GATE ELECTRONICS & COMMUNICATION

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1 Eighth Editio GATE ELECTRONICS & COMMUNICATION Sigals ad Systems Vol 7 of 0 RK Kaodia Ashish Murolia NODIA & COMPANY

2 GATE Electroics & Commuicatio Vol 7, 8e Sigals ad Systems RK Kaodia & Ashish Murolia Copyright By NODIA & COMPANY Iformatio cotaied i this book has bee obtaied by author, from sources believes to be reliable. However, either NODIA & COMPANY or its author guaratee the accuracy or completeess of ay iformatio herei, ad NODIA & COMPANY or its author shall be resposible for ay error, omissios, or damages arisig out of use of this iformatio. This book is published with the uderstadig that NODIA & COMPANY ad its author are supplyig iformatio but are ot attemptig to reder egieerig or other professioal services. MRP NODIA & COMPANY B 8, Dhashree Ist, Cetral Spie, Vidyadhar Nagar, Jaipur 009 Ph : , equiry@odia.co.i Prited by Nodia ad Compay, Jaipur

3 Preface to the Series For almost a decade, we have bee receivig tremedous resposes from GATE aspirats for our earlier books: GATE Multiple Choice Questios, GATE Guide, ad the GATE Cloud series. Our first book, GATE Multiple Choice Questios (MCQ), was a compilatio of objective questios ad solutios for all subjects of GATE Electroics & Commuicatio Egieerig i oe book. The idea behid the book was that Gate aspirats who had just completed or about to fiish their last semester to achieve his or her B.EB.Tech eed oly to practice aswerig questios to crack GATE. The solutios i the book were preseted i such a maer that a studet eeds to kow fudametal cocepts to uderstad them. We assumed that studets have leared eough of the fudametals by his or her graduatio. The book was a great success, but still there were a large ratio of aspirats who eeded more preparatory materials beyod just problems ad solutios. This large ratio maily icluded average studets. Later, we perceived that may aspirats could t develop a good problem solvig approach i their B.EB.Tech. Some of them lacked the fudametals of a subject ad had difficulty uderstadig simple solutios. Now, we have a idea to ehace our cotet ad preset two separate books for each subject: oe for theory, which cotais brief theory, problem solvig methods, fudametal cocepts, ad poitstoremember. The secod book is about problems, icludig a vast collectio of problems with descriptive ad stepbystep solutios that ca be uderstood by a average studet. This was the origi of GATE Guide (the theory book) ad GATE Cloud (the problem bak) series: two books for each subject. GATE Guide ad GATE Cloud were published i three subjects oly. Thereafter we received a immese umber of s from our readers lookig for a complete study package for all subjects ad a book that combies both GATE Guide ad GATE Cloud. This ecouraged us to preset GATE Study Package (a set of 0 books: oe for each subject) for GATE Electroic ad Commuicatio Egieerig. Each book i this package is adequate for the purpose of qualifyig GATE for a average studet. Each book cotais brief theory, fudametal cocepts, problem solvig methodology, summary of formulae, ad a solved questio bak. The questio bak has three exercises for each chapter: ) Theoretical MCQs, ) Numerical MCQs, ad ) Numerical Type Questios (based o the ew GATE patter). Solutios are preseted i a descriptive ad stepbystep maer, which are easy to uderstad for all aspirats. We believe that each book of GATE Study Package helps a studet lear fudametal cocepts ad develop problem solvig skills for a subject, which are key essetials to crack GATE. Although we have put a vigorous effort i preparig this book, some errors may have crept i. We shall appreciate ad greatly ackowledge all costructive commets, criticisms, ad suggestios from the users of this book. You may write to us at rajkumar. kaodia@gmail.com ad ashish.murolia@gmail.com. Ackowledgemets We would like to express our sicere thaks to all the coauthors, editors, ad reviewers for their efforts i makig this project successful. We would also like to thak Team NODIA for providig professioal support for this project through all phases of its developmet. At last, we express our gratitude to God ad our Family for providig moral support ad motivatio. We wish you good luck! R. K. Kaodia Ashish Murolia

4 SYLLABUS GATE Electroics & Commuicatios: Defiitios ad properties of Laplace trasform, cotiuoustime ad discretetime Fourier series, cotiuoustime ad discretetime Fourier Trasform, DFT ad FFT, trasform. Samplig theorem. Liear TimeIvariat (LTI) Systems: defiitios ad properties; causality, stability, impulse respose, covolutio, poles ad eros, parallel ad cascade structure, frequecy respose, group delay, phase delay. Sigal trasmissio through LTI systems. IES Electroics & Telecommuicatio Classificatio of sigals ad systems: System modellig i terms of differetial ad differece equatios; State variable represetatio; Fourier series; Fourier trasforms ad their applicatio to system aalysis; Laplace trasforms ad their applicatio to system aalysis; Covolutio ad superpositio itegrals ad their applicatios; trasforms ad their applicatios to the aalysis ad characterisatio of discrete time systems; Radom sigals ad probability, Correlatio fuctios; Spectral desity; Respose of liear system to radom iputs. **********

5 CONTENTS CHAPTER CONTINUOUS TIME SIGNALS. CONTINUOUS TIME AND DISCRETE TIME SIGNALS. SIGNALCLASSIFICATION.. Aalog ad Discrete Sigals.. Determiistic ad Radom Sigal.. Periodic ad Aperiodic Sigal..4 Eve ad Odd Sigal..5 Eergy ad Power Sigal 4. BASIC OPERATIONS ON SIGNALS 5.. Additio of Sigals 5.. Multiplicatio of Sigals 5.. Amplitude Scalig of Sigals 5..4 TimeScalig 5..5 TimeShiftig 6..6 TimeReversalFoldig 7..7 Amplitude Iverted Sigals 8.4 MULTIPLE OPERATIONS ON SIGNALS 8.5 BASIC CONTINUOUS TIME SIGNALS 9.5. The UitImpulse Fuctio 9.5. The UitStep Fuctio.5. The UitRamp Fuctio.5.4 Uit Rectagular Pulse Fuctio.5.5 Uit Triagular Fuctio.5.6 Uit Sigum Fuctio The Sic Fuctio 4.6 MATHEMATICAL REPRESENTATION OF SIGNALS 5 EXERCISE. 6 EXERCISE. 4 EXERCISE. 44 EXERCISE.4 49 SOLUTIONS. 56 SOLUTIONS. 79 SOLUTIONS. 84 SOLUTIONS.4 85

