { t } = ------------ s 0-0---------- s s s 2 (2.40) 0Since the upper limit of integration in (2.40) produces an indeterminate form, we apply L Hpi-tals rule*, that is,d (t) st t dt 1lim te = lim ----- = lim --------------- = lim -------- = 0 - - t t est t dst t se st(e ) dt 1Evaluating the second term of (2.40), we obtain L { t } = ---- 2sThus, we have obtained the transform pair1 t --- - (2.41) s2for > 0 .2.3.3 The Laplace Transform of t u0 ( t ) nBefore deriving the Laplace transform of this function, we digress to review the gamma or gener-alized factorial function ( n ) which is an improper integral but converges (approaches a limit) forall n > 0 . It is defined as f( x)*Often, the ratio of two functions, such as ---------- , for some value of x, say a, results in an indeterminate form. To work -g(x) f( x ) around this problem, we consider the limit lim ---------- , and we wish to find this limit, if it exists. To find this limit, we use - xa g(x)d-d- LHpitals rule which states that if f ( a ) = g ( a ) = 0 , and if the limit ----- f ( x ) ----- g ( x ) as x approaches a exists, then,dxdx f(x) lim ---------- = lim ----- f ( x ) ----- g ( x )-d-d- xa g( x)x a dxdx Improper integrals are two types and these are:ba.a f ( x ) dx where the limits of integration a or b or both are infinite bb.a f ( x ) dx where f(x) becomes infinite at a value x between the lower and upper limits of integration inclusive.Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 215Copyright Orchard Publications 57. Chapter 2 The Laplace Transformation n 1 x(n) = 0 x e dx(2.42)We will now derive the basic properties of the gamma function, and its relation to the well knownfactorial function n! = n ( n 1 ) ( n 2 ) 3 2 1The integral of (2.42) can be evaluated by performing integration by parts. Thus, in (2.42) we letxn1 u = e and dv = xThen,nxdu = e dx and v = x ----- nand (2.42) is written as n x 1n x0 x ex e ( n ) = ------------ - + ---dx (2.43) n x=0 nWith the condition that n > 0 , the first term on the right side of (2.43) vanishes at the lower limitx = 0 . It also vanishes at the upper limit as x . This can be proved with L Hpitals rule bydifferentiating both numerator and denominator m times, where m n . Then,m m1d n d n1 n x nmx m1 nx x ex dxdx lim ------------ = lim ------- = lim ------------------- = lim ------------------------------------ = - - - x nx nex x d m xm1xx d mnem1 nedx dxnm = lim n ( n 1 ) ( n 2 ) ( n m + 1 )x - = lim ( n 1 ) ( n 2 ) ( n m + 1 - = 0 ------------------------------------------------------------------------------------ -------------------------------------------------------------------) xnex x mn x x eTherefore, (2.43) reduces to 1 n x ( n ) = -- n- 0 x e dxand with (2.42), we have n 1 x1n x (n) = 0xe dx = --n - 0 x e dx(2.44)By comparing the integrals in (2.44), we observe that216Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth EditionCopyright Orchard Publications 58. The Laplace Transform of Common Functions of Time(n ) = (n + 1)-------------------- -(2.45) norn ( n ) = ( n + 1 )(2.46)It is convenient to use (2.45) for n < 0 , and (2.46) for n > 0 . From (2.45), we see that ( n )becomes infinite as n 0 .For n = 1 , (2.42) yields x x (1) = 0 e dx = e 0 = 1(2.47)and thus we have obtained the important relation, (1) = 1 (2.48)From the recurring relation of (2.46), we obtain(2) = 1 (1) = 1 ( 3 ) = 2 ( 2 ) = 2 1 = 2!(2.49) ( 4 ) = 3 ( 3 ) = 3 2 = 3!and in general ( n + 1 ) = n!