GATE Signals and Systems by Kanodia

Eighth Edition

GATE

ELECTRONICS & COMMUNICATION

Signals and Systems Vol 7 of 10

RK Kanodia Ashish Murolia

NODIA & COMPANY

GATE Electronics & Communication Vol 7, 8e Signals and Systems RK 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 NODIA & 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 of use of this information. This book is published with the understanding that NODIA & COMPANY and its author are supplying information but are not attempting to render engineering or other professional services. MRP 690.00

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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, GATE Guide, and the GATE Cloud series. Our first book, GATE Multiple Choice Questions (MCQ), was a compilation of objective questions and solutions for all subjects of GATE Electronics & Communication Engineering in one book. The 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 practice answering questions to crack GATE. The solutions in the book were presented in such a manner that a student needs to know fundamental concepts to understand them. We assumed that students have learned enough of the fundamentals by his or her graduation. The book was a great success, but still there were a large ratio of aspirants who needed more preparatory materials beyond just problems and solutions. This large ratio mainly included 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, which contains brief theory, problem solving methods, fundamental concepts, and points-to-remember. The second book is 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 GATE Guide (the theory book) and GATE Cloud (the problem bank) series: two books for each subject. GATE Guide and GATE Cloud were published in three subjects only. Thereafter we received an immense number of emails from our readers looking for a complete study package for all subjects and a book that combines both GATE Guide and GATE Cloud. This encouraged us to present GATE Study Package (a set of 10 books: one for each subject) for GATE Electronic and Communication Engineering. 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 Type Questions (based on the new GATE pattern). Solutions are presented in a descriptive and step-by-step manner, which are easy to understand for all aspirants. We believe that each book of GATE Study Package helps a student learn fundamental concepts and develop problem solving skills for a subject, which are key essentials to crack GATE. Although we have put a vigorous effort in preparing this book, some errors may have crept in. We shall appreciate and greatly acknowledge all constructive comments, criticisms, and suggestions from the users of this book. You may write to us at rajkumar. [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 NODIA for providing professional support for this project through all phases of its development. At last, we express our gratitude to God and our Family for providing moral support and motivation. We wish you good luck ! R. K. Kanodia Ashish Murolia

SYLLABUS GATE Electronics & Communications: Definitions and properties of Laplace transform, continuous-time and discrete-time Fourier series, continuous-time and 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 and cascade 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; State variable representation; Fourier series; Fourier transforms and their application to system analysis; Laplace transforms 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 signals and probability, Correlation functions; Spectral density; Response of linear system to random inputs. **********

CONTENTS CHAPTER 1

CONTINUOUS TIME SIGNALS

1.1

CONTINUOUS - TIME AND DISCRETE - TIME SIGNALS

1.2

SIGNAL-CLASSIFICATION

1.2.1 1.2.2 1.2.3 1.2.4 1.2.5 1.3

1

Analog and Discrete Signals 1 Deterministic and Random Signal Periodic and Aperiodic Signal Even and Odd Signal 3 Energy and Power Signal 4

BASIC OPERATIONS ON SIGNALS

5

1.3.1 1.3.2 1.3.3 1.3.4 1.3.5 1.3.6 1.3.7

5 5 5

Addition of Signals Multiplication of Signals Amplitude Scaling of Signals Time-Scaling 5 Time-Shifting 6 Time-Reversal/Folding 7 Amplitude Inverted Signals

1.4

MULTIPLE OPERATIONS ON SIGNALS

1.5

BASIC CONTINUOUS TIME SIGNALS

1.5.1 1.5.2 1.5.3 1.5.4 1.5.5 1.5.6 1.5.7 1.6

1 2

8 8 9

The Unit-Impulse Function 9 The Unit-Step Function 12 The Unit-Ramp Function 12 Unit Rectangular Pulse Function Unit Triangular Function 13 Unit Signum Function 14 The Sinc Function 14

MATHEMATICAL REPRESENTATION OF SIGNALS

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

13

15

1

CHAPTER 2 2.1

CONTINUOUS TIME SYSTEMS

CONTINUOUS TIME SYSTEM & CLASSIFICATION

2.1.1 2.1.2 2.1.3 2.1.4 2.1.5 2.1.6 2.2

93

Linear and Non-Linear System 93 Time-Varying and Time-Invariant system 93 Systems With and Without Memory (Dynamic and Static Systems) Causal and Non-causal Systems 94 Invertible and Non-Invertible Systems 94 Stable and Unstable systems 94

LINEAR TIME INVARIANT SYSTEM

2.2.1 2.2.2

95

Impulse Response and The Convolution Integral Properties of Convolution Integral 96

95

2.3

STEP RESPONSE OF AN LTI SYSTEM

2.4

PROPERTIES OF LTI SYSTEMS IN TERMS OF IMPULSE RESPONSE

2.4.1 2.4.2 2.4.3 2.4.4 2.5

Memoryless LTI System Causal LTI System 101 Invertible LTI System 102 Stable LTI System 102

Systems in Parallel Configuration System in Cascade 103

CORRELATION

2.6.1 2.6.2 2.6.3 2.7

2.8

103

109

TIME DOMAIN ANALYSIS OF CONTINUOUS TIME SYSTEMS

Natural Response or Zero-input Response Forced Response or Zero-state Response The Total Response 111

BLOCK DIAGRAM REPRESENTATION

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 3.1

103

103

Cross-Correlation 103 Auto-Correlation 105 Correlation and Convolution

2.7.1 2.7.2 2.7.3

112

DISCRETE TIME SIGNALS Representation of Discrete Time signals

109

110 111

INTRODUCTION TO DISCRETE TIME SIGNALS 203

3.1.1

101

101

IMPULSE RESPONSE OF INTER-CONNECTED SYSTEMS

2.5.1 2.5.2 2.6

100

203

94

3.2

SIGNAL CLASSIFICATION

3.2.1 3.2.2 3.2.3 3.3

204

Periodic and Aperiodic DT Signals Even and Odd DT Signals 205 Energy and Power Signals 207

BASIC OPERATIONS ON DT SIGNALS

3.3.1 3.3.2 3.3.3 3.3.4 3.3.5 3.3.6 3.3.7

207

Addition of DT Signals 208 Multiplication of DT Signal 208 Amplitude scaling of DT Signals 208 Time-Scaling of DT Signals 208 Time-Shifting of DT Signals 209 Time-Reversal (folding) of DT signals Inverted DT Signals 211

3.4

MULTIPLE OPERATIONS ON DT SIGNALS

3.5

BASIC DISCRETE TIME SIGNALS

212

3.5.1 3.5.2 3.5.3 3.5.4 3.5.5 3.5.6

212 213 213 214 214

3.6

Discrete Impulse Function Discrete Unit Step Function Discrete Unit-ramp Function Unit-Rectangular Function Unit-Triangular Function Unit-Signum Function 215 216

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 285

Linear and Non-linear Systems 285 Time-Varying and Time-Invariant Systems 285 System With and Without Memory (Static and Dynamic Systems) Causal and Non-Causal System 286 Invertible and Non-Invertible Systems 286 Stable and Unstable System 286

LINEAR-TIME INVARIANT DISCRETE SYSTEM 287

4.2.1 4.2.2 4.3

211

DISCRETE TIME SYSTEM & CLASSIFICATION

4.1.1 4.1.2 4.1.3 4.1.4 4.1.5 4.1.6 4.2

210

MATHEMATICAL REPRESENTATION OF DT SIGNALS USING IMPULSE OR STEP FUNCTION

EXERCISE 3.1

4.1

204

Impulse Response and Convolution Sum Properties of Convolution Sum 288

STEP RESPONSE OF AN LTI SYSTEM

292

287

286

215

4.4

PROPERTIES OF DISCRETE LTI SYSTEM IN TERMS OF IMPULSE RESPONSE

4.4.1 4.4.2 4.4.3 4.4.4 4.4.5 4.5

Memoryless LTID System Causal LTID System 293 Invertible LTID System Stable LTID System FIR and IIR Systems 294

293 294

IMPULSE RESPONSE OF INTERCONNECTED SYSTEMS

4.5.1 4.5.2 4.6

293

Systems in Parallel System in Cascade

CORRELATION

4.6.1 4.6.2 4.6.3 4.6.4 4.6.5

295 295

296

Cross-Correlation 296 Auto-Correlation 296 Properties of Correlation 297 Relationship Between Correlation and Convolution 299 Methods to Solve Correlation 299

4.7

DECONVOLUTION

4.8

RESPONSE OF LTID SYSTEMS IN TIME DOMAIN

4.8.1 4.8.2 4.8.3 4.9

Natural Response or Zero Input Response Forced Response or Zero State Response 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

