Transforms andApplications Primerfor Engineerswith Examplesand MATLABElectrical Engineering Primer SeriesSeries EditorAlexander D. PoularikasUniversity of AlabamaHuntsville, Alabama Transforms and Applications Primer for Engineers with Examples and MATLAB, Alexander D. Poularikas Discrete Random Signal Processing and Filtering Primer with MATLAB, Alexander D. Poularikas Signals and Systems Primer with MATLAB, Alexander D. Poularikas Adaptive Filtering Primer with MATLAB, Alexander D. Poularikas and Zayed M. Ramadan CRC Press is an imprint of theTaylor & Francis Group, an informa businessBoca Raton London New YorkAlexander D. PoularikasTransforms andApplications Primerfor Engineerswith Examplesand MATLABMATLAB and Simulink are trademarks of The MathWorks, Inc. and are used with permission. The Math-Works does not warrant the accuracy of the text of exercises in this book. This books use or discussion of MATLAB and Simulink software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB and Simulink software.CRC PressTaylor & Francis Group6000 Broken Sound Parkway NW, Suite 300Boca Raton, FL 33487-2742 2010 by Taylor and Francis Group, LLCCRC Press is an imprint of Taylor & Francis Group, an Informa businessNo claim to original U.S. Government worksPrinted in the United States of America on acid-free paper10 9 8 7 6 5 4 3 2 1International Standard Book Number-13: 978-1-4200-8932-5 (Ebook-PDF)This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint.Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmit-ted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers.For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged.Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe.Visit the Taylor & Francis Web site athttp://www.taylorandfrancis.comand the CRC Press Web site athttp://www.crcpress.com ContentsPreface .......................................................................................................................... ixAuthor .......................................................................................................................... xi1 Signals and Systems ..................................................................................... 1-11.1 Introduction............................................................................................................. 1-11.2 Signals ....................................................................................................................... 1-11.3 Circuit Elements and Equation ........................................................................ 1-131.4 Linear Mechanical and Rotational Mechanical Elements .......................... 1-211.4.1 Linear Mechanical Systems .................................................................. 1-211.4.2 Rotational Mechanical Systems ........................................................... 1-221.5 Discrete Equations and Systems ...................................................................... 1-231.6 Digital Simulation of Analog Systems ............................................................ 1-261.7 Convolution of Analog Signals ........................................................................ 1-261.8 Convolution of Discrete Signals ....................................................................... 1-292 Fourier Series ................................................................................................. 2-12.1 Introduction............................................................................................................. 2-12.2 Fourier Series in a Complex Exponential Form.............................................. 2-12.3 Fourier Series in Trigonometric Form .............................................................. 2-22.3.1 Differentiation of the Fourier Series ..................................................... 2-22.3.2 Integration of the Fourier Series ............................................................ 2-32.4 Waveform Symmetries .......................................................................................... 2-32.5 Some Additional Features of Periodic Continuous Functions ..................... 2-32.5.1 Power Content: Parsevals Theorem..................................................... 2-32.5.2 Output of an LTI System When the InputIs a Periodic Function .............................................................................. 2-42.5.3 Transmission without Distortion .......................................................... 2-42.5.4 Band-Limited Periodic Signals ............................................................... 2-52.5.5 Sum and Difference of Functions .......................................................... 2-52.5.6 Product of Two Functions ...................................................................... 2-5v2.5.7 Convolution of Two Functions ......................................................... 2-62.5.8 Gibbs Phenomenon............................................................................. 2-62.5.9 Fourier Series of the Comb Function .............................................. 2-73 Fourier Transforms ...................................................................................... 3-13.1 IntroductionFourier Transform.................................................................... 3-13.2 Other Forms of Fourier Transform ................................................................. 3-13.2.1 f(t) Is a Complex Function ................................................................. 3-13.2.2 Real Time Functions ............................................................................ 3-23.2.3 Imaginary Time Functions ................................................................. 3-23.2.4 f(t) Is Even.............................................................................................. 3-33.2.5 f(t) Odd ................................................................................................. 3-33.2.6 Odd and Even Representations ......................................................... 3-33.2.7 Causal-Time Functions ....................................................................... 3-43.3 Fourier Transform Examples ............................................................................ 3-53.4 Fourier Transform Properties ........................................................................... 3-83.5 Examples on Fourier Properties ....................................................................... 3-83.6 FT Examples of Singular Functions .............................................................. 3-123.7 Duration of a Signal and the Uncertainty Principle ................................. 3-373.8 Applications to Linear-Time Invariant Systems ........................................ 3-383.9 Applications to Communication Signals ..................................................... 3-473.10 Signals, Noise, and Correlation ...................................................................... 3-503.11 Average Power Spectra, Random Signals, InputOutput Relations ...... 3-513.12 FT in Probability Theory................................................................................. 3-533.12.1 Characteristic Function .................................................................... 3-553.12.2 Joint Cumulative Distribution Function ...................................... 3-553.12.3 Characteristic Function of Two Variables ................................... 3-564 Relatives to the Fourier Transform .......................................................... 4-14.1 Innite Fourier Sine Transform ....................................................................... 4-14.2 Innite Fourier Cosine Transform................................................................... 4-14.3 Applications to Boundary-Value Problems ................................................... 4-94.4 Finite Sine Fourier Transform and Finite CosineFourier Transform ........................................................................................... 4-154.5 Two-Dimensional Fourier Transform.......................................................... 4-184.5.1 Two-Dimensional Convolution...................................................... 4-214.5.2 Two-Dimensional Correlation........................................................ 4-214.5.3 Theorems of Two-Dimensional Functions .................................. 4-225 Sampling of Continuous Signals ............................................................... 5-15.1 Fundamentals of Sampling ................................................................................ 5-15.2 The Sampling Theorem ...................................................................................... 5-6vi Contents6 Discrete-Time Transforms ......................................................................... 6-16.1 Discrete-Time Fourier Transform.................................................................... 6-16.1.1 Approximating the Fourier Transform ........................................... 6-16.1.2 Symmetry Properties of the DTFT................................................... 6-56.2 Summary of DTFT Properties .......................................................................... 6-56.3 DTFT of Finite Time Sequences ....................................................................... 6-76.3.1 Windowing ............................................................................................. 6-96.4 Frequency Response of LTI Discrete Systems ............................................ 6-116.5 Discrete Fourier Transform ............................................................................ 6-136.6 Summary of DFT Properties .......................................................................... 6-156.7 Multirate Digital Signal Processing and Spectra ........................................ 6-276.7.1 Down Sampling (or Decimation) .................................................. 6-286.7.2 Frequency Domain of Down-Sampled Signals ........................... 6-306.7.3 Interpolation (Up-Sampling) by a Factor U................................ 6-346.7.4 Frequency Domain Characterizationof Up-Sampled Signals ..................................................................... 6-35Appendix ......................................................................................................................... 6-386.A.1 Proofs of DTFT Properties ........................................................................... 6-386.A.2 Proofs of DFT Properties .............................................................................. 6-406.A.3 Fast Fourier Transform.................................................................................. 6-436.A.3.1 Decimation in Time Procedure .................................................... 6-437 Laplace Transform ....................................................................................... 7-17.1 One-Sided Laplace Transform .......................................................................... 7-17.2 Summary of the Laplace Transform Properties ........................................... 7-47.3 Systems Analysis: Transfer Functions of LTI Systems ................................ 7-87.4 Inverse Laplace Transform ............................................................................. 7-197.5 Problem Solving with Laplace Transform................................................... 7-267.5.1 Ordinary Differential Equations ..................................................... 7-267.5.2 Partial Differential Equations .......................................................... 7-397.6 Frequency Response of LTI Systems ............................................................ 7-497.7 Pole Location and the Stability of LTI Systems ......................................... 7-577.8 Feedback for Linear Systems .......................................................................... 7-607.9 Bode Plots ........................................................................................................... 7-717.10 *Inversion Integral ............................................................................................ 7-757.11 *Complex Integration and the Bilateral Laplace Transform................... 7-867.12 *State Space and State Equations .................................................................. 7-887.12.1 State Equations in Phase Variable Form...................................... 7-907.12.2 Time Response Using State Space Representation .................... 7-987.12.3 Solution Using the Laplace Transform....................................... 7-1027.12.4 State Space Transfer Function ...................................................... 7-1057.12.5 Impulse and Step Response ........................................................... 7-106Contents vii8 The z-Transform .......................................................................................... 8-18.1 The z-Transform .................................................................................................. 8-18.2 Convergence of the z-Transform ..................................................................... 8-58.3 Properties of the z-Transform........................................................................ 8-118.4 z-Transform Pairs ............................................................................................. 8-208.5 Inverse z-Transform ......................................................................................... 8-218.5.1 Partial Fraction Expansion .............................................................. 8-218.5.2 *Inverse Transform by Integration ................................................ 8-288.5.3 *Residues for Simple Poles .............................................................. 8-288.5.4 *Residues for Multiple Poles ........................................................... 8-288.5.5 *Residues for Simple Poles Not Factorable ................................. 8-298.6 Transfer Function ............................................................................................. 8-318.6.1 Higher Order Transfer Functions .................................................. 8-378.6.1.1 Stability ................................................................................ 8-398.6.1.2 Causality .............................................................................. 8-398.7 Frequency Response of First-Order Discrete Systems .............................. 8-398.7.1 Phase Shift in Discrete Systems ...................................................... 8-458.8 Frequency Response of Higher Order Digital Systems ............................ 8-468.9 z-Transform Solution of First-Order Difference Equations .................... 8-498.10 Higher Order Difference Equations .............................................................. 8-538.10.1 Method of Undetermined Coefcients ......................................... 8-598.11 *LTI Discrete-Time Dynamical Systems ...................................................... 8-648.12 *z-Transform and Random Processes .......................................................... 8-698.12.1 Power Spectral Densities .................................................................. 8-698.12.2 Linear Discrete-Time Filters ........................................................... 8-718.12.3 Optimum Linear Filtering ............................................................... 8-728.13 *Relationship between the Laplace and z-Transforms ............................. 8-748.14 *Relationship to the Fourier Transform ...................................................... 8-78Appendix ......................................................................................................................... 8-799 *Hilbert Transforms..................................................................................... 9-19.1 Denition ............................................................................................................... 9-19.2 Hilbert Transforms, Properties, and the Analytic Signal ............................ 9-29.3 Hilbert Transform Properties and Hilbert Pairs ........................................ 9-15Appendix A: Functions of a Complex Variable ............................................. A-1Appendix B: Series and Summations................................................................ B-1Appendix C: Denite Integrals ........................................................................... C-1Appendix D: Suggestions and Explanations for MATLAB1Use ............ D-1Index ....................................................................................................................... IN-1viii PrefacePrefaceThis book presents the most common and useful mathematical transforms for studentsand practicing engineers. It can be considered as a companion for students and a handyreference for practicing engineers who will need to use transforms in their work.The Laplace transform, which undoubtedly is the most familiar example, is basic to thesolution of initial value problems. The Fourier transform, being suited to solving bound-ary-value problems, is basic to the frequency spectrum analysis of time-varying signals.For discrete signals, we develop the z-transform and its uses. The purpose of this book isto develop the most important integral transforms and present numerous exampleselucidating their use. Laplace and Fourier transforms are by far the most widely andmost useful of all integral transforms. For this reason, they have been given a moreextensive treatment in this book when compared to other books on the same subject.This book is primarily written for seniors, rst-year graduate students, and practicingengineers and scientists. To comprehend some of the topics, the reader should have abasic knowledge of complex variable theory. Advanced topics are indicated by a star (*).The book contains several appendices to complement the main subjects. The extensivetables of the transforms are the most important contributions in this book. Anotherimportant contribution is the inclusion of an ample number of examples drawn fromseveral disciplines. The included examples help the readers understand any of the trans-forms and give them the condence to use it. Furthermore, it includes, wherever needed,MATLAB1functions and Book MATLAB functions developed by the author, which areincluded in the text.MATLAB is a registered trademark of The MathWorks, Inc. For product information,please contact:The MathWorks, Inc.3 Apple Hill Drive Natick, MA 01760-2098 USATel: 508 647 7000Fax: 508-647-7001E-mail: info@mathworks.comWeb: www.mathworks.comixAuthorAlexander D. Poularikas received his PhD from the University of Arkansas, Fayetteville,and became a professor at the University of Rhode Island, Kingston. He became thechairman of the engineering department at the University of Denver, Colorado, and thenbecame the chairman of the electrical and computer engineering department at theUniversity of Alabama in Huntsville.Dr. Poularikas has authored seven books and has edited two. He has served as theeditor in chief of the Signal Processing series (19931997) with Artech House, and is nowthe editor in chief of the Electrical Engineering and Applied Signal Processing series aswell as the Engineering and Science Primer series (1998 to present) with Taylor &Francis. He was a Fulbright scholar, is a lifelong senior member of the IEEE, and is amember of Tau Beta Pi, Sigma Nu, and Sigma Pi. In 1990 and in 1996, he received theOutstanding Educators Award of the IEEE, Huntsville Section. He is now a professoremeritus at the University of Alabama in Huntsville.Dr. Poularikas has authored, coauthored, and edited the following books:Electromagnetics, Marcel Dekker, New York, 1979.Electrical Engineering: Introduction and Concepts, Matrix Publishers, Beaverton, OR,1982.Workbook, Matrix Publishers, Beaverton, OR, 1982.Signals and Systems, Brooks=Cole, Boston, MA, 1985.Elements of Signals and Systems, PWS-Kent, Boston, MA, 1988.Signals and Systems, 2nd edn., PWS-Kent, Boston, MA, 1992.The Transforms and Applications Handbook, CRC Press, Boca Raton, FL, 1995.The Handbook for Formulas and Tables for Signal Processing, CRC Press, Boca Raton,FL, 1998, 2nd edn. (2000), 3rd edn. (2009).Adaptive Filtering Primer with MATLAB, Taylor & Francis, Boca Raton, FL, 2006.Signals and Systems Primer with MATLAB, Taylor & Francis, Boca Raton, FL, 2007.Discrete Random Signal Processing and Filtering Primer with MATLAB, Taylor &Francis, Boca Raton, FL, 2009.xi1Signals and Systems1.1 IntroductionThe term systems, in general, has many meanings such as electronic systems, biologicalsystems, communication systems, etc. The same is true for the term signals, since we talkabout optical signals, intelligence signals, radio signals, bio-signals, etc.The two terms mentioned above can have the following three interpretations: (1) Anelectric system is considered to be made of resistors, inductors, capacitors, and energysources. Signals are the currents and voltages in the electric system. The signals are afunction of time and they are related by a set of equations that are the product of physicallaws (Kirchhoff s voltage and current laws). (2) We interpret the system based on themathematical function it performs. For example, a resistor is a multiplier, an inductor is adifferentiator, and a capacitor is an integrator. The signals are the result of the rules of theinterconnected elements of the system. (3) If the operations can be performed digitallyand in real time, then the analog system can be substituted by a computer. The system,under these circumstances, is a digital device (computer) whose input and output aresequences of numbers. Figure 1.1 illustrates three systems and their responses. The toppart of the gure represents the ability of a lter to clear a signal from a superimposednoise. The middle part of the gure shows how a feedback conguration affects an inputpulse. This is known as the step response of systems. The bottom part of the gure showshow a rectier and a lter can produce a DC (direct current) source when the input is asinusoidal signal as the one present in power transmission lines.In addition to analog systems, we also have digital ones. These systems deal only withdiscrete signals and are presented later on in the book. Abasic, but sophisticated, instrumentis the analog-to-digital (A=D) converter, which most instruments nowadays contain.1.2 SignalsA signal is a function representing a physical quantity. This can be a current, a voltage,heart signals (EKG), velocities of motion, music signals, economic time series, etc. In thischapter, we will concentrate only on one-dimensional signals, although images, forexample, are two-dimensional signals.A continuous-time signal is a function whose domain is every point in a speciedinterval.1-1x(t) =s(t) +v(t)Filtery(t) =s(t)Ox(t)y(t)yo(t)Plantt+ O1t0yo(t)u(t) u(t)O2FeedbackOFilter RectiferOcos(0t)Constant0.511.50.40.60.810 20 40 60 80 100 1201.510.500 20 40 60 80 100 12010.80.60.40.200.2101224680 20 40 60 80 100 120 140200.200.20.40.60.810.40.50.60.70.80.912 0 2 4 6 8 1010.80.60.42 0 2 4 6 8 1000.10.20.3t|cos(0t)|FIGURE 1.11-2TransformsandApplicationsPrimerforEngineerswithExamplesandMATLAB1A discrete-time signal is a function whose domain is a set of integers. Therefore, thistype of signal is a sequence of numbers denoted by {x(n)}. It is understood that thediscrete-time signal is often formed by sampling a continuous-time signal x(t). In thiscase and for equidistance samples, we writex(n) x(nT) T sampling interval (1:1)Figure 1.2 shows a transformation from a continuous-time signal to a discrete-timesignal.Some important and useful functions are given in Table 1.1.If the above analog signals are sampled every T seconds, then we will obtain thecorresponding discrete ones.Approximation of a derivativeFrom Figure 1.3, we observe that we can approximate the samples y(nT) of the derivativey(t) x0(t) of the signal x(t) for a sufciently small T as follows:x0(t) x(t) x(t T)T (1:2)y(nT) x0(nT) x(nT) x(nT T)T 1TDx(nT) (1:3)We observe that as T !0, the approximate derivative of x(t), indicated by the inclinationof line A, comes closer and closer to the exact one, indicated by the inclination of line B.Approximation of an integralThe approximation of an integral with its discrete form is shown in Figure 1.4. Therefore,we writey(nT)