(2.50)for n = 1, 2, 3, The formula of (2.50) is a noteworthy relation; it establishes the relationship between the ( n )function and the factorial n!We now return to the problem of finding the Laplace transform pair for t u 0 t , that is,n n st0 tn L { t u0 t } =edt(2.51)To make this integral resemble the integral of the gamma function, we let st = y , or t = y s ,and thus dt = dy s . Now, we rewrite (2.51) as n y e y d y = ----------1- n y(n + 1)0 0 y e n L { t u0 t } =--- -- - -n! -dy = ------------------- = ---------- s sn+1n+1 n+1ss sTherefore, we have obtained the transform pairSignals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 217Copyright Orchard Publications 59. Chapter 2 The Laplace Transformationn n! t u 0 ( t ) ----------n+1-(2.52) sfor positive integers of n and > 0 .2.3.4 The Laplace Transform of the Delta Function ( t )We apply the definition st L { ( t )} = 0 ( t ) edtand using the sifting property of the delta function,* we obtain st s ( 0 ) L { ( t )} = 0 ( t ) edt = e = 1Thus, we have the transform pair(t) 1 (2.53)for all .2.3.5 The Laplace Transform of the Delayed Delta Function ( t a )We apply the definition st L {(t a)} = 0 ( t a ) e dtand again, using the sifting property of the delta function, we obtain st as L {(t a)} = 0 ( t a ) edt = eThus, we have the transform pair as(t a) e (2.54)for > 0 .* The sifting property of the ( t ) is described in Subsection 1.4.2, Chapter 1, Page 113.218Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth EditionCopyright Orchard Publications 60. The Laplace Transform of Common Functions of Time at2.3.6 The Laplace Transform of e u 0 ( t )We apply the definition dt = ---------- e at at st ( s + a )t ( s + a )t 0 e 0 e1 1L {eu0 ( t ) } = e dt =-= ---------- - s+as+a0 Thus, we have the transform pair at 1 e u 0 ( t ) -----------(2.55) s+afor > a . n at2.3.7 The Laplace Transform of t e u 0 ( t )For this derivation, we will use the transform pair of (2.52), i.e.,n n! t u 0 ( t ) ----------n+1-(2.56) sand the frequency shifting property of (2.14), that is, at ef(t) F(s + a ) (2.57)Then, replacing s with s + a in (2.56), we obtain the transform pairn atn! - t eu 0 ( t ) ------------------------n+1 (2.58)(s + a)where n is a positive integer, and > a . Thus, for n = 1 , we obtain the transform pair at 1teu 0 ( t ) ------------------2 (2.59)(s + a)for > a .For n = 2 , we obtain the transform2 at2! t e u 0 ( t ) ----------------- - 3 (2.60) (s + a)and in general,Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition219Copyright Orchard Publications 61. Chapter 2 The Laplace Transformationn at n! t e u 0 ( t ) ------------------------ n+1- (2.61) (s + a)for > a .2.3.8 The Laplace Transform of sin t u 0 ( t )We apply the definitiona st stL { sin t u 0 ( t ) } =0( sin t ) e dt = lima 0 ( sin t ) e dtand from tables of integrals* ax e sin bx dx = e ( a sin bx b cos bx - ) ax-----------------------------------------------------2 2 a +bThen, sta e ( s sin t cos t -)L { sin t u 0 ( t ) } = lim ---------------------------------------------------------- a2s + 2 0 ase ( s sin a cos a )= lim ------------------------------------------------------------- + ---------------- = ------------------- a 2s +22s + 22 s +2Thus, we have obtained the transform pairsin t u 0 t ----------------2-2(2.62) s +for > 0 .2.3.9 The Laplace Transform of cos t u 0 ( t )We apply the definition a st st L { cos t u 0 ( t ) } =0( cos t ) edt = lima 0 ( cos t ) edt1- jt jt at 1-* This can also be derived from sin t = ---- ( e e) , and the use of (2.55) where e u 0 ( t ) ---------- . By the linearity j2 s+aproperty, the sum of these terms corresponds to the sum of their Laplace transforms. Therefore, 1-L [ sin tu 0 ( t ) ] = ---- 1 - 1 ------------- -------------- = -----------------j2 s j s + j s2 + 2220Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth EditionCopyright Orchard Publications 62. The Laplace Transform of Common Functions of Timeand from tables of integrals* axe ( acos bx + b sin bx ) axe cos bx dx = ------------------------------------------------------2 2 a +bThen, st ae ( s cos t + sin t )L { cos t u 0 ( t ) } = lim ---------------------------------------------------------- -a 2 s + 2 0 as= lim e ( s cos a + sin a ) + ---------------- = ------------------------------------------------------------------------------s -s -a2s +22s + 22 s +2Thus, we have the fransform pairscos t u 0 t ---------------- 2 - 2 (2.63)s +for > 0 . at2.3.10 The Laplace Transform of e sin t u 0 ( t )From (2.62), sin tu 0 t ---------------- 2 - 2s +Using the frequency shifting property of (2.14), that is, at ef(t) F(s + a ) (2.64)we replace s with s + a , and we obtain atesin t u 0 ( t ) ------------------------------ 2 - 2 (2.65)(s + a) + for > 0 and a > 0 . 