5.1

300

BLOCK DIAGRAM REPRESENTATION

CHAPTER 5

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

5.3

REGION OF CONVERGENCE

375

375

376

Poles and Zeros of Rational Laplace Transforms Properties of ROC 377

THE INVERSE LAPLACE TRANSFORM

5.4.1 5.4.2

302

375

THE EXISTENCE OF LAPLACE TRANSFORM

5.4

301

303

5.2

5.3.1 5.3.2

300

THE LAPLACE TRANSFORM

INTRODUCTION

5.1.1 5.1.2

295

376

382

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

292

5.5

PROPERTIES OF THE LAPLACE TRANSFORM 384

5.5.1 5.5.2 5.5.3 5.5.4 5.5.5 5.5.6 5.5.7 5.5.8 5.5.9 5.5.10 5.5.11 5.5.12 5.5.13 5.6

Linearity 384 Time Scaling 384 Time Shifting 385 Shifting in the s -domain(Frequency Shifting) Time Differentiation 386 Time Integration 387 Differentiation in the s -domain 388 Conjugation Property 389 Time Convolution 389 s -Domain Convolution 390 Initial value Theorem 390 Final Value Theorem 391 Time Reversal Property 391

ANALYSIS OF CONTINUOUS LTI SYSTEMS USING LAPLACE TRANSFORM

5.6.1 5.6.2 5.7

Response of LTI Continuous Time System Impulse Response and Transfer Function Causality 395 Stability 395 Stability and Causality Parallel Connection Cascaded Connection Feedback Connection

395 395

395 396 396

BLOCK DIAGRAM REPRESENTATION OF CONTINUOUS LTI SYSTEM

5.9.1 5.9.2 5.9.3 5.9.4

Direct Form I structure Direct Form II structure Cascade Structure 401 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 6.1

393 394

SYSTEM FUNCTION FOR INTERCONNECTED LTI SYSTEMS

5.8.1 5.8.2 5.8.3 5.9

393

STABILITY AND CAUSALITY OF CONTINUOUS LTI SYSTEM USING LAPLACE TRANSFORM

5.7.1 5.7.2 5.7.3 5.8

386

THE Z-TRANSFORM

INTRODUCTION

6.1.1

397 399

493

The Bilateral or Two-Sided z -transform

493

397

394

6.1.2

The Unilateral or One-sided z -transform

6.2

EXISTENCE OF

6.3

REGION OF CONVERGENCE

6.3.1 6.3.2 6.4

6.5

6.6

6.9

500

Partial Fraction Method 502 Power Series Expansion Method

z -TRANSFORM

494

503

503

Linearity 503 Time Shifting 504 Time Reversal 505 Differentiation in the z -domain Scaling in z -Domain 506 Time Scaling 506 Time Differencing 507 Time Convolution 508 Conjugation Property 508 Initial Value Theorem 509 Final Value Theorem 509

506

z -TRANSFORM

Response of LTI Continuous Time System Impulse Response and Transfer Function Causality 513 Stability 513 Stability and Causality

514

BLOCK DIAGRAM REPRESENTATION

514

6.8.1 6.8.2 6.8.3 6.8.4

515 516

Direct Form I Realization Direct Form II Realization Cascade Form 517 Parallel Form 518

RELATIONSHIP BETWEEN

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

s-PLANE & z -PLANE

511

511 512

STABILITY AND CAUSALITY OF LTI DISCRETE SYSTEMS USING

6.7.1 6.7.2 6.7.3 6.8

494

ANALYSIS OF DISCRETE LTI SYSTEMS USING

6.6.1 6.6.2 6.7

z -TRANSFORM

PROPERTIES OF

6.5.1 6.5.2 6.5.3 6.5.4 6.5.5 6.5.6 6.5.7 6.5.8 6.5.9 6.5.10 6.5.11

494

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

THE INVERSE

6.4.1 6.4.2

z -TRANSFORM

494

518

z -TRANSFORM

513

CHAPTER 7 7.1

THE CONTINUOUS TIME FOURIER TRANSFORM

DEFINITION 607

7.1.1 7.1.2 7.1.3 7.2

Magnitude and Phase Spectra 607 Existence of Fourier transform 607 Inverse Fourier Transform 608

SPECIAL FORMS OF FOURIER TRANSFORM

7.2.1 7.2.2 7.2.3 7.2.4 7.3

Real-valued Even Symmetric Signal 608 Real-valued Odd Symmetric Signal 610 Imaginary-valued Even Symmetric Signal Imaginary-valued Odd Symmetric Signal

PROPERTIES OF FOURIER TRANSFORM

7.3.1 7.3.2 7.3.3 7.3.4 7.3.5 7.3.6 7.3.7 7.3.8 7.3.9 7.3.10 7.3.11 7.3.12 7.3.13 7.3.14 7.3.15 7.3.16 7.4

7.5

610 611

612

Linearity 612 Time Shifting 612 Conjugation and Conjugate Symmetry Time Scaling 613 Differentiation in Time-Domain 614 Integration in Time-Domain 614 Differentiation in Frequency Domain 615 Frequency Shifting 615 Duality Property 615 Time Convolution 616 Frequency Convolution 616 Area Under x (t) 617 Area Under X (jw) 617 Parseval’s Energy Theorem 618 Time Reversal 618 Other Symmetry Properties 619

612

ANALYSIS OF LTI CONTINUOUS TIME SYSTEM USING FOURIER TRANSFORM

620

7.4.1 7.4.2

620

Transfer Function & Impulse Response of LTI Continuous System Response of LTI Continuous system using Fourier Transform 620

RELATION BETWEEN FOURIER AND LAPLACE TRANSFORM

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 8.1

608

THE DISCRETE TIME FOURIER TRANSFORM

DEFINITION 705

621

8.1.1 8.1.2 8.1.3

Magnitude and Phase Spectra Existence of DTFT 705 Inverse DTFT 705

705

8.2

SPECIAL FORMS OF DTFT

8.3

PROPERTIES OF DISCRETE-TIME FOURIER TRANSFORM

8.3.1 8.3.2 8.3.3 8.3.4 8.3.5 8.3.6 8.3.7 8.3.8 8.3.9 8.3.10 8.3.11 8.3.12 8.3.13 8.4

706

Linearity 707 Periodicity 707 Time Shifting 708 Frequency Shifting 708 Time Reversal 708 Time Scaling 709 Differentiation in Frequency Domain 710 Conjugation and Conjugate Symmetry Convolution in Time Domain 711 Convolution in Frequency Domain Time Differencing 712 Time Accumulation 712 Parseval’s Theorem 713 Transfer Function & Impulse Response Response of LTI DT system using DTFT

RELATION BETWEEN THE DTFT & THE

8.6

DISCRETE FOURIER TRANSFORM (DFT)

8.6.1

FAST FOURIER TRANSFORM (FFT)

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

716

716

Linearity 716 Periodicity 717 Conjugation and Conjugate Symmetry Circular Time Shifting 718 Circular Frequency Shift 719 Circular Convolution 719 Multiplication 720 Parseval’s Theorem 721 Other Symmetry Properties 721 722

714

714 714

715

Inverse Discrete Fourier Transform (IDFT)

PROPERTIES OF DFT

8.7.1 8.7.2 8.7.3 8.7.4 8.7.5 8.7.6 8.7.7 8.7.8 8.7.9 8.8

711

Z -TRANSFORM

8.5

8.7

710

ANALYSIS OF LTI DISCRETE TIME SYSTEM USING DTFT

8.4.1 8.4.2

707

717

715

SOLUTIONS 8.4

CHAPTER 9 9.1

770

THE CONTINUOUS TIME FOURIER SERIES

INTRODUCTION TO CTFS

9.1.1 9.1.2 9.1.3

775

Trigonometric Fourier Series Exponential Fourier Series Polar Fourier Series 779

775 778

9.2

EXISTENCE OF FOURIER SERIES

780

9.3

PROPERTIES OF EXPONENTIAL CTFS

780

9.3.1 9.3.2 9.3.3 9.3.4 9.3.5 9.3.6 9.3.7 9.3.8 9.3.9 9.3.10 9.3.11

Linearity 780 Time Shifting 781 Time Reversal Property 781 Time Scaling 782 Multiplication 782 Conjugation and Conjugate Symmetry Differentiation Property 783 Integration in Time-Domain 784 Convolution Property 784 Parseval’s Theorem 785 Frequency Shifting 786

783

9.4

AMPLITUDE & PHASE SPECTRA OF PERIODIC SIGNAL

9.5

RELATION BETWEEN CTFT & CTFS

9.5.1 9.5.2 9.6

787

787

CTFT using CTFS Coefficients 787 CTFS Coefficients as Samples of CTFT

787

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

EXERCISE 9.1

790

EXERCISE 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

10.3

PROPERTIES OF DTFS

861

10.3.1 10.3.2 10.3.3

862 862

Linearity 862 Periodicity Time-Shifting

861

788

10.3.4 10.3.5 10.3.6 10.3.7 10.3.8 10.3.9 10.3.10 10.3.11 10.3.12 10.3.13

Frequency Shift 863 Time-Reversal 863 Multiplication 864 Conjugation and Conjugate Symmetry Difference Property 865 Parseval’s Theorem 865 Convolution 866 Duality 867 Symmetry 867 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