nTT0x(t)dt

nTnTTx(t)dt (1:4)10 0 1010.500.51ts(t)=cos 0.4 t10 0 1010.500.51ns(n)=cos (0.4 n0.25)FIGURE 1.2Signals and Systems 1-3which becomesy(nT) y(nT T)

nTnTTx(t)dt (1:5)TABLE 1.1 Some Useful Mathematical Functions in Analog and Discrete Format1. Signum functionsgn(t) 1 t > 00 t 01 t < 1

; sgn(nT) 1 nT > 00 nT 01 nT < 0

2. Step functionu(t) 1212sgn(t) 1 t > 00 t < 0

; u(nT) 1 nT > 00 nT < 0

3. Ramp functionr(t)

t1u(x)dx tu(t); r(nT) nTu(nT)4. Pulse functionpa(t) u(t a) u(t a) 1 jt j < a0 jt j > a

; pa(nT) u(nT mT) u(nT mT)5. Triangular pulseLa(t) 1 jtja jt j < a0 jt j > a

; La(nT) 1 jnTjmT jnTj < mT0 jnTj > mT

6. Sinc functionsinca(t) sin att 1< t < 1; sinca(nT) sin anTnT7. Gaussian functionga(t) eat21< t < 18. Error functionerf (t) 2pp

t0ex2dx 2pp1n0(1)nt2n1n!(2n 1)properties: erf (1) 1, erf (0) 0, erf (t) erf (t)erfc(t) complementary error function 1 erf (t) 2pp

1tex2dx9. Exponential and double exponentialf (t) etu(t) t ! 0; f (t) ejtj 1< t < 1f (nT) enTu(nT) nT ! 0; f (nT) ejnTj 1< nT < 1Note: T, sampling time; n, integer.1-4 Transforms and Applications Primer for Engineers with Examples and MATLAB1Approximating the integral in the above equation by the rectangle shown in Figure 1.4,we obtain its approximate discrete form:y(nT) y(nT T) Tx(nT) n 0, 1, 2, . . . (1:6)Trigonometric functionsOf special interest in the study of linear systems is the class of sine and cosine functions:a cos vt b sin vt r cos (vt w)These functions are periodic with a period 2p=v and a frequency f v=2pcycles=s or Hz.x(nT)Ax(nTT) BnT nTTTtFIGURE 1.3x(t) Errorx(nT)TnT nTT t 0FIGURE 1.4Signals and Systems 1-5Complex signalsSignals representing physical quantities are, in general, real. However, in many cases it isconvenient to consider complex signals and to use their real or imaginary parts torepresent physical quantities. One of these signals is the complex exponential ejvt.This function can be dened by its power series ex 1 x x22!x33! :ejvt 1 jvt ( jvt)22! ( jvt)33! ( jvt)nn! (1:7)The sum of two sine functions with the same frequency is also a sine function:a cos vt b sin vt r cos (vt w) (1:8)The discrete form of a sine function isx(nT) cos vnTBy separating the real and the imaginary parts of (1.7), we obtainejvt cos vt j sin vt (1:9)This fundamental identity can also be used to dene the complex exponential exp( jvt)and to derive all its properties in terms of the properties of trigonometric functions.We observe that exp(jvt) is a complex number with unity amplitude and phase vt.The sample value of the complex exponential isx(nT) ejvnTThis function is a geometric series whose ratio ejvTis a complex number of unitamplitude.From (1.9), it follows thate(ajv)t eat( cos vt j sin vt) (1:10)Therefore, if s a jv is a complex number, then estis a complex signal whose real parteatcos vt and imaginary part eatsin vt are exponentially decreasing (a 0) sine functions.From (1.9), we obtainejvt cos vt j sin vt1-6 Transforms and Applications Primer for Engineers with Examples and MATLAB1Adding and subtracting the last equation from (1.9), we nd Eulers formula:cos vt ejvtejvt2 sin vt ejvtejvt2j (1:11)A general complex signal x(t) is a function of the formx(t) x1(t) jx2(t)where x1(t) and x2(t) are the real functions of the real variable t. The derivative of x(t) is acomplex signal given bydx(t)dt dx1(t)dt jdx2(t)dtand, in general, for any s, real or complex, we havedestdt sest(1:12)Impulse (delta) functionAn important function in science and engineering is the impulse function also known asDiracs delta function. The signal is represented graphically in Figure 1.5. The deltafunction is not an ordinary one. Therefore, some fundamental properties of these types offunctions, and specically those of the delta function are presented so that the reader usesit appropriately.Property 1.1 The impulse function d(t) is a signal with a unit area and is zero outsidethe point at the origin:11d(t)dt 1d(t) 0 t 6 0

(1:13)Property 1.2 The impulse function is the derivative ofthe step function u(t):d(t) du(t)dt (1:14)(t)1t 0FIGURE 1.5Signals and Systems 1-7Property 1.3 The area of the product w(t)d(t) equals w(0) for any regular functionthat is continuous at the origin:

