Signals and Systems by Kanodia-150707084730-Lva1-App6892

8/16/2019 Signals and Systems by Kanodia-150707084730-Lva1-App6892

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Eighth Edition

GATEELECTRONICS & COMMUNICATION

Signals and SystemsVol 7 of 10

RK Kanodia Ashish Murolia

NODIA & COMPANY


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GATE Electronics & Communication Vol 7, 8eSignals and SystemsRK Kanodia & Ashish Murolia

Copyright © By NODIA & COMPANY

Information contained in this book has been obtained by author, from sources believes to be reliable. However,neither NODI A & COMPANY nor its author guarantee the accuracy or completeness of any information herein,and NODIA & COMPANY nor its author shall be responsible for any error, omissions, or damages arising out ofuse of this information. This book is published with the understanding that NODIA & COMPANY and its authorare supplying information but are not attempting to render engineering or other professional services.

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Preface to the Series

For almost a decade, we have been receiving tremendous responses from GATE aspirants for our earlier books:GATE Multiple Choice Questions, GAT E Guide, and the GATE Cloud series. Our first book, GAT E MultipleChoice Questions (MCQ), was a compilation of objective questions and solutions for all subjects of GATEElectronics & Communication Engineering in one book. T he idea behind the book was that Gate aspirants who

had just completed or about to finish their last semester to achieve his or her B.E/ B.Tech need only to practiceanswering questions to crack GAT E. T he solutions in the book were presented in such a manner that a studentneeds to know fundamental concepts to understand them. We assumed that students have learned enough ofthe fundamentals by his or her graduation. T he book was a great success, but still there were a large ratio ofaspirants who needed more preparatory materials beyond just problems and solutions. This large ratio mainlyincluded average students.

Later, we perceived that many aspirants couldn’t develop a good problem solving approach in their B.E/ B.Tech.Some of them lacked the fundamentals of a subject and had difficulty understanding simple solutions. Now,we have an idea to enhance our content and present two separate books for each subject: one for theory, whichcontains brief theory, problem solving methods, fundamental concepts, and points-to-remember. T he second bookis about problems, including a vast collection of problems with descriptive and step-by-step solutions that can

be understood by an average student. This was the origin of GAT E Guide (the theory book) and GAT E Cloud (the problem bank) series: two books for each subject. GAT E Guide and GAT E Cloud were published in threesubjects only.

Thereafter we received an immense number of emails from our readers looking for a complete study packagefor all subjects and a book that combines both GAT E Guide and GAT E Cloud . T his encouraged us to presentGAT E Study Package (a set of 10 books: one for each subject) for GAT E Electronic and CommunicationEngineering. Each book in this package is adequate for the purpose of qualifying GATE for an average student.Each book contains brief theory, fundamental concepts, problem solving methodology, summary of formulae,and a solved question bank. The question bank has three exercises for each chapter: 1) Theoretical MCQs, 2)Numerical MCQs, and 3) Numerical T ype Questions (based on the new GATE pattern). Solutions are presentedin a descriptive and step-by-step manner, which are easy to understand for all aspirants.

We believe that each book of GAT E Study Package helps a student learn fundamental concepts and developproblem solving skills for a subject, which are key essentials to crack GATE. Although we have put a vigorouseffort in preparing this book, some errors may have crept in. We shall appreciate and greatly acknowledge allconstructive comments, criticisms, and suggestions from the users of this book. You may write to us at [email protected] and [email protected].

Acknowledgements

We would like to express our sincere thanks to all the co-authors, editors, and reviewers for their efforts in

making this project successful. We would also like to thank Team NODI A for providing professional support forthis project through all phases of its development. At last, we express our gratitude to God and our Family forproviding moral support and motivation.

We wish you good luck !R. K . KanodiaAshish Murolia


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SYLLABUS

GATE Electronics & Communications:

Definitions and properties of Laplace transform, continuous-time and discrete-time Fourier series, continuous-timeand discrete-time Fourier Transform, DFT and FFT, z-transform. Sampling theorem. Linear Time-Invariant (LTI)Systems: definitions and properties; causality, stability, impulse response, convolution, poles and zeros, parallel andcascade structure, frequency response, group delay, phase delay. Signal transmission through LTI systems.

IES Electronics & Telecommunication

Classification of signals and systems: System modelling in terms of differential and difference equations; Statevariable representation; Fourier series; Fourier transforms and their application to system analysis; Laplacetransforms and their application to system analysis; Convolution and superposition integrals and their applications;Z-transforms and their applications to the analysis and characterisation of discrete time systems; Random signalsand probability, Correlation functions; Spectral density; Response of linear system to random inputs.

**********


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CONTENTS

CHAPTER 1 CONTINUOUS TIME SIGNALS

1.1 CONTINUOUS - TIME AND DISCRETE - TIME SIGNALS 1

1.2 SIGNAL-CLASSIFICATION 1

1.2.1 Analog and Discrete Signals 11.2.2 Deterministic and Random Signal 11.2.3 Periodic and Aperiodic Signal 21.2.4 Even and Odd Signal 31.2.5 Energy and Power Signal 4

1.3 BASIC OPERATIONS ON SIGNALS 5

1.3.1 Addition of Signals 51.3.2 Multiplication of Signals 51.3.3 Amplitude Scaling of Signals 51.3.4 Time-Scaling 51.3.5 Time-Shifting 61.3.6 Time-Reversal/ Folding 71.3.7 Amplitude Inverted Signals 8

1.4 MULTIPLE OPERATIONS ON SIGNALS 8

1.5 BASIC CONTINUOUS TIME SIGNALS 9

1.5.1 The Unit-Impulse Function 91.5.2 The Unit-Step Function 121.5.3 The Unit-Ramp Function 121.5.4 Unit Rectangular Pulse Function 131.5.5 Unit Triangular Function 131.5.6 Unit Signum Function 141.5.7 The Sinc Function 14

1.6 MATHEMATICAL REPRESENTATION OF SIGNALS 15

EXERCISE 1.1 16

EXERCISE 1.2 41

EXERCISE 1.3 44

EXERCISE 1.4 49

SOLUTIONS 1.1 56

SOLUTIONS 1.2 79

SOLUTIONS 1.3 84

SOLUTIONS 1.4 85


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CHAPTER 2 CONTINUOUS TIME SYSTEMS

2.1 CONTINUOUS TIME SYSTEM & CLASSIFICATION 93

2.1.1 Linear and Non-Linear System 932.1.2 Time-Varying and Time-Invariant system 932.1.3 Systems With and Without Memory (Dynamic and Static Systems) 942.1.4 Causal and Non-causal Systems 94

2.1.5 Invertible and Non-Invertible Systems 942.1.6 Stable and Unstable systems 94

2.2 LINEAR TIME INVARIANT SYSTEM 95

2.2.1 Impulse Response and The Convolution Integral 952.2.2 Properties of Convolution Integral 96

2.3 STEP RESPONSE OF AN LTI SYSTEM 100

2.4 PROPERTIES OF LTI SYSTEMS IN TERMS OF IMPULSE RESPONSE 101

2.4.1 Memoryless LT I System 1012.4.2 Causal LT I System 1012.4.3 Invertible LT I System 1022.4.4 Stable LT I System 102

2.5 IMPULSE RESPONSE OF INTER-CONNECTED SYSTEMS 103

2.5.1 Systems in Parallel Configuration 1032.5.2 System in Cascade 103

2.6 CORRELATION 103

2.6.1 Cross-Correlation 1032.6.2 Auto-Correlation 1052.6.3 Correlation and Convolution 109

2.7 TIME DOMAIN ANALYSIS OF CONTINUOUS TIME SYSTEMS 109

2.7.1 Natural Response or Zero-input Response 1102.7.2 Forced Response or Zero-state Response 1112.7.3 The Total Response 111

2.8 BLOCK DIAGRAM REPRESENTATION 112

EXERCISE 2.1 114

EXERCISE 2.2 133

EXERCISE 2.3 135

EXERCISE 2.4 138

SOLUTIONS 2.1 149

SOLUTIONS 2.2 179

SOLUTIONS 2.3 186

SOLUTIONS 2.4 187

CHAPTER 3 DISCRETE TIME SIGNALS

3.1 INTRODUCTION TO DISCRETE TIME SIGNALS 203

3.1.1 Representation of Discrete T ime signals 203


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3.2 SIGNAL CLASSIFICATION 204

3.2.1 Periodic and Aperiodic DT Signals 2043.2.2 Even and Odd DT Signals 2053.2.3 Energy and Power Signals 207

3.3 BASIC OPERATIONS ON DT SIGNALS 207

3.3.1 Addition of DT Signals 208

3.3.2 Multiplication of DT Signal 2083.3.3 Amplitude scaling of DT Signals 2083.3.4 T ime-Scaling of DT Signals 2083.3.5 Time-Shifting of DT Signals 2093.3.6 Time-Reversal (folding) of DT signals 2103.3.7 Inverted DT Signals 211

3.4 MULTIPLE OPERATIONS ON DT SIGNALS 211

3.5 BASIC DISCRETE TIME SIGNALS 212

3.5.1 Discrete Impulse Function 2123.5.2 Discrete Unit Step Function 213

3.5.3 Discrete Unit-ramp Function 2133.5.4 Unit-Rectangular Function 2143.5.5 Unit-Triangular Function 2143.5.6 Unit-Signum Function 215