898

SOLUTIONS 10.3

903

SOLUTIONS 10.4

904

864

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

11.5

ALIASING

11.6

SIGNAL RECONSTRUCTION

11.7

SAMPLING OF BAND-PASS SIGNALS

907

907

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

908 909

***********

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CHAPTER 6 Page 493 Chap 6

The Z-Transform

THE Z-TRANSFORM

6.1

INTRODUCTION

As we studied in previous chapter, the Laplace transform is an important tool for analysis of continuous time signals and systems. Similarly, z -transforms enables us to analyze discrete time signals and systems in the z -domain. Like, the Laplace transform, it is also classified as bilateral z -transform and unilateral z -transform. The bilateral or two-sided z -transform is used to analyze both causal and non-causal LTI discrete systems, while the unilateral z -transform is defined only for causal signals.

i. n o c . a i d o n . w w w

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) = Z {x [n]} =

3

/ x [n] z

-n

(6.1.1)

n =- 3

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 = re jW where r is the magnitude of z and W is the angle of z . This corresponds to a 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 ]

Z

X (z)

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GATE STUDY PACKAGE Electronics & Communication 10 Subject-wise books by R. K. Kanodia General Aptitude Engineering Mathematics Networks Electronic Devices Analog Electronics Digital Electronics Signals & Systems Control Systems Communication Systems Electromagnetics Page 494 Chap 6

6.1.2

The Unilateral or One-sided z -transform The z -transform for causal signals and systems is referred to as the unilateral z -transform. For a causal sequence x [n] = 0 , for n < 0 Therefore, the unilateral z -transform is defined as

The Z-Transform

X (z) =

3

/ x [ n] z

-n

(6.1.2)

n=0

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) 3

/ x [ n] z

X [z] =

-n

n =- 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.

i. n

Substituting z = re jW 3

/ x [n] (re

X [z] =

3

/ {x [n] r

X [z] =

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

} e-jWn

o n

3

/ x [ n] r

. w w n =- 3

-n

a i d

n =- 3

)

o .c

n =- 3

or,

6.3

jW -n

-n

13

(6.2.1)

REGION OF CONVERGENCE

The existence of z -transform is given from equation (6.2.1). The values of r for which x [n] r-n is absolutely summable is referred to as region of convergence. Since,z = re jW so r = z . Therefore we conclude that the range of values of the variable z for which the sum in equation (6.1.1) converges is called the region of convergence. This can be explained through the following examples.

w 6.3.1

Poles and Zeros of Rational z -transforms The most common form of z -transform is a rational function. Let X (z) be the z -transform of sequence x [n] , expressed as a ratio of two polynomials N (z) and D (z). N (z) X (z) = D (z) The roots of numerator polynomial i.e. values of z for which X (z) = 0 is referred to as zeros of X (z). The roots of denominator polynomial for which X (z) = 3 is referred to as poles of X (z). The 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) = 2 z z - 5z + 6 z = (z - 2) (z - 3)

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

Page 495 Chap 6 The Z-Transform

Figure 6.3.1 Pole-zero plot of X ^z h NOTE : 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

i. n o c . a i d o n . w w w

Properties of ROC

The various properties of ROC are summarized as follows. These properties can 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) =

Put z = re jW

X (z) = X (re jW) =

3

/ x [n] z-n

n =- 3 3

/ x [n] r-n e-jWn n =- 3

X (z) converges for those values of z for which x [n] r-n is absolutely summbable that is 3

/ x [n] r-n < 3

n =- 3

Thus, convergence is dependent only on r , where, r = z The equation z = re jW , describes a circle in z -plane. Hence the ROC will consists 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. We know that X (z) will be infinite at pole, and, therefore X (z) does not converge at 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 ‘finite duration sequence’. Consider a finite duration sequence x [n] shown in figure 6.3.2a; x [n] is non-zero only for some interval N1 # n # N2 .

Figure 6.3.2a A Finite Duration Sequence

o .c

The z -transform of x [n] is defined as X (z) =

N2

i. n

a i d

/ x [n] z-n n = N1

This summation converges for all finite values of z . If N1 is negative and N2 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 . Similarly positive 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.

o n

. w w

NOTE : Both N1 and N 2 can be either positive or negative.

w

PROPERTY 4

If x [n] is a right-sided sequence, and if the circle z = r0 is in the ROC, then all values of z for which z > r0 will also be in the ROC.

PROOF :

A sequence which is zero prior to some finite time is called the right-sided sequence. Consider a right-sided sequence x [n] shown in figure 6.3.2b; that is; x [n] = 0 for n < N1 .

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 _i.e. z = r0 i . From the condition of convergence we can write 3

/ x [n] z-n n =- 3 3

/

n =- 3

Page 497 Chap 6 The Z-Transform

1 or z > r 0 r0 Thus, we conclude that if the circle z = r0 is in the ROC, then all values of z for which z > r0 will also be in the ROC. The ROC of a rightsided sequence is illustrated in figure 6.3.2c. =

/

-n

1

i. n o c . a i d o n . w w w 1

1

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 = r0 is in the ROC, then all values of z for which z < r0 will also be in the ROC. PROOF :

A sequence which is zero after some finite time interval is called a ‘leftsided signal’. Consider a left-sided signal x [n] shown in figure 6.3.2d; that is x [n] = 0 for n > N2 .

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GATE STUDY PACKAGE Electronics & Communication 10 Subject-wise books by R. K. Kanodia General Aptitude Engineering Mathematics Networks Electronic Devices Analog Electronics Digital Electronics Signals & Systems Control Systems Communication Systems Electromagnetics Page 498 Chap 6 The Z-Transform

Figure 6.3.2d A left-sided sequence

Let z -transform of x [n] converges for some values of z _i.e. z = r0 i. From the condition of convergence we write 3

/ x [n] z-n or

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

w

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 = r0 is in the ROC, then the ROC consists of a ring in the z -plane that includes the circle z = r0

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PROOF :

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

Page 499 Chap 6 The Z-Transform

Figure 6.3.2f A Two - Sided Sequence

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

i. n o c . a i d o n . w w w

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

The z -transform of x [n] converges for the values of z for which the transform of both xR [n] and xL [n] converges. From property 4, the ROC of a right-sided sequence is a region which is bounded on the inside by a circle and extending outward to infinity i.e. | z | > r1 . From property 5, the ROC of a left sided sequence is bounded on the outside by a circle and extending inward to zero i.e. | z | < r2 . So the ROC of combined signal includes intersection of both ROCs which is ring in the z -plane. The ROC for the right-sided sequence xR [n], the left-sequence xL [n] and their combination which is a two sided sequence x [n] are shown in figure 6.3.2h.

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GATE STUDY PACKAGE Electronics & Communication 10 Subject-wise books by R. K. Kanodia General Aptitude Engineering Mathematics Networks Electronic Devices Analog Electronics Digital Electronics Signals & Systems Control Systems Communication Systems Electromagnetics Page 500 Chap 6 The Z-Transform

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 by poles or extends to infinity.

i. n

PROOF :

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

o .c

PROPERTY 8

a i d

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

o n

. w w

PROOF :

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

w

PROPERTY 9

If the z -transform X (z) of x [n] is rational and x [n] is a left-sided sequence then 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 smallest magnitude 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 their respective 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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Sample Chapter of Signals TABLE 6.1 :

z -Transform of Basic Discrete Time Signals

DT sequence x [n] d [ n]

1. 2.

d [n - n 0]

3.

u [n] an u [n]

4.

n-1

a

5.

6.

7.

and Systems (Vol-7, GATE Study Package)

u [n - 1]

z -transform 1

Page 501 Chap 6

ROC

The Z-Transform

entire z -plane

z-n

entire z -plane, except z = 0

0

1 = z 1 - z-1 z - 1

z >1

1 z -1 = z - a 1 - az

z > a

z-1 = 1 1 - az-1 z - a

z > a

i. n o c . a i d o n . w w w nu [n]

n

na u [ n]

z-1 z = (1 - z-1) 2 (z - 1) 2

z >1

az-1 az -1 2 = (z - a) 2 (1 - az )

z >a

1 - z-1 cos W 0 or 1 - 2z-1 cos W 0 + z-2

8.

cos (W0 n) u [n]

z [z - cos W0] z - 2z cos W0 + 1

z >1

2

z-1 sin W0 or 1 - 2z-1 cos W0 + z-2

9.

sin (W0 n) u [n]

z sin W0 z2 - 2z cos W0 + 1

z >1

1 - az-1 cos W0 1 - 2az-1 cos W0 + a2 z-2 10.

an cos (W0 n) u [n]

z [z - a cos W0] or 2 z - 2az cos W0 + a2

z > a

az-1 sin W0 1 - 2az-1 cos W0 + a2 z-2 11.

an sin (W0 n) u [n]

or

az sin W0 z - 2az cos W0 + a2

z >a

2

A + Bz-1 1 + 2gz-1 + a2 z-2

12.

ran sin (W0 n + q) u [n] with a ! R or

z (Az + B) 2 z + 2g z + g 2

z # a (a)