6 CHAPTER CONTINUOUS TIME SYSTEMS. CONTINUOUS TIME SYSTEM & CLASSIFICATION 9.. Liear ad NoLiear System 9.. TimeVaryig ad TimeIvariat system 9.. Systems With ad Without Memory (Dyamic ad Static Systems) Causal ad Nocausal Systems Ivertible ad NoIvertible Systems Stable ad Ustable systems 94. LINEAR TIME INVARIANT SYSTEM 95.. Impulse Respose ad The Covolutio Itegral 95.. Properties of Covolutio Itegral 96. STEP RESPONSE OF AN LTI SYSTEM 00.4 PROPERTIES OF LTI SYSTEMS IN TERMS OF IMPULSE RESPONSE 0.4. Memoryless LTI System 0.4. Causal LTI System 0.4. Ivertible LTI System Stable LTI System 0.5 IMPULSE RESPONSE OF INTERCONNECTED SYSTEMS 0.5. Systems i Parallel Cofiguratio 0.5. System i Cascade 0.6 CORRELATION 0.6. CrossCorrelatio 0.6. AutoCorrelatio Correlatio ad Covolutio 09.7 TIME DOMAIN ANALYSIS OF CONTINUOUS TIME SYSTEMS Natural Respose or eroiput Respose 0.7. Forced Respose or erostate Respose.7. The Total Respose.8 BLOCK DIAGRAM REPRESENTATION EXERCISE. 4 EXERCISE. EXERCISE. 5 EXERCISE.4 8 SOLUTIONS. 49 SOLUTIONS. 79 SOLUTIONS. 86 SOLUTIONS.4 87 CHAPTER DISCRETE TIME SIGNALS. INTRODUCTION TO DISCRETE TIME SIGNALS 0.. Represetatio of Discrete Time sigals 0

7 . SIGNAL CLASSIFICATION 04.. Periodic ad Aperiodic DT Sigals 04.. Eve ad Odd DT Sigals 05.. Eergy ad Power Sigals 07. BASIC OPERATIONS ON DT SIGNALS 07.. Additio of DT Sigals 08.. Multiplicatio of DT Sigal 08.. Amplitude scalig of DT Sigals TimeScalig of DT Sigals TimeShiftig of DT Sigals TimeReversal (foldig) of DT sigals 0..7 Iverted DT Sigals.4 MULTIPLE OPERATIONS ON DT SIGNALS.5 BASIC DISCRETE TIME SIGNALS.5. Discrete Impulse Fuctio.5. Discrete Uit Step Fuctio.5. Discrete Uitramp Fuctio.5.4 UitRectagular Fuctio UitTriagular Fuctio UitSigum Fuctio 5.6 MATHEMATICAL REPRESENTATION OF DT SIGNALS USING IMPULSE OR STEP FUNCTION 5 EXERCISE. 6 EXERCISE. 4 EXERCISE. 44 EXERCISE.4 47 SOLUTIONS. 49 SOLUTIONS. 7 SOLUTIONS. 8 SOLUTIONS.4 8 CHAPTER 4 DISCRETE TIME SYSTEMS 4. DISCRETE TIME SYSTEM & CLASSIFICATION Liear ad Noliear Systems TimeVaryig ad TimeIvariat Systems System With ad Without Memory (Static ad Dyamic Systems) Causal ad NoCausal System Ivertible ad NoIvertible Systems Stable ad Ustable System LINEARTIME INVARIANT DISCRETE SYSTEM Impulse Respose ad Covolutio Sum Properties of Covolutio Sum STEP RESPONSE OF AN LTI SYSTEM 9

8 4.4 PROPERTIES OF DISCRETE LTI SYSTEM IN TERMS OF IMPULSE RESPONSE Memoryless LTID System Causal LTID System Ivertible LTID System Stable LTID System FIR ad IIR Systems IMPULSE RESPONSE OF INTERCONNECTED SYSTEMS Systems i Parallel System i Cascade CORRELATION CrossCorrelatio AutoCorrelatio Properties of Correlatio Relatioship Betwee Correlatio ad Covolutio Methods to Solve Correlatio DECONVOLUTION RESPONSE OF LTID SYSTEMS IN TIME DOMAIN Natural Respose or ero Iput Respose Forced Respose or ero State Respose Total Respose BLOCK DIAGRAM REPRESENTATION 0 EXERCISE EXERCISE 4. 7 EXERCISE 4. 0 EXERCISE 4.4 SOLUTIONS 4. 9 SOLUTIONS 4. 5 SOLUTIONS 4. 6 SOLUTIONS CHAPTER 5 THE LAPLACE TRANSFORM 5. INTRODUCTION The Bilateral or TwoSided Laplace Trasform The Uilateral Laplace Trasform THE EXISTENCE OF LAPLACE TRANSFORM REGION OF CONVERGENCE Poles ad eros of Ratioal Laplace Trasforms Properties of ROC THE INVERSE LAPLACE TRANSFORM Iverse Laplace Trasform Usig Partial Fractio Method Iverse Laplace Trasform Usig Covolutio Method 8

9 5.5 PROPERTIES OF THE LAPLACE TRANSFORM Liearity Time Scalig Time Shiftig Shiftig i the sdomai(frequecy Shiftig) Time Differetiatio Time Itegratio Differetiatio i the sdomai Cojugatio Property Time Covolutio sdomai Covolutio Iitial value Theorem Fial Value Theorem Time Reversal Property ANALYSIS OF CONTINUOUS LTI SYSTEMS USING LAPLACE TRANSFORM Respose of LTI Cotiuous Time System Impulse Respose ad Trasfer Fuctio STABILITY AND CAUSALITY OF CONTINUOUS LTI SYSTEM USING LAPLACE TRANSFORM Causality Stability Stability ad Causality SYSTEM FUNCTION FOR INTERCONNECTED LTI SYSTEMS Parallel Coectio Cascaded Coectio Feedback Coectio BLOCK DIAGRAM REPRESENTATION OF CONTINUOUS LTI SYSTEM Direct Form I structure Direct Form II structure Cascade Structure Parallel Structure 40 EXERCISE EXERCISE EXERCISE 5. 4 EXERCISE SOLUTIONS SOLUTIONS SOLUTIONS SOLUTIONS CHAPTER 6 THE TRANSFORM 6. INTRODUCTION The Bilateral or TwoSided trasform 49

10 6.. The Uilateral or Oesided trasform EXISTENCE OF TRANSFORM REGION OF CONVERGENCE Poles ad eros of Ratioal trasforms Properties of ROC THE INVERSE TRANSFORM Partial Fractio Method Power Series Expasio Method PROPERTIES OF TRANSFORM Liearity Time Shiftig Time Reversal Differetiatio i the domai Scalig i Domai Time Scalig Time Differecig Time Covolutio Cojugatio Property Iitial Value Theorem Fial Value Theorem ANALYSIS OF DISCRETE LTI SYSTEMS USING TRANSFORM Respose of LTI Cotiuous Time System Impulse Respose ad Trasfer Fuctio STABILITY AND CAUSALITY OF LTI DISCRETE SYSTEMS USING TRANSFORM Causality Stability Stability ad Causality BLOCK DIAGRAM REPRESENTATION Direct Form I Realiatio Direct Form II Realiatio Cascade Form Parallel Form RELATIONSHIP BETWEEN splane & PLANE 58 EXERCISE EXERCISE EXERCISE EXERCISE SOLUTIONS SOLUTIONS SOLUTIONS SOLUTIONS