3.6 MATHEMATICAL REPRESENTATION OF DT SIGNALS USING IMPULSE OR STEP FUNCTION 215

EXERCISE 3.1 216

EXERCISE 3.2 241

EXERCISE 3.3 244

EXERCISE 3.4 247

SOLUTIONS 3.1 249

SOLUTIONS 3.2 273

SOLUTIONS 3.3 281

SOLUTIONS 3.4 282

CHAPTER 4 DISCRETE TIME SYSTEMS

4.1 DISCRETE TIME SYSTEM & CLASSIFICATION 285

4.1.1 Linear and Non-linear Systems 2854.1.2 Time-Varying and Time-Invariant Systems 2854.1.3 System With and Without Memory (Static and Dynamic Systems) 2864.1.4 Causal and Non-Causal System 2864.1.5 Invertible and Non-Invertible Systems 2864.1.6 Stable and Unstable System 286

4.2 LINEAR-TIME INVARIANT DISCRETE SYSTEM 287

4.2.1 Impulse Response and Convolution Sum 2874.2.2 Properties of Convolution Sum 288

4.3 STEP RESPONSE OF AN LTI SYSTEM 292


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4.4 PROPERTIES OF DISCRETE LTI SYSTEM IN TERMS OF IMPULSE RESPONSE 292

4.4.1 Memoryless LT ID System 2934.4.2 Causal LT ID System 2934.4.3 Invertible LT ID System 2934.4.4 Stable LT ID System 2944.4.5 FIR and I IR Systems 294

4.5 IMPULSE RESPONSE OF INTERCONNECTED SYSTEMS 2954.5.1 Systems in Parallel 2954.5.2 System in Cascade 295

4.6 CORRELATION 296

4.6.1 Cross-Correlation 2964.6.2 Auto-Correlation 2964.6.3 Properties of Correlation 2974.6.4 Relationship Between Correlation and Convolution 2994.6.5 Methods to Solve Correlation 299

4.7 DECONVOLUTION 300

4.8 RESPONSE OF LTID SYSTEMS IN TIME DOMAIN 300

4.8.1 Natural Response or Zero Input Response 3014.8.2 Forced Response or Zero State Response 3024.8.3 Total Response 302


EXERCISE 4.1 304

EXERCISE 4.2 317

EXERCISE 4.3 320

EXERCISE 4.4 323

SOLUTIONS 4.1 329

SOLUTIONS 4.2 353

SOLUTIONS 4.3 361

SOLUTIONS 4.4 362

CHAPTER 5 THE LAPLACE TRANSFORM

5.1 INTRODUCTION 375

5.1.1 The Bilateral or Two-Sided Laplace Transform 3755.1.2 The Unilateral Laplace Transform 375

5.2 THE EXISTENCE OF LAPLACE TRANSFORM 3755.3 REGION OF CONVERGENCE 376

5.3.1 Poles and Zeros of Rational Laplace Transforms 3765.3.2 Properties of ROC 377

5.4 THE INVERSE LAPLACE TRANSFORM 382

5.4.1 Inverse Laplace Transform Using Partial Fraction Method 3835.4.2 Inverse Laplace Transform Using Convolution Method 383


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5.5 PROPERTIES OF THE LAPLACE TRANSFORM 384

5.5.1 Linearity 3845.5.2 Time Scaling 3845.5.3 Time Shifting 3855.5.4 Shifting in the s -domain(Frequency Shifting) 3865.5.5 Time Differentiation 3865.5.6 Time Integration 3875.5.7 Differentiation in the s -domain 3885.5.8 Conjugation Property 3895.5.9 Time Convolution 3895.5.10 s -Domain Convolution 3905.5.11 Initial value Theorem 3905.5.12 Final Value Theorem 3915.5.13 Time Reversal Property 391

5.6 ANALYSIS OF CONTINUOUS LTI SYSTEMS USING LAPLACE TRANSFORM 393

5.6.1 Response of LT I Continuous Time System 393

5.6.2 Impulse Response and Transfer Function 3945.7 STABILITY AND CAUSALITY OF CONTINUOUS LTI SYSTEM USING LAPLACE TRANSFORM 394

5.7.1 Causality 3955.7.2 Stability 3955.7.3 Stability and Causality 395

5.8 SYSTEM FUNCTION FOR INTERCONNECTED LTI SYSTEMS 395

5.8.1 Parallel Connection 3955.8.2 Cascaded Connection 3965.8.3 Feedback Connection 396

5.9 BLOCK DIAGRAM REPRESENTATION OF CONTINUOUS LTI SYSTEM 397

5.9.1 Direct Form I structure 3975.9.2 Direct Form II structure 3995.9.3 Cascade Structure 4015.9.4 Parallel Structure 402

EXERCISE 5.1 404

EXERCISE 5.2 417

EXERCISE 5.3 422

EXERCISE 5.4 426

SOLUTIONS 5.1 442

SOLUTIONS 5.2 461

SOLUTIONS 5.3 473

SOLUTIONS 5.4 474

CHAPTER 6 THE Z-TRANSFORM

6.1 INTRODUCTION 493

6.1.1 The Bilateral or Two-Sided z -transform 493


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6.1.2 The Unilateral or One-sided z -transform 494

6.2 EXISTENCE OF z -TRANSFORM 494

6.3 REGION OF CONVERGENCE 494

6.3.1 Poles and Zeros of Rational z -transforms 4946.3.2 Properties of ROC 495

6.4 THE INVERSE z -TRANSFORM 500

6.4.1 Partial Fraction Method 5026.4.2 Power Series Expansion Method 503

6.5 PROPERTIES OF z -TRANSFORM 503

6.5.1 Linearity 5036.5.2 Time Shifting 5046.5.3 Time Reversal 5056.5.4 Differentiation in the z -domain 5066.5.5 Scaling in z -Domain 5066.5.6 Time Scaling 5066.5.7 Time Differencing 5076.5.8 Time Convolution 5086.5.9 Conjugation Property 5086.5.10 Initial Value Theorem 5096.5.11 Final Value Theorem 509

6.6 ANALYSIS OF DISCRETE LTI SYSTEMS USING z -TRANSFORM 5116.6.1 Response of LT I Continuous Time System 5116.6.2 Impulse Response and Transfer Function 512

6.7 STABILITY AND CAUSALITY OF LTI DISCRETE SYSTEMS USING z -TRANSFORM 5136.7.1 Causality 513

6.7.2 Stability 5136.7.3 Stability and Causality 514


6.8.1 Direct Form I Realization 5156.8.2 Direct Form II Realization 5166.8.3 Cascade Form 5176.8.4 Parallel Form 518

6.9 RELATIONSHIP BETWEEN s -PLANE & z -PLANE 518

EXERCISE 6.1 520

EXERCISE 6.2 536

EXERCISE 6.3 538

EXERCISE 6.4 541

SOLUTIONS 6.1 554

SOLUTIONS 6.2 580

SOLUTIONS 6.3 586

SOLUTIONS 6.4 587


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CHAPTER 7 THE CONTINUOUS TIME FOURIER TRANSFORM

7.1 DEFINITION 607

7.1.1 Magnitude and Phase Spectra 6077.1.2 Existence of Fourier transform 6077.1.3 Inverse Fourier Transform 608

7.2 SPECIAL FORMS OF FOURIER TRANSFORM 608

7.2.1 Real-valued Even Symmetric Signal 6087.2.2 Real-valued Odd Symmetric Signal 6107.2.3 Imaginary-valued Even Symmetric Signal 6107.2.4 Imaginary-valued Odd Symmetric Signal 611

7.3 PROPERTIES OF FOURIER TRANSFORM 612

7.3.1 Linearity 6127.3.2 Time Shifting 6127.3.3 Conjugation and Conjugate Symmetry 6127.3.4 Time Scaling 6137.3.5 Differentiation in Time-Domain 6147.3.6 Integration in Time-Domain 6147.3.7 Differentiation in Frequency Domain 6157.3.8 Frequency Shifting 6157.3.9 Duality Property 6157.3.10 Time Convolution 6167.3.11 Frequency Convolution 6167.3.12 Area Under ( )x t 6177.3.13 Area Under ( )X j w 6177.3.14 Parseval’s Energy Theorem 6187.3.15 Time Reversal 6187.3.16 Other Symmetry Properties 619

7.4 ANALYSIS OF LTI CONTINUOUS TIME SYSTEM USING FOURIER TRANSFORM 620

7.4.1 Transfer Function & Impulse Response of LT I Continuous System 6207.4.2 Response of LT I Continuous system using Fourier Transform 620

7.5 RELATION BETWEEN FOURIER AND LAPLACE TRANSFORM 621

EXERCISE 7.1 622

EXERCISE 7.2 634

EXERCISE 7.3 641

EXERCISE 7.4 645

SOLUTIONS 7.1 658

SOLUTIONS 7.2 672

SOLUTIONS 7.3 688

SOLUTIONS 7.4 689

CHAPTER 8 THE DISCRETE TIME FOURIER TRANSFORM

8.1 DEFINITION 705


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8.1.1 Magnitude and Phase Spectra 7058.1.2 Existence of DT FT 7058.1.3 Inverse DT FT 705

8.2 SPECIAL FORMS OF DTFT 706

8.3 PROPERTIES OF DISCRETE-TIME FOURIER TRANSFORM 707

8.3.1 Linearity 707

8.3.2 Periodicity 7078.3.3 Time Shifting 7088.3.4 Frequency Shifting 7088.3.5 Time Reversal 7088.3.6 Time Scaling 7098.3.7 Differentiation in Frequency Domain 7108.3.8 Conjugation and Conjugate Symmetry 7108.3.9 Convolution in T ime Domain 7118.3.10 Convolution in Frequency Domain 7118.3.11 Time Differencing 712