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GATE STUDY PACKAGE Electronics & Communication 10 Subject-wise books by R. K. Kanodia General Aptitude Engineering Mathematics Networks Electronic Devices Analog Electronics Digital Electronics Signals & Systems Control Systems Communication Systems Electromagnetics x [n] = 1 X (z) zn - 1 dz 2pj This method involves the contour integration, so difficult to solve. There are other commonly used methods to evaluate the inverse z -transform given as follows 1. Partial fraction method 2. Power series expansion

Page 502 Chap 6

#

The Z-Transform

6.4.1

Partial Fraction Method If X (z) is a rational function of z then it can be expressed as follows. N (z) X (z) = D (z) It is convenient if we consider X (z) /z rather than X (z) to obtain the inverse z -transform by partial fraction method. Let p1 , p2 , p 3 ....pn are the roots of denominator polynomial, also the poles of X (z). Then, using partial fraction method X (z) /z can be expressed as X (z) = A1 + A2 + A 3 + ... + An z - p1 z - p 2 z - p 3 z - pn z X (z) = A1 z + A2 z + ... + z z - p1 z - p2 z - pn Now, the inverse z -transform of above equation can be obtained by comparing each term with the standard z -tranform pair given in table 6.1. The values of coefficients A1 , A2 , A 3 ....An depends on whether the poles are real & distinct or repeated or complex. Three cases are given as follows

i. n

o .c

a i d

o n

Case I : Poles are Simple and Real

. w w

X (z) /z can be expanded in partial fraction as X (z) = A1 + A2 + A 3 + ... + An z - p1 z - p 2 z - p 3 z - pn z where A1 , A2 ,... An are calculated as follows X (z) A1 = (z - p1) z z=p X (z) A2 = (z - p2) z z=p In general,

w

(6.4.1)

1

2

Ai = (z - pi) X (z) z = p

(6.4.2) i

Case II : If Poles are Repeated In this case X (z) /z has a different form. Let pk be the root which repeats l times, then the expansion of equation must include terms X (z) = A1k + A2k 2 + ... z - pk (z - pk ) z (6.4.3) + Aik i + ... + Alk l (z - pk ) (z - pk ) The coefficient Aik are evaluated by multiplying both sides of equation (6.4.3) by (z - pk ) l , differentiating (l - i) times and then evaluating the resultant equation at z = pk . Thus, l-i X (z) (6.4.4) Cik = 1 d l - i :(z - pk ) l z D (l - i) dz z = pk Buy Online: shop.nodia.co.in

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Case III : Complex Poles If X (z) has complex poles then partial fraction of the X (z) /z can be expressed as ) X (z) (6.4.5) = A1 + A 1 ) z z - p1 z - p 1 where A 1) is complex conjugate of A1 and p) 1 is complex conjugate of z1 . The coefficients are obtained by equation (6.4.2)

6.4.2

Page 503 Chap 6 The Z-Transform

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 3

/ x [ n] z

X (z) =

-n

n =- 3

Now, X (z) is expanded in the following form X (z) = .. + x [- 2] z2 + x [- 1] z1 + x [0] + x [1] z-1 + x [2] z-2 + ... To obtain inverse z -transform(i.e. x [n]), represent the given X (z) in the form of above power series. Then by comparing we can get

i. n o c . a i d o n . w w w x [n] = {...x [- 2], x [- 1], x [0], x [1], x [2], ...}

6.5

PROPERTIES OF

-TRANSFORM

The unilateral and bilateral z -transforms possess a set of properties, which are useful in the analysis of DT signals and systems. The proofs of properties are given for bilateral transform only and can be obtained in a similar way for 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 their z transform. x 1 [ n]

Z

X1 (z),

with ROC: R1

x 2 [ n]

Z

X2 (z),

with ROC: R2

ax1 [n] + bx2 [n]

Z

Let

and

aX1 (z) + bX2 (z), with ROC: at least R1 + R2 for both unilateral and bilateral z -transform. then,

PROOF :

The z -transform of signal {ax1 [n] + bx2 [n]} is given by equation (6.1.1) as follows 3

/ {ax [n] + bx [n]} z

Z {ax1 [n] + bx2 [n]} =

1

n =- 3 3

=a

/ x [n] z 1

-n

2

-n

+b

n =- 3

3

/ x [n] z 2

-n

n =- 3

= aX1 (z) + bX2 (z) Hence, ax1 [n] + bx2 [n]

Z

aX1 (z) + bX2 (z)

ROC : Since, the z -transform X1 (z) is finite within the specified ROC, R1 . Buy Online: shop.nodia.co.in

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GATE STUDY PACKAGE Electronics & Communication 10 Subject-wise books by R. K. Kanodia General Aptitude Engineering Mathematics Networks Electronic Devices Analog Electronics Digital Electronics Signals & Systems Control Systems Communication Systems Electromagnetics Similarly, X2 (z) is finite within its ROC, R2 . Therefore, the linear combination aX1 (z) + bX2 (z) should be finite at least within region R1 + R2 .

Page 504 Chap 6 The Z-Transform

NOTE : In certain cases, due to the interaction between x1 [n] and x 2 [n], which may lead to cancellation of 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 R1 and R 2 , the z-transform of ax1 [n] + bx 2 [n] does not exist.

6.5.2

Time Shifting For the bilateral z -transform x [n]

Z

X (z),

x [n - n 0]

Z

z-n X (z),

If then

with ROC Rx

0

and x [n + n 0] zn X (z), with ROC : Rx except for the possible deletion or addition of z = 0 or z = 3. Z

0

i. n

PROOF :

o .c

The bilateral z -transform of signal x [n - n 0] is given by equation (6.1.1) as follows 3

/ x [n - n ] z

a i d

Z {x [n - n 0]} =

0

n =- 3

-n

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

o n

Z {x [n - n 0]} =

. w w

w

=

3

/ x [a] z

- (n 0 + a)

a =- 3 3

/ x [a] z

-n 0 - a

= z-n

z

3

/ x [a] z

0

a =- 3

-a

a =- 3

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

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

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

TIME SHIFTING FOR UNILATERAL

0

z -TRANSFORM

For the unilateral z -transform If then and

x [n]

Z

X (z),

x [n - n 0]

Z

z-n e X (z) +

x [n + n 0]

Z

0

zn e X (z) 0

with ROC Rx n0

/ x [- m] z

m=1 n0 - 1

/ x [m] z

-m

m

o,

o,

m=0

with ROC : Rx except for the possible deletion or addition of z = 0 or z = 3. PROOF :

The unilateral z -transform of signal x [n - n 0] is given by equation (6.1.2) as follows Buy Online: shop.nodia.co.in

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Sample Chapter of Signals Z {x [n - n 0]} =

and Systems (Vol-7, GATE Study Package)

3

/ x [n - n ] z

Page 505 Chap 6

-n

0

n=0

Multiplying RHS by z and z-n n0

Z {x [n - n 0]} =

The Z-Transform

0

3

/ x [n - n ] z

-n n 0 -n 0

0

z z

n=0

= z-n

3

/ x [n - n ] z

0

- (n - n 0)

0

n=0

Substituting n - n 0 = a Limits; when n " 0, a " - n 0 when n " + 3, a " + 3 Now, Z {x [n - n 0]} = z-n = z-n or, or,

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

3

/ x [a] z

0

-a

a =- n 0 -1

/ x [a] z

0

a =- n 0 3

/ x [a] z

0

+ z-n

-a

-a

+ z-n

3

0

/ x [a] z

-a

a=0 -1 0

/ x [a] z

-a

i. n o c . a i d o n . w w w Z {x [n - n 0]} = z-n

a=0 3

/ x [a] z

0

-a

+ z-n

a =- n 0 n0

0

a=0

/ x [- a] z

a

a=1

Changing the variables as a " n and a " m in first and second summation respectively Z {x [n - n 0]} = z-n

3

/ x [n] z

0

-n

+ z-n

n=0

n0

0

/ x [- m] z

m

m=1

= z-n X [z] + z-n 0

n0

/ x [- m] z

0

m

m=1

In similar way, we can also prove that x [n + n 0]

z e X (z) n0

Z

n0 - 1

/ x [m] z

-m

o

m=0

6.5.3

Time Reversal

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

Z

x [- n]

Z

If then for bilateral z -transform.

X (z), X b 1 l, z

with ROC : Rx with ROC : 1/Rx

PROOF :

The bilateral z -transform of signal x [- n] is given by equation (6.1.1) as follows 3

/ x [ - n] z

Z {x [- n]} =

-n

n =- 3

Substituting - n = k on the RHS, we get Z {x [- n]} =

-3

/ x [ k] z

k=3

Hence,

x [- n]

Z

k

Xb 1 l z

=

3

/ x [k] (z k =- 3

-1 -k

)

= Xb 1 l z

ROC : z-1 ! Rx or z ! 1/Rx

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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 result by - z in the z -domain.