11 CHAPTER 7 THE CONTINUOUS TIME FOURIER TRANSFORM 7. DEFINITION Magitude ad Phase Spectra Existece of Fourier trasform Iverse Fourier Trasform SPECIAL FORMS OF FOURIER TRANSFORM Realvalued Eve Symmetric Sigal Realvalued Odd Symmetric Sigal Imagiaryvalued Eve Symmetric Sigal Imagiaryvalued Odd Symmetric Sigal 6 7. PROPERTIES OF FOURIER TRANSFORM Liearity Time Shiftig Cojugatio ad Cojugate Symmetry Time Scalig Differetiatio i TimeDomai Itegratio i TimeDomai Differetiatio i Frequecy Domai Frequecy Shiftig Duality Property Time Covolutio Frequecy Covolutio Area Uder xt () Area Uder Xjw ( ) Parseval s Eergy Theorem Time Reversal Other Symmetry Properties ANALYSIS OF LTI CONTINUOUS TIME SYSTEM USING FOURIER TRANSFORM Trasfer Fuctio & Impulse Respose of LTI Cotiuous System Respose of LTI Cotiuous system usig Fourier Trasform RELATION BETWEEN FOURIER AND LAPLACE TRANSFORM 6 EXERCISE 7. 6 EXERCISE EXERCISE EXERCISE SOLUTIONS SOLUTIONS SOLUTIONS SOLUTIONS CHAPTER 8 THE DISCRETE TIME FOURIER TRANSFORM 8. DEFINITION 705

12 8.. Magitude ad Phase Spectra Existece of DTFT Iverse DTFT SPECIAL FORMS OF DTFT PROPERTIES OF DISCRETETIME FOURIER TRANSFORM Liearity Periodicity Time Shiftig Frequecy Shiftig Time Reversal Time Scalig Differetiatio i Frequecy Domai Cojugatio ad Cojugate Symmetry Covolutio i Time Domai Covolutio i Frequecy Domai Time Differecig Time Accumulatio Parseval s Theorem ANALYSIS OF LTI DISCRETE TIME SYSTEM USING DTFT Trasfer Fuctio & Impulse Respose Respose of LTI DT system usig DTFT RELATION BETWEEN THE DTFT & THE TRANSFORM DISCRETE FOURIER TRANSFORM (DFT) Iverse Discrete Fourier Trasform (IDFT) PROPERTIES OF DFT Liearity Periodicity Cojugatio ad Cojugate Symmetry Circular Time Shiftig Circular Frequecy Shift Circular Covolutio Multiplicatio Parseval s Theorem Other Symmetry Properties FAST FOURIER TRANSFORM (FFT) 7 EXERCISE EXERCISE EXERCISE EXERCISE SOLUTIONS SOLUTIONS SOLUTIONS

13 SOLUTIONS CHAPTER 9 THE CONTINUOUS TIME FOURIER SERIES 9. INTRODUCTION TO CTFS Trigoometric Fourier Series Expoetial Fourier Series Polar Fourier Series EXISTENCE OF FOURIER SERIES PROPERTIES OF EXPONENTIAL CTFS Liearity Time Shiftig Time Reversal Property Time Scalig Multiplicatio Cojugatio ad Cojugate Symmetry Differetiatio Property Itegratio i TimeDomai Covolutio Property Parseval s Theorem Frequecy Shiftig AMPLITUDE & PHASE SPECTRA OF PERIODIC SIGNAL RELATION BETWEEN CTFT & CTFS CTFT usig CTFS Coefficiets CTFS Coefficiets as Samples of CTFT RESPONSE OF AN LTI CT SYSTEM TO PERIODIC SIGNALS USING FOURIER SERIES 788 EXERCISE EXERCISE EXERCISE EXERCISE SOLUTIONS SOLUTIONS SOLUTIONS SOLUTIONS CHAPTER 0 THE DISCRETE TIME FOURIER SERIES 0. DEFINITION AMPLITUDE AND PHASE SPECTRA OF PERIODIC DT SIGNALS PROPERTIES OF DTFS Liearity Periodicity TimeShiftig 86

14 0..4 Frequecy Shift TimeReversal Multiplicatio Cojugatio ad Cojugate Symmetry Differece Property Parseval s Theorem Covolutio Duality Symmetry Time Scalig 867 EXERCISE EXERCISE EXERCISE EXERCISE SOLUTIONS SOLUTIONS SOLUTIONS SOLUTIONS CHAPTER SAMPLING AND SIGNAL RECONSTRUCTION. THE SAMPLING PROCESS 905. THE SAMPLING THEOREM 905. IDEAL OR IMPULSE SAMPLING NYQUIST RATE OR NYQUIST INTERVAL ALIASING SIGNAL RECONSTRUCTION SAMPLING OF BANDPASS SIGNALS 909 EXERCISE. 9 EXERCISE. 99 EXERCISE. 9 EXERCISE.4 95 SOLUTIONS. 98 SOLUTIONS. 97 SOLUTIONS. 94 SOLUTIONS.4 94 ***********

15 Electroics & Commuicatio Sample Chapter of Sigals ad Systems (Vol7, GATE Study Package) Page 49 CHAPTER Chap 6 The Trasform THE TRANSFORM 6. INTRODUCTION As we studied i previous chapter, the Laplace trasform is a importat tool for aalysis of cotiuous time sigals ad systems. Similarly, trasforms eables us to aalye discrete time sigals ad systems i the domai. Like, the Laplace trasform, it is also classified as bilateral trasform ad uilateral trasform. The bilateral or twosided trasform is used to aalye both causal ad ocausal LTI discrete systems, while the uilateral trasform is defied oly for causal sigals. NOTE : The properties of trasform are similar to those of the Laplace trasform. 6.. The Bilateral or TwoSided trasform The trasform of a discretetime sequece x, [ ] is defied as X () { x [ ]} x [ ] (6..) Where, X () is the trasformed sigal ad represets the trasformatio. is a complex variable. I polar form, ca be expressed as re jw where r is the magitude of ad W is the agle of. This correspods to a circle i plae with radius r as show i figure 6.. below Figure 6.. plae NOTE : The sigal x [ ] ad its trasform X() are said to form a trasform pair deoted as x [ ] X ( ) *Shippig Free* Buy Olie: shop.odia.co.i *Maximum Discout*

16 0 Subjectwise books by R. K. Kaodia Electroics & Commuicatio Geeral Aptitude Egieerig Mathematics Networks Electroic Devices Aalog Electroics Digital Electroics Sigals & Systems Cotrol Systems Commuicatio Systems Electromagetics Page 494 The Trasform 6.. The Uilateral or Oesided trasform The trasform for causal sigals ad systems is referred to as the uilateral trasform. For a causal sequece x [ ] 0, for < 0 Therefore, the uilateral trasform is defied as X () x [ ] (6..) 0 NOTE : For causal sigals ad systems, the uilateral ad bilateral trasform are the same. 6. EXISTENCE OF TRANSFORM Cosider the bilateral trasform give by equatio (6..) X [] x [ ] The trasform exists whe the ifiite sum i above equatio coverges. For this summatio to be coverged x [ ] must be absolutely summable. Substitutig re jw or, X [] x [ ]( re j W ) X [] { xr [ ] } e jw Thus for existece of trasform X () < xr [ ] (6..) 6. REGION OF CONVERGENCE The existece of trasform is give from equatio (6..). The values of r for which xr [ ] is absolutely summable is referred to as regio of covergece. Sice, re jw so r. Therefore we coclude that the rage of values of the variable for which the sum i equatio (6..) coverges is called the regio of covergece. This ca be explaied through the followig examples Poles ad eros of Ratioal trasforms The most commo form of trasform is a ratioal fuctio. Let X () be the trasform of sequece x [ ], expressed as a ratio of two polyomials N () ad D. () N () X () D () The roots of umerator polyomial i.e. values of for which X () 0is referred to as eros of X. () The roots of deomiator polyomial for which X () is referred to as poles of X. () The represetatio of X () through its poles ad eros i the plae is called poleero plot of X. () For example cosider a ratioal trasfer fuctio X () give as H () ( )( ) *Shippig Free* Buy Olie: shop.odia.co.i *Maximum Discout*