11w(t)d(t)dt w(0) (1:15)Property 1.4 The delta function can be written as a limit:d(t) limv(t) !0 (1:16)where v(t) is a family of functions with the unit area vanishing outside the interval2, 2 :

=2=2ve(t)dt 1 v(t) 0 for t < 2 and t > 2 (1:17)Figure 1.6 shows the approximation of the delta function by the pulse and the sufcientlysmall .We can show (see Prob) that the impulse function is even. Hence,d(t) d(t) (1:18)The impulse function d(t t0) is centered at t0 of area one. Therefore, from (1.18), weobtaind(t t0) d(t0t) (1:19)p(t)u(t)1 112t t/2/2 /2/2FIGURE 1.61-8 Transforms and Applications Primer for Engineers with Examples and MATLAB1Using Property 1.2 above, we writed(t t0) du(t t0)dt (1:20)Based on the above, the derivative of the function shown in Figure 1.7a is that shown inFigure 1.7b.Considering Property 1.3, and taking into consideration the evenness of the deltafunction, we write

11y(x)d(t x)dx y(t) (1:21)The above integral is also known as the convolution integral. Therefore, we state thatthe convolution of a function with a delta function reproduces the function. Let usconsider the function y(t t0) to be convolved with the shifted delta function d(t a).From (1.21), we write

11y(x t0)d(t x a)dx y(t t0a)The identity (1.21) is basic. We can use it, for example, to dene the derivative of thedelta function. Because the two sides of the equation are functions of t, we can differen-tiate with respect to t to obtainx(t)x(t)2321t3t212t10 t2 t3 tt t112(b) (a)FIGURE 1.7Signals and Systems 1-9

11y(x)d0(t x)dx y0(t) (1:22)Thus, the derivative of the delta function is such that the area of the product y(x)d0(t x),considered as a function of x, equals y0(t). With t 0, (1.22) yields

11y(x)d0(x)dx y0(0) (1:23)From calculus we know that when a function is even its derivative is an odd function.Hence,d0(t) d0(t) (1:24)Inserting (1.24) in (1.23) and changing the dummy variable from x to t, we nd

11y(t)d0(t)dt y0(0) (1:25)Additional delta functional properties are given in Table 1.2.TABLE 1.2 Delta Functional Properties1. d(at) 1jajd(t)2. d t t0a jajd(t t0)3. d(at t0) 1jajd t t0a 4. d(t t0) d(t t0)5. d(t) d(t); d(t) even function6. 11 d(t)f (t)dt f (0)7.11 d(t t0)f (t) f (t0)8. f (t)d(t) f (0)d(t)9. f (t)d(t t0) f (t0)d(t t0)10. td(t) 011. 11 Ad(t)dt 11 Ad(t t0)dt A12. f (t)*d(t) convolution 11 f (t t)d(t)dt f (t)13. d(t t1)*d(t t2) 11 d(t t1)d(t t t2)dt d[t (t1t2)]14.NnN d(t nT)*NnN d(t nT) 2Nn2N (2N 1 jnj)d(t nT)1-10 Transforms and Applications Primer for Engineers with Examples and MATLAB1TABLE 1.2 (continued) Delta Functional Properties15.11dd(t)dt f (t)dt df (0)dt16.11dd(t t0)dt f (t)dt df (t0)dt17.11dnd(t)dtn f (t)dt (1)n dnf (0)dtn18. f (t)dd(t)dt df (0)dt d(t) f (0)dd(t)dt19. t dd(t)dt d(t)20. tn dmd(t)dtm (1)nn!d(t), m n(1)n m!mn!dmnd(t)dtmn , m > n0, m < n

21.11dd(t)dt 0, dd(t)dt odd function22. f (t)*dd(t)dt df (t)dt23. f (t)dnd(t)dtn nk0 (1)k n!k!(n k)!dkf (0)dtkdnkd(t)dtnk24. @d(yt)@y 1y2 d(t)25. d(t) du(t)dt26. dnd(t)dtn (1)n dnd(t)dtn , dnd(t)dtn is even if n is even, and odd if n is odd: 27. (sin at)dd(t)dt ad(t)28. dd(t)dt d2u(t)dt229. d(t) du(t)dt30. d(t t0) du(t t0)dt31. dsgn(t)dt 2d(t)32. d[r(t)] nd(t tn)dr(tn)dt

, tn zeros of r(t), dr(tn)dt 6 033. dd[r(t)]dt ndd(t tn)dtdr(t)dtdr(tn)dt

, tn zeros of r(t), dr(tn)dt 6 0, dr(t)dt 6 034. d(sin t) 1n1 d(t np)35. d(t21) 12d(t 1) 12d(t 1)36. d(t2a2) 12a[d(t a) d(t a)](continued)Signals and Systems 1-11TABLE 1.2 (continued) Delta Functional Properties37. d(t) lim!0et2=pp38. d(t) limv!1sin vtpt39. d(t) lim!01pt2240. d(t) 12p

11cos vt dv41. df (t)dt ddt [tu(t) (t 1)u(t 1) u(t 1)] td(t) u(t) (t 1)d(t 1) u(t 1) d(t 1)42. combT(t) 1n1 d(t nT), f (t)combT(t) 1n1 f (nT)d(t nT)COMBv0(v) F{combT(t)} v01n1 d(v nv0), v0 2pTddt ([2 u(t)] cos t) ddt (2 cos t u(t) cos t) 2 sin t d(t) cos t u(t) sin t (u(t) 2) sint d(t)ddt u t p2 u(t p) sin t d t p2 d(t p) sin t u t p2 u(t p) cos t d t p2 u t p2 u(t p) cos tExampleThe values of the following integrals are

11e2tsin 4t d2d(t)dt2 dt (1)2 d2dt2 [e2tsin 4t]jt0 2 2 4 16

11(t32t 3) dd(t 1)dt 2d2d(t 2)dt2 dt

11(t32t 3)dd(t 1)dt2 dt2

11(t32t 3)d2d(t 2)dt2 dt (1)(3t22)jt1(1)22(6t)jt2 5 24 19.ExampleThe values of the following integrals are

40e4td(2t 3)dt

40e4td 2 t 32 dt 12

40e4td t 32 dt 12e43212e61-12 Transforms and Applications Primer for Engineers with Examples and MATLAB1The comb functionThe comb function is represented mathematically as follows:combT(t) 1n1d(t nT) (1:26)This function is used extensively for studying the sampling of signals. Figure 1.8 showsthe comb function pictorially.1.3 Circuit Elements and EquationIn this text, we use the idealized model of physical devices, passive or active, which isspecied in terms of its terminal properties. In Figure 1.9, we show the passive and activeelements of electrical circuits.A circuit or network is a combination of connected elements and external sources.The inputs are the sources (voltage or current) and the outputs are voltages or currentsacross elements and through elements, respectively. A network is a special form of ananalog system since its inputs and outputs are continuous signals.+ ++ + + v(t)v(t)i(t)R L C e(t) is(t)FIGURE 1.9CombT (t). .1t 0 T 2T 3T T 2T 3TFIGURE 1.8Signals and Systems 1-13The state of a network at a certain time (taking the time t 0 for simplicity) is the setof all the voltages across the capacitors and all the currents through the inductors. If weknow the initial state of a network at t 0 and all its inputs at t 0, then we candetermine all its responses for an all-time t !0. If all the currents through the inductorsand all the voltages across the capacitors are zero, the network is at a zero initial state. Ifthe network is at a zero initial state, then its response is known as the zero-stateresponse. If all the sources are zero, then its response is called the zero-input response.The zero-input response is due to the energy stored in the network.The voltages and the currents of the passive elements areResistorv(t) Ri(t) i(t) Gv(t) G 1R (1:27)Inductorv(t) Ldi(t)dt i(t) 1L

t0v(x)dx i(0) (1:28)Capacitori(t) C dv(t)dt v(t) 1C

t0i(x)dx v(0) (1:29)Voltage sourcee(t) known, independent of i(t)Current sourceis(t) known, independent of v(t)Initial conditionsKnowing the initial conditions of a network (voltages across the capacitors and the currentsthrough the inductors) it is sufcient to nd its response for t !0. In this text we assume thatthere is a continuation of the initial conditions which means that the currents through theinductors or the voltages across the capacitors are the same at t(0) and t(0).Impulse responseThe following simple example will elucidate how a network responds to an impulse inputsource. Let the input voltage of a simple RL series circuit be a delta function as shown inFigure 1.10. Kirchhoff s voltage law of a network loop isLdi(t)dt Ri(t) d(t) (1:30)1-14 Transforms and Applications Primer for Engineers with Examples and MATLAB1Integrating the above equation from (0) to (0), and taking into consideration that i(t)is a continuous function, we obtainL

00di(t)dt dt R

00i(t)dt

00d(t)dt or L[i(0) i(0)] R0 1or L[i(0) i(0)] 1 (1:31)Since the input impulse function is a discontinuous one, the current is also a discon-tinuous function with a discontinuity such that L[i(0) i(0)] is equal to 1. If, inaddition, the system (here the network) is causal, i(0) 0 and hence i(0) 1=L.Therefore, if the circuit is in the zero state and it is connected to a delta function source,the current i(t) changes instantly from zero to 1=L.Derived initial conditionsDerived initial conditions are determined from the circuit equations, and, in general,depend also on the sources. Let us assume that there is an initial current, i(0) i0, in theRL circuit shown in Figure 1.10. In addition, let the voltage source be a constant, v(t) V.In this case, the solution of (1.30) with a constant voltage is the functioni(t) VR (1 eRt=L)....zero-state response i0eRt=L....zero-input response(1:32)This result will be derived in Chapter 7.For an RC series circuit, Kirchhoff s mesh voltage law results inRi(t) vc(t) v(t) or Ri(t) 1C