8.3.12 Time Accumulation 7128.3.13 Parseval’s Theorem 713

8.4 ANALYSIS OF LTI DISCRETE TIME SYSTEM USING DTFT 714

8.4.1 Transfer Function & Impulse Response 7148.4.2 Response of LT I DT system using DT FT 714

8.5 RELATION BETWEEN THE DTFT & THE Z -TRANSFORM 7158.6 DISCRETE FOURIER TRANSFORM (DFT) 715

8.6.1 Inverse Discrete Fourier Transform (I DFT) 716

8.7 PROPERTIES OF DFT 716

8.7.1 Linearity 7168.7.2 Periodicity 7178.7.3 Conjugation and Conjugate Symmetry 7178.7.4 Circular T ime Shifting 7188.7.5 Circular Frequency Shift 7198.7.6 Circular Convolution 7198.7.7 Multiplication 7208.7.8 Parseval’s Theorem 7218.7.9 Other Symmetry Properties 721

8.8 FAST FOURIER TRANSFORM (FFT) 722

EXERCISE 8.1 724

EXERCISE 8.2 735

EXERCISE 8.3 739

EXERCISE 8.4 742

SOLUTIONS 8.1 746

SOLUTIONS 8.2 760

SOLUTIONS 8.3 769


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SOLUTIONS 8.4 770

CHAPTER 9 THE CONTINUOUS TIME FOURIER SERIES

9.1 INTRODUCTION TO CTFS 775

9.1.1 Trigonometric Fourier Series 7759.1.2 Exponential Fourier Series 778

9.1.3 Polar Fourier Series 7799.2 EXISTENCE OF FOURIER SERIES 780

9.3 PROPERTIES OF EXPONENTIAL CTFS 780

9.3.1 Linearity 7809.3.2 Time Shifting 7819.3.3 Time Reversal Property 7819.3.4 Time Scaling 7829.3.5 Multiplication 7829.3.6 Conjugation and Conjugate Symmetry 7839.3.7 Differentiation Property 783

9.3.8 Integration in Time-Domain 7849.3.9 Convolution Property 7849.3.10 Parseval’s Theorem 7859.3.11 Frequency Shifting 786

9.4 AMPLITUDE & PHASE SPECTRA OF PERIODIC SIGNAL 787

9.5 RELATION BETWEEN CTFT & CTFS 787

9.5.1 CT FT using CT FS Coefficients 7879.5.2 CTFS Coefficients as Samples of CTFT 787

9.6 RESPONSE OF AN LTI CT SYSTEM TO PERIODIC SIGNALS USING FOURIER SERIES 788

EXERCISE 9.1 790EXERCISE 9.2 804

EXERCISE 9.3 806

EXERCISE 9.4 811

SOLUTIONS 9.1 824

SOLUTIONS 9.2 840

SOLUTIONS 9.3 844

SOLUTIONS 9.4 845

CHAPTER 10 THE DISCRETE TIME FOURIER SERIES

10.1 DEFINITION 861

10.2 AMPLITUDE AND PHASE SPECTRA OF PERIODIC DT SIGNALS 861

10.3 PROPERTIES OF DTFS 861

10.3.1 Linearity 86210.3.2 Periodicity 86210.3.3 Time-Shifting 862


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10.3.4 Frequency Shift 86310.3.5 Time-Reversal 86310.3.6 Multiplication 86410.3.7 Conjugation and Conjugate Symmetry 86410.3.8 Difference Property 86510.3.9 Parseval’s Theorem 865

10.3.10 Convolution 86610.3.11 Duality 86710.3.12 Symmetry 86710.3.13 Time Scaling 867

EXERCISE 10.1 870

EXERCISE 10.2 880

EXERCISE 10.3 882

EXERCISE 10.4 884

SOLUTIONS 10.1 886

SOLUTIONS 10.2 898SOLUTIONS 10.3 903

SOLUTIONS 10.4 904

CHAPTER 11 SAMPLING AND SIGNAL RECONSTRUCTION

11.1 THE SAMPLING PROCESS 905

11.2 THE SAMPLING THEOREM 905

11.3 IDEAL OR IMPULSE SAMPLING 905

11.4 NYQUIST RATE OR NYQUIST INTERVAL 907

11.5 ALIASING 90711.6 SIGNAL RECONSTRUCTION 908

11.7 SAMPLING OF BAND-PASS SIGNALS 909

EXERCISE 11.1 911

EXERCISE 11.2 919

EXERCISE 11.3 922

EXERCISE 11.4 925

SOLUTIONS 11.1 928

SOLUTIONS 11.2 937

SOLUTIONS 11.3 941

SOLUTIONS 11.4 942

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Page 493Chap 6

The Z-Transform

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GATE STUDY PACKAGE Electronics & Communication

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6.1 INTRODUCTION

As we studied in previous chapter, the Laplace transform is an important toolfor analysis of continuous time signals and systems. Similarly, z -transformsenables us to analyze discrete time signals and systems in the z -domain.

Like, the Laplace transform, it is also classified as bilateral z -transformand unilateral z -transform.

The bilateral or two-sided z -transform is used to analyze both causaland non-causal LTI discrete systems, while the unilateral z -transform isdefined only for causal signals.

NOTE : The properties of z -transform are similar to those of the Laplace transform.

6.1.1 The Bilateral or Two-Sided z -transform

The z -transform of a discrete-time sequence [ ]x n , is defined as

( )X z { [ ]} [ ]x n x n z Z n n

= =3

3

-

=-/ (6.1.1)

Where, ( )X z is the transformed signal and Z represents the z -transformation. z is a complex variable. In polar form, z can be expressed as

z r e j = W

where r is the magnitude of z and W is the angle of z . This corresponds toa circle in z plane with radius r as shown in figure 6.1.1 below

Figure 6.1.1 z -plane

NOTE : The signal [ ]x n and its z -transform ( )X z are said to form a z -transform pair denoted as

[ ] ( )x n X z Z

CHAPTER 6THE Z-TRANSFORM


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Page 494Chap 6The Z-Transform

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6.1.2 The Unilateral or One-sided z -transform

The z -transform for causal signals and systems is referred to as the unilateralz -transform. For a causal sequence [ ]x n 0= , for 0n <

Therefore, the unilateral z -transform is defined as

( )X z [ ]x n z n n 0

=3

-

=/ (6.1.2)

NOTE :For causal signals and systems, the unilateral and bilateral z -transform are the same.

6.2 EXISTENCE OF -TRANSFORM

Consider the bilateral z -transform given by equation (6.1.1)

[ ]X z [ ]x n z n n

=3

3

-

=-/

The z -transform exists when the infinite sum in above equation converges.For this summation to be converged [ ]x n z n - must be absolutely summable.

Substituting z r e j = W

[ ]X z [ ]( )x n re j n

n =

3

3

W -

=-/or, [ ]X z { [ ] }x n r e n j n

n

=3

3

W - -

=-/

Thus for existence of z -transform ( )X z < 3

[ ]x n r n n 3

3

-

=-/ 31 (6.2.1)

6.3 REGION OF CONVERGENCE

The existence of z -transform is given from equation (6.2.1). The values of r forwhich [ ]x n r n - is absolutely summable is referred to as region of convergence.Since, z r e j = W so r z = . T herefore we conclude that the range of values ofthe variable z for which the sum in equation (6.1.1) converges is called theregion of convergence. T his can be explained through the following examples.

6.3.1 Poles and Zeros of Rational z -transforms

The most common form of z -transform is a rational function. Let ( )X z bethe z -transform of sequence [ ]x n , expressed as a ratio of two polynomials

( )N z and ( )D z .

( )X z ( )( )

D z N z =

The roots of numerator polynomial i.e. values of z for which ( ) 0X z = isreferred to as zeros of ( )X z . The roots of denominator polynomial for which( )X z 3= is referred to as poles of ( )X z . T he representation of ( )X z through

its poles and zeros in the z -plane is called pole-zero plot of ( )X z .For example consider a rational transfer function ( )X z given as

( )H z z z

z 5 62

=- +

( )( )z z z

2 3=

- -


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Page 495Chap 6

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Now, the zeros of ( )X z are roots of numerator that is z 0= and poles areroots of equation ( )( )z z 2 3 0- - = which are given as z 2= and z 3= . Thepoles and zeros of ( )X z are shown in pole-zero plot of figure 6.3.1.

Figure 6.3.1 Pole-zero plot of X z ^ hNOTE :In pole-zero plot poles are marked by a small cross ‘ # ’ and zeros are marked by a small dot‘o’ as shown in figure 6.3.1.

6.3.2 Properties of ROC The various properties of ROC are summarized as follows. These propertiescan be proved by taking appropriate examples of different DT signals.

PROPERTY 1

The ROC is a concentric ring in the z -plane centered about the origin.

PROOF :

The z -transform is defined as

( )X z [ ]x n z n n

=3

3

-

=-/

Put z r e j = W

( ) ( )X z X r e j = W [ ]x n r e n j n n

=3

3

W - -

=-/

( )X z converges for those values of z for which [ ]x n r n - is absolutely summbablethat is

[ ]x n r n n 3

3

-

=-/ < 3

Thus, convergence is dependent only on r , where, r z = The equation z r e j = W , describes a circle in z -plane. Hence the ROC willconsists of concentric rings centered at zero.

PROPERTY 2

The ROC cannot contain any poles.

PROOF :

ROC is defined as the values of z for which z -transform ( )X z converges. Weknow that ( )X z will be infinite at pole, and, therefore ( )X z does not convergeat poles. Hence the region of convergence does not include any pole.


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PROPERTY 3

If [ ]x n is a finite duration two-sided sequence then the ROC is entire z -plane except at z 0= and z 3= .

PROOF :

A sequence which is zero outside a finite interval of time is called ‘finiteduration sequence’. Consider a finite duration sequence [ ]x n shown in figure6.3.2a; [ ]x n is non-zero only for some interval N n N 1 2# # .

Figure 6.3.2a A Finite Duration Sequence

The z -transform of [ ]x n is defined as

( )X z [ ]x n z n n N

N

1

2

= -=/

This summation converges for all finite values of z . If N 1 is negative andN 2 is positive, then ( )X z will have both positive and negative powers of z .