17 Electroics & Commuicatio Sample Chapter of Sigals ad Systems (Vol7, GATE Study Package) Now, the eros of X () are roots of umerator that is 0 ad poles are roots of equatio ( )( ) 0 which are give as ad. The poles ad eros of X () are show i poleero plot of figure 6... Page 495 The Trasform Figure 6.. Poleero plot of X^ h NOTE : I poleero plot poles are marked by a small cross # ad eros are marked by a small dot o as show i figure Properties of ROC The various properties of ROC are summaried as follows. These properties ca be proved by takig appropriate examples of differet DT sigals. PROPERTY The ROC is a cocetric rig i the plae cetered about the origi. PROOF : The trasform is defied as Put re jw X () Xre ( j W ) X () x [ ] xr [ ] e j W X () coverges for those values of for which xr [ ] is absolutely summbable that is xr [ ] < Thus, covergece is depedet oly o r, where, r The equatio re jw, describes a circle i plae. Hece the ROC will cosists of cocetric rigs cetered at ero. PROPERTY The ROC caot cotai ay poles. PROOF : ROC is defied as the values of for which trasform X () coverges. We kow that X () will be ifiite at pole, ad, therefore X () does ot coverge at poles. Hece the regio of covergece does ot iclude ay pole. *Shippig Free* Buy Olie: shop.odia.co.i *Maximum Discout*

18 0 Subjectwise books by R. K. Kaodia Electroics & Commuicatio Geeral Aptitude Egieerig Mathematics Networks Electroic Devices Aalog Electroics Digital Electroics Sigals & Systems Cotrol Systems Commuicatio Systems Electromagetics Page 496 The Trasform PROPERTY If x [ ] is a fiite duratio twosided sequece the the ROC is etire plae except at 0 ad. PROOF : A sequece which is ero outside a fiite iterval of time is called fiite duratio sequece. Cosider a fiite duratio sequece x [ ] show i figure 6..a; x [ ] is oero oly for some iterval N # # N. Figure 6..a A Fiite Duratio Sequece The trasform of x [ ] is defied as N X () x [ ] N This summatio coverges for all fiite values of. If N is egative ad N is positive, the X () will have both positive ad egative powers of. The egative powers of becomes ubouded (ifiity) if " 0. Similarly positive powers of becomes ubouded (ifiity) if ". So ROC of X () is etire plae except possible 0 ador. NOTE : Both N ad N ca be either positive or egative. PROPERTY 4 If x [ ] is a rightsided sequece, ad if the circle r 0 is i the ROC, the all values of for which > r 0 will also be i the ROC. PROOF : A sequece which is ero prior to some fiite time is called the rightsided sequece. Cosider a rightsided sequece x [ ] show i figure 6..b; that is; x [ ] 0 for < N. Figure 6..b A Right Sided Sequece *Shippig Free* Buy Olie: shop.odia.co.i *Maximum Discout*

19 Electroics & Commuicatio Sample Chapter of Sigals ad Systems (Vol7, GATE Study Package) Let the trasform of x [ ] coverges for some value of _ i.e. r0i. From the coditio of covergece we ca write x [ ] < x [ ] r0 < The sequece is right sided, so limits of above summatio chages as x [ ] r0 < (6..) N Now if we take aother value of as r with r< r0, the xr [ ] decays faster tha xr [ ] 0 for icreasig. Thus we ca write x [ ] x [ ] r0 r0 N N x [ ] r r 0 a k (6..) N 0 From equatio (6..) we kow that xr [ ] 0 is absolutely summable. Let, it is bouded by some value M x, the equatio (6..) becomes as x [ ] M N r # x a k (6..) N 0 The right had side of above equatio coverges oly if or r > > r0 0 Thus, we coclude that if the circle r 0 is i the ROC, the all values of for which > r 0 will also be i the ROC. The ROC of a rightsided sequece is illustrated i figure 6..c. Page 497 The Trasform Figure 6..c ROC of a rightsided sequece PROPERTY 5 If x [ ] is a leftsided sequece, ad if the circle r 0 is i the ROC, the all values of for which < r 0 will also be i the ROC. PROOF : A sequece which is ero after some fiite time iterval is called a leftsided sigal. Cosider a leftsided sigal x [ ] show i figure 6..d; that is x [ ] 0 for > N. *Shippig Free* Buy Olie: shop.odia.co.i *Maximum Discout*

20 0 Subjectwise books by R. K. Kaodia Electroics & Commuicatio Geeral Aptitude Egieerig Mathematics Networks Electroic Devices Aalog Electroics Digital Electroics Sigals & Systems Cotrol Systems Commuicatio Systems Electromagetics Page 498 The Trasform Figure 6..d A leftsided sequece Let trasform of x [ ] coverges for some values of _ ie.. r 0 i. From the coditio of covergece we write x [ ] < or x [ ] r0 < (6..4) The sequece is left sided, so the limits of summatio chages as N x [ ] r0 < (6..5) Now if take aother value of as r, the we ca write N x [ ] x [ ] r r N N 0 0 x [ ] r r 0 0 a k (6..6) From equatio (6..4), we kow that xr [ ] 0 is absolutely summable. Let it is bouded by some value M x, the equatio (6..6) becomes as N N x [ ] M r 0 # x a k The above summatio coverges if r0 > (because is icreasig egatively), so < r 0 will be i ROC. The ROC of a leftsided sequece is illustrated i figure 6..e. Figure 6..e ROC of a Left Sided Sequece PROPERTY 6 If x [ ] is a twosided sigal, ad if the circle r 0 is i the ROC, the the ROC cosists of a rig i the plae that icludes the circle r 0 *Shippig Free* Buy Olie: shop.odia.co.i *Maximum Discout*

21 Electroics & Commuicatio Sample Chapter of Sigals ad Systems (Vol7, GATE Study Package) PROOF : A sequece which is defied for ifiite extet for both > 0 ad < 0 is called twosided sequece. A twosided sigal x [ ] is show i figure 6..f. Page 499 The Trasform Figure 6..f A Two Sided Sequece For ay time N 0, a twosided sequece ca be divided ito sum of leftsided ad rightsided sequeces as show i figure 6..g. Figure 6..g A Two Sided Sequece Divided ito Sum of a Left Sided ad Right Sided Sequece The trasform of x [ ] coverges for the values of for which the trasform of both xr [ ] ad xl [ ] coverges. From property 4, the ROC of a rightsided sequece is a regio which is bouded o the iside by a circle ad extedig outward to ifiity i.e. > r. From property 5, the ROC of a left sided sequece is bouded o the outside by a circle ad extedig iward to ero i.e. < r. So the ROC of combied sigal icludes itersectio of both ROCs which is rig i the plae. The ROC for the rightsided sequece xr [ ], the leftsequece xl [ ] ad their combiatio which is a two sided sequece x [ ] are show i figure 6..h. *Shippig Free* Buy Olie: shop.odia.co.i *Maximum Discout*