1 jt jt* We can use the relation cos t = -- ( e + e- ) and the linearity property, as in the derivation of the transform of2d-sin t on the footnote of the previous page. We can also use the transform pair ---- f ( t ) sF ( s ) f ( 0 ) ; this is the time dtdifferentiationpropertyof (2.16).Applyingthis transformpair for this derivation,weobtain1 d-1 d-1 sL [ cos tu 0 ( t ) ] = L --- ---- sin tu 0 ( t ) = --- L ---- sin tu 0 ( t ) = --- s ----------------- = ----------------- - - - dt dt s2 + 2 s + 22Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition221Copyright Orchard Publications 63. Chapter 2 The Laplace Transformation2.3.11 The Laplace Transform of e at cos t u 0 ( t )From (2.63), s cos t u 0 ( t ) ----------------2-2 s +and using the frequency shifting property of (2.14), we replace s with s + a , and we obtain at s+a ecos t u 0 ( t ) ------------------------------ 2 - 2 (2.66)(s + a) + for > 0 and a > 0 .For easy reference, we have summarized the above derivations in Table 2.2.TABLE 2.2 Laplace Transform Pairs for Common Functionsf (t)F( s)1 u0 ( t )1s2 t u0 ( t )1s 23nt u0 ( t )n!---------- - n+1s4 (t)15 (t a)e as6 at 1- eu0 ( t )----------s+a7 n atn!t eu0 ( t ) ------------------------ -n+1(s + a)8 sin t u 0 ( t ) ---------------- - 2 2s +9 cos t u 0 ( t ) s---------------- - 2 2s +10e at sin t u 0 ( t ) ------------------------------ -22(s + a) + 11e at cos t u 0 ( t )s+a------------------------------ -22(s + a) + 222Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth EditionCopyright Orchard Publications 64. The Laplace Transform of Common Waveforms2.4 The Laplace Transform of Common WaveformsIn this section, we will present procedures for deriving the Laplace transform of common wave-forms using the transform pairs of Tables 1 and 2. The derivations are described in Subsections2.4.1 through 2.4.5 below.2.4.1 The Laplace Transform of a PulseThe waveform of a pulse, denoted as f P ( t ) , is shown in Figure 2.1. fP ( t ) A0at Figure 2.1. Waveform for a pulseWe first express the given waveform as a sum of unit step functions as weve learned in Chapter1. Then, fP ( t ) = A [ u0 ( t ) u0 ( t a ) ](2.67)From Table 2.1, Page 213, asf ( t a )u 0 ( t a ) eF(s)and from Table 2.2, Page 222 u0 ( t ) 1 sThus, Au 0 ( t ) A sand as A Au 0 ( t a ) e--- -sThen, in accordance with the linearity property, the Laplace transform of the pulse of Figure 2.1isA as A A asA [ u 0 ( t ) u 0 ( t a ) ] --- e --- = --- ( 1 e )- - - s s s2.4.2 The Laplace Transform of a Linear SegmentThe waveform of a linear segment, denoted as f L ( t ) , is shown in Figure 2.2.Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 223Copyright Orchard Publications 65. Chapter 2 The Laplace Transformation fL ( t )1t012Figure 2.2. Waveform for a linear segmentWe must first derive the equation of the linear segment. This is shown in Figure 2.3. fL ( t ) t1 1t0 1 2 Figure 2.3. Waveform for a linear segment with the equation that describes itNext, we express the given waveform in terms of the unit step function as follows: f L ( t ) = ( t 1 )u 0 ( t 1 )From Table 2.1, Page 213, as f ( t a )u 0 ( t a ) eF(s)and from Table 2.2, Page 222, 1- tu 0 ( t ) ---2sTherefore, the Laplace transform of the linear segment of Figure 2.2 is s 1 ( t 1 )u 0 ( t 1 ) e ---- (2.68) s22.4.3 The Laplace Transform of a Triangular WaveformThe waveform of a triangular waveform, denoted