01i(x)dx 1C

t0i(x)dx v(t) orRi(t) 1C

t0i(x)dx vc(0) v(t) (1:33)v(t) =(t)eRt/L1i(t)=i(t)i(t)LRv(t)v(t)+t tSystemLFIGURE 1.10Signals and Systems 1-15where vc(t) is the voltage across the capacitor. This equation can be cast into an ordinarydifferential equation by differentiating both sides. Hence,Rdi(t)dt 1C i(t) dv(t)dt (1:34)To solve (1.34), we must nd the initial value of the current, i(0). Setting t 0, we obtainRi(0) vc(0) v(0) or i(0) v(0) vc(0)RThis is the derived initial condition, and it depends not only on the initial (state) voltageacross the capacitor but also on the initial value of the voltage source.As another example, Kirchhoff s mesh equation for a series RLC circuit with an initialcurrent i(0) through the inductor and the initial voltage vc(0) across the capacitor isLdi(t)dt Ri(t) 1C

t0i(x)dx vc(0) v(t) i(0) (1:35)Taking the derivative with respect to the independent variable t, we ndLd2i(t)dt2 Rdi(t)dt 1C i(t) dv(t)dt (1:36)Since the above equation is a second-order differential equation of the dependent variablei, we must nd, in addition to its initial value i(0), the initial value of its derivative i0(0).Setting t 0 in (1.35) and assuming that v(t) does not have a discontinuity at t 0, weobtainLi0(0) Ri(0) vc(0) v(0) or i0(0) 1L[v(0) vc(0) Ri(0)]which is a derived initial condition.State equations of an RLC series circuitThe state variables are the current i(t) through the inductor and the voltage vc(t) acrossthe capacitor and they satisfy the following two rst-order differential equations:C dvc(t)dt i(t) vc(0)Ldi(t)dt Ri(t) vc(t) v(t) i(0)(1:37)1-16 Transforms and Applications Primer for Engineers with Examples and MATLAB1Node and state equations of the circuit in Figure 1.11The circuit in Figure 1.11 has two variables: the voltage v1(t) across the capacitor and thecurrent i(t) through the inductor. The initial voltage across the capacitor is v1(0) andthe initial current through the inductor is i(0).Node equations (the algebraic sum of currents at a node should be equal to zero)For the node equation we use as primary unknowns, the node voltages v1(t) and v2(t).Hence,C dv1(t)dt v2(t)R is(t) v1(0)Ldi(t)dt Ri(t) v1(t) 0 or LRdv2(t)dt v2(t) v1(t) 0 v2(0) Ri(0)(1:38)State equationsC dv1(t)dt i(t) is(t) v1(0)Ldi(t)dt Ri(t) v1(t) 0 i(0)(1:39)State equations for the circuit in Figure 1.12State equationsL1di1(t)dt R1i1(t) vc(t) v(t)L2di2(t)dt R2i2(t) vc(t) 0i1(t) i2(t) C dvc(t)dt 0(1:40)Block diagrams of systemsCircuit diagrams describe the structure of a network. However, the block diagramsdescribe the terminal properties of the network (system). Inside the block we presentis(t)i(t)C RLv1(t) v2(t)FIGURE 1.11Signals and Systems 1-17different identiers that will characterize the system operation. In general, we introducein the block a script O to represent a general operator that operates on the input to theblock to produce the output. In Figure 1.13, we show the block-diagram representationof, a general system, a differentiator, a multiplier, an integrator, and a pick-off point.The signicance of s is given later in Chapter 7. Note that at the pick-off point the inputquantity appears in all the branches without any variation of its magnitude.Figure 1.14a depicts three basic ways that systems can be congured. It is assumed thatthe terminal properties of each system remain unchanged (no loading effect takes place).Figure 1.14b shows the equivalent inputoutput of the cascade and the parallel andthe feedback congurations. In Figure 1.14a and for the rst system, we obtainy1(t) O1x(t) or y(t) O2y1(t) O2O1x(t). The second expression characterizes therst system of part (b) of the gure. For the second system of part (a) of the gure, we ndy1(t) O1x(t) and y2(t) O2x(t), and, therefore, y(t) y1(t) y2(t) [O1O2]x(t)which characterizes the second system of part (b) of the gure. For the feedback system ofpart (a), we obtain: y1(t) x(t) O2y(t) or y(t) O1[x(t) O2y(t)]. Solving y(t), andkeeping in mind that we do not perform divisions with the operators but only use theirinverse form, we nd y(t) [1 O1O2]1O1x(t). This expression characterizes the thirdR1R2L1 L2i1(t)Cv(t)+ +i2(t)vc(t)FIGURE 1.12t0y(t) = x()ddx(t)y(t) =dt1sSx(t)Pick-off pointy(t) =ax(t)y(t) =O{x(t)}Multiplierx(t)IntegratorDifferentiatorx(t)x(t)x(t)Oax(t)x(t)x(t)FIGURE 1.131-18 Transforms and Applications Primer for Engineers with Examples and MATLAB1system of part (b) of the gure. If we substitute the two operators with constants a and b,the transfer functions of the three systems areHc ab, Hp a b, Hf a1 ab (1:41)Table 1.3 represents block-diagram transformations.++O1O2O1 O1+O2O1O1O2O2O2[1 O1O2]1O1x(t) x(t)x(t)x(t)x(t)x(t)y2(t)y1(t)y1(t)y1(t)y(t)y(t) y(t)y(t)y(t)y(t)(b)(a)FIGURE 1.14TABLE 1.3 Block-Diagram Transformations of Systems(c)(b)(a)Pick-of pointSummation pointTwo blocks incascade++xxa b abx yyy =x vxxvxx(continued)Signals and Systems 1-19TABLE 1.3 (continued) Block-Diagram Transformations of SystemsFeedback loopSpecial case of unitfeedback loop(e)(d)+ a yx y a1 ax ++ 1ayx +x y 11 aMoving a pick-offpoint aheadComplete feedbackloopMoving a pick-offpoint behindx yy(h)(g)(f )aaab++x yy xx xyax1/ax yyaaa1 abx y1-20 Transforms and Applications Primer for Engineers with Examples and MATLAB11.4 Linear Mechanical and RotationalMechanical ElementsThe linear mechanical systems with their equivalent circuit characterizations are shownin Figure 1.15. The rotating fundamental mechanical systems and their equivalent circuitcharacterizations are shown in Figure 1.16. The terminal properties of these signals aregiven below.1.4.1 Linear Mechanical SystemsDamperf (t) Dv(t), v(t) 1Df (t), D damping constant (N s=m)f (t) force (N), v(t) velocity (m=s)(1:42)TABLE 1.3 (continued) Block-Diagram Transformations of SystemsMoving a summingpoint aheadMoving a summingpoint behind(i)(j)aayvvy++++xx ya1/a v++xyva+a+xf (t)v(t)v(t)f (t)Referencevg=0=velocity(ground)f (t)v1v2v2x1 x2v1v2 v1v =v1v2v =v1v2DOilPhysicalsystemf (t)+++(a)MM(b) (c)FIGURE 1.15Signals and Systems 1-21Springf (t) Kx(t), v(t) 1Kdf (t)dt , x(t) displacement (m)K spring constant (N=m)(1:43)Massf (t) M dv(t)dt M d2x(t)dt2 Newton kg m s2(N)v(t) 1M

t1f (x)dx M mass(kg)(1:44)From the above equations, we observe the following analogies between the circuitelements and the linear mechanical elements: the mass and the capacitor, the springand the inductor, and the damper and the resistor.1.4.2 Rotational Mechanical SystemsDamperT(t) Dv(t), D damping costant (N s m=rad)v(t) 1DT(t); T torque (N m)(1:45)+Viscous fluid (c)2 1 +TTT TT T+1, 1 2, 2=12=12++(a) (b)JJFIGURE 1.161-22 Transforms and Applications Primer for Engineers with Examples and MATLAB1SpringT(t) Ku(t) K

t1v(x)dx, K spring constant (N m=rad)v(t) 1DdT(t)dt(1:46)Moment of inertiaT(t) J dv(t)dt J d2u(t)dt2 , J polar moment of inertia (kg m2)v(t) 1J

t1T(x)dx(1:47)For the rotation elements, we observe the following analogies between these elementsand the circuit elements: the damper and the resistor, the inductor and the spring, andthe mass and the moment of inertia. The current in circuits, the force in linear mech-anical systems, and the torque in rotational mechanical elements are the throughvariables. The voltage in circuits, the velocity in linear mechanical systems, and theangular velocity in rotational mechanical systems are the across variables.1.5 Discrete Equations and SystemsA discrete system is a process that relates the input discrete-time signal x(n) (or x(nT))with the discrete-time output signal y(n) (or y(nT)). The elements with which we creatediscrete systems are shown in Figure 1.17. The special symbol z1indicates the delaywhich we discuss in Chapter 8. The delay element has a memory. This means that theoutput at any particular time depends on the value of the input one unit earlier. In thediscrete systems, we also have pick-off points as we have dened them in the circuits case.A simple rst-order system is dened by the following discrete equation:y(n) 2y(n 1) x(n) (1:48)Its block-type representation and its solution is shown in Figure 1.18. This is a recursiveequation, and its solution is found by iteration, assuming (or dening), of course, itsx(n) x(n) x(n)a x(n) x(n1)ay(n)Digital system Delay element Multiplierz1FIGURE 1.17Signals and Systems 1-23initial conditions. If we set y(1) 0 and the input function to be a delta function,we obtainy(0) 2y(1) d(0) 0 1 1y(1) 2y(0) d(1) 2 0 2y(2) 2y(1) d(2) 2(2) 0 4y(3) 2y(2) d(3) 2(4) 0 8...The state of a discrete system at a certain time n0 is the set of values of the outputsqi(n 1) of all delay elements at n n0. Therefore, if we know the state of the system atn n0 and all its sources for n >n0, then we can determine all its responses for anyn >n0.The initial state of a system is its state at n 0, where this time of origin is taken forconvenience. Hence, the initial state of a system is the set of values qi(1) of the inputsqi(n) to all delay elements at n 1. If the system is at the zero state, then its responsesfor n >0 are called zero-state responses. We therefore conclude that its responses aredue only to its inputs (external sources). On the other hand, if all external sources arezero, its responses are only due to the energy sources of the system and they are calledzero-input responses.State equationsState variables are the inputs qi(n) to all the delay elements (or any linear transformationof these signals). The state variables are determined from the state equations resultingfrom the rules of the interconnected elementary systems. The state equations become asystem of a specic number of equations equal to the number of the delay elementspresent in the system. To nd the solution, besides the input sources, we need the initialconditions which are the values of the state variables at n 1. It is apparent, forexample, that the second-order discrete systemy(n) 3:5y(n 1) 5y(n 2) x(n) (1:49)nnx(n)x(n) =(n)y(n)y(n)8242z12+FIGURE 1.181-24 Transforms and Applications Primer for Engineers with Examples and MATLAB1which is shown in the block-diagrammatic form in Figure 1.19, has the following statevariables representation:q1(n) 3:5q2(n) 5q2(n 1) x(n)q2(n) q1(n 1) 0q1(1) y(1)q2(1) y(2)

initial conditions(1:50)Recursive and non-recursive systemsIf a discrete (difference) equation, which represents the system, has one input and anoutput with additional delayed outputs, it is called a recursive one. We also call thesesystems innite impulse systems (IIR). The difference equation of (1.49) represents arecursive system and it is shown in Figure 1.19.If, however, we have the discrete system representation by the equationy(n) b0x(n) b1x(n 1) b2x(n 2) (1:51)we say that the system is not recursive. This type of system is also called the niteimpulse response system (FIR). Figure 1.20 shows such a system. Note the feedbackconguration of the IIR systems and the forward conguration of the FIR systems.y(n)z1z1y(n2), q2(n1)q2(n), y(n1), q1(n1)x(n)3.55++q1(n), y(n)q1(n), y(n)FIGURE 1.19x(n) y(n)b0 ++b2b1z1z1x(n1)x(n2)FIGURE 1.20Signals and Systems 1-251.6 Digital Simulation of Analog SystemsSince the physical systems are represented mathematically by differential and integro-differential equations, we must approximate derivatives and integrals (see (1.3) and (1.6)).The approximations are derived by interrogating Figures 1.3 and 1.4. A second-orderderivative is approximated in the formd2y(t)dt2 y(nT) 2y(nT T) y(nT 2T)T2 (1:52)To have the solution of a second-order differential equation, we must have the value ofthe derivative at t 0. Therefore, we must substitute the analog derivative with anequivalent discrete one. Hence, we writedy(t)dt

t0dy(0)dt y(0T) y(0T T)Tory(T) y(0) T dy(0)dt (1:53)1.7 Convolution of Analog SignalsThe convolution operation on functions is one of the most useful operations encounteredin the study of signals and systems. The importance of the convolution integral insystems studies stems from the fact that a knowledge of the output of the system to animpulse (delta) function excitation allows us to nd its output to any input function(subject to some mild restrictions).To help us develop the convolution integral, let us begin with the properties of thedelta function. Based on the delta properties, we writef (t)