22 0 Subjectwise books by R. K. Kaodia Electroics & Commuicatio Geeral Aptitude Egieerig Mathematics Networks Electroic Devices Aalog Electroics Digital Electroics Sigals & Systems Cotrol Systems Commuicatio Systems Electromagetics Page 500 The Trasform Figure 6..h ROC of a leftsided sequece, a rightsided sequece ad two sided sequece PROPERTY 7 If the trasform X () of x [ ] is ratioal, the its ROC is bouded by poles or exteds to ifiity. PROOF : The expoetial DT sigals also have ratioal trasform ad the poles of X () determies the boudaries of ROC. PROPERTY 8 If the trasform X () of x [ ] is ratioal ad x [ ] is a rightsided sequece the the ROC is the regio i the plae outside the outermost pole i.e. ROC is the regio outside a circle with a radius greater tha the magitude of largest pole of X. () PROOF : This property ca be be proved by takig property 4 ad 7 together. PROPERTY 9 If the trasform X () of x [ ] is ratioal ad x [ ] is a leftsided sequece the the ROC is the regio i the plae iside the iermost pole i.e. ROC is the regio iside a circle with a radius equal to the smallest magitude of poles of X. () PROOF : This property ca be be proved by takig property 5 ad 7 together. Trasform of Some Basic Fuctios trasform of basic fuctios are summaried i the Table 6. with their respective ROCs. 6.4 THE INVERSE TRANSFORM Let X () be the trasform of a sequece x. [ ] To obtai the sequece x [ ] from its trasform is called the iverse trasform. The iverse trasform is give as *Shippig Free* Buy Olie: shop.odia.co.i *Maximum Discout*

23 Electroics & Commuicatio Sample Chapter of Sigals ad Systems (Vol7, GATE Study Package) TABLE 6. : Trasform of Basic Discrete Time Sigals DT sequece x [ ] trasform ROC Page 50 The Trasform. d[ ] etire plae. d[ 0 ] 0 etire plae, except 0. u [ ] > 4. a u [ ] a a > a 5. a u [ ] a a 6. u[ ] ( ) ( ) a a 7. a u[ ] ( a ) ( a) 8. cos( u ) [ ] W 0 cos W0 cos W + 0 > a 9. si( u ) [ ] W 0 or > > a [ cos W ] > 0 cos W + si W0 cos W0 + 0 si W0 cos W + 0 or > 0. a cos( W 0 u ) [ ]. a si( W 0 u ) [ ] a cos W0 a cos W + a or 0 [ a cos W0] acos W + a a si W0 a cos W + a or 0 a si W0 acos W + a 0 0 > a > a. ra si( W 0+ q) u[ ] with a! R A+ B + g + a or A ( + B) + g+ g ( a) # a *Shippig Free* Buy Olie: shop.odia.co.i *Maximum Discout*

24 0 Subjectwise books by R. K. Kaodia Electroics & Commuicatio Geeral Aptitude Egieerig Mathematics Networks Electroic Devices Aalog Electroics Digital Electroics Sigals & Systems Cotrol Systems Commuicatio Systems Electromagetics Page 50 The Trasform # x [ ] X () d j p This method ivolves the cotour itegratio, so difficult to solve. There are other commoly used methods to evaluate the iverse trasform give as follows. Partial fractio method. Power series expasio 6.4. Partial Fractio Method If X () is a ratioal fuctio of the it ca be expressed as follows. N () X () D () It is coveiet if we cosider X () rather tha X () to obtai the iverse trasform by partial fractio method. Let p, p, p...p are the roots of deomiator polyomial, also the poles of X. () The, usig partial fractio method X () ca be expressed as X () A A A... A p p p p X () A A... p p p Now, the iverse trasform of above equatio ca be obtaied by comparig each term with the stadard traform pair give i table 6.. The values of coefficiets A, A, A...A depeds o whether the poles are real & distict or repeated or complex. Three cases are give as follows Case I : Poles are Simple ad Real X () ca be expaded i partial fractio as X () A A A... A (6.4.) p p p p where A, A,... A are calculated as follows X () A ( p) X () A ( p) I geeral, A i ( p) X( ) i p i p p (6.4.) Case II : If Poles are Repeated I this case X () has a differet form. Let p k be the root which repeats l times, the the expasio of equatio must iclude terms X () Ak Ak +... p + k ( pk) Aik... A ( pk) i lk (6.4.) ( pk) l The coefficiet A ik are evaluated by multiplyig both sides of equatio (6.4.) l by ( p k ), differetiatig ( l i) times ad the evaluatig the resultat equatio at p k. Thus, li C d X () ik li: ( p ) ( l i) d D (6.4.4) k l p k *Shippig Free* Buy Olie: shop.odia.co.i *Maximum Discout*

25 Electroics & Commuicatio Sample Chapter of Sigals ad Systems (Vol7, GATE Study Package) Case III : Complex Poles If X () has complex poles the partial fractio of the X () ca be expressed as ) X () A A + p ) (6.4.5) p where A ) is complex cojugate of A ad p ) is complex cojugate of. The coefficiets are obtaied by equatio (6.4.) Page 50 The Trasform 6.4. Power Series Expasio Method Power series method is also coveiet i calculatig the iverse trasform. The trasform of sequece x [ ] is give as X () x [ ] Now, X () is expaded i the followig form X( ).. + x[ ] + x[ ] + x[0] + x[] + x[] +... To obtai iverse trasform(i.e. x), [ ] represet the give X () i the form of above power series. The by comparig we ca get x [ ] {... x[ ], x[ ], x[0], x[], x[],...} 6.5 PROPERTIES OF TRANSFORM 6.5. Liearity The uilateral ad bilateral trasforms possess a set of properties, which are useful i the aalysis of DT sigals ad systems. The proofs of properties are give for bilateral trasform oly ad ca be obtaied i a similar way for the uilateral trasform. Like Laplace trasform, the liearity property of trasform states that, the liear combiatio of DT sequeces i the time domai is equivalet to liear combiatio of their trasform. Let x [ ] X (), with ROC: R ad x [ ] X (), with ROC: R the, ax[ ] + bx[ ] ax() + bx(), with ROC: at least R for both uilateral ad bilateral trasform. + R PROOF : The trasform of sigal { ax[ ] + bx[ ]} is give by equatio (6..) as follows { ax [ ] + bx [ ]} { ax [ ] + bx [ ]} a x [ ] + b x [ ] ax () + bx () Hece, ax [ ] + bx [ ] ax () + bx () ROC : Sice, the trasform X () is fiite withi the specified ROC, R. *Shippig Free* Buy Olie: shop.odia.co.i *Maximum Discout*