as f T ( t ) , is shown in Figure 2.4. fT ( t ) 11 t 02 Figure 2.4. Triangular waveformThe equations of the linear segments are shown in Figure 2.5.224Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth EditionCopyright Orchard Publications 66. The Laplace Transform of Common WaveformsfT ( t )1 tt+2 12t0Figure 2.5. Triangular waveform with the equations of the linear segmentsNext, we express the given waveform in terms of the unit step function.fT ( t ) = t [ u0 ( t ) u0 ( t 1 ) ] + ( t + 2 ) [ u0 ( t 1 ) u0 ( t 2 ) ]= tu 0 ( t ) tu 0 ( t 1 ) tu 0 ( t 1 ) + 2u 0 ( t 1 ) + tu 0 ( t 2 ) 2u 0 ( t 2 )Collecting like terms, we obtain f T ( t ) = tu 0 ( t ) 2 ( t 1 )u 0 ( t 1 ) + ( t 2 )u 0 ( t 2 )From Table 2.1, Page 213, asf ( t a )u 0 ( t a ) e F(s)and from Table 2.2, Page 222, 1- tu 0 ( t ) ---2sThen,1- s 1 2s 1tu 0 ( t ) 2 ( t 1 )u 0 ( t 1 ) + ( t 2 )u 0 ( t 2 ) --- 2e --- + e --- - 2 - 2 2 s s sor 1- s 2s tu 0 ( t ) 2 ( t 1 )u 0 ( t 1 ) + ( t 2 )u 0 ( t 2 ) --- ( 1 2e + e )2sTherefore, the Laplace transform of the triangular waveform of Figure 2.4 is 1-s 2f T ( t ) --- ( 1 e )2(2.69)s2.4.4 The Laplace Transform of a Rectangular Periodic WaveformThe waveform of a rectangular periodic waveform, denoted as f R ( t ) , is shown in Figure 2.6. Thisis a periodic waveform with period T = 2a , and we can apply the time periodicity propertyT s 0 f ( ) e dL { f ( ) } = -------------------------------- sT 1eSignals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 225Copyright Orchard Publications 67. Chapter 2 The Laplace Transformationwhere the denominator represents the periodicity of f ( t ) . fR ( t )At 0 a2a3a AFigure 2.6. Rectangular periodic waveformFor this waveform, 2a a2a 1 st 1 st stL { f R ( t ) } = -------------------1e 2as-0fR ( t ) edt = ------------------- 1e 2as - 0 Aedt +a( A ) edt sta st 2a = ------------------- e - + ------- A -e - 2as ---------- 1es 0saA as 2as asL { f R ( t ) } = --------------------------- ( e + 1 + e 2as- e )s(1 e ) as2A as= --------------------------- ( 1 2e + e- 2as A(1 e ) ) = ------------------------------------------------- 2as as ass(1 e )s(1 + e )(1 e ) as as 2 as 2 as 2 as 2A (1 e ) A e eee= --- ---------------------- = --- --------------------------------------------------------------- - - s ( 1 + e as ) s e as 2 e as 2 + e as 2 e as 2 as 2 as 2 as 2 Aee eA sinh ( as 2 )= --- ------------- ------------------------------- = --- ----------------------------- - - -- s e as 2e as 2 +e as 2s cosh ( as 2 ) f R ( t ) --- tanh ---- Aas -2 - (2.70)s2.4.5 The Laplace Transform of a HalfRectified Sine WaveformThe waveform of a half-rectified sine waveform, denoted as f HW ( t ) , is shown in Figure 2.7. This isa periodic waveform with period T = 2a , and we can apply the time periodicity propertyT s 0 f ( ) ed L { f ( ) } = ------------------------------- - sT1e226Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth EditionCopyright Orchard Publications 68. Using MATLAB for Finding the Laplace Transforms of Time Functionswhere the denominator represents the periodicity of f ( t ) . f HW ( t )2 3 4 5 Figure 2.7. Half-rectified sine waveform*For this waveform,2 1 st 1 st L { f HW ( t ) } = --------------------1e 2s - 0f( t)e dt = -------------------- 1e 2s- 0 sin t e dt st s1 - e ( s sin t cos t - ) 1 ( 1 + e )- = -------------------- ----------------------------------------- 2s 2= ------------------ ------------------------- 2 2s 1es +1(s + 1) (1 e )0 s 1(1 + e )L { f HW ( t ) } = ------------------ -----------------------------------------------2 s s (s + 1) (1 + e )(1 e ) 1f HW ( t ) ------------------------------------------2 s (2.71) (s + 1)(1 e )2.5 Using MATLAB for Finding the Laplace Transforms of Time FunctionsWe can use the MATLAB function laplace to find the Laplace transform of a time function. Forexamples, please typehelp laplacein MATLABs Command prompt.We will be using this function extensively in the subsequent chapters of this book.