11f (t)d(t t)dt (1:54)Observe that, as far as the integral is concerned, the time t is a parameter (constant forthe integral although it can take any value) and the integration is with respect to t. Ournext step is to represent the integral with its equivalent approximate form, the summa-tion form, by dividing the t axis into intervals of DT, then the above integral isrepresented approximately by the sumfa(t) limDT!01n1f (nDT)d(t nDT)DT (1:55)1-26 Transforms and Applications Primer for Engineers with Examples and MATLAB1As DT goes to zero and n increases to innity, the product nDT takes the value of t, DTbecomes dt and the summation becomes integral, thus recapturing (1.54).Note: The function f (t) has been approximated with an innite sum of shifted deltafunctions equal to nDT and their area is equal to f (nDT)DT.We dene the response of a causal (system that reacts after being excited) and an LTIsystem to a delta function excitation by h(t), known as the impulse response of thesystem. If the input to the system is d(t) the output is h(t), and when the input is d(tt0)then the output is h(t t0). Further, we dene the output of a system by g(t) if its input isf (t). Based on the denitions discussed so far, it is obvious that if the input to the systemis fa(t), the output is a sum of impulse functions shifted identically to the shifts of theinput delta functions of the summation, and, therefore, the output is equal tog(t) limDT!01n1f (nDT)h(t nDT)DTIn the limit, as DT approaches zero, the summation becomes an integral of the formg(t)

11f (t)h(t t)dt (1:56)This is the convolution integral for any two functions f(t) and h(t).Convolution is a general mathematical operation, and for any two real-valued func-tions, their convolution, indicated mathematically by the asterisk between the functions,is given byg(t) Df (t)*h(t)

11f (t)h(t t)dt

11f (t t)h(t)dt (1:57)Note: Equation 1.57 tells us the following: given two functions in the time domain t, wend their convolution g(t) by doing the following steps: (1) rewrite one of the functions inthe t domain by just setting wherever there is t, the variable t; the shape of the function isidentical to that in the t domain; (2) to the second function substitute t-t wherever there is t;this produces a function in the t domain which is ipped (the minus sign in front of t) andshifted by t (positive values of t shift the function to the right and negative values shift thefunction to the left); (3) multiply these two functions and nd another function of t, since t isa parameter and a constant as far as the integration is concerned; and (4) next nd the areaunder the product function whose value is equal to the output of the convolution at t (in ourcase it is g(t)). By introducing the innite values of ts, from minus innity to innity, weobtain the output function g(t).Signals and Systems 1-27f (t) =etu(t) f (t) =e0.5tu(t)1 1t t (a)1x = a1att (b)f () =eu()f() h(t ) =ee0.5(t )u()u(t )Area=g(t)= ee0.5(t)u()u(t)d= ee0.5(t )dt0h(t ) =e0.5 u(t )2t2(c)g1(t) =2e0.5tu(t)g(t) =g1(t) +g2(t)g2(t) =2etu(t)FIGURE 1.211-28 Transforms and Applications Primer for Engineers with Examples and MATLAB1From the convolution integral, we observe that one of the functions does not changewhen it is mapped from the t to t domain. The second function is reversed or folded over(mirrored with respect to the vertical axis) in the t domain and it is shifted by an amount t,which is just a parameter in the integrand. Figure 1.21a and b shows two functions in the tand t domains, respectively. We now writeg(t) f (t)*h(t)

11etu(t)e0:5(tt)u(t t)dt

t0ete0:5(tt)dt e0:5t

t0e0:5tdt 2(e0:5tet)Figure 1.21c shows the results of the convolution.1.8 Convolution of Discrete SignalsAs we have indicated in the above section, the convolution of continuous signals isdened as follows:g(t)

11f (x)h(t x)dx (1:58)The above equation is approximated as follows:g(t)

11f (x)h(t x)dx 1m1

mTmTTf (x)h(t x)dx 1m1Tf (mT)h(t mT)org(nT) T1m1f (mT)h(nT mT) n 0, 1, 2, . . . m 0, 1, 2, . . . (1:59)For T1, the above convolution equation becomesg(n) 1m1f (m)h(n m) n 0, 1, 2, . . . m 0, 1, 2, . . . (1:60)If the input function to the system is the delta functiond(nT) 1 n 00 n 6 0

d(nT mT) 1 n m0 n 6 m

(1:61)then, (1.60) gives g(n) h(n).Additional properties of the convolution process are shown in Table 1.4.Signals and Systems 1-29TABLE 1.4 Convolution Properties1. Commutativeg(t)

11f (t)h(t t)dt

11f (t t)h(t)dt2. Distributiveg(t) f (t)*[h1(t) h2(t)] f (t)*h1(t) f (t)*h2(t)3. Associative[f (t)*h1(t)]*h2(t) f (t)*[h1(t)*h2(t)]4. Shift invarianceg(t) f (t)*h(t)g(t t0) f (t t0)*h(t)

11f (t t0)h(t t)dt5. Area propertyAf area of f (t),mf

11tf (t)dt first momentKf mfAf center of gravityAg AfAh, Kg Kf Kh6. Scalingg(t) f (t)*h(t)f ta *h ta jajg ta 7. Complex valued functionsg(t) f (t)*h(t) [fr(t)*hr(t) fi(t)*hi(t)] j[fr(t)*hi(t) fi(t)*hr(t)]8. Derivativeg(t) f (t)*dd(t)dt df (t)dt9. Moment expansiong(t) mh0f (t) mh1f(1)(t) mh22! f(1)(t) (1)n1n 1! mh(n1) f(n1)(t) Enmhk

11tkh(t)dtEn (1)nmhnn! f(n)(t t0), t0 constant in the interval of integration10. Fourier transformF{f (t)*h(t)} F(v) H(v)(continued)1-30 Transforms and Applications Primer for Engineers with Examples and MATLAB1ExamplesExample 1.1It is desired to plot the functions x(t) 2u(2 t), x(t) u(t 1) 2u(t 3), andx(t) 2d(t 1) d(t 1). These functions are plotted in Figure E.1.1. &TABLE 1.4 (continued) Convolution Properties11. Inverse Fourier transform12p

11F(v)H(v)ejvtdv

11f (t)h(t t)dt12. Band-limited functiong(t)

11f (t)h(t t)dt 1n1 Tf (nT)hs(t nT)hs(t) 12p

ssH(v)ejvtdv, f (t) s band limited 0, jtj > s13. Cyclical convolutionx(n) y(n) N1m0 x( (n m) mod N)y(m)14. Discrete-timex(n)*y(n) 1m1 x(n m)y(m)15. Sampledx(nT)*y(nT) T1m1 x(nT mT)y(mT)whereH(ejv) 1n1 h(n)ejvn:x(t) =2(t) (t +2) x(t) =u(t 1) 2u(t 3) x(t) =2u(2t)t t t 3 1 11221122FIGURE E.1.1Signals and Systems 1-31Example 1.2The evaluation of integrals, involving delta functions, is shown in the equations below:

54(t22)[d(t) 3d(t 2)]dt

54(t22)d(t)dt

543(t22)d(t 2)dt 218 20

34t2[d(t 2) d(t) d(t 5)]dt

34t2d(t 2)dt

34t2d(t)dt

34t2d(t 5)dt 400 4&Example 1.3A series RLC circuit is shown in Figure E.1.2 driven by the voltage source v(t). The circuithas two state variables: the capacitor voltage vc(t) and the inductor current i(t), with theinitial conditions vc(0) and i(0). The circuit has one mesh and one mesh current thatsatises Kirchhoffs voltage law:Ldi(t)dt Ri(t) 1C

t0i(x)dx vc(0) v(t) i(0) (1:62)We next, reduce the above integrodifferential equation into a differential equation bydifferentiation:Ld2i(t)dt2 Rd(i)dt 1C i(t) dv(t)dt (1:63)To solve (1.63), we need the initial conditions i(0) and its derivative at zero time i0(0)since this is an equation of the second order. The i(0) is the given initial state of thevc(t)++LCi(t)v(t)RFIGURE E.1.21-32 Transforms and Applications Primer for Engineers with Examples and MATLAB1system. To nd the second initial condition, we must set t 0 in (1.62). The substitutiongivesLi0(0) Ri(0) 1C

00i(x)dx vc(0) v(0) or Li0(0) Ri(0) vc(0) v(0)The above equation gives the desired initial condition:i0(0) 1L [v(0) vc(0) Ri(0)]State equationsThe current through the capacitor based on Kirchhoffs law is equal to the current i(t).Second, the algebraic sum of the voltages in the mesh should be equal to zero. Hence,C dvc(t)dt i(t) vc(0)Ldi(t)dt Ri(t) vc(t) v(t) i(0)(1:64)and this is a system of two rst-order differential equations. &Example 1.4Let the circuit (system) shown in Figure E.1.3 have the initial conditions vc(0), i1(0), andi2(0) of its state variables. To nd the state equations, we sum algebraically the voltagesin the two loops and the currents at the node. Hence,State equationsL1di1(t)dt R1i1(t) vc(t) v(t)L2di2(t)dt R2i2(t) vc(t) 0i1(t) i2(t) C dvc(t)dt 0(1:65)R2L2L1R1C vc(t)i2(t) i1(t)v(t)++FIGURE E.1.3Signals and Systems 1-33Mesh equationsR1i1(t) L1di1(t)dt 1C

t0i1(x)dx 1C

t0i2(x)dx vc(0) v(t)1C

t0i1(x)dx L2di2(t)dt R2i2(t) 1C

t0i2(x)dx vc(0) 0(1:66)&Example 1.5It is desired to create the block-diagram representation of the following differentialequation and its equivalent discrete representation. The equation is3dy(t)dt y v(t) (1:67)Its discrete representation isy(nT) 11 T3y(nT T) T3 1 T3 v(nT) (1:68)The block-diagram representation of the above two equations are given in Figure E.1.4.&Example 1.6It is required to nd the differential equation of the linear mechanical system shown inFigure E.1.5a with respect to the distance traveled by the mass. This system is a roughrepresentation, for example, of the spring-shock absorber system of a car. From thegure, the motion of the mass that is subjected to a spring and a damping force isdescribed by the equationf (t) fM(t) fK(t) fD(t) 0 (1:69)+s3+v(nT) v(t) y(t) y(nT)TT3 13z11T13FIGURE E.1.41-34 Transforms and Applications Primer for Engineers with Examples and MATLAB1orMdv(t)dt K