The Z-Transform

If

Z

X (z), dX (z) Z then , nx [n] -z dz For both unilateral and bilateral z -transforms. x [n]

with ROC : Rx with ROC : Rx

PROOF :

The bilateral z -transform of signal x [n] is given by equation (6.1.1) as follows 3

/ x [ n] z

X (z) =

-n

n =- 3

Differentiating both sides with respect to z gives 3 3 -n dX (z) = x [n] dz = x [n] (- nz-n - 1) dz dz n =- 3 n =- 3 Multiplying both sides by - z , we obtain 3 dX (z) -z = nx [n] z-n dz n =- 3 dX (z) Z Hence, nx [n] -z dz ROC : This operation does not affect the ROC.

6.5.5

/

/

/

o .c

i. n

a i d

o n

Scaling in z -Domain

. w w

Multiplication of a time sequence with an exponential sequence an corresponds to scaling in z -domain by a factor of a . If

Z

X (z), Z then a n x [n] X a z k, a for both unilateral and bilateral transform. x [n]

w

with ROC : Rx with ROC : a Rx

PROOF :

The bilateral z -transform of signal x [n] is given by equation (6.1.1) as 3

/ a x [n] z

Z {an x [n]} =

n

-n

=

n =- 3

3

/ x [n] [a

-1

z] -n

n =- 3

a x [n] Xa z k a 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). n

6.5.6

Z

Time Scaling As we discussed in Chapter 2, there are two types of scaling in the DT domain decimation(compression) and interpolation(expansion). Time Compression Since the decimation (compression) of DT signals is an irreversible process (because some data may lost), therefore the z -transform of x [n] and its Buy Online: shop.nodia.co.in

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

Page 507 Chap 6

Time Expansion In the discrete time domain, time expansion of sequence x [n] is defined as x [n/k] if n is a multiple of integer k (6.5.1) x k [ n] = ) 0 otherwise Time-scaling property of z -transform is derived only for time expansion which is given as If then

x [n]

Z

X (z),

x k [ n]

Z

Xk (z) = X (zk ),

The Z-Transform

with ROC : Rx with ROC : (Rx ) 1/k

for both the unilateral and bilateral z -transform. PROOF :

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

i. n o c . a i d o n . w w w Z {xk [n]} =

3

/ x [n] z k

-n

n=0

= xk [0] + xk [1] z-1 + ... + xk [k] z-k + xk [k + 1] z- (k + 1) + ...xk [2k] z-2k + ...

Since the expanded sequence xk [n] is zero everywhere except when n is a multiple of k . This reduces the above transform as follows Z {xk [n]} = xk [0] + xk [k] z-k + xk [2k] z-2k + xk [3k] z-3k + ... As defined in equation 6.5.1, interpolated sequence is x k [ n] n=0 xk [0] n=k x k [k] n = 2k xk [2k] Thus, we can write

= x [n/k] = x [0], = x [1] = x [2]

Z {xk [n]} = x [0] + x [1] z-k + x [2] z-2k + x [3] z-3k + ... =

3

/ x [n] (z )

k -n

= X (zk )

n=0

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

6.5.7

Time Differencing If

x [n]

Z

X (z),

with ROC : Rx -1

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

PROOF :

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

3

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

-n

n =- 3

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GATE STUDY PACKAGE Electronics & Communication 10 Subject-wise books by R. K. Kanodia General Aptitude Engineering Mathematics Networks Electronic Devices Analog Electronics Digital Electronics Signals & Systems Control Systems Communication Systems Electromagnetics Page 508 Chap 6

3

/

=

n =- 3

The Z-Transform

3

/ x [ n - 1] z

x [n] z-n -

-n

n =- 3

In the second summation, substituting n - 1 = r 3

/ x [ n] z

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

-n

n =- 3 3

3

/ x [r] z

-

- (r + 1)

r =- 3

/ x [ n] z

-n

3

/ x [r] z

- z-1

n =- 3

-r

r =- 3 -1

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

6.5.8

(1 - z-1) X (z)

Z

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

x 1 [ n]

Z

X1 (z),

ROC : R1

and

x 2 [ n]

Z

X2 (z),

ROC : R2

i. n

then the convolution property states that x 1 [ n] * x 2 [ n]

Z

X1 (z) X2 (z),

ROC : at least R1 + R2

o .c

for both unilateral and bilateral z -transforms.

a i d

PROOF :

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

o n

x 1 [ n] * x 2 [ n] =

. w w

3

/ x [k] x [n - k] 1

2

k =- 3

Taking the z -transform of both sides gives

w

x 1 [ n] * x 2 [ n]

3

3

/ / x [k ] x [n - k ] z

Z

1

-n

2

n =- 3 k =- 3

Interchanging the order of the two summations, we get x 1 [ n] * x 2 [ n]

3

3

/ x [k ] / x [n - k ] z

Z

1

-n

2

k =- 3

n =- 3

Substituting n - k = a in the second summation

or

6.5.9

x [n ] * x 2 [n ]

Z

x [n ] * x 2 [n ]

Z

x 1 [ n] * x 2 [ n]

Z

3

3

/ x [k ] / x [a] z 1

k =- 3 3

e

- (a + k)

2

a =- 3

/ x [k] z 1

-k

k =- 3

oe

3

/ x [a] z

-a

2

a =- 3

o

X1 (z) X2 (z)

Conjugation Property x [n]

Z

X (z),

with ROC : Rx

then

x [n]

Z

X (z ),

with ROC : Rx

If x [n] is real, then

X (z) = X )(z ))

If

)

)

)

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PROOF :

Page 509 Chap 6

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

3

/

/ 6x [n] (z ) 3

x)[n] z-n =

n =- 3

@

) -n )

(6.5.2)

The Z-Transform

n =- 3

Let z -transform of x [n] is X (z) X (z) =

3

/ x [ n] z

-n

n =- 3

by taking complex conjugate on both sides of above equation X)(z) =

3

/ [ x [ n] z

-n )

]

n =- 3

Replacing z " z), we will get X)(z)) =

/ 6x [n] (z ) 3

) -n

@

)

(6.5.3)

n =- 3

Comparing equation (6.5.2) and (6.5.3) Z {x)[n]} = X)(z)) For real x [n],

(6.5.4)

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

i. n o c . a i d o n . w w w Z {x)[n]} =

3

/ x [ n] z

-n

= X (z)

(6.5.5)

n =- 3

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

6.5.10 Initial Value Theorem

Z

If x [n] then initial-value theorem states that,

X (z),

with ROC : Rx

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) =

3

/ x [ n] z

-n

n=0

= x [0] + x [1] z-1 + x [2] z-2 + ... Taking limit as z " 3 on both sides we get lim X (z) = lim (x [0] + x [1] z-1 + x [2] z-2 + ...) = x [0] z"3 x [0] = lim X (z)

z"3

z"3

6.5.11 Final Value Theorem If x [n] then final-value theorem states that

Z

X (z),

with ROC : Rx

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

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

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PROOF :

= lim

Z {x [n + 1]} - Z {x [n]} (6.5.6)

k

/ {x [n + 1] - x [n]} z

-n

k"3 n=0

From the time shifting property of unilateral z -transform discussed in section 6.5.2 x [n + n 0] For n 0 = 1

x [ n + 1]

Z

Z

zn e X (z) 0

z e X (z) -

n0 - 1

/ x [m] z

m=0 0

-m

/ x [m] z

o

o

-m

m=0

x [ n + 1] z ^X (z) - x [0]h Put above transformation in the equation (6.5.6) Z

k

zX [z] - zx [0] - X [z] = lim

/ (x [n + 1] - x [n]) z

(z - 1) X [z] - zx [0] = lim

/ (x [n + 1] - x [n]) z

-n

k"3 n=0 k

-n

k"3 n=0

Taking limit as z " 1 on both sides we get lim (z - 1) X [z] - x [0] = lim z"1

i. n

k

/ x [n + 1] - x [n]

k"3 n=0

o .c

lim (z - 1) X [z] - x [0] = lim {(x [1] - x [0]) + (x [2] - x [1]) + (x [3] - x [2]) + ... z"1 k"3 ... + (x [k + 1] - x [k])

a i d

lim (z - 1) X [z] - x [0] = x [3] - x [0] z"1 Hence, x [3] = lim (z - 1) X (z) z"1

o n

Summary of Properties

. w w

Let,

w

x [n]

Z

X (z),

with ROC Rx

x 1 [ n]

Z

X1 (z),

with ROC R1

x 2 [ n]

Z

X2 (z),

with ROC R2

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

ax1 [n] + bx2 [n] aX1 (z) + bX2 (z)

Time shifting (bilateral or non-causal)

x [n - n 0]

z-n X (z)

x [n + n 0]

zn X (z)

at least R1 + R2 Rx except for the possible deletion or addition of z = 0 or z = 3

0

0

z-n ^X (z) 0

x [n - n 0] Time shifting (unilateral or causal)

n0

/ x [- m] z

+

m

m=1

zn ^X (z) 0

x [n + n 0]