* This waveform was produced with the following MATLAB script:t=0:pi/64:5*pi; x=sin(t); y=sin(t2*pi); z=sin(t4*pi); plot(t,x,t,y,t,z); axis([0 5*pi 0 1])Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition227Copyright Orchard Publications 69. Chapter 2 The Laplace Transformation2.6 Summary The twosided or bilateral Laplace Transform pair is defined as stL {f(t)}= F(s ) = f ( t ) edt + j 11- j stL { F ( s ) } = f ( t ) = ------- F ( s ) e ds2j1where L { f ( t ) } denotes the Laplace transform of the time function f ( t ) , L { F ( s ) } denotesthe Inverse Laplace transform, and s is a complex variable whose real part is , and imaginarypart , that is, s = + j . The unilateral or onesided Laplace transform defined as st stL {f(t)}= F(s) = t 0f(t)edt = 0 f ( t ) e dt We denote transformation from the time domain to the complex frequency domain, and viceversa, asf(t) F(s) The linearity property states that c1 f1 ( t ) + c2 f2 ( t ) + + cn fn ( t ) c 1 F1 ( s ) + c2 F2 ( s ) + + cn Fn ( s ) The time shifting property states that as f ( t a )u 0 ( t a ) e F(s) The frequency shifting property states that at ef(t) F(s + a ) The scaling property states that f ( at ) -- F - 1 s - -a a The differentiation in time domain property states that d f( t ) = ---- f ( t ) sF ( s ) f ( 0 ) -dtAlso,228Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth EditionCopyright Orchard Publications 70. Summary d 2-------- f ( t ) s 2 F ( s ) sf ( 0 ) f( 0 )2dt d3------- f ( t ) s 3 F ( s ) s 2 f ( 0 ) sf( 0 ) f( 0 )-3dt and in general n d------- f ( t ) s n F ( s ) s n 1 f ( 0 ) s n 2 f( 0 ) f-n1 (0 )ndtwhere the terms f ( 0 ), f( 0 ), f( 0 ) , and so on, represent the initial conditions. The differentiation in complex frequency domain property states that d tf ( t ) ---- F ( s ) -dsand in general, n n ndt f ( t ) ( 1 ) ------- F ( s )-nds The integration in time domain property states that t F(s) f (0 ) f ( ) d ---------- + ------------ s s - The integration in complex frequency domain property states that f(t) -------- ts F ( s ) dsprovided that the limit lim f ( t ) exists.--------t0t The time periodicity property states that T st 0 f ( t ) edt f ( t + nT ) ----------------------------- sT-1e The initial value theorem states that lim f ( t ) = lim sF ( s ) = f ( 0 ) t0sSignals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition229Copyright Orchard Publications 71. Chapter 2 The Laplace Transformation The final value theorem states thatlim f ( t ) = lim sF ( s ) = f ( ) ts0 Convolution in the time domain corresponds to multiplication in the complex frequencydomain, that is,f 1 ( t )*f 2 ( t ) F 1 ( s )F 2 ( s ) Convolution in the complex frequency domain divided by 1 2j , corresponds to multiplica-tion in the time domain. That is,1f 1 ( t )f 2 ( t ) ------- F 1 ( s )*F 2 ( s ) - 2j The Laplace transforms of some common functions of time are shown in Table 2.1, Page 213 The Laplace transforms of some common waveforms are shown in Table 2.2, Page222 We can use the MATLAB function laplace to find the Laplace transform of a time function230 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright Orchard Publications 72. Exercises2.7 Exercises1. Derive the Laplace transform of the following time domain functions: a. 12 b. 6u0 ( t ) c. 24u 0 ( t 12 ) d. 5tu 0 ( t )e. 4t u 0 ( t )52. Derive the Laplace transform of the following time domain functions: 5t 7 5t a. j8 b. j5 90 c. 5e u0 ( t )d. 8t eu0 ( t ) e. 15 ( t 4 )3. Derive the Laplace transform of the following time domain functions: a. ( t + 3t + 4t + 3 )u 0 ( t )b. 3 ( 2t 3 ) ( t 3 )3 2 c. ( 3 sin 5t )u 0 ( t ) d. ( 5 cos 3t )u 0 ( t ) e. ( 2 tan 4t )u 0 ( t ) Be careful with this! Comment and you may skip derivation.4. Derive the Laplace transform of the following time domain