v(t)dt Dv(t) f (t) (1:70)Since the velocity is related to the displacement x by the relation v(t) dx(t)=dt, thisequation takes the formd2x(t)dt2 DMdx(t)dt KMx(t) 1Mf (t) (1:71)Because the velocity is an across variable, the velocity of the mass with respect to theground, Figure E.1.5b represents the circuit representation of the system. We observe thatthe force is a through variable and the system is a node equivalent type circuit. &Example 1.7The system shown in Figure E.1.6 represents an idealized model of a stiff human limb asa step in assessing the passive control process of locomotive action. We try to nd themovement of the system if the input torque is an exponential function. During themovement, we characterize the friction by the friction constant D. Furthermore, weassume that the initial conditions are zero, u(0) du(0)=dt 0.Applying DAlemberts principle, which requires that the algebraic sum of thetorques must be equal to zero at a node, we writeT(t) Tg(t) TD(t) TJ(t) (1:72)whereT(t) input torqueTg(t) gravity torque Mgl sin u(t)TD(t) frictional torque Dv(t) Ddu(t)dtTJ(t) inertial torque Jdv(t)dt Jd2u(t)dt2fK DM vf MDvfKfMfDvg=0(a) (b)FIGURE E.1.5Signals and Systems 1-35Therefore, the equation that describes the system isJd2u(t)dt2 Ddu(t)dt Mgl sin u(t) T(t) (1:73)The above equation is nonlinear owing to the presence of the sin u(t) term in theexpression of the gravity torque. To create a linear equation, we must assume thatthe system does not deect much and the deection angle stays below 308. Underthese conditions, (1.73) becomesJ d2u(t)dt2 Ddu(t)dt Mglu T(t) (1:74)SystemInput OutputlMgsin (t)(t)(t) (t)(t)(t)Mg(a)0 1 2 3 4 5 600.050.10.150.20.25t s(t); (nT)(b)Continuous caseDiscrete case T=0.1Discrete case T=0.5FIGURE E.1.61-36 Transforms and Applications Primer for Engineers with Examples and MATLAB1This is a second-order differential equation and, hence, its solution must contain twoarbitrary constants, the values of which are determined from specied initial conditions.For the specic constants J 1, D2, and Mgl 2, the above equation becomesd2u(t)dt2 2du(t)dt 2u(t) etu(t) (1:75)We must rst nd the homogeneous solution from the homogeneous equation (theabove equation equal to zero). If we assume a solution of the form uh(t) Cest, thesolution requirements iss22s 2 0from which we nd the roots s11 j and s21 j. Therefore, the homogeneoussolution isuh(t) C1es1tC2es2t(1:76)where Cis are arbitrary unknown constants to be found by the initial conditions.To nd the particular solution, we assume a trial solution of the form up(t) Aetfort !0. Introducing the assumed solution in (1.75), we ndAet2Aet2Aet etor A 1The total solution isu(t) uh(t) up(t) C1es1tC2es2tett ! 0Applying, next, the initial conditions in the above equation, we nd the followingsystem of equations:u(0) C1C21 0du(0)dt C1s1C2s21 0Solving the unknown constants, we obtain C1(1 s2)=(s1s2), C2(1 s1)=(s1s2).Introducing, next, these constants into the total solution and the two roots, we ndu(t) 12etejt12etejtet (1 cos t)e1t ! 0 (1:77)The digital simulation of (1.75) is deduced by employing (1.3), (1.52), and (1.53). Hence,u(nT) 2u(nT T) u(nT 2T)T2 2u(nT) u(nT T)T 2u(nT) enTn 0, 1:2, . . .(1:78)Signals and Systems 1-37After rearranging the above equation, we obtainu(nT) a(2 2T)u(nT T) au(nT 2T) aT2enTa 11 2T 2T2 , n 0, 1, 2, . . .(1:79)Using (1.53), we obtain that u(T) 0. Next, introducing this value and the initialcondition u(0T) 0 in (1.78), we nd u(2T) T2. The following m-le produces thedesired output for the continuous case and for the two different sampling values,T 0.5 and T 0.1.Book MATLAB1m-le for the Example 1.7: ex_1_5_1%Book m-file for the Example 1.7: ex_1_5_1t0:0.1:5.5;th(1cos(t)).*exp(t);T10.5;N15.5=T1;T20.1;N25.5=T2;a11=(12*T12*T1^2);a21=(12*T22*T2^2);thd1(2)0;thd1(1) T1^2;thd2(2) 0;thd2(1)T2^2;for n0:N1thd1(n3)a1*(22*T1)*thd1(n2)a1*thd1(n1)T1^2*a1*exp(n*T1);end;for n0:N2thd2(n3)a2*(22*T2)*thd2(n2)a2*thd2(n1)T2^2*a2*exp(n*T2);end;plot([0:55],th,'k');hold on;stem([0:5:N1*5],thd1(1,3:14),'k');hold on;stem([0:N2],thd2(1,3:58),'k'); &Example 1.8It is desired to write the state equations for the system shown in Figure E.1.7 andexpress the output y(n) in terms of the state variables. From the gure, we obtainq1(n) 3q2(n) 2q2(n 1) x(n)q2(n) q1(n 1)y(n) 5q1(n) 4q2(n) &Example 1.9The convolution of the exponential function f(t) exp(t)u(t) and the pulse symmetricfunction p2(t) of width 4 is given by1-38 Transforms and Applications Primer for Engineers with Examples and MATLAB1g(t)

11[u(x 2) u(x 2)]e(tx)u(t x)dx et

11u(x 2)exu(t x)dx et

11u(x 2)exu(t x)dx g1(t) g2(t)For t 0 (2:73)u(x, 0) f (x) 0 < x < L (2:74)To ndthe particular solutions of (2.72) that satisfy (2.73), we assume a solutionof the formu(x, t) X(x)T(t). Substituting the assumed solution into differential equation and separat-ingthefactors withdependence only onx andthefactors dependent only ont andsettingthe ratio equal to a constant a, we obtain the following ordinary differential equations:X00(x) aX(x) 0 T00(t) akT(t) 0 (2:75)If the function X(x)T(t) is to satisfy conditions (2.73), thenX(0, 0) 0 X(L, 0) 0 (2:76)The solution of the rst differential equation is found from its characteristic equationD2a 0 to be X(x, 0) Aeap xBe ap x. Applying the boundary condition X(0, 0) 0we obtain A B, and hence, the solution isX(x, 0) 2Aeap xeap x2 C eap xeap x2 C sinh x apThis solution can satisfy the second condition of (2.76) ifa n2n 1, 2, 3, . . .Then X(x, 0) C sin xn2p jC sin nx Dsin nx. The solution of the second equationof (2.75) is T(t) Een2kt. Therefore, the solutions of the PDF with the boundaryconditions (2.73) are in the form u(x, t) X(x)T(t):Bnen2ktsin nx n 1, 2, 3, . . . (2:77)where the constants Bn are arbitrary. Since an innite number of solutions will convergeto f (x) and since any solution satises the PDF and its boundary conditions, then thegeneral solution isu(x, t) 1n1Bnen2ktsin nx (2:78)Fourier Series 2-21For t 0, the general solution becomesu(x, 0) 1n1Bn sin nxThe above equation is a sine Fourier series expansion of u(x, 0) and, therefore, thecoefcients areBn 2L

L0f (x) sin nx dx n 1, 2, 3, . . . (2:79)Therefore, the nal form of the solution isu(x, t) 2L1n1en2ktsin nx

L0f (x0) sin nx0 dx0 n 1, 2, 3, . . . (2:80)&2-22 Transforms and Applications Primer for Engineers with Examples and MATLAB13Fourier Transforms3.1 IntroductionFourier TransformNot all functions are Fourier transformable. However, the Dirichlet conditions providesufciency conditions for a function f(t) to be Fourier transformable. These are1. 11jf (t)jdt < 12. f(t) has nite maxima and minima within any nite interval3. f(t) has a nite number of discontinuities within any nite intervalIf these conditions are met, the function can be transformed uniquely. Although the deltafunction does not obey the above conditions, it belongs, however, to a specic set offunctions known as generalized functions and it is transformable. The Fourier transform(FT) pair is dened as follows:F{f (t)} DF(v)

11f (t)ejvtdt direct Fourier transform (FT)F1{F(v)} Df (t) 12p

11F(v)ejvtdv inverse Fourier transform (IFT)(3:1)3.2 Other Forms of Fourier Transform3.2.1 f(t) Is a Complex FunctionIf f(t) fr(t) jfi(t), where the two functions are real, then we haveF(v)

11[ fr(t) cos vt fi(t) sin vt]dt j

11[ fr(t) sin vt fi(t) cos vt]dt (3:2)3-1Therefore,F(v) R(v) jX(v)R(v)

11[ fr(t) cos vt fi(t) sin vt]dtX(v)

11[ fr(t) sin vt fi(t) cos vt]dt(3:3)From (3.1) and Eulers expansion ejvtcos vt j sin vt, we obtain the inverseformulas:fr(t) 12p

11[R(v) cos vt X(v) sin vt]dv (3:4)fi(t) 12p

11[R(v) sin vt X(v) cos vt]dv (3:5)3.2.2 Real Time FunctionsIf f(t) is real, thenR(v)

11f (t) cos vt dt X(v)

11f (t) sin vt dt (3:6)From (3.6), we obtain the relationsR(v) R(v) even X(v) X(v) odd, F(v) F*(v) (3:7)Therefore, the inverse transform isfr(t) Df (t) 12p

11[R(v) cos vt X(v) sin vt]dv (3:8)3.2.3 Imaginary Time FunctionsIf f(t) jfi(t), thenR(v)

11fi(t) sin vt dt X(v)

11fi(t) cos vt dt (3:9)3-2 Transforms and Applications Primer for Engineers with Examples and MATLAB1From the above equation, we haveR(v) R(v) X(v) X(v) F(v) F*(v) (3:10)3.2.4 f(t) Is EvenIf f(t) f(t), then f(t) cos vt is even and f(t) sin vt is odd with respect to t. Hence(see (3.6)),R(v) 2

10f (t) cos vt dt X(v) 0 (3:11)From (3.11) and (3.8), we obtainf (t) 1p

10R(v) cos vt dv (3:12)3.2.5 f(t) OddIf f(t) f(t), thenR(v) 0 X(v) 2

10f (t) sin vt dt (3:13)The inversion of (3.8) becomesf (t) 1p

10X(v) sin vt dv (3:14)3.2.6 Odd and Even RepresentationsAny function can be decomposed into an odd and an even function as follows:fe(t) f (t) f (t)2 fo(t) f (t) f (t)2 (3:15)Hence,f (t) fe(t) fo(t) (3:16)Fourier Transforms 3-3The following relations also hold (Table 3.1)F(v) R(v) jX(v)

11fe(t) cos vt dt j

11fo(t) cos vt dt DFe(v) Fo(v)R(v) 2

10fe(t) cos vt dt X(v) 2

10fo(t) sinvt dtfe(t) 1p

10R(v) cos vt dv fo(t) 1p

10X(v) sinvt dv(3:17)3.2.7 Causal-Time FunctionsA function is causal if it is zero for negative t:f (t) 0 t < 0 (3:18)For t >0, we have f(t) 0 and, therefore, from (3.15) we obtainf (t) 2fe(t) 2fo(t) t > 0 (3:19)From (3.19) and the bottom expressions of (3.17), we obtainf (t) 2p