-

n0 - 1

/ x [m] z

-m

o

Rx except for the possible deletion or addition of z = 0 or z = 3

o

m=0

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Time domain

z -transform

Time reversal

x [- n]

Xb 1 l z

Differentiation in z domain

nx [n]

-z

Scaling in z domain

a n x [n]

Xa z k a

Time scaling (expansion)

xk [n] = x [n/k] X (zk )

Page 511 Chap 6

ROC

The Z-Transform

Time differencing

1/Rx

dX (z) dz

Rx

a Rx (Rx ) 1/k

-1

x [n] - x [n - 1] (1 - z ) X (z)

Rx , except for the possible deletion of the origin

i. n o c . a i d o n . w w w

Time convolution

x 1 [ n] * x 2 [ n]

X1 (z) X2 (z)

at least R1 + R2

Conjugations

x [n]

X (z )

Rx

x [0] = lim X (z)

provided for n < 0

x [n] = 0

provided exists

x [3]

)

)

Initial-value theorem

)

z"3

x [3] = lim x [n]

Final-value theorem

n"3

= lim (z - 1) X (z) z"1

6.6

ANALYSIS OF DISCRETE LTI SYSTEMS USING

z -TRANSFORM

The z -transform is very useful tool in the analysis of discrete LTI system. As the Laplace transform is used in solving differential equations which describe continuous LTI systems, the z -transform is used to solve difference equation which describe the discrete LTI systems. Similar to Laplace transform, for CT domain, the z -transform gives transfer function of the LTI 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 LTI system is always described by a linear constant coefficient difference equation given as follows N

M

/ a y [n - k ] = / b x [n - k ] k

k=0

k

k=0

aN y [n - N] + aN - 1 y [n - (N - 1)] + ....... + a1 y [n - 1] + a0 y [n]

(6.6.1) = bM x [n - M] + bM - 1 x [n - (M - 1)] + ..... + b1 x [n - 1] + b0 x [n] where, N is order of the system. Z The time-shift property of z -transform x [n - n 0] z-n X (z), is used to solve the above difference equation which converts it into an algebraic equation. By taking z -transform of above equation 0

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GATE STUDY PACKAGE Electronics & Communication 10 Subject-wise books by R. K. Kanodia General Aptitude Engineering Mathematics Networks Electronic Devices Analog Electronics Digital Electronics Signals & Systems Control Systems Communication Systems Electromagnetics aN z-N Y (z) + aN - 1 z- (N - 1) Y (z)] + ....... + a1 z-1 + a0 Y (z)

Page 512 Chap 6

= bM z-M X (z) + bM - 1 z- (M - 1) X (z) + ..... + b1 z-1 X (x) + b0 X (z)

The Z-Transform

-N M-1 Y (z) = bM z N + bM - 1 zN - 1 + ..... + b1 + b 0 X (z) + ..... + a1 + a 0 aN z + aN - 1 z this equation can be solved for Y (z) to find the response y [n].The solution or total response y [n] consists of two parts as discussed below.

1. Zero-input Response or Free Response or Natural Response The zero input response yzi [n] is mainly due to initial output in the system. The zero-input response is obtained from system equation (6.6.1) when input x [n] = 0 . By substituting x [n] = 0 and y [n] = yzi [n] in equation (6.6.1), we get =0 On taking z -transform of the above equation with given initial conditions, we can form an equation for Yzi (z). The zero-input response yzi [n] is given by inverse z -transform of Yzi (z). aN y [n - N] + aN - 1 y [n - (N - 1)] + ....... + a1 y [n - 1] + a0 y [n]

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

2. Zero-State Response or Forced Response

i. n

o .c

The zero-state response yzs [n] is the response of the system due to input signal and with zero initial conditions. The zero-state response is obtained from the difference equation (6.6.1) governing the system for specific input signal x [n] for n $ 0 and with zero initial conditions. Substituting y [n] = yzs [n] in equation (6.6.1) we get,

a i d

o n

aN yzs [n - N] + aN - 1 yzs [n - (N - 1)] + ....... + a1 yzs [n - 1] + a0 yzs [n] = bM x [n - M] + bM - 1 x [n - (M - 1)] + ..... + b1 x [n - 1] + b0 x [n]

. w w

Taking z -transform of the above equation with zero initial conditions for output (i.e., y [- 1] = y [- 2] ... = 0 we can form an equation for Yzs (z). The zero-state response yzs [n] is given by inverse z -transform of Yzs (z).

w

NOTE : The zero state response is also called the forced response of the system and it is denoted as y F [n].

Total Response The total response y [n] is the response of the system due to input signal and initial output. The total response can be obtained in following two ways : Taking z -transform of equation (6.6.1) with non-zero initial conditions for both input and output, and then substituting for X (z) we can form an equation for Y (z). The total response y [n] is given by inverse Laplace transform of Y (s). Alternatively, that total response y [n] is given by sum of zero-input response yzi [n] and zero-state response yzs [n]. Therefore total response, y [n] = yzi [n] + yzs [n]

6.6.2

Impulse Response and Transfer Function System function or transfer function is defined as the ratio of the z -transform of the output y [n] and the input x [n] with zero initial conditions. L Z Let x [n] Y (z) is the output of an LTI X (z) is the input and y [n] L discrete time system having impulse response h (n) H (z). The response Buy Online: shop.nodia.co.in

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

Page 513 Chap 6 The Z-Transform

Y (z) = X (z) H (z) Y (z) H (z) = X (z) where, H (z) is defined as the transfer function of the system. It is the z -transform of the impulse response. Alternatively we can say that the inverse z -transform of transfer function is the impulse response of the system. Impulse response Y (z) h [n] = Z-1 {H (z)} = Z-1 ) X (z) 3 6.7

STABILITY AND CAUSALITY OF LTI DISCRETE SYSTEMS USING -TRANSFORM

z

i. n o c . a i d o n . w w w

z -transform is also used in characterization of LTI discrete systems. In this section, we derive a z -domain condition to check the stability and causality of 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 impulse response h [n] = 0 , for n < 0 and it is therefore right-sided. The ROC of such a system H (z) is the exterior of a circle. If H (z) is rational then the system is said to be causal if 1. The ROC is the exterior of a circle outside the outermost pole ; and 2. 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 LTI discrete-time system is said to be BIBO stable if the impulse response h [n] is summable. That is 3

/

h [ n] < 3

n =- 3

z -transform of h [n] is given as H (z) =

3

/ h [n] z

-n

n =- 3 jW

Let z = e (which describes a unit circle in the z -plane), then H [e jW] =

3

/ h [n] e n =- 3 3

# =

-jWn

/ h [n] e

n =- 3 3

/

-jWn

h [ n] < 3

n =- 3

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

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STABILITY OF LTI DISCRETE SYSTEM

The Z-Transform

An LTI system is stable if the ROC of its system function H (z) contains the 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 causal and stable theses two conditions are satisfied if all the poles are within the unit circle in the z -plane. STABILITY AND CAUSALITY OF LTI DISCRETE SYSTEM

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

6.8

i. n

o .c

BLOCK DIAGRAM REPRESENTATION

In z -domain, the input-output relation of an LTI discrete time system is represented by the transfer function H (z) ,which is a rational function of z , as shown in equation Y (z) H (z) = X (z) M M-1 M-2 = b 0 z N + b1 z N - 1 + b2 z N - 2 + ... + bM - 1 z + bM + a2 z + ... + aN - 1 z + aN a 0 z + a1 z where, N = Order of the system, M # N and a 0 = 1 The above transfer function is realized using unit delay elements, unit advance elements, adders and multipliers. Basic elements of block diagram with their z -domain representation is shown in table 6.3.

a i d

o n

. w w

w

TABLE 6.3 : Basic Elements of Block Diagram

Elements of Block diagram

Time Domain Representation

s -domain Representation

Adder

Constant multiplier Unit delay element Unit advance element

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

6.8.1

Page 515 Chap 6 The Z-Transform

Direct Form I Realization Consider the difference equation governing the discrete time system with a 0 = 1, y [n] + a1 y [n - 1] + a2 y [n - 2] + .... + aN y [n - N] = b 0 x [n] + b1 x [n - 1] + b2 x [n - 2] + ... + bM x [n - M] Taking Z transform of the above equation we get, Y (z)=- a1 z-1 Y (z) - a2 z-2 Y (z) - ... - aN z-N Y (z) + (6.8.1) b 0 X (z) + b1 z-1 X (z) + b2 z-2 X (z) + ... + bM z-M X (z) 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-I structure. This structure uses separate delay elements for both input and output of the system. So, this realization uses more memory.

i. n o c . a i d o n . w w w Figure 6.8.1a General structure of direct form-realization

For example consider a discrete LTI system which has the following impulse response Y (z) 1 + 2z-1 + 2z-2 H (z) = = X (z) 1 + 4z-1 + 3z-2 Y (z) + 4z-1 Y (z) + 3z-2 Y (z) = 1X (z) + 2z-1 X (z) + 2z-2 X (z) Comparing with standard form of equation (6.8.1), we get a1 = 4 , a2 = 3 and b 0 = 1, b1 = 2 , b2 = 2 . Now put these values in general structure of Direct form-I realization we get