26 0 Subjectwise books by R. K. Kaodia Electroics & Commuicatio Geeral Aptitude Egieerig Mathematics Networks Electroic Devices Aalog Electroics Digital Electroics Sigals & Systems Cotrol Systems Commuicatio Systems Electromagetics Page 504 The Trasform Similarly, X () is fiite withi its ROC, R. Therefore, the liear combiatio ax () + bx () should be fiite at least withi regio R + R. NOTE : I certai cases, due to the iteractio betwee x [ ] ad x [ ], which may lead to cacellatio of certai terms, the overall ROC may be larger tha the itersectio of the two regios. O the other had, if there is o commo regio betwee R ad R, the trasform of ax [ ] + bx [ ] does ot exist Time Shiftig For the bilateral trasform If x [ ] X (), with ROC R x 0 the x [ 0 ] X(), ad x [ + 0 ] 0 X(), with ROC : R x except for the possible deletio or additio of 0 or. PROOF : The bilateral trasform of sigal x [ 0 ] is give by equatio (6..) as follows 0 { x [ ]} x [ ] Substitutig 0 a o RHS, we get 0 a { x [ ]} x[ a] ( 0 + a ) x[ a] 0 a 0 x[ a] a { x [ ]} 0 X[] Similarly we ca prove { x [ + ]} 0 X[] 0 a a ROC : The ROC of shifted sigals is altered because of the terms which affects the roots of the deomiator i X. () or, 0 0 TIME SHIFTING FOR UNILATERAL TRANSFORM For the uilateral trasform If x [ ] X (), with ROC R x the x [ 0 ] 0 0 m ex() + x[ m] o, m 0 0 m ad x [ + 0 ] ex() x[ m] o, m 0 with ROC : R x except for the possible deletio or additio of 0 or. PROOF : The uilateral trasform of sigal x [ 0 ] is give by equatio (6..) as follows *Shippig Free* Buy Olie: shop.odia.co.i *Maximum Discout*

27 Electroics & Commuicatio Sample Chapter of Sigals ad Systems (Vol7, GATE Study Package) 6.5. Time Reversal 0 0 { x [ ]} x [ ] Multiplyig RHS by ad 0 0 { x [ ]} x [ ] 0 Substitutig 0 a 0 0 Limits; whe " 0, a " 0 whe " +, a " x[ ] ( 0 ) 0 Now, { x [ ]} x[ a] 0 a a 0 a a 0 x[ ] + x[ a] a 0 a a a 0 a 0 0 or, { x [ ]} x[ a] + x[ a] 0 a 0 a a or, { x [ ]} x[ a] + x[ a] 0 a 0 a Chagig the variables as a " ad a " m i first ad secod summatio respectively 0 0 { x [ ]} x[ ] + x[ m] 0 0 m 0 m 0 m 0 0 X[] + x[ m] I similar way, we ca also prove that 0 m 0 x [ + 0 ] ex() x[ m] m 0 Time reversal property states that time reflectio of a DT sequece i time domai is equivalet to replacig by i its trasform. 0 m o a Page 505 The Trasform If x [ ] X (), with ROC : R x the x[ ] X b l, with ROC : Rx for bilateral trasform. PROOF : The bilateral trasform of sigal x[ ] is give by equatio (6..) as follows { x[ ]} x[ ] Substitutig k o the RHS, we get { x[ ]} xk [] k k xk []( ) X b l Hece, x[ ] k ROC :! R or! R x X b l x k *Shippig Free* Buy Olie: shop.odia.co.i *Maximum Discout*

28 0 Subjectwise books by R. K. Kaodia Electroics & Commuicatio Geeral Aptitude Egieerig Mathematics Networks Electroic Devices Aalog Electroics Digital Electroics Sigals & Systems Cotrol Systems Commuicatio Systems Electromagetics Page 506 The Trasform Differetiatio i the domai This property states that multiplicatio of time sequece x [ ] with correspods to differetiatio with respect to ad multiplicatio of result by i the domai. If x [ ] X (), with ROC : R x the x[ ] dx() d with ROC : R x For both uilateral ad bilateral trasforms. PROOF : The bilateral trasform of sigal x [ ] is give by equatio (6..) as follows X () x [ ] Differetiatig both sides with respect to gives dx() x [ ] d x [ ]( ) d d Multiplyig both sides by, we obtai dx() x[ ] d () Hece, x[ ] dx d ROC : This operatio does ot affect the ROC Scalig i Domai Multiplicatio of a time sequece with a expoetial sequece a correspods to scalig i domai by a factor of a. If x [ ] X (), with ROC : R x the ax [ ] X a a k, with ROC : ar x for both uilateral ad bilateral trasform. PROOF : Time Scalig The bilateral trasform of sigal x [ ] is give by equatio (6..) as { ax [ ]} ax [ ] x [ ][ a ] ax [ ] X a a k ROC : If is a poit i the ROC of X () the the poit a is i the ROC of Xa. () As we discussed i Chapter, there are two types of scalig i the DT domai decimatio(compressio) ad iterpolatio(expasio). Time Compressio Sice the decimatio (compressio) of DT sigals is a irreversible process (because some data may lost), therefore the trasform of x [ ] ad its *Shippig Free* Buy Olie: shop.odia.co.i *Maximum Discout*

29 Electroics & Commuicatio Sample Chapter of Sigals ad Systems (Vol7, GATE Study Package) decimated sequece y [ ] xa [ ] ot be related to each other. Time Expasio I the discrete time domai, time expasio of sequece x [ ] is defied as xk [ ] if is a multiple of iteger k xk [ ] ) (6.5.) 0 otherwise Timescalig property of trasform is derived oly for time expasio which is give as Page 507 The Trasform If x [ ] X (), with ROC : R x k the xk [ ] Xk () X( ), with ROC : ( x) for both the uilateral ad bilateral trasform. R k PROOF : The uilateral trasform of expaded sequece xk [ ] is give by { x k [ ]} x [ ] k 0 k xk[0] + xk[] xk[ k] ( k+ ) k + x [ k+ ] +... x [ k] +... k k Sice the expaded sequece xk [ ] is ero everywhere except whe is a multiple of k. This reduces the above trasform as follows k k k { x k [ ]} x[0] + x[ k ] + x[ k ] + x[ k ] +... k k As defied i equatio 6.5., iterpolated sequece is xk [ ] xk [ ] 0 xk [ 0] x[ 0], k xk [] k x[] k xk [ k] x[ ] Thus, we ca write { x k [ ]} x[0] x[] k x[] k x[] k k k x [ ]( ) X ( ) k NOTE : Time expasio of a DT sequece by a factor of k correspods to replacig as k i its trasform. k Time Differecig If x [ ] X (), with ROC : R x the x [ ]x [ ] ( ) X( ), with the ROC : R x except for the possible deletio of 0, for both uilateral ad bilateral trasform. PROOF : The trasform of x [ ]x [ ] is give by equatio (6..) as follows { x [ ] x [ ]} { x [ ]x [ ]} *Shippig Free* Buy Olie: shop.odia.co.i *Maximum Discout*