The negative powers of z becomes unbounded (infinity) if z 0" . Similarlypositive powers of z becomes unbounded (infinity) if z " 3 . So ROC of

( )X z is entire z -plane except possible z 0= and/or z 3= .

NOTE :Both N 1 and N 2 can be either positive or negative.

PROPERTY 4

If [ ]x n is a right-sided sequence, and if the circle z r 0= is in the ROC,then all values of z for which z r > 0 will also be in the ROC.

PROOF :

A sequence which is zero prior to some finite time is called the r i ght-si dedsequence . Consider a right-sided sequence [ ]x n shown in figure 6.3.2b; that is; [ ]x n 0= for n N < 1.

Figure 6.3.2b A Right - Sided Sequence


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Let the z -transform of [ ]x n converges for some value of z z r i.e. 0=_ i. From the condition of convergence we can write [ ]x n z n

n 3

3

-

=-/ < 3

[ ]x n r n n

03

3

-

=-/ < 3

The sequence is right sided, so limits of above summation changes as

[ ]x n r n n N

01

3

-

=/ < 3 (6.3.1)Now if we take another value of z as z r 1= with r r >0

0

Thus, we conclude that if the circle z r 0= is in the ROC, then allvalues of z for which z r > 0 will also be in the ROC. The ROC of a right-sided sequence is illustrated in figure 6.3.2c.

Figure 6.3.2c ROC of a right-sided sequence

PROPERTY 5

If [ ]x n is a left-sided sequence, and if the circle z r 0= is in the ROC,then all values of z for which z r < 0 will also be in the ROC.

PROOF :

A sequence which is zero after some finite time interval is called a ‘left-sided signal’. Consider a left-sided signal [ ]x n shown in figure 6.3.2d; that is

[ ] 0x n = for n N > 2.


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Figure 6.3.2d A left-sided sequence

Let z -transform of [ ]x n converges for some values of z . .i e z r 0=_ .From the condition of convergence we write [ ]x n z n

n 3

3

-

=-/ < 3

or [ ]x n r n n

03

3

-

=-/ < 3 (6.3.4)

The sequence is left sided, so the limits of summation changes as

[ ]x n r n

n

N

0

2

3

-

=-/ <3

(6.3.5)Now if take another value of z as z r 1= , then we can write

[ ]x n z n n

N 2

3

-

=-/ [ ]x n z r r n n n

n

N

0 0

2

=3

- -

=-/

[ ]x n r z r n n

n

N

00

2

=3

-

=- a k/ (6.3.6)From equation (6.3.4), we know that [ ]x n r n 0- is absolutely summable. Let itis bounded by some value M x , then equation (6.3.6) becomes as

[ ]x n z n n

N 2

3

-

=-/ M z r x

n

n

N 0

2

#3=- a k/

The above summation converges if 1z r >0 (because n is increasing

negatively), so z r < 0 will be in ROC. The ROC of a left-sided sequence is illustrated in figure 6.3.2e.

Figure 6.3.2e ROC of a Left - Sided Sequence

PROPERTY 6

If [ ]x n is a two-sided signal, and if the circle z r 0= is in the ROC, thenthe ROC consists of a ring in the z -plane that includes the circle z r 0=


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Sample Chapter of Signals and Systems (Vol-7, GATE Study Package)PROOF :

A sequence which is defined for infinite extent for both 0n > and 0n < iscalled ‘two-sided sequence’. A two-sided signal [ ]x n is shown in figure 6.3.2f.

Figure 6.3.2f A Two - Sided Sequence

For any time N 0, a two-sided sequence can be divided into sum of left-sided and right-sided sequences as shown in figure 6.3.2g.

Figure 6.3.2g A Two Sided Sequence Divided into Sum of a Left - Sided and Right - SidedSequence

The z -transform of [ ]x n converges for the values of z for which thetransform of both [ ]x n R and [ ]x n L converges. From property 4, the ROC of aright-sided sequence is a region which is bounded on the inside by a circle andextending outward to infinity i.e. | |z r > 1. From property 5, the ROC of a leftsided sequence is bounded on the outside by a circle and extending inward tozero i.e. | |z r < 2. So the ROC of combined signal includes intersection of bothROCs which is ring in the z -plane.

The ROC for the right-sided sequence [ ]x n R , the left-sequence [ ]x n L andtheir combination which is a two sided sequence [ ]x n are shown in figure6.3.2h.


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Figure 6.3.2h ROC of a left-sided sequence, a right-sided sequence and two sided sequence

PROPERTY 7

If the z -transform ( )X z of [ ]x n is rational, then its ROC is bounded bypoles or extends to infinity.

PROOF :

The exponential DT signals also have rational z -transform and the poles of( )X z determines the boundaries of ROC.

PROPERTY 8

If the z -transform ( )X z of [ ]x n is rational and [ ]x n is a right-sided sequencethen the ROC is the region in the z -plane outside the outermost polei.e. ROC is the region outside a circle with a radius greater than themagnitude of largest pole of ( )X z .

PROOF :

This property can be be proved by taking property 4 and 7 together.

PROPERTY 9

If the z -transform ( )X z of [ ]x n is rational and [ ]x n is a left-sided sequencethen the ROC is the region in the z -plane inside the innermost pole i.e.ROC is the region inside a circle with a radius equal to the smallestmagnitude of poles of ( )X z .

PROOF :

This property can be be proved by taking property 5 and 7 together.

-Transform of Some Basic Functions

Z-transform of basic functions are summarized in the Table 6.1 with theirrespective ROCs.

6.4 THE INVERSE -TRANSFORM

Let ( )X z be the z -transform of a sequence [ ]x n . To obtain the sequence[ ]x n from its z -transform is called the inverse z -transform. The inverse z

-transform is given as


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TABLE 6.1 : z -Transform of Basic Discrete Time Signals

DT sequence [ ]x n z -transform ROC

1. [ ]n d 1 entire z -plane

2. [ ]n n 0d -z n 0- entire z -plane,

except z 0=

3. [ ]u n z z z

11

11- =

-- 1z >

4. [ ]u n n a z z z

11

1a a- =

-- z > a

5. [ 1]u n n 1a -- z z

z 11

1

1

a a- =

---

z > a

6. [ ]nu n ( ) ( )z z

z z

1 11 21

2- =

---

1z >

7. [ ]n u n n a ( ) ( )z z

z z

1 1 21

2aa

aa

- =

---

z > a

8. ( ) [ ]cos n u n 0W cos

cosz z

z 1 2

11

02

10

W W

- +-

- -

-

or

[ ]coscos

z z z z

2 12 00

W W

- +- 1z >

9. ( ) [ ]sin n u n 0W

cossin

z z z

1 2 1 0 21

0

W W

- +- --

or

cossin

z z z 2 12 0

0

W W

- +1z >

10. ( ) [ ]cos n u n n 0a W cos

cosz z

z 1 2

11

02 2

10

a aa

W W

- +-

- -

-

or [ ]cos

cosz z

z z 22 0 2

0

a aa

W W

- +- z > a

11. ( ) [ ]sin n u n n 0a W cos

sinz z

z 1 2 1 0 2 2

10

a aa

W W

- +- --

or cossin

z z z

22 0 20

a aa

W W

- +

z > a

12. ( ) [ ]sinr n u n n

0a qW + with R !a

z z A Bz

1 2 1 2 21

g a+ ++

- -

-

or ( )z z

z A z B 22 2g g + +

+ z ( )a # a


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[ ]x n ( ) j X z z dz 21 n 1p

= - # This method involves the contour integration, so difficult to solve. There

are other commonly used methods to evaluate the inverse z -transform givenas follows1. Partial fraction method2. Power series expansion

6.4.1 Partial Fraction Method

If ( )X z is a rational function of z then it can be expressed as follows.

( )X z ( )( )

D z N z =

It is convenient if we consider ( )/X z z rather than ( )X z to obtain theinverse z -transform by partial fraction method.

Let p 1, p 2, p 3.... p n are the roots of denominator polynomial, also thepoles of ( )X z . T hen, using partial fraction method ( )/X z z can be expressedas

( )z X z ...z p

Az p

Az p

Az p

An

n

1

1

2

2

3

3=-

+-

+-

+ +-

( )X z ...A z p z A z p

z z p

z n

11

22

= - + - + + -

Now, the inverse z -transform of above equation can be obtained by comparingeach term with the standard z -tranform pair given in table 6.1. T he values ofcoefficients A 1, A 2, A 3.... A n depends on whether the poles are real & distinctor repeated or complex. Three cases are given as follows

Case I : Poles are Simple and Real

( )/X z z can be expanded in partial fraction as

( )z X z ...z p

Az p

Az p

Az p

An

n

1

1

2

2

3

3= - + - + - + + - (6.4.1)

where A 1, A 2,... A n are calculated as follows

A 1 ( ) ( )z p z X z

z p 1

1

= -=

A 2 ( )( )

z p z X z

z p 2

2

= -=

In general, A i ( ) ( )z p X z i

z p i = -

= (6.4.2)

Case II : If Poles are RepeatedIn this case ( )/X z z has a different form. Let p k be the root which repeats l times, then the expansion of equation must include terms

( )z X z

( )

...z p A

z p A

k

k

k

k 12

2=-

+-

+

( )

...( )z p

Az p

Ak

i ik

k l

lk +-

+ +-

(6.4.3)

The coefficient A ik are evaluated by multiplying both sides of equation (6.4.3)by ( )z p k l - , differentiating ( )l i - times and then evaluating the resultantequation at z p k = .