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GATE STUDY PACKAGE Electronics & Communication 10 Subject-wise books by R. K. Kanodia General Aptitude Engineering Mathematics Networks Electronic Devices Analog Electronics Digital Electronics Signals & Systems Control Systems Communication Systems Electromagnetics Page 516 Chap 6 The Z-Transform

Figure 6.8.1b

6.8.2

Direct Form II Realization Consider the general difference equation governing a discrete LTI system y [n] + a1 y [n - 1] + a2 y [n - 2] + .... + aN y [n - N] = b 0 x [n] + b1 x [n - 1] + b2 x [n - 2] + ... + bM x [n - M] Taking Z transform of the above equation we get,

i. n

Y (z)=- a1 z-1 Y (z) - a2 z-2 Y (z) - ... - aN z-N Y (z) + b 0 X (z) + b1 z-1 X (z) + b2 z-2 X (z) + ... + bM z-M X (z) It can be simplified as, Y (z) 61 + a1 z-1 + a2 z-2 + ... + aN z-N @ = X (z) 6b 0 + b1 z-1 + b2 z-2 + ... + bM z-M @ Y (z) W (z) Y (z) Let, = X (z) X (z) # W (z) where, W (z) 1 (6.8.2) = X (z) 1 + a1 z-1 + a2 z-2 + ... + aN z-N Y (z) (6.8.3) = b 0 + b1 z-1 + b2 z-2 + ... + bM z-M W (z) Equation (6.8.2) can be simplified as, W (z) + a1 z-1 W (z) + a2 z-2 W (z) + ... + aN z-N W (z) = X (z) (6.8.4) W (z) = X (z) - a1 z-1 W (z) - a2 z-2 W (z) - ... - aN z-N W (z) Similarly by simplifying equation (6.8.3), we get (6.8.5) Y (z) = b 0 W (z) + b1 z-1 W (z) + b2 z-2 W (z) + ... + bM z-M W (z) Equation (6.8.4) and (6.8.5) can be realized together by a direct structure called direct form-II structure as shown in figure 6.8.2a. It uses less number of delay elements then the Direct Form I structure. For example, consider the same transfer function H (z) which is discussed above Y (z) 1 + 2z-1 + 2z-2 H (z) = = X (z) 1 + 4z-1 + 3z-2 Y (z) Y (z) W (z) Let = X (z) W (z) # X (z) W (z) 1 where, , = -1 X (z) 1 + 4z + 3z-2 Y (z) = 1 + 2z-1 + 2z-2 W (z)

o .c

a i d

o n

. w w

w

W (z) = X (z) - 4z-1 W (z) - 3z-2 W (z) Y (z) = 1W (z) + 2z-1 W (z) + 2z-2 W (z) Comparing these equations with standard form of equation (6.8.4) and (6.8.5), we have a1 = 4 , a2 = 3 and b 0 = 1, b1 = 2, b2 = 2 . Substitute these so, and

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

Page 517 Chap 6 The Z-Transform

i. n o c . a i d o n . w w w

Figure 6.8.2a General structure of direct form-II realization

Figure 6.8.2b

6.8.3

Cascade Form The transfer function H (z) of a discrete time system can be expressed as a product of several transfer functions. Each of these transfer functions is realized in direct form-I or direct form II realization and then they are cascaded. Consider a system with transfer function (b + bk1 z-1 + bk2 z-2) (bm0 + bm1 z-1 + bm2 z-2) H (z) = k0 (1 + ak1 z-1 + ak2 z-2) (1 + am1 z-1 + am2 z-2) = H1 (z) H2 (z) -1 -2 where H1 (z) = bk0 + bk1 z-1 + bk2 z-2 1 + ak1 z + ak2 z

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GATE STUDY PACKAGE Electronics & Communication 10 Subject-wise books by R. K. Kanodia General Aptitude Engineering Mathematics Networks Electronic Devices Analog Electronics Digital Electronics Signals & Systems Control Systems Communication Systems Electromagnetics -1 -2 H2 (z) = bm0 + bm1 z-1 + bm2 z-2 1 + am1 z + am2 z Realizing H1 (z) and H2 (z) in direct form II and cascading we obtain cascade form of the system function H (z) as shown in figure 6.8.3.

Page 518 Chap 6 The Z-Transform

Figure 6.8.3 Cascaded form realization of discrete LTI system

6.8.4

i. n

Parallel Form

o .c

The transfer function H (z) of a discrete time system can be expressed as the sum of several transfer functions using partial fractions. Then the individual transfer functions are realized in direct form I or direct form II realization and connected in parallel for the realization of H (z). Let us consider the transfer function c1 c2 cN H (z) = c + -1 + -1 + ...... 1 - p1 z 1 - pz z 1 - pn z-1 Now each factor in the system is realized in direct form II and connected in parallel as shown in figure 6.8.4.

a i d

o n

. w w

w

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 Buy Online: shop.nodia.co.in

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

Page 519 Chap 6 The Z-Transform

3

/ x (nT) d (t - nT) n =- 3

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

3

/ x (nT) L {d (t - nT)} n =- 3 3

/ X (nT) e

-nTs

(6.9.1)

n =- 3

The z -transform of x [n] is given by X (z) =

3

/ x [ n] z

-n

(6.9.2)

n =- 3

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

sT

x [n] = x (nT)

i. n o c . a i d o n . w w w ***********

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EXERCISE 6.1

Page 520 Chap 6 The Z-Transform

MCQ 6.1.1

The z -transform and its ROC of a discrete time sequence n n 1 2z - 1 2 z-2 2 (C)

MCQ 6.1.2

MCQ 6.1.3

2z , 2z - 1

-1 (D) 2z , z-1

z 1 2

The ROC of z -transform of the discrete time sequence x [n] = ^ 12 h| n | is (A) 12 < z < 2 (B) z > 2 (C) - 2 < z < 2 (D) z < 12

i. n

n n Consider a discrete-time signal x [n] = b 1 l u [n] + b 1 l u [- n - 1]. The ROC 3 2 of its z -transform is (B) z < 1 (A) 3 < z < 2 2 (C) z > 1 (D) 1 < z < 1 3 3 2

o .c

MCQ 6.1.4

a i d

o n

For a signal x [n] = [an + a-n] u [n], the ROC of its z -transform would be (A) z > min e a , 1 o a (C) z > max e a , 1 o a

MCQ 6.1.5

. w w

w

(B) z > a (D) z < a

Match List I (discrete time sequence) with List II (z -transform) and choose the correct answer using the codes given below the lists: List-I (Discrete Time Sequence)

List-II (z -Transform)

P.

u [n - 2]

1.

1 , z-2 (1 - z-1)

Q.

- u [- n - 3]

2.

- z-1 , 1 - z-1

R.

u [n + 4]

3.

1 , z-4 (1 - z-1)

S.

u [- n]

4.

z-2 , 1 - z-1

Codes : P (A) 1 (B) 2 (C) 4 (D) 4

Q 4 4 1 2

R 2 1 3 3

z 1

S 3 3 2 1

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MCQ 6.1.7

and Systems (Vol-7, GATE Study Package)

The z -transform of signal x [n] = e jnp u [n] is (B) z , ROC : z > 1 (A) z , ROC : z > 1 z+1 z-j (C) 2 z , ROC : z < 1 (D) 1 , ROC : z < 1 z+1 z +1

Page 521 Chap 6 The Z-Transform

Consider the pole zero diagram of an LTI system shown in the figure which corresponds to transfer function H (z).

Match List I (The impulse response) with List II (ROC which corresponds to above diagram) and choose the correct answer using the codes given below: {Given that H (1) = 1}

i. n o c . a i d o n . w w w List-I (Impulse Response)

P.