30 0 Subjectwise books by R. K. Kaodia Electroics & Commuicatio Geeral Aptitude Egieerig Mathematics Networks Electroic Devices Aalog Electroics Digital Electroics Sigals & Systems Cotrol Systems Commuicatio Systems Electromagetics Page 508 The Trasform x [ ] x [ ] I the secod summatio, substitutig r { x [ ] x [ ]} x [ ] xr [ ] ( r+ ) r r x [ ] xr [ ] X () X () Hece, x [ ]x [ ] ( ) X( ) r Time Covolutio Time covolutio property states that covolutio of two sequece i time domai correspods to multiplicatio i domai. Let x [ ] X (), ROC : R ad x [ ] X (), ROC : R the the covolutio property states that x[ ] * x[ ] X() X(), ROC : at least R + R for both uilateral ad bilateral trasforms. PROOF : As discussed i chapter 4, the covolutio of two sequeces is give by k x[ ] * x[ ] x[] k x[ k] Takig the trasform of both sides gives k x[ ] * x[ ] x[] k x[ k] Iterchagig the order of the two summatios, we get k x[ ] * x[ ] x[] k x[ k] Substitutig k a i the secod summatio k a k e oe k a x [ ]* x[ ] x [] k x [ ] ( a+ k a ) or x [ ]* x[ ] x [] k x [ a] x[ ] * x[ ] X() X() a o Cojugatio Property If x [ ] X (), with ROC : R x ) the x [ ] If x [ ] is real, the X () X ) ( ) ) X ) ( ) ), with ROC : R x *Shippig Free* Buy Olie: shop.odia.co.i *Maximum Discout*

31 Electroics & Commuicatio Sample Chapter of Sigals ad Systems (Vol7, GATE Study Package) PROOF : ) The trasform of sigal x [ ] is give by equatio (6..) as follows { x ) ) [ ]} x [ ] ) ) 6 x [ ]( (6.5.) Let trasform of x [ ] is X () X () x [ ] by takig complex cojugate o both sides of above equatio ) X () [ x [ ] ] ) Replacig " ), we will get ) ) ) ) X ( ) 6 x [ ]( (6.5.) Comparig equatio (6.5.) ad (6.5.) { x ) [ ]} X ) ( ) ) (6.5.4) Page 509 The Trasform ) For real x, [ ] x [ ] x [ ], so { x ) [ ]} x [ ] X ( ) (6.5.5) Comparig equatio (6.5.4) ad (6.5.5) X () X ) ( ) ) Iitial Value Theorem If x [ ] X (), with ROC : R x the iitialvalue theorem states that, x[ 0 ] lim X () " The iitialvalue theorem is valid oly for the uilateral Lapalce trasform PROOF : For a causal sigal x [ ] X () x [ ] 0 x[0] + x[] + x[] +... Takig limit as " o both sides we get lim X () lim( x[0] + x[] + x[] +...) " " x[ 0 ] lim X () " x[ 0] 6.5. Fial Value Theorem If x [ ] X (), with ROC : R x the fialvalue theorem states that x[ ] lim( ) X( ) " Fial value theorem is applicable if X () has o poles outside the uit circle. This theorem ca be applied to either the uilateral or bilateral trasform. *Shippig Free* Buy Olie: shop.odia.co.i *Maximum Discout*

32 0 Subjectwise books by R. K. Kaodia Electroics & Commuicatio Geeral Aptitude Egieerig Mathematics Networks Electroic Devices Aalog Electroics Digital Electroics Sigals & Systems Cotrol Systems Commuicatio Systems Electromagetics Page 50 The Trasform PROOF : { x [ + ]} { x [ ]} lim { x [ + ] x [ ]} k 0 (6.5.6) k " From the time shiftig property of uilateral trasform discussed i sectio x [ + 0 ] ex() x[ m] m 0 0 m For 0 x [ + ] ex() x[ m] m o m 0 x [ + ] X ^ ()x0 [] h Put above trasformatio i the equatio (6.5.6) k k " 0 k X[] x[0] X[] lim ( x [ + ] x [ ]) ( ) X[ ] x[0] lim ( x [ + ] x [ ]) k " 0 Takig limit as " o both sides we get lim( ) X[ ] x[0] lim x [ + ] x [ ] " k k " 0 lim( ) X[ ] x[0] lim{( x[] x[0]) + ( x[ ] x[ ]) + ( x[ ] x[ ]) +... " k " lim( ) X[ ] x[0] x[ ] x[ 0] " Hece, x[ ] lim( ) X( ) Summary of Properties " o... + ( xk [ + ] xk [ ]) Let, x [ ] X (), with ROC R x x [ ] X (), with ROC R x [ ] X (), with ROC R The properties of trasforms are summaried i the followig table. TABLE 6. Properties of trasform Properties Time domai trasform ROC Liearity ax [ ] + bx [ ] ax () + bx () at least R + R Time shiftig (bilateral or ocausal) x [ 0 ] 0 X() R x except for the possible deletio x [ + 0 ] 0 X() or additio of 0or Time shiftig (uilateral or causal) x [ 0 ] x [ + 0 ] 0 X() 0 + x[ m] m o R x except for the m possible deletio or additio of 0 ^ X() 0or ^ 0 m 0 xm [ ] m o *Shippig Free* Buy Olie: shop.odia.co.i *Maximum Discout*

33 Electroics & Commuicatio Sample Chapter of Sigals ad Systems (Vol7, GATE Study Package) Properties Time domai trasform ROC Time reversal x[ ] X b l R x Page 5 The Trasform Differetiatio i domai x[ ] dx() R x d Scalig i domai ax [ ] X a a k ar x Time scalig (expasio) xk [ ] x[ k] X ( k ) ( ) R x k Time differecig Time covolutio ) Cojugatios x [ ] Iitialvalue theorem Fialvalue theorem x [ ]x [ ] ( ) X( ) possible deletio of R x, except for the the origi x[ ] * x[ ] X() X() at least R + R ( ) X ) ) R x provided x [ ] 0 x[ 0] lim X( ) " for < 0 x[ ] lim x [ ] " lim( ) X( ) 6.6 ANALYSIS OF DISCRETE LTI SYSTEMS USING TRANSFORM " provided x[ ] exists The trasform is very useful tool i the aalysis of discrete LTI system. As the Laplace trasform is used i solvig differetial equatios which describe cotiuous LTI systems, the trasform is used to solve differece equatio which describe the discrete LTI systems. Similar to Laplace trasform, for CT domai, the trasform gives trasfer fuctio of the LTI discrete systems which is the ratio of the trasform of the output variable to the trasform of the iput variable. These applicatios are discussed as follows 6.6. Respose of LTI Cotiuous Time System As discussed i chapter 4 (sectio 4.8), a discretetime LTI system is always described by a liear costat coefficiet differece equatio give as follows N M ay k [ k] bx k [ k] k 0 k 0 a y[ N] + a y[ ( N )] a y[ ] + a y[ ] N N 0 bmx[ M] + bm x[ ( M )] bx[ ] + b0x[ ] (6.6.) where, N is order of the system. The timeshift property of trasform x [ ] 0 0 X ( ), is used to solve the above differece equatio which coverts it ito a algebraic equatio. By takig trasform of above equatio *Shippig Free* Buy Olie: shop.odia.co.i *Maximum Discout*