Thus,

C ik ( )

( ) ( )l i dz

d z p z X z 1

l i

l i

k l

z p k

=-

---

=: D (6.4.4)


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Case III : Complex PolesIf ( )X z has complex poles then partial fraction of the ( )/X z z can be expressedas

( )z X z z p

Az p

A1

1

1

1=-

+- )

)

(6.4.5)

where A 1) is complex conjugate of A 1 and p 1

) is complex conjugate of z 1. Thecoefficients are obtained by equation (6.4.2)

6.4.2 Power Series Expansion Method

Power series method is also convenient in calculating the inverse z -transform. The z -transform of sequence [ ]x n is given as

( )X z [ ]x n z n n

=3

3

-

=-/

Now, ( )X z is expanded in the following form

( ) .. [ 2] [ 1] [0] [1] [2] ...X z x z x z x x z x z 2 1 1 2= + - + - + + + +- -

To obtain inverse z -transform(i.e. [ ]x n ), represent the given ( )X z in theform of above power series. T hen by comparing we can get [ ]x n {... [ 2], [ 1], [0], [1], [2], ...}x x x x x = - -

6.5 PROPERTIES OF -TRANSFORM

The unilateral and bilateral z -transforms possess a set of properties, whichare useful in the analysis of DT signals and systems. The proofs of propertiesare given for bilateral transform only and can be obtained in a similar wayfor the unilateral transform.

6.5.1 Linearity

Like Laplace transform, the linearity property of z transform states that,the linear combination of DT sequences in the time domain is equivalent to

linear combination of theirz transform.

Let [ ]x n 1 ( )X z 1Z

, with ROC: R 1and [ ]x n 2 ( )X z 2

Z , with ROC: R 2

then, [ ] [ ]ax n bx n 1 2+ ( ) ( )aX z bX z 1 2Z

+ ,with ROC: at least R R 1 2+

for both unilateral and bilateral z -transform.

PROOF :

The z -transform of signal { [ ] [ ]}ax n bx n 1 2+ is given by equation (6.1.1) asfollows

{ [ ] [ ]}ax n bx n Z 1 2+ { [ ] [ ]}ax n bx n z n

n 1 2= +

3

3

=-

-/ [ ] [ ]a x n z b x n z n

n n

n 1 2= +

3

3

3

3

-

=- =-

-/ / ( ) ( )aX z bX z 1 2= +

Hence, [ ] [ ]ax n bx n 1 2+ ( ) ( )aX z bX z 1 2Z

+

ROC : Since, the z -transform ( )X z 1 is finite within the specified ROC, R 1.


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Similarly, ( )X z 2 is finite within its ROC, R 2. T herefore, the linear combination( ) ( )aX z bX z 1 2+ should be finite at least within region R R 1 2+ .

NOTE :In certain cases, due to the interaction between [ ] x n 1 and [ ]x n 2 , which may lead to cancellationof certain terms, the overall ROC may be larger than the intersection of the two regions.On the other hand, if there is no common region between R 1 and R 2 , the z-transform of

[ ] [ ]ax n bx n 1 2+ does not exist.

6.5.2 Time Shifting

For the bilateral z -transformIf [ ]x n ( )X z Z , with ROC R x then [ ]x n n 0- ( )z X z n

Z 0- ,

and [ ]x n n 0+ ( )z X z n Z 0 ,

with ROC : R x except for the possible deletion or addition of z 0= orz 3= .

PROOF :

The bilateral z -transform of signal [ ]x n n 0- is given by equation (6.1.1) asfollows

{ [ ]}x n n Z 0- [ ]x n n z n n

0= -3

3

-

=-/

Substituting n n 0 a- = on RHS, we get

{ [ ]}x n n Z 0- [ ]x z ( )n 0a=3

3

a

a

- +

=-/

[ ]x z z n 0a=3

3

a

a

- -

=-/ [ ]z x z n 0 a=

3

3

a

a

- -

=-/

{ [ ]}x n n Z 0- [ ]z X z n 0= -

Similarly we can prove{ [ ]}x n n Z 0+ [ ]z X z n 0=

ROC : T he ROC of shifted signals is altered because of the terms orz z n n 0 0- ,which affects the roots of the denominator in ( )X z .

TIME SHIFTING FOR UNILATERAL z -TRANSFORM

For the unilateral z -transformIf [ ]x n ( )X z

Z , with ROC R x

then [ ]x n n 0- ( ) [ ]z X z x m z n m m

n

1

Z 00

+ --

=e o/ ,and [ ]x n n 0+ ( ) [ ]z X z x m z n m

m

n

0

1Z 00

- -

=

-

e o/ ,with ROC : R x except for the possible deletion or addition of z 0= orz 3= .

PROOF :

The unilateral z -transform of signal [ ]x n n 0- is given by equation (6.1.2) asfollows


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{ [ ]}x n n Z 0- [ ]x n n z n n

00

= -3

-

=/

Multiplying RHS by andz z n n 0 0-

{ [ ]}x n n Z 0- [ ]x n n z z z n n n n

00

0 0= -3

- -

=/

[ ]z x n n z ( )n n n n

00

0 0= -3

- - -

=/

Substituting n n 0- a=Limits; 0,when n " n 0"a - ,when n " 3+ " 3a +

Now, { [ ]}x n n Z 0- [ ]z x z n n

0

0

a=3

a

a

- -

=-/

[ ] [ ]z x z z x z n n

n 1

0

0

0

0a a= +3

a

a

a

a

- -

=-

-- -

=/ /

or, { [ ]}x n n Z 0- [ ] [ ]z x z z x z n n n 0

10 0

0

a a= +3

a

a

a

a

- -

=

- -

=-

-

/ /or, { [ ]}x n n Z 0- [ ] [ ]z x z z x z n n

n

0 1

0 0

0

a a= + -3

a

a

a

a

- -

=

-

=/ /

Changing the variables as n "a and m "a in first and second summationrespectively

{ [ ]}x n n Z 0- [ ] [ ]z x n z z x m z n n n m m

n

n 10

0 0

0

= + -3

- - -

==//

[ ] [ ]z X z z x m z n n m m

n

1

0 0

0

= + -- -

=/

In similar way, we can also prove that

[ ]x n n 0+ ( ) [ ]z X z x m z n m m

n

0

1Z 0

0

- -

=

-

e o/6.5.3 Time Reversal

Time reversal property states that time reflection of a DT sequence in timedomain is equivalent to replacing z by 1/ z in its z -transform.

If [ ]x n ( )X z Z

, with ROC : R x

then [ ]x n - 1X z Z b l, with ROC : / R 1 x

for bilateral z -transform.

PROOF :

The bilateral z -transform of signal [ ]x n - is given by equation (6.1.1) asfollows

{ [ ]}x n Z - [ ]x n z n n

= -3

3

-

=-/Substituting n k - = on the RHS, we get

{ [ ]}x n Z - [ ]x k z k k

=3

3

=

-

/ [ ]( )x k z k

k 1=3

3

=-

- -/ X z 1= b lHence, [ ]x n - 1X z

Z b lROC : /orz R z R 1x x 1 ! !-


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6.5.4 Differentiation in the z -domain

This property states that multiplication of time sequence [ ]x n with n corresponds to differentiation with respect to z and multiplication of resultby z - in the z -domain.

If [ ]x n ( )X z Z , with ROC : R x

then [ ]nx n ( )

z dz dX z Z

- , with ROC : R x For both unilateral and bilateral z -transforms.

PROOF :

The bilateral z -transform of signal [ ]x n is given by equation (6.1.1) as follows

( )X z [ ]x n z n

n =3

3

=-

-/Differentiating both sides with respect to z gives

( )dz dX z [ ] [ ]( )x n dz

dz x n nz n

n n

n

1= = -3

3

3

3

=-

-- -

=-/ /

Multiplying both sides by z - , we obtain

( )z dz dX z - [ ]nx n z n

n

=3

3

-

=-/

Hence, [ ]nx n ( )z dz dX z Z -

ROC : T his operation does not affect the ROC.

6.5.5 Scaling in z -Domain

Multiplication of a time sequence with an exponential sequence a n correspondsto scaling in z -domain by a factor of a .

If [ ]x n ( )X z Z , with ROC : R x

then [ ]a x n n X a z Z a k, with ROC : a R x for both unilateral and bilateral transform.

PROOF :

The bilateral z -transform of signal [ ]x n is given by equation (6.1.1) as

{ [ ]}a x n Z n [ ]a x n z n n n

=3

3

-

=-/ [ ][ ]x n a z n

n

1=3

3

- -

=-/

[ ]a x n n X a z Z a k

ROC : If z is a point in the ROC of ( )X z then the point a z is in the ROC

of ( / )X z a .

6.5.6 Time Scaling

As we discussed in Chapter 2, there are two types of scaling in the DTdomain decimation(compression) and interpolation(expansion).

Time CompressionSince the decimation (compression) of DT signals is an irreversible process(because some data may lost), therefore the z -transform of [ ]x n and its


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decimated sequence [ ] [ ]y n x an = not be related to each other.

Time ExpansionIn the discrete time domain, time expansion of sequence [ ]x n is defined as

[ ]x n k [ / ] intif isamultipleof eger

otherwisex n k n k

0= ) (6.5.1)

Time-scaling property of z -transform is derived only for time expansionwhich is given as

If [ ]x n ( )X z Z

, with ROC : R x

then [ ]x n k ( ) ( )X z X z k k Z

= , with ROC : ( )R /x k 1 for both the unilateral and bilateral z -transform.