functions: 5t a. 3t ( sin 5t )u 0 ( t ) b. 2t ( cos 3t )u 0 ( t ) c. 2e2 sin 5t 3t d. 8e cos 4t e. ( cos t ) ( t 4 )5. Derive the Laplace transform of the following time domain functions: 2t a. 5tu 0 ( t 3 ) b. ( 2t 5t + 4 )u 0 ( t 3 ) c. ( t 3 )e2 u0 ( t 2 )2(t 2) 3t d. ( 2t 4 )eu0 ( t 3 )e. 4te( cos 2t )u 0 ( t )6. Derive the Laplace transform of the following time domain functions: a. d ( sin 3t ) b. d ( 3e )c. d ( t cos 2t ) d. d ( e sin 2t ) e. d ( t e ) 4t2 2t2 2tdt dtdt dt dt7. Derive the Laplace transform of the following time domain functions: sin -tt sin sin atcos e - a. --------tb.--------- d0 -c. ----------- t - d.t---------- d- e. ------ dt Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition231Copyright Orchard Publications 73. Chapter 2 The Laplace Transformation8. Derive the Laplace transform for the sawtooth waveform f ST ( t ) below. f ST ( t )A a3at2a9. Derive the Laplace transform for the fullrectified waveform f FR ( t ) below.f FR ( t )23 4 Write a simple MATLAB script that will produce the waveform above.232Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth EditionCopyright Orchard Publications 74. Solutions to EndofChapter Exercises2.8 Solutions to EndofChapter Exercises1. From the definition of the Laplace transform or from Table 2.2, Page 222, we obtain: 12s 245! a 12 sb. 6 s c. e d. 5 s e. 4 ----2 ----- --6 s s2. From the definition of the Laplace transform or from Table 2.2, Page 222, we obtain: 5 7! 4s a. j8 s b. 5 s c. ----------- d. 8 ------------------ e. 15es+58(s + 5)3.3 2! 4 a. From Table 2.2, Page 222, and the linearity property, we obtain 3! + ------------- + ---- + 3 -----43- 2 --- s ss s 3s b. 3 ( 2t 3 ) ( t 3 ) = 3 ( 2t 3 )t=3 ( t 3 ) = 9 ( t 3 ) and 9 ( t 3 ) 9e22 5s sin 4t 4 (s + 2 ) c. 3 --------------- d. 5 --------------- e. 2 tan 4t = 2 ------------- 2 --------------------------- = 8 2-2 2 - 2 2 2 - ---s +5 s +3cos 4ts (s + 2 ) s This answer for part (e) looks suspicious because 8 s 8u 0 ( t ) and the Laplace transform is unilateral, that is, there is onetoone correspondence between the time domain and the complex frequency domain. The fallacy with this procedure is that we assumed that if f1 ( t ) F1 ( s ) f 1 ( t ) F 1 ( s ) and f 2 ( t ) F 2 ( s ) , we cannot conclude that ---------- ------------ . For this exercise-- f2 ( t ) F2 ( s ) 1 f 1 ( t ) f 2 ( t ) = sin 4t ------------- , and as weve learned, multiplication in the time domain corre-cos 4t sponds to convolution in the complex frequency domain. Accordingly, we must use the st Laplace transform definition 0 ( 2 tan 4t )e dt and this requires integration by parts. We skip this analytical derivation. The interested reader may try to find the answer with the MAT- LAB script syms s t; 2*laplace(sin(4*t)/cos(4*t))4. From (2.22), Page 26, nn nd t f ( t ) ( 1 ) ------- F ( s ) - n ds a. 5 ( 2s )3 ( 1 ) ---- --------------- = 3 ----------------------- = ----------------------- 1 d530s - - - - 2ds s + 5 2 22 2 ( s + 25 ) 2 ( s + 25 )Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 233Copyright Orchard Publications 75. Chapter 2 The Laplace Transformation b. d s +9 222 2 d s + 3 s ( 2s )2 ( 1 ) ------- --------------- = 2 ---- ---------------------------------- = 2 ---- -------------------- 2 d s- -----2 22ds 2 (s + 9) ds 22ds s + 3 2(s + 9)22 2 2 ( s + 9 ) ( 2s ) 2 ( s + 9 ) ( 2s ) ( s + 9 - ) = 2 ------------------------------------------------------------------------------------------------- 4 2(s + 9)22 3 3 ( s + 9 ) ( 2s ) 4s ( s + 9 - ) 2s 18s + 4s 36s = 2 ------------------------------------------------------------------- = 2 -------------------------------------------------------3 3 -2 2 (s + 9) (s + 9) 32 2s 54s 2s ( s 27 ) 4s ( s 2 27 ) = 2 ---------------------- = 2 -------------------------- = -------------------------- 33 -- 223(s + 9)(s + 9) 2(s + 9) c. 2510------------------