10R(v) cos vt dv 2p

10X(v) sin vt dv t > 0 (3:20)From (3.19), we obtainF(v) 2

11fe(t) cos vt dt 2Fe(v) F(v) 2j

11fo(t) sin vt dt 2Fo(v) (3:21)TABLE 3.1 Properties of f(t) and F(v)Property of f(t) Characteristics of F(v)f(t) fe(t) Real transform and evenf(t) fo(t) Imaginary transform and oddf(t) real Even real part and odd imaginary partf(t) Complex and no symmetry Complex transform and no symmetryf(t) Complex even or odd Complex and even transform orcomplex and odd3-4 Transforms and Applications Primer for Engineers with Examples and MATLAB1From the inverse transform and the relationships of even and odd real functions, weobtainf (t) 12p

112Fe(v)ejvtdv 1p

11Fe(v) cos vt dv 1p

11R(v) cos vt dv t > 0f (t) 12p

112Fo(v)ejvtdv jp

11Fo(v) sin vt dv 1p

11X(v) sin vt dv t > 0(3:22)Furthermore, the two functions F(v) and X(v) of a causal function are related to eachother by the equations (Table 3.2)R(v) 1pP

11X(t)v tdtX(v) 1pP

11R(t)v tdt(3:23)These are Hilbert transform relationships and the script P, in front of the integrals,denotes the principal value of the integral (see Chapter 9).3.3 Fourier Transform ExamplesExample 3.1The FT of the pulse function is given byF(v)

11pa(t)ejvtdt

aaejvtdt ejvaejvajv 2sin vav (3:24)The signal and the amplitude spectrum are shown in Figure E.3.1. The amplitude isgiven by jF(v)j 2 sin vav

. &Example 3.2The FT of f(t) eatu(t) a >0 isF(v)

11eatu(t)ejvtdt

10eatejvtdt 1ajv 1a2v2p ej tan1(v=a) 1a2v2p ej tan1(v=a)(3:25)Fourier Transforms 3-5TABLE 3.2 Fourier Transform PairsF(v) F{f (t)}, f (t) F1{F(v)}General time functionF{ f (t)} DF(v) 11 f (t)ejvtdt F1{F(v)} D f (t) 12p11 F(v)ejvtdvComplex function: f(t) fr(t) jfi(t)F(v) 11 [ fr(t) cos vt fi(t) sin vt]dt j11 [ fr(t) sin vt fi(t) cos vt]dtF(v) R(v) jX(v)R(v) 11 [ fr(t) cos vt fi(t) sin vt]dtX(v) 11 [ fr(t) sin vt fi(t) cos vt]dtfr(t) 12p11 [R(v) cos vt X(v) sin vt]dvfi(t) 12p11 [R(v) sin vt X(v) cos vt]dvReal time function: f(t) fr(t)R(v) 11 f (t) cos vt dtX(v) 11 f (t) sin vt dtfr(t) Df (t) 12p11 [R(v) cos vt X(v) sin vt]dvImaginary time function: f(t) jfi(t)R(v) 11 fi(t) sin vt dt X(v) 11 fi(t) cos vt dtReal and even functionR(v) 210 f (t) cos vt dt X(v) 0f (t) 1p10 R(v) cos vt dvReal and odd functionR(v) 0 X(v) 210 f (t) sin vt dtf (t) 1p10 X(v) sin vt dvOdd and even representation: f (t) fe(t) fo(t) f (t)f (t)2 f (t)f (t)2F(v) R(v) jX(v) 11 fe(t) cos vt dt j11 fo(t) cos vt dt DFe(v) Fo(v)R(v) 210 fe(t) cos vt dt X(v) 210 fo(t) sin vt dtfe(t) 1p10 R(v) cos vt dv fo(t) 1p10 X(v) sin vt dvCausal-time functions: f(t) 0 t < 0F(v) 211 fe(t) cos vt dt 2Fe(v) F(v) 2j11 fo(t) sin vt dt 2Fo(v)f (t) 12p11 2Fe(v)ejvtdv 1p11 Fe(v) cos vt dv 1p11 R(v) cos vt dv t > 0f (t) 12p11 2Fo(v)ejvtdv jp11 Fo(v) sin vt dv 1p11 X(v) sin vt dv t > 0Hilbert transforms of causal-time functionsR(v) 1pP11X(t)vtdtX(v) 1pP11R(t)vtdtScript P stands for the principal value of the integrals.3-6 Transforms and Applications Primer for Engineers with Examples and MATLAB1The amplitude and the phase spectrums are given byA(v) 1a2v2p f(v) tan1(v=a)The signal, the amplitude, and the phase spectra are shown in Figure E.3.2. &Example 3.3In this example, we obtain the FT of the function f(t) sgn(t)pa(t). Therefore,F(v)

11sgn(t)pa(t)ejvtdt

aasgn(t)ejvtdt

0aejvtdt

a0ejvtdt 2j1 cos vav(3:26)&0 1 21/aA()()00.5eat u(t)1t sf(t)20 0 2021012 rad/sAmplitude, phaseFIGURE E.3.2t a a1pa(t)1510.500.511.522.510 5 0 rad/sMagnitude of F()5 102/a/a/a2a15FIGURE E.3.1Fourier Transforms 3-7Example 3.4The FT of f (t) eajtj 1< t < 1isF(v)

01eatejvtdt

10eatejvtdt 2aa2v2 (3:27)&Example 3.5The FT of f (t) sgn(t)eajtj 1< t < 1 isF(v)

01eatejvtdt

10eatejvtdt 2jva2v2 (3:28)&3.4 Fourier Transform PropertiesThe FT properties are given in Table 3.3.3.5 Examples on Fourier PropertiesExample 3.6If in the inverse FT formula of a function f(t) we interchange time to frequency andfrequency to time and then we minus the omegas, Property #3 in Table 3.3 is obtained.Applying this property, for example, we nd the relation2aa2t2F$ 2peajvjThis property is useful to nd additional FTs of functions, most of the times difcult toobtain. &Example 3.7To prove the scaling Property #4 of Table 3.3, we proceed as follows:

11f (at)ejvtdt 1jaj

11f (x)ejvx=adx 1jajF va &3-8 Transforms and Applications Primer for Engineers with Examples and MATLAB1Example 3.8Based on the properties of the delta function (see Section 1.2), we ndF{d(t)} DD(v)

11d(t)ejvtdt ejv0 1 (3:29)Therefore, in connection with the symmetry Property #3 of Table 3.3 we obtainF{a}

11aejvtdt a2pd(v) a constant (3:30)&Example 3.9From the relation cos v0t 12(ejvtejvt) and the shifting Property #2 of Table 3.3, weobtain the FT of a modulated signal:f (t) cos v0t F$12[F(v v0) F(v v0)] (3:31)TABLE 3.3 Commonly Used Fourier Properties1. Linearity af (t) bh(t) F$aF(v) bH(v)2. Time shifting f (t a) F$ejavF(v)3. Symmetry F(t) F$2pf (v), F(v) F$f (t)1 F$ 2pd(v) 2pd(v)

4. Time scaling f (at) F$1jajF va 5. Time reversal f (t) F$F(v) (real time functions)6. Frequency shifting ejv0t F$F(v v0)7. Modulation f (t) cos v0t F$ 12[F(v v0) F(v v0)]f (t) sin v0t F$ 12j[F(v v0) F(v v0)]

8. Time differentiation df (t)dtF$jvF(v) dntdtnF$(jv)nF(v)9. Frequency differentiation(jt)f (t) F$dF(v)dv(jt)nf (t) F$dnF(v)dvn

10. Time convolution f (t)*h(t) F$F(v)H(v)11. Frequency convolution f (t)h(t) F$ 12pF(v)*H(v) 12p11 F(x)H(v x)dx12. Correlation f (t)(t) 11 f (x)h*(x t)dx F$jF(v)H*(v)j13. Central ordinate f (0) 12p11 F(v)dv F(0) 11 f (t)dt14. Parsevals theorem11jf (t)j2dt 12p11jF(v)j2dv11 f (t)h*(t)dt 12p11 F(v)H*(v)dv

Fourier Transforms 3-9Therefore, since pa(t) F$ 2 sin av=v then by (3.31) we obtainpa(t) cos v0t F$sin a(v v0)v v0sin a(v v0)v v0(3:32)&Example 3.10The FT of the convolution of two functions is given byf (t)*h(t) F$

11ejvt

11f (x)h(t x)dx

dt

11f (x)

11ejv(yx)h(x)dxdy

11f (x)ejvxdx

11h(y)ejvydy F(v)H(v)where we set (t x) y (see Table 3.3, Property #10). &Example 3.11Applying the convolution property to an LTI system whose impulse response is h(t) andits input is f(t), we obtain its output spectrum to be equal to F(v)H(v). &Example 3.12The FT of the pulse with height B and width a is F{Bpa(t)} 2B sin lrbav)=v. Theconvolution of the pulse to itself is equal to the triangle:Bpa(t)*Bpa(t) B22a(1 t) 1 < t < 0B22a(1 t) 0 < t < 1

(3:33)Therefore, the FT of the convolution (the triangle) is equal toF(v) 2Bsin avv 2 4B2 sin2avv2 B2L2sin2 L2vL2v24 B2 2a2aL2sin2 L2vL2v24 ALsin2 L2vL2v24(3:34)where rst we substituted L 2a and next B22a A. For these substitutions and thespectrum see Figure E.3.3. &3-10 Transforms and Applications Primer for Engineers with Examples and MATLAB1Example 3.13To nd the FT of the function f (t) sin att , we use the symmetry property and, thus, weobtainpa(t) F$2 sin avv2 sin attF$2ppa(v) or sin atptF$pa(v)(3:35)&Example 3.14To obtain Parsevals identity, we proceed as follows:

11jf (t)j2dt

11f (t)f *(t)dt

11f (t) 12p

11F(v)ejvtdv

*dt 12p

11F*(v)

11f (t)ejvtdt

dv 12p

11F*(v)F(v)dv 12p

11jF(v)j2dv &Btt a aB22a=A2a=L2aBpa(t) * Bpa(t)Bpa(t)2000.10.20.30.40.50.60.70.80.9115 10 52/L2/a/a2/a/a2/L 4/LALB2a0 rad/s5 10 15 20 200.500.511.5215 10 5 0 rad/sMagnitude of F()Magnitude of F()5 10 15 20FIGURE E.3.3Fourier Transforms 3-11Example 3.15Suppose the FT F(v) of a function f(t) is truncated above to jvj a. This would implythat we have obtained the band-limited function:Fa(v) F(v) jvj < a0 jvj > a F(v)pa(v) (3:36)Therefore, the IFT of both sides of (3.36), taking into consideration (3.35) and theconvolution Property #11 of Table 3.3, we obtainfa(t)

11f (x)sin a(t x)p(t x) dx f (t)* sin atptF$F(v)pa(v) (3:37)&3.6 FT Examples of Singular FunctionsThe basic properties of the delta function are

11d(t t0)f(t)dt f(t0)

11dnd(t t0)dtn f(t)dt (1)n dnf(t0)dtn (3:38)where f(t) is an arbitrary function and continuous at a given point t0 (Table 3.4).Example 3.16Based on the above properties of the delta function, its FT isD(v)

11d(t)ejvtdt ejv0 1 d(t) F$1 (3:39)Using the symmetry property we obtain the relationsd(t) F$11 F$2pd(v)(3:40)3-12 Transforms and Applications Primer for Engineers with Examples and MATLAB1From the second relation of (3.39), we also obtaind(t) 12p