PROOF :

The unilateral z -transform of expanded sequence [ ]x n k is given by

{ [ ]}x n Z k [ ]x n z k n n 0

=3

-

=/

[0] [1] ... [ ]x x z x k z k k k k 1= + + +- -

[ 1] ... [2 ] ...x k z x k z ( )k k k k 1 2+ + + +- + -

Since the expanded sequence [ ]x n k is zero everywhere except when n is amultiple of k . T his reduces the above transform as follows { [ ]}x n Z k [0] [ ] [2 ] [3 ] ...x x k z x k z x k z k k k k k k k 2 3= + + + +- - -

As defined in equation 6.5.1, interpolated sequence is [ ]x n k [ / ]x n k =n 0= [ ]x 0k [ ]x 0= ,n k = [ ]x k k [1]x =n k 2= [ ]x k 2k [ ]x 2=

Thus, we can write { [ ]}x n Z k [0] [1] [2] [3] ...x x z x z x z k k k 2 3= + + + +- - -

[ ]( ) ( )x n z X z k n k n 0

= =3

-

=/

NOTE : T ime expansion of a DT sequence by a factor of k corresponds to replacing z as z k in its z -transform.

6.5.7 Time Differencing

If [ ]x n ( )X z Z

, with ROC : R x

then [ ] [ ]x n x n 1- - (1 ) ( )z X z 1Z - - , with the ROC: R x except for the possible deletion of z 0= , for both unilateral andbilateral transform.

PROOF :

The z -transform of [ ] [ ]x n x n 1- - is given by equation (6.1.1) as follows

{ [ ] [ 1]}x n x n Z - - { [ ] [ ]}x n x n z 1 n n

= - -3

3

-

=-/


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[ ] [ ]x n z x n z 1n n

n

n

= - -3

3

3

3

-

=-

-

=-/ /

In the second summation, substituting n r 1- =

{ [ ] [ 1]}x n x n Z - - [ ] [ ]x n z x r z ( )n n

r

r

1= -3

3

3

3

-

=-

- +

=-/ /

[ ] [ ]x n z z x r z n n

r

r

1= -3

3

3

3

-

=-

- -

=-/ /

( ) ( )X z z X z 1= - -

Hence, [ ] [ ]x n x n 1- - (1 ) ( )z X z 1Z

- -

6.5.8 Time Convolution

Time convolution property states that convolution of two sequence in timedomain corresponds to multiplication in z -domain.

Let [ ]x n 1 ( )X z 1Z , ROC : R 1

and [ ]x n 2 ( )X z 2Z

, ROC : R 2then the convolution property states that

[ ] [ ]x n x n 1 2* ( ) ( )X z X z 1 2Z , ROC : at least R R 1 2+

for both unilateral and bilateral z -transforms.

PROOF :

As discussed in chapter 4, the convolution of two sequences is given by

[ ] [ ]x n x n 1 2* [ ] [ ]x k x n k k

1 2= -3

3

=-/

Taking the z -transform of both sides gives

[ ] [ ]x n x n 1 2* [ ] [ ]x k x n k z n

k n 1 2

Z -

3

3

3

3

-

=-=-//

Interchanging the order of the two summations, we get

[ ] [ ]x n x n 1 2* [ ] [ ]x k x n k z k n

n 1 2

Z -

3

3

3

3

=- =-

-/ /Substituting n k a- = in the second summation

[ ] [ ]x n x n 2* [ ] [ ]x k x z ( )

k

k 1 2

Z a

3

3

3

3

a

a=-

- +

=-/ /

or [ ] [ ]x n x n 2* [ ] [ ]x k z x z k

k 1 2

Z a

3

3

3

3

a

a

-

=-

-

=-e e o o/ / [ ] [ ]x n x n 1 2* ( ) ( )X z X z 1 2

Z

6.5.9 Conjugation Property

If [ ]x n ( )X z Z

, with ROC : R x then [ ]x n ) ( )X z

Z ) ) , with ROC : R x

If [ ]x n is real, then ( )X z ( )X z = ) )


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Sample Chapter of Signals and Systems (Vol-7, GATE Study Package)PROOF :

The z -transform of signal [ ]x n ) is given by equation (6.1.1) as follows

{ [ ]}x n Z ) [ ]x n z n n

= )3

3

-

=-/ [ ]( )x n z n

n

= ) )

3

3

-

=- 6 @/ (6.5.2)Let z -transform of [ ]x n is ( )X z

( )X z [ ]x n z n n

=3

3

-

=-/

by taking complex conjugate on both sides of above equation ( )X z ) [ [ ] ]x n z n

n

= )3

3

-

=-/

Replacing z z " ) , we will get

( )X z ) ) [ ]( )x n z n n

= ))

3

3

-

=- 6 @/ (6.5.3)Comparing equation (6.5.2) and (6.5.3) { [ ]}x n Z ) ( )X z = ) ) (6.5.4)

For real [ ]x n , [ ]x n ) [ ]x n = , so

{ [ ]}x n Z ) [ ] ( )x n z X z n n

= =3

3

-

=-/ (6.5.5)

Comparing equation (6.5.4) and (6.5.5) ( )X z ( )X z = ) )

6.5.10 Initial Value Theorem

If [ ]x n ( )X z Z

, with ROC : R x then initial-value theorem states that,

[ ]x 0 ( )lim X z z

=" 3

The initial-value theorem is valid only for the unilateral Lapalce transform

PROOF :

For a causal signal [ ]x n

( )X z [ ]x n z n n 0

=3

-

=/

[0] [1] [2] ...x x z x z 1 2= + + +- -

Taking limit as z " 3 on both sides we get

( )lim X z z " 3

( [0] [1] [2] ...)lim x x z x z z

1 2= + + +" 3

- - [ ]x 0=

[ ]x 0 ( )lim X z z

=" 3

6.5.11 Final Value Theorem

If [ ]x n ( )X z Z , with ROC : R x then final-value theorem states that

[ ]x 3 ( 1) ( )lim z X z z 1

= -"

Final value theorem is applicable if ( )X z has no poles outside the unitcircle. This theorem can be applied to either the unilateral or bilateral z -transform.


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PROOF :

{ [ 1]} { [ ]}x n x n Z Z + - { [ ] [ ]}lim x n x n z 1k

n

k n

0

= + -" 3

=

-/(6.5.6)

From the time shifting property of unilateral z -transform discussed in section6.5.2

[ ]x n n 0+ ( ) [ ]z X z x m z n m m

n

0

1Z 0

0

- -

=

-

e o/For n 10 = [ ]x n 1+ ( ) [ ]z X z x m z m m 0

0Z - -

=e o/ [ ]x n 1+ ( ) [ ]z X z x 0Z -^ hPut above transformation in the equation (6.5.6) [ ] [0] [ ]zX z zx X z - - ( [ ] [ ])lim x n x n z 1

k

n

n

k

0

= + -" 3

-

=/

( 1) [ ] [0]z X z zx - - ( [ ] [ ])lim x n x n z 1k

n

n

k

0

= + -" 3

-

=/

Taking limit as z 1" on both sides we get

( 1) [ ] [0]lim z X z x z 1

- -"

[ ] [ ]lim x n x n 1k n

k

0

= + -" 3 =/( 1) [ ] [0]lim z X z x

z 1- -

" {( [1] [0]) ( [ ] [ ]) ( [ ] [ ]) ...lim x x x x x x 2 1 3 2

k = - + - + - +

" 3

... ( [ ] [ ])x k x k 1+ + - ( 1) [ ] [0]lim z X z x z 1

- -"

[ ] [ ]x x 03= -

Hence, [ ]x 3 ( 1) ( )lim z X z z 1

= -"

Summary of Properties

Let, [ ]x n ( )X z Z

, with ROC R x [ ]x n 1 ( )X z 1

Z , with ROC R 1 [ ]x n 2 ( )X z 2

Z , with ROC R 2

The properties of z -transforms are summarized in the following table. TABLE 6.2 Properties of z -transform

Properties Time domain z -transform ROC

Linearity [ ] [ ]ax n bx n 1 2+ ( ) ( )aX z bX z 1 2+ at least R R 1 2+

Time shifting(bilateral ornon-causal)

[ ]x n n 0- ( )z X z n 0- R x except for thepossible deletionor addition of

0orz z 3= =[ ]x n n 0+ ( )z X z n 0

Time shifting(unilateral orcausal)

[ ]x n n 0-( )z X z n 0-

[̂ ]x m z m m

n

1

0

+ -= o/ R x except for thepossible deletion

or addition of0orz z 3= =

[ ]x n n 0+( )z X z n 0^[ ]x m z m

m

n

0

10- -

=

-

o/


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Properties Time domain z -transform ROC

Time reversal [ ]x n - X z 1b l / R 1 x

Differentiationin z domain [ ]

nx n ( )z dz dX z - R x

Scaling in z domain [ ]a x n

n X a

z a k a R x Time scaling(expansion)

[ ] [ / ]x n x n k k = ( )X z k ( )R /x k 1

Timedifferencing [ ] [ ]

x n x n 1- - ( ) ( )z X z 1 1- -R x , except for thepossible deletion ofthe origin

Timeconvolution [ ] [ ]

x n x n 1 2* ( ) ( )X z X z 1 2 at least R R 1 2+

Conjugations [ ]x n ) ( )X z ) ) R x

Initial-valuetheorem [ ] ( )lim

x X z 0z

=" 3

provided [ ]x n 0= for n 0<

Final-valuetheorem

[ ][ ]

( ) ( )

lim

lim

x

x n

z X z 1n

z 1

3

=

= -"

"

3provided [ ]x 3 exists

6.6 ANALYSIS OF DISCRETE LTI SYSTEMS USING z -TRANSFORM

The z -transform is very useful tool in the analysis of discrete LT I system. Asthe Laplace transform is used in solving differential equations which describecontinuous LTI systems, the z -transform is used to solve difference equationwhich describe the discrete LTI systems.

Similar to Laplace transform, for CT domain, the z -transform givestransfer function of the LT I discrete systems which is the ratio of the z -transform of the output variable to the z -transform of the input variable.