34 0 Subjectwise books by R. K. Kaodia Electroics & Commuicatio Geeral Aptitude Egieerig Mathematics Networks Electroic Devices Aalog Electroics Digital Electroics Sigals & Systems Cotrol Systems Commuicatio Systems Electromagetics Page 5 The Trasform N ( N ) a Y ( ) + a Y ( )] a + ay ( ) N N ( ) bm M M X ( ) + bm X ( ) b Xx ( ) + bx 0 ( ) Y () bm N M bm... b b X () an N N + an a+ a0 this equatio ca be solved for Y () to fid the respose y.the [ ] solutio or total respose y [ ] cosists of two parts as discussed below.. eroiput Respose or Free Respose or Natural Respose The ero iput respose yi [ ] is maily due to iitial output i the system. The eroiput respose is obtaied from system equatio (6.6.) whe iput x [ ] 0. By substitutig x [ ] 0 ad y [ ] y [ ] i equatio (6.6.), we get i any[ N] + an y[ ( N )] ay[ ] + a0y[ ] 0 O takig trasform of the above equatio with give iitial coditios, we ca form a equatio for Yi (). The eroiput respose yi [ ] is give by iverse trasform of Y () i. 0 NOTE : The ero iput respose is also called the atural respose of the system ad it is deoted as y [ ] N.. erostate Respose or Forced Respose The erostate respose ys [ ] is the respose of the system due to iput sigal ad with ero iitial coditios. The erostate respose is obtaied from the differece equatio (6.6.) goverig the system for specific iput sigal x [ ] for $ 0 ad with ero iitial coditios. Substitutig y [ ] y [ ] i equatio (6.6.) we get, s a y [ N] + a y [ ( N )] a y [ ] + a y [ ] N s N s s 0 s bmx[ M] + bmx[ ( M )] bx[ ] + b0x[ ] Takig trasform of the above equatio with ero iitial coditios for output (i.e., y[ ] y[ ]... 0 we ca form a equatio for Ys (). The erostate respose ys [ ] is give by iverse trasform of Ys (). NOTE : The ero state respose is also called the forced respose of the system ad it is deoted as y [ ] F. Total Respose The total respose y [ ] is the respose of the system due to iput sigal ad iitial output. The total respose ca be obtaied i followig two ways : Takig trasform of equatio (6.6.) with oero iitial coditios for both iput ad output, ad the substitutig for X () we ca form a equatio for Y. () The total respose y [ ] is give by iverse Laplace trasform of Ys. () Alteratively, that total respose y [ ] is give by sum of eroiput respose yi [ ] ad erostate respose ys [ ]. Therefore total respose, y [ ] y [ ] + y [ ] 6.6. Impulse Respose ad Trasfer Fuctio i s System fuctio or trasfer fuctio is defied as the ratio of the trasform of the output y [ ] ad the iput x [ ] with ero iitial coditios. L Let x [ ] X ( ) is the iput ad y [ ] Y ( ) is the output of a LTI L discrete time system havig impulse respose h ( ) H ( ). The respose *Shippig Free* Buy Olie: shop.odia.co.i *Maximum Discout*

35 Electroics & Commuicatio Sample Chapter of Sigals ad Systems (Vol7, GATE Study Package) y [ ] of the discrete time system is give by covolutio sum of iput ad impulse respose as y [ ] x [ ]* h [ ] By applyig covolutio property of trasform we obtai Y () XH () () Y () H () X () where, H () is defied as the trasfer fuctio of the system. It is the trasform of the impulse respose. Alteratively we ca say that the iverse trasform of trasfer fuctio is the impulse respose of the system. Impulse respose Y () h [ ] { H ( )} ) X () 6.7 STABILITY AND CAUSALITY OF LTI DISCRETE SYSTEMS USING TRANSFORM 6.7. Causality trasform is also used i characteriatio of LTI discrete systems. I this sectio, we derive a domai coditio to check the stability ad causality of a system directly from its trasfer fuctio. A liear timeivariat discrete time system is said to be causal if the impulse respose h [ ] 0, for < 0 ad it is therefore rightsided. The ROC of such a system H () is the exterior of a circle. If H () is ratioal the the system is said to be causal if. The ROC is the exterior of a circle outside the outermost pole ; ad. The degree of the umerator polyomial of H () should be less tha or equal to the degree of the deomiator polyomial. Page 5 The Trasform 6.7. Stability A LTI discretetime system is said to be BIBO stable if the impulse respose h [ ] is summable. That is h [ ] < trasform of h [ ] is give as H () h [ ] Let e jw (which describes a uit circle i the plae), the He [ jw ] he [ ] j # W he [ ] j W h [ ] < which is the coditio for the stability. Thus we ca coclude that *Shippig Free* Buy Olie: shop.odia.co.i *Maximum Discout*

36 0 Subjectwise books by R. K. Kaodia Electroics & Commuicatio Geeral Aptitude Egieerig Mathematics Networks Electroic Devices Aalog Electroics Digital Electroics Sigals & Systems Cotrol Systems Commuicatio Systems Electromagetics Page 54 The Trasform STABILITY OF LTI DISCRETE SYSTEM A LTI system is stable if the ROC of its system fuctio H () cotais the uit circle 6.7. Stability ad Causality As we discussed previously, for a causal system with ratioal trasfer fuctio H, () the ROC is outside the outermost pole. For the BIBO stability the ROC should iclude the uit circle. Thus, for the system to be causal ad stable theses two coditios are satisfied if all the poles are withi the uit circle i the plae. STABILITY AND CAUSALITY OF LTI DISCRETE SYSTEM A LTI discrete time system with the ratioal system fuctio H () is said to be both causal ad stable if all the poles of H () lies iside the uit circle. 6.8 BLOCK DIAGRAM REPRESENTATION I domai, the iputoutput relatio of a LTI discrete time system is represeted by the trasfer fuctio H (),which is a ratioal fuctio of, as show i equatio Y () H () X () M M M b 0 b b... bm bm N N N a 0 + a + a an+ an where, N Order of the system, M # N ad a0 The above trasfer fuctio is realied usig uit delay elemets, uit advace elemets, adders ad multipliers. Basic elemets of block diagram with their domai represetatio is show i table TABLE 6. : Basic Elemets of Block Diagram Elemets of Block diagram Time Domai Represetatio sdomai Represetatio Adder Costat multiplier Uit delay elemet Uit advace elemet *Shippig Free* Buy Olie: shop.odia.co.i *Maximum Discout*

37 Electroics & Commuicatio Sample Chapter of Sigals ad Systems (Vol7, GATE Study Package) The differet types of structures for realiig discrete time systems are same as we discussed for the cotiuoustime system i the previous chapter. Page 55 The Trasform 6.8. Direct Form I Realiatio Cosider the differece equatio goverig the discrete time system with a 0, y [ ] + ay [ ] + ay [ ] an y [ N] bx 0 [ ] + bx [ ] + bx [ ] bm x [ M] Takig trasform of the above equatio we get, N Y () a Y ()a Y ()... a Y () + b0x() + b X() + b X() bm M X() (6.8.) The above equatio of Y () ca be directly represeted by a block diagram as show i figure 6.8.a. This structure is called direct formi structure. This structure uses separate delay elemets for both iput ad output of the system. So, this realiatio uses more memory. N Figure 6.8.a Geeral structure of direct formrealiatio For example cosider a discrete LTI system which has the followig impulse respose Y () H () + + X () Y () 4 Y () Y () X () X () + X () Comparig with stadard form of equatio (6.8.), we get a 4, a ad b0, b, b. Now put these values i geeral structure of Direct formi realiatio we get *Shippig Free* Buy Olie: shop.odia.co.i *Maximum Discout*