11ejvtdv 12p

11cos vt dv (3:41)Furthermore,d(t t0) F$ejvt0)ejv0t F$d(v v0)cos v0t 12(ejv0tejv0t) F$p[d(v v0) d(v v0)](3:42)&Example 3.17The FT of the sgn(t) issgn(t) lima!0eatt > 00 t = 0eatt < 0

)F{sgn(t)} lima!0

01eatejvtdt

10eatejvtdt

2jv(3:43)&Example 3.18From sin at=t F$ppa(v) Parsevals relation gives

11sin2att2 dt 12p

aap2dv ap (3:44)&Example 3.19The FT of the unit step function isF{u(t)} F 1212sgn(t) pd(v) 1jv)F{ejv0tu(t)} pd(v v0) 1j(v v0) (3:45)&Fourier Transforms 3-13Example 3.20To nd the FT of the combT(t) function (see Figure 1.8), we rst expand the periodiccomb function into its FS and then we take its FT. Hence,combT(t) 1n1anejnv0tan1T

T=2T=2d(t)ejnv0tdt 1Tejnv001T;F{combT(t)} COMBv0(v) 1T1n1

11ejnv0tejvtdt 1T1n1

11ej(vnv0)tdt2pT1n1d(vnv0) v02pT (3:46)Therefore, the FT of a comb function is another comb function in the frequencydomain whose amplitudes are 2p=T and whose location is at points nv0n2p=T forn . . . ,1, 2, 0, 1, 2, . . . &Example 3.21The FT of a periodic pulse function with period T is given by (see Figure E.3.4)F{f (t)} F{f0(t)*combT(t)} 2 sin avv2pT1n1d(v nv0)4pT1n1sin anv0nv0d(v nv0) T > 2a (3:47)The FT of the pulse is F{f0(t)} DF0(v) 2 sin va=v and the FT of the periodic functionf(t) is shown to be discrete as was anticipated since the function is periodic. &Example 3.22 (Gibbs phenomenon)Let us truncate the spectrum U(v) of the unit step:Uv0(v) U(v) jvj v00 elsewher or Uv0(v) U(v)pv0(v)

(3:48)The approximate reconstruction of the unit step function, ua(t), isua(t) F1{U(v)pv0(v)} u(t)*F1{pv0(v)} u(t)*sin v0tptv0p

t1sin v0xv0x dx 1p

01sin yy dy 1p

v0t0sin yy dy 12 1pSi(v0t) (3:49)where Si( ) is the sine integral. Properties of Si(x) include: Si(x) Si(x) and Si(1) Si(1) p=2. The following MATLAB1Book program produces the Figure E.3.5.3-14 Transforms and Applications Primer for Engineers with Examples and MATLAB1*tta aTcombT(t)f0(t)COMB0()0X11tf (t)=1a aT150.500.511.5210 5 0 rad/sF0 ()5 102/a/a2a15=201510.500.511.53.532.5210 5 0 rad/s5 10F0()2/a4a/T20 15FIGURE E.3.42 1 0 1 2 31040.200.20.40.60.811.20tua(t)FIGURE E.3.5Fourier Transforms 3-15TABLE 3.4 Fourier Transforms of Some Common Functionsf(t) F(w) F[f (t)]jvjpa(t) 2v sin (av)(a jtj)pa(t) 2 sin a2v v 21 t2p p1(t) pvJ1(v)(1t2)a1p1(t) G(a) pp 2jvj a12Ja12(jvj)sgn(t)pa(t) 2j1cos (av)vcos p2at pa(t) 4pap24a2v2 cos (av)Rect(b,g)(t) jv[ejgvejbv]e(ajb)tu(t) 1ajbjvtn1e(ajb)tu(t) G(y)(ajbjv)ye(ajb)tu(t) 1(ajbjv)(1)y 1e(ajb)tu(t) G(y)(ajbjv)yeajtj 2aa2v2sgn(t) eajtj 2jva2v2eat2 pa

exp 14av2 eat2bt pa

exp 14av2j b2av b24a eltajbet p(a jb)l1jvcsc (pl jpv)sech (at) pasech p2av ejat2 pa

exp j 14a(v2ap) eut2 juj a

j sgn(b)juj a

(w=u a jb) 1jujp2

exp 14uv2 1t sin (at) ppa(v)1t sin (at) 2 p2 (2a jvj)p2a(v)1jtj sin (at) j sgn(v) ln jvjajvja

1 2pd(v)tnjn2pd(n)(v)ejbt2pd(v b)d(t b) ejbvd(n)(t) (jv)nsin (at) jp[d(v a) d(v a)]cos (at) p[d(v a) d(v a)]sin (at2) pa

sin 14a(v2ap) cos (at2) pa

cos 14a(v2ap) eat2cos (nt2) 1jujp2

exp a4juj2 v2 (w=u a jn) juj a

cos nv24juj2 juj a

sin nv24juj2 3-16 Transforms and Applications Primer for Engineers with Examples and MATLAB1TABLE 3.4 (continued) Fourier Transforms of Some Common Functionsf(t) F(w) f [f (t)]jvjeat2sin(nt2) 1jujp2

exp a4jujv2 (w=u a jn) juj a

cos nv24juj2 juj a

sin nv24juj2 comba(t) 2pa comb2pa(v)jsin(at)j 1k1414k2 d(v 2ak)jcos(at)j 1k1 ( 1)k 414k2 d(v 2ak)saw(t) j 1n1n60(1)n 2nd(v np)1m1 pa(t mn) 4pan d(v) 1k1k602k sin 2pkan d v 2pkn (w=2a n)sgn(t) j 2vu(t) pd(v) j 1v1t jpsgn(v)tnjp(jv)n1(n1)! sgn(v)jtj 2v2tnsgn(t) (j)n1 2(n!)vn1ramp(t) jpd0(v) 1v2tnu(t)jnpd(n)(v) n! jv n1jtj1=2 2pp jvj1=2jtjl12G(l) cos lp2 jvjlJ0(t) 21v2p p1(v)Y0(jtj) 2v21p [1 p1(v)]J2n(t) 2 cos [2narcsin(v)]1v2p p1(v)J2n1(t) 2j sin[(2n1)arcsin(v)]1v2p p1(v)Jn(t) 2(j)nTn(v)1v2p p1(v)1tn Jn(t) 2(1v2)n12135(2n1)p1(v)jtja12Ja12(jtj)2ppG(a)1v22 a1p1(v)1t Jn(t) (j)n 2jn1 v2p Un1(v)p1(v)t1=2Jn12(t) (j)n 2pp Pn(v)p1(v)sgn(t)J0(t) j 2v21p sgn(v)[1 p1(v)]J0(t)u(t) p1(v)jsgn(v)[1p1(v)]j1v2jpNotes: a, b, g, l, n, and n denote real numbers with a > 0, 0 < l < 1, n > 0, and n1,2, 3, . . . G(x), the Gamma function; Pn(x), the nth Legendre polynomial; Jn, the Besselfunction of the rst kind of order n; Yn, the Bessel function of the second kind of order n;Tn(x), the nth Chebyshev polynomial of the rst kind; Un(x), the nth Chebyshevpolynomial of the second kind; saw(t) is the saw function.Fourier Transforms 3-17TABLE 3.5 Graphical Representation of Some Fourier Transformsf (x) F(y)f (x) 1=2p 11 F(y)e1xydy FT F(y) f g F(y) 11 f (x)e1xydx FT f (x) f g 1/a AAexp (a2x2)[Gaussian]2a AaA ppa exp y2=4a2 [Gaussian] (2:38)1/aAAexp ajxj 2Aaa2a2y2[Lorentzian]a2Aa(2:39)1/aAAexp (ax) [x > 0]0 [x < 0]A/2aaA/aaA a iya2y2 (2:40)3-18TransformsandApplicationsPrimerforEngineerswithExamplesandMATLAB11/aAAexp (ax) [x > 0]Aexp ajxj [x < 0]A/aa2iA ya2y2 (2:41)A2/yoA exp(iyox a|x|)A2/yo2A/aayo2Aaa2a2(y y0)2 (2:42)A2/yoAcos y0x exp ajxj ~A/aayoAaa2a2(y y0)2 a2a2(y y0)2 Aa2a2a2y20y2 a2y20y2 24a2y2 (2:43)(continued)FourierTransforms3-19TABLE 3.5 (continued) Graphical Representation of Some Fourier Transformsf (x) F(y)A2/yoAsin y0x exp ajxj yoa~A/aiAaa2a2(y y0)2 a2a2(y y0)2 iAa4a2yy0a2y20y2 24a2y2 (2:44)2/yo[x >0][x 0]0 [x < 0]A/2aayoaA/4ayoaA2aa2(y y0)2 aa2(y y0)2 i y0ya2(y0y)2 y0ya2(y0y)2 A a a2y20y2 iy a2y2y20 a2y20y2 24a2y2 (2:46)A2/yoAsin y0x exp (ax) [x > 0]0 [x < 0]~A/4a yoaaa~A/2ayoA2y0ya2(y0y)2 y0ya2(y0y)2 i aa2(y0y)2 aa2(y0y)2 Ay01a2y20y2 i2ay (2:47)(continued)FourierTransforms3-21TABLE 3.5 (continued) Graphical Representation of Some Fourier Transformsf (x) F(y)AL LA jxj < L 0 jxj > L 2AL2/L 2/L2Asin Lyy (2:48)baSL LAA [a < x < b]0 [x < a; x > b]2AL 2AL2/S2/L 2/L2/S2Asin Lyy exp (iSy) A (sin by sin ay) i(cos ay cos by)y 2A (sin Ly cos Sy) i(sin Ly sin Sy)y iAy exp (iby) exp (iay) (2:49)3-22TransformsandApplicationsPrimerforEngineerswithExamplesandMATLAB1AS S2L 2LA (S L) < jxj < (S L) 0 [otherwise]4AL 2/S2/L4Acos Sy sin Lyy (2:50)AL L2/yoAL L2/yo[|x | L]2ALyo2/L 2/L2Asin L(y0y) f g(y0y) (2:51)AL L2/yoAcos y0x jxj < L 0 jxj > L yo yo2/LALA sin L(y y0)(y y0) sin L(y y0)(y y0) (2:52)(continued)FourierTransforms3-23TABLE 3.5 (continued) Graphical Representation of Some Fourier Transformsf (x) F(y)AL L2/yoAsin y0x jxj < L 0 jxj > L ~ALyoyo2/LiA sin L(y y0)(y y0) sin L(y y0)(y y0) (2:53)/yoAAcos y0x jxj < p=2y0 0 jxj > p=2y0 2A/yo6yo 4yo 2A y0y20y2 cos py2y0 See (2:52) with L p=2y0 (2:54)L LAA 1 jxjL jxj < L 0 jxj > L 2/L 4/LALAL sin Ly=2 Ly=2 2(2:55)3-24TransformsandApplicationsPrimerforEngineerswithExamplesandMATLAB1LLAAxL jxj < L 0 jxj > L 2/L2iAy cos Ly sin LyLy (2:56)L LAAjxjL jxj < L 0 jxj > L 2/LAL2AL sin LyLy 2 sin Ly=2 Ly 2 (2:57)2/yoA2/yoAexp (iy0x)yo2A2pAd(y y0): (2:58)(continued)FourierTransforms3-25TABLE