These applications are discussed as follows

6.6.1 Response of LTI Continuous Time System

As discussed in chapter 4 (section 4.8), a discrete-time LT I system is always

described by a linear constant coefficient difference equation given as follows [ ]a y n k k

k

N

0

-=/ [ ]b x n k k

k

M

0

= -=/

[ ] [ ( 1)] ....... [ 1] [ ]a y n N a y n N a y n a y n N N 1 1 0- + - - + + - +- [ ] [ ( 1)] ..... [ 1] [ ]b x n M b x n M b x n b x n M M 1 1 0= - + - - + + - +- (6.6.1)

where, N is order of the system. The time-shift property of z -transform [ ] ( )x n n z X z n 0

Z 0- - , is used

to solve the above difference equation which converts it into an algebraicequation. By taking z -transform of above equation


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( ) ( )] ....... ( )a z Y z a z Y z a z a Y z ( )N N N N 1 1 1 1 0+ + + +- - - - -

( ) ( ) ..... ( ) ( )b z X z b z X z b z X x b X z ( )M M M M 1 1 1 1 0= + + + +- - - - -

( )( )

X z Y z

..........

a z a z a a b z b z b b

N N

N N

M N

M M

11

1 0

11

1 0=+ + + ++ + + +

--

--

-

this equation can be solved for ( )Y z to find the response [ ]y n .T he solution ortotal response [ ]y n consists of two parts as discussed below.

1. Zero-input Response or Free Response or Natural Response The zero input response [ ]y n zi is mainly due to initial output in the system. The zero-input response is obtained from system equation (6.6.1) when input

[ ] 0x n = .By substituting [ ] 0x n = and [ ] [ ]y n y n zi = in equation (6.6.1), we get [ ] [ ( 1)] ....... [ 1] [ ]a y n N a y n N a y n a y n N N 1 1 0- + - - + + - +- 0=

On taking z -transform of the above equation with given initial conditions,we can form an equation for ( )Y z zi . T he zero-input response [ ]y n zi is given byinverse z -transform of ( )Y z zi .

NOTE : The zero input response is also called the natural response of the system and it is denotedas [ ]y n N .

2. Zero-State Response or Forced Response The zero-state response [ ]y n zs is the response of the system due to inputsignal and with zero initial conditions. T he zero-state response is obtainedfrom the difference equation (6.6.1) governing the system for specific inputsignal [ ]x n for 0n $ and with zero initial conditions.Substituting [ ] [ ]y n y n zs = in equation (6.6.1) we get,

[ ] [ ( 1)] ....... [ 1] [ ]a y n N a y n N a y n a y n N zs N zs zs zs 1 1 0- + - - + + - +- [ ] [ ( 1)] ..... [ 1] [ ]b x n M b x n M b x n b x n M M 1 1 0= - + - - + + - +-

Taking z -transform of the above equation with zero initial conditions foroutput (i.e., [ ] [ ]...y y 1 2 0- = - = we can form an equation for ( )Y z zs .

The zero-state response [ ]y n zs is given by inverse z -transform of ( )Y z zs .

NOTE : The zero state response is also called the forced response of the system and it is denoted as

[ ]y n F .

Total Response

The total response [ ]y n is the response of the system due to input signal andinitial output. T he total response can be obtained in following two ways :

Taking z -transform of equation (6.6.1) with non-zero initial conditionsfor both input and output, and then substituting for ( )X z we can forman equation for ( )Y z . T he total response [ ]y n is given by inverse Laplacetransform of ( )Y s .

Alternatively, that total response [ ]y n is given by sum of zero-inputresponse [ ]y n zi and zero-state response [ ]y n zs .

Therefore total response, [ ]y n [ ] [ ]y n y n zi zs = +

6.6.2 Impulse Response and Transfer Function

System function or transfer function is defined as the ratio of the z -transformof the output [ ]y n and the input [ ]x n with zero initial conditions.Let [ ] ( )x n X z

Z is the input and [ ] ( )y n Y z

L is the output of an LT I

discrete time system having impulse response ( ) ( )h n H z L . T he response


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[ ]y n of the discrete time system is given by convolution sum of input andimpulse response as [ ]y n [ ] [ ]x n h n = *By applying convolution property of z -transform we obtain

( )Y z ( ) ( )X z H z =

( )H z ( )( )

X z Y z =

where, ( )H z is defined as the transfer function of the system. I t is the z -transform of the impulse response.

Alternatively we can say that the inverse z -transform of transfer functionis the impulse response of the system.Impulse response

[ ]h n { ( )} ( )( )

H z X z Y z Z Z 1 1= =- - ) 3

6.7 STABILITY AND CAUSALITY OF LTI DISCRETE SYSTEMS USING z-TRANSFORM

z -transform is also used in characterization of LT I discrete systems. In thissection, we derive a z -domain condition to check the stability and causalityof a system directly from its z -transfer function.

6.7.1 Causality

A linear time-invariant discrete time system is said to be causal if the impulseresponse [ ]h n 0= , for n 0< and it is therefore right-sided. T he ROC of sucha system ( )H z is the exterior of a circle. I f ( )H z is rational then the systemis said to be causal if

1. The ROC is the exterior of a circle outside the outermost pole ; and2. The degree of the numerator polynomial of ( )H z should be less than

or equal to the degree of the denominator polynomial.

6.7.2 Stability

An LT I discrete-time system is said to be BIBO stable if the impulse response[ ]h n is summable. T hat is

[ ]h n n 3

3

=-/ < 3

z -transform of [ ]h n is given as

( )H z [ ]h n z n n

=3

3

-

=-/

Let z e j

= W

(which describes a unit circle in the z -plane), then [ ]H e j W [ ]h n e j n

n

=3

3

W -

=-/

[ ]h n e j n n

#3

3

W -

=-/

[ ]h n <n

3=3

3

=-/

which is the condition for the stability. Thus we can conclude that


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STABILITY OF LTI DISCRETE SYSTEM

An LT I system is stable if the ROC of its system function ( )H z containsthe unit circle z 1=

6.7.3 Stability and Causality

As we discussed previously, for a causal system with rational transfer function( )H z , the ROC is outside the outermost pole. For the BIBO stability the

ROC should include the unit circle z 1= . Thus, for the system to be causaland stable theses two conditions are satisfied if all the poles are within theunit circle in the z -plane.

STABILITY AND CAUSALITY OF LTI DISCRETE SYSTEM

An LTI discrete time system with the rational system function ( )H z issaid to be both causal and stable if all the poles of ( )H z lies inside theunit circle.

6.8 BLOCK DIAGRAM REPRESENTATION

In z -domain, the input-output relation of an LT I discrete time system isrepresented by the transfer function ( )H z ,which is a rational function of z ,as shown in equation

( )H z ( )( )

X z Y z =

......

a z a z a z a z a b z b z b z b z b

N N N N N

M M M M M

0 11

22

1

0 11

22

1=+ + + + ++ + + + +

- --

- --

where, N = Order of the system, M N # and a 10 = The above transfer function is realized using unit delay elements, unit

advance elements, adders and multipliers. Basic elements of block diagramwith their z -domain representation is shown in table 6.3.

TABLE 6.3 : Basic Elements of Block Diagram

Elementsof Blockdiagram

Time Domain Representation s -domain Representation

Adder

Constantmultiplier

Unit delayelement

Unitadvanceelement


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The different types of structures for realizing discrete time systems are sameas we discussed for the continuous-time system in the previous chapter.

6.8.1 Direct Form I Realization

Consider the difference equation governing the discrete time system witha 10 = ,

[ ] [ 1] [ 2] .... [ ]y n a y n a y n a y n N N 1 2+ - + - + + - [ ] [ 1] [ 2] ... [ ]b x n b x n b x n b x n M M 0 1 2= + - + - + + - Taking Z transform of the above equation we get,

( )Y z ( ) ( ) ... ( )a z Y z a z Y z a z Y z N N 1 1 2 2=- - - - +- - -

( ) ( ) ( ) ... ( )b X z b z X z b z X z b z X z M M 0 1 1 2 2+ + + +- - - (6.8.1) The above equation of ( )Y z can be directly represented by a block

diagram as shown in figure 6.8.1a. This structure is called direct form-Istructure. T his structure uses separate delay elements for both input andoutput of the system. So, this realization uses more memory.

Figure 6.8.1a General structure of direct form-realization

For example consider a discrete LT I system which has the following

impulse response ( )H z ( )

( )X z Y z

z z z z

1 4 31 2 2

1 2

1 2= =

+ ++ +

- -

- -

( ) ( ) 3 ( ) 1 ( ) 2 ( ) ( )Y z z Y z z Y z X z z X z z X z 4 21 2 1 2+ + = + +- - - -

Comparing with standard form of equation (6.8.1), we get a 41 = , a 32 = and b 10 = , b 21 = , b 22 = . Now put these values in general structure of Directform-I realization we get


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Figure 6.8.1b

6.8.2 Direct Form II Realization

Consider the general difference equation governing a discrete LTI system[ ] [ 1] [ 2] .... [ ]y n a y n a y n a y n N N 1 2+ - + - + + -

[ ] [ 1] [ 2] ... [ ]b x n b x n b x n b x n M M 0 1 2= + - + - + + - Taking Z transform of the above equation we get,

( )Y z ( ) ( ) ... ( )a z Y z a z Y z a z Y z N N 1 1 2 2=- - - - +- - -

( ) ( ) ( ) ... ( )b X z b z X z b z X z b z X z M M

0 11

22

+ + + +- - -

It can be simplified as,( ) ... ( ) ...Y z a z a z a z X z b b z b z b z 1 N N M M 1 1 2 2 0 1 1 2 2+ + + + = + + + +- - - - - -6 6@

Let, ( )( )

X z Y z ( )

( )( )( )

X z W z

W z Y z

#=

where,

( )( )

X z W z

...a z a z a z 11

N N

11

22= + + + +- - -

(6.8.2)

( )( )

W z Y z ...b b z b z b z M M 0 1 1 2 2= + + + +- - - (6.8.3)

Equation (6.8.2) can be simplified as,( ) ( ) ( ) ... ( ) ( )W z a z W z a z W z a z W z X z N N 1 1 2 2+ + + + =- - -

( ) ( ) ( ) ( ) ... ( )W z X z a z W z a z W z a z W z N N

11

22

= - - - -- - -

(6.8.4)Similarly by simplifying equation (6.8.3), we get( ) ( ) ( ) ( ) ... ( )Y z b W z b z W z b z W z b z W z M M 0 1 1 2 2= + + + +- - - (6.8.5)

Equation (6.8.4) and (6.8.5) can be realized together by a direct structurecalled direct form-I I structure as shown in figure 6.8.2a. It uses less numberof delay elements then the Direct Form I structure.

For example, consider the same transfer function ( )H z which is discussedabove

( )H z ( )( )

X z Y z

z z z z

1 4 31 2 2

1 2

1 2= =

+ ++ +

- -

- -

Let ( )( )

X z Y z ( )

( )( )( )

W z Y z

X z W z

#=

where, ( )( )

X z W z

z z 1 4 311 2= + +- -

,

( )( )

W z Y z z z 1 2 21 2= + +- -

so, ( )W z ( ) ( ) ( )X z z W z z W z 4 31 2= - -- -

and ( )Y z ( ) ( ) ( )W z z W z z W z 1 2 21 2= + +- -

Comparing these equations with standard form of equation (6.8.4) and(6.8.5), we have a 41 = , a 32 = and , ,b b b 1 2 20 1 2= = = . Substitute these


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values in general structure of Direct form II , we get as shown in figure 6.8.2b

Figure 6.8.2a General structure of direct form-I I realization

Figure 6.8.2b

6.8.3 Cascade Form

The transfer function ( )H z of a discrete time system can be expressed asa product of several transfer functions. Each of these transfer functions is

realized in direct form-I or direct form II realization and then they arecascaded.Consider a system with transfer function

( )H z ( )( )

( )( )a z a z a z a z

b b z b z b b z b z 1 1k k m m k k k m m m

11

22

11

22

0 11

22

0 11

22

=+ + + ++ + + +

- - - -

- - - -

( ) ( )H z H z 1 2=

where ( )H z 1 a z a z

b b z b z 1 k k k k k

11

22

0 11

22

=+ ++ +

- -

- -


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( )H z 2 a z a z

b b z b z 1 m m m m m

11

22

0 11

22

=+ +

+ +- -

- -

Realizing ( ) ( )andH z H z 1 2 in direct form II and cascading we obtaincascade form of the system function ( )H z as shown in figure 6.8.3.

Figure 6.8.3 Cascaded form realization of discrete LTI system

6.8.4 Parallel Form

The transfer function ( )H z of a discrete time system can be expressed as thesum of several transfer functions using partial fractions. T hen the individualtransfer functions are realized in direct form I or direct form II realizationand connected in parallel for the realization of ( )H z . L et us consider thetransfer function

( )H z ......c p z

c p z c

p z c

1 1 1z n N

11

11

21= + -

+-

+-- - -

Now each factor in the system is realized in direct form II and connectedin parallel as shown in figure 6.8.4.

Figure 6.8.4 Parallel form realization of discrete LTI system

6.9 RELATIONSHIP BETWEEN s -PLANE & -PLANE

There exists a close relationship between the Laplace and z -transforms. We


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know that a DT sequence [ ]x n is obtained by sampling a CT signal ( )x t witha sampling interval T , the CT sampled signal ( )x t s is written as follows

( )x t s ( ) ( )x nT t nT n

d = -3

3

=-/

where ( )x nT are sampled value of ( )x t which equals the DT sequence [ ]x n . Taking the Laplace transform of ( )x t s , we have

( ) { ( )}X s L x t s = ( ) { ( )}x nT L t nT n

d = -3

3

=-/ ( )X nT e n T s n

=3

3

-

=-/ (6.9.1)

The z -transform of [ ]x n is given by

( )X z [ ]x n z n n

=3

3

-

=-/ (6.9.2)

Comparing equation (6.9.1) and (6.9.2) ( )X s ( )X z

z e sT =

= [ ] ( )x n x nT =

***********


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EXERCISE 6.1

MCQ 6.1.1 The z -transform and its ROC of a discrete time sequence

[ ]x n , 0,

n n

21

0 0<

n

$

= - b l*will be(A) ,z

z z 2 12

21>- (B) ,z

z z 2 21 a ae o (B) z > a(C) ,maxz 1> a ae o (D) z < a

MCQ 6.1.5Match List I (discrete time sequence) with List I I ( z -transform) and choosethe correct answer using the codes given below the lists:

List-I (Discrete Time Sequence) List-II ( z -Transform)

P. [ 2]u n - 1.( )

, 1z z

z 11 1

2

- --

Codes : P Q R S(A) 1 4 2 3(B) 2 4 1 3(C) 4 1 3 2(D) 4 2 3 1


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MCQ 6.1.6 The z -transform of signal [ ] [ ]x n e u n jn = p is(A) , : 1ROCz

z z 1 >

+ (B) , : 1ROCz j z z >-

(C) , : 1ROCz

z z 1

, then signal [ ]x n would be(A) [ ( ) ( ) [ ]u n 2 3 1n n - - (B) [ ( ) ( ) ] [ ]u n 2 3 1 1n n - + - - -(C) ( ) [ ] ( ) [ ]u n u n 2 3 1 1n n - - - - - (D) [ ( ) ] [ ]u n 2 3 1n +


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MCQ 6.1.11 If ROC of ( )X z is z 1 3< < , the signal [ ]x n would be(A) [ ( ) ( ) ] [ ]u n 2 3 1n n - - (B) [ ( ) ( ) ] [ ]u n 2 3 1 1n n - + - - -(C) ( ) [ ] ( ) [ ]u n u n 2 3 1 1n n - - - - - (D) [ ( ) ( ) ] [ ]u n 2 3 1 1n n + - - -

MCQ 6.1.12 Consider a DT sequence [ ]x n [ ] [ ]x n x n 1 2= + where, [ ]x n 1 ( . ) [ ]u n 0 7 1n = - and[ ] ( 0.4) [ 2]x n u n n 2 = - - . The region of convergence of z -transform of [ ]x n is

(A) . .z 0 4 0 7< < (B) .z 0 7>

(C) .z 0 4<

(D) none of theseMCQ 6.1.13 The z -transform of a DT signal [ ]x n is ( )X z

z z z

8 2 12=

- -. What will be the

z -transform of [ ]x n 4- ?

(A)( ) ( )

( )z z

z 8 4 2 4 1

42+ - + -

+ (B)

z z z

8 2 125

- -

(C)z z

z 128 8 1

42 - -

(D)z z z 8 2

15 4 3- -

MCQ 6.1.14 Let [ ], [ ]x n x n 1 2 and [ ]x n 3 be three discrete time signals and ( ), ( )X z X z 1 2 and( )X z 3 are their z -transform respectively given as

( )X z 1 ( )( . )z z

z

1 0 5

2=

- - ,

( )X z 2 ( )( . )z z z

1 0 5=

- -

and ( )X z 3 ( )( . )z z 1 0 51=

- -

Then [ ], [ ]x n x n 1 2 and [ ]x n 3 are related as(A) [ ] [ ] [ ]x n x n x n 2 11 2 3- = - = (B) [ ] [ ] [ ]x n x n x n 2 11 2 3+ = + =(C) [ ] [ ] [ ]x n x n x n 1 21 2 3= - = - (D) [ ] [ ] [ ]x n x n x n 1 11 2 3+ = - =

MCQ 6.1.15 The z -transform of the discrete time signal [ ]x n shown in the figure is

(A)z

z 1

k

1- --

(B)z

z 1

k

1+ --

(C)z z

11 k

1--

-

-

(D)z z

11 k

1-+

-

-

MCQ 6.1.16 Consider the unilateral z -transform pair [ ] ( )x n X z z z

1Z

=- . The z

-transform of [ ]x n 1- and [ ]x n 1+ are respectively

(A) z z

12

- , z 11- (B) z 1

1- , z

z 1

2

-

(C) z 11- , z

z 1- (D) z

z 1- , z

z 1

2

-

MCQ 6.1.17 A discrete time causal signal [ ]x n has the z -transform

( )X z . , : 0.4ROCz z z 0 4

>=-

The ROC for z -transform of the even part of [ ]x n will be


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(A) same as ROC of ( )X z (B) . .z 0 4 2 5< <

(C) .z 0 2> (D) .z 0 8>

MCQ 6.1.18 Match List I (Discrete time sequence) with List I I ( z -transform) and selectthe correct answer using the codes given below the lists.

List-I (Discrete time sequence) List-II ( z -transform)

P. ( ) [ ]n u n 1 n - 1.( )

, :ROCz

z z 1

1>1 21- --

Q. [ ]nu n 1- - - 2.( )

, : 1ROCz

z 1

1 >1+ -

R. ( ) [ ]u n 1 n - 3.( )

, : 1ROCz

z z 1

1 21

-+ -

-

Codes : P Q R S(A) 4 1 2 3(B) 4 3 2 1(C) 3 1 4 2(D) 2 4 1 3

MCQ 6.1.19 A discrete time sequence is defined as [ ]x n ( 2) [ 1]u n n n 1= - - -- . T he z -transform of [ ]x n is(A) , :log ROCz z 2

121

:S 2 [ ]x n 2+ :

Signals and Systems by Kanodia-150707084730-Lva1-App6892

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