List-II (ROC)

[(- 4) 2n + 6 (3) n] u [n]

1. 2.

z >3

n

3.

z 0.7 (C) z < 0.4 (D) none of these The z -transform of a DT signal x [n] is X (z) = 2 z . What will be the 8z - 2z - 1 z -transform of x [n - 4] ? (z + 4) 2 8 (z + 4) - 2 (z + 4) - 1 4z (C) 128z2 - 8z - 1

z5 8z2 - 2z - 1 (D) 5 1 4 8z - 2z - z3 (B)

(A)

MCQ 6.1.14

Let x1 [n], x2 [n] and x 3 [n] be three discrete time signals and X1 (z), X2 (z) and X 3 (z) are their z -transform respectively given as z2 , X1 (z) = (z - 1) (z - 0.5) z X2 (z) = (z - 1) (z - 0.5) 1 and X 3 (z) = (z - 1) (z - 0.5) Then x1 [n], x2 [n] and x 3 [n] are related as (A) x1 [n - 2] = x2 [n - 1] = x 3 [n] (B) x1 [n + 2] = x2 [n + 1] = x 3 [n] (C) x1 [n] = x2 [n - 1] = x 3 [n - 2] (D) x1 [n + 1] = x2 [n - 1] = x 3 [n]

i. n

o .c

MCQ 6.1.15

a i d

o n

. w w

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

w

z-k 1 - z-1 -k (C) 1 - z-1 1-z (A)

MCQ 6.1.16

MCQ 6.1.17

z-k 1 + z-1 -k (D) 1 + z-1 1-z

(B)

Z

Consider the unilateral z -transform pair x [n] -transform of x [n - 1] and x [n + 1] are respectively

X (z) =

(A)

z2 , 1 z-1 z-1

(B)

1 , z2 z-1 z-1

(C)

1 , z z-1 z-1

(D)

z , z2 z-1 z-1

z . The z z-1

A discrete time causal signal x [n] has the z -transform z , ROC : z > 0.4 X (z) = z - 0.4 The ROC for z -transform of the even part of x [n] will be Buy Online: shop.nodia.co.in

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(A) same as ROC of X (z) (C) z > 0.2

Page 523 Chap 6 The Z-Transform

MCQ 6.1.18

Match List I (Discrete time sequence) with List II (z -transform) and select the correct answer using the codes given below the lists. List-I (Discrete time sequence) P.

n (- 1) n u [n]

1.

z-1 , ROC : z > 1 (1 - z-1) 2

Q.

- nu [- n - 1]

2.

1 , ROC : z > 1 (1 + z-1)

R. (- 1) n u [n]

3.

z-1 , ROC : z < 1 (1 - z-1) 2

S.

4.

-

nu [n]

Q 1 3 1 4

R 2 2 4 1

S 3 1 2 3

A discrete time sequence is defined as x [n] = n1 (- 2) -n u [- n - 1]. The z -transform of x [n] is (B) log bz - 1 l, ROC : z < 1 (A) log bz + 1 l, ROC : z < 1 2 2 2 2 (C) log (z - 2), ROC : z > 2

MCQ 6.1.20

MCQ 6.1.21

z-1 , ROC : z > 1 (1 + z-1) 2

i. n o c . a i d o n . w w w

Codes : P (A) 4 (B) 4 (C) 3 (D) 2 MCQ 6.1.19

List-II (z -transform)

(D) log (z + 2), ROC : z < 2

Z

Consider a z -transform pair x [n] X (z) with ROC Rx . The z transform n and its ROC for y [n] = a x [n] will be (A) X a z k, ROC : a Rx a

(B) X (z + a), ROC : Rx

(C) z-a X (z), ROC : Rx

(D) X (az), ROC : a Rx

Let X (z) be the z -transform of a causal signal x [n] = an u [n] with ROC : z > a . Match the discrete sequences S1, S2, S 3 and S 4 with ROC of their z -transforms R1, R2 and R 3 . Sequences

ROC

S 1 : x [ n - 2]

R1 :

z >a

S 2 : x [ n + 2]

R2 :

z 1.2 (z - 1.2) 3

1.

Non causal but stable

Q. H (z) =

z2 , ROC : z < 1.2 (z - 1.2) 3

2.

Neither causal nor stable

R. H (z) =

z4 , ROC : z < 0.8 (z - 0.8) 3

3.

Causal but not stable

S. H (z) =

z3 , ROC : z > 0.8 (z - 0.8) 3

4.

Both causal and stable

w

Codes : P (A) 4 (B) 1 (C) 3 (D) 3 MCQ 6.1.39

List-II (Property of system)

Q 2 4 1 2

R 1 2 2 1

S 3 3 4 4

The transfer function of a DT feedback system is P H (z) = 1 + Pa z k z - 0.9 The range of P , for which the system is stable will be (A) - 1.9 < P < - 0.1 (B) P < 0 (C) P > - 1 (D) P > - 0.1 or P < - 1.9

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Sample Chapter of Signals MCQ 6.1.40

MCQ 6.1.41

MCQ 6.1.42

MCQ 6.1.43

Consider three stable LTI systems S1, S2 and S 3 whose transfer functions are z- 1 S1 : H (z) = 2 1 2 3 2z + 2 z - 16 S2 : H (z) = 2 -3 z 1+-12 4 -3z - 2z + 3 +z 1 + 12 z-2 - 43 z-1 S 3 : H (z) = -1 z ^1 - 13 z-1h^1 - 12 z-1h Which of the above systems is/are causal? (A) S1 only (B) S1 and S2 (C) S1 and S 3 (D) S1, S2 and S 3 The z -transform of d [n - k], k > 0 is (A) zk , z > 0 (C) zk , z ! 0

(B) z-k , z > 0 (D) z-k , z ! 0

The z -transform of d [n + k], k > 0 is (A) z-k , z ! 0 (C) z-k , all z

(B) zk , z ! 0 (D) zk , all z

MCQ 6.1.46

MCQ 6.1.47

The Z-Transform

i. n o c . a i d o n . w w w

1 , 1 - z-1 z , (C) 1 - z-1

MCQ 6.1.45

Page 527 Chap 6

The z -transform of u [n] is (A)

MCQ 6.1.44

and Systems (Vol-7, GATE Study Package)

1 , 1 - z-1 z , (D) 1 - z-1

z >1

(B)

z 0.25 z (z - 0.25) z5 - (0.25) 5 (C) 3 (D) , z < 0.25 z (z - 0.25) n The z -transform of b 1 l u [- n] is 4 (A) 4z , z > 1 4z - 1 4 1 , z >1 (C) 1 - 4z 4

The z -transform of 3n u [- n - 1] is (A) z , z > 3 3-z (C) 3 , z > 3 3-z

z >1

z5 - (0.25) 5 , z > 0. 5 z 4 (z - 0.25) z5 - (0.25) 5 , all z z 4 (z - 0.25)

4z , 4z - 1 1 , (D) 1 - 4z

(B)

z 1 , x [n] = ; n1- 1 - b - 1 l E u [n] 2 3 2 n (b) z < 1 , x [n] = ; -n -11 + b - 1 l E u [- n + 1] 3 3 2 n (c) 1 < z < 1 , x [n] =- n1- 1 u [- n - 1] - b - 1 l u [n] 3 2 3 2 Correct solution are (A) (a) and (b) (B) (a) and (c) (C) (b) and (c) (D) (a), (b), (c) Buy Online: shop.nodia.co.in

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Sample Chapter of Signals MCQ 6.1.65

The X (z) has poles at z = 12 and z =- 1. If x [1] = 1, x [- 1] = 1, and the ROC includes the point z = 34 . The time signal x [n] is (A) (C)

MCQ 6.1.66

and Systems (Vol-7, GATE Study Package)

2

1 u [n] + u [- n + 1] 2n - 1

If x [n] is right-sided, X (z) has a signal pole and x [0] = 2, x [2] = 1 , then x [n] 2 is u [ - n] 2n - 1 u [ - n] (C) n + 1 2

u [n] 2n - 1 u [ - n] (D) a n + 1 2 (B)

(A)

MCQ 6.1.67

The Z-Transform

(B) 1n u [n] - (- 1) n u [- n - 1] 2 (D) 1n u [n] + u [- n + 1] 2

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

Page 531 Chap 6

n n The z -transform of b 1 l u [n] + b 1 l u [- n - 1] is 2 4 1 1 (A) , 1< z r0 will also be in the ROC (B) the values of z for which z < r0 will also be in the ROC (C) both (A) & (B) (D) none of these

MCQ 6.3.9

The ROC does not contain any (A) poles (C) zeros

(B) 1’s (D) none

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Sample Chapter of Signals MCQ 6.3.10

and Systems (Vol-7, GATE Study Package)

Let x [n] X (z) be a z -transform pair. If x [n] = d [n], then the ROC of X (z) is (A) z < 1 (B) z > 1 (C) entire z -plane (D) none of the above Z

MCQ 6.3.11

The ROC of z -transform of unit-step sequence u [n], is (A) entire z -plane (B) z < 1 (C) z > 1 (D) none of the above

MCQ 6.3.12

The (A) (B) (C) (D)

MCQ 6.3.13

Which of the following statement about ROC is not true ? (A) ROC never lies exactly at the boundary of a circle (B) ROC consists of a circle in the z -plane centred at the origin (C) ROC of a right handed finite sequence is the entire z -plane except z = 0 (D) ROC contains both poles and zeroes

Page 539 Chap 6 The Z-Transform

ROC of the unilateral z -transform of an is z > a z < a z 1

i. n o c . a i d o n . w w w

MCQ 6.3.14

The z -transform of unit step sequence is (A) 1 (B) z -1 1 (C) z -z 1 (D) 0

MCQ 6.3.15

The ROC for the z -transform of the sequence x [n] = u [- n] is (A) z > 0 (B) z < 1 (C) z > 1 (D) does not exist

MCQ 6.3.16

Let x [n] X (z), then unilateral z -transform of sequence x1 [n] = x [n - 1] will be (A) X1 (z) = z-1 X (z) + x [0] (B) X1 (z) = z-1 X (z) - x [1] (C) X1 (z) = z-1 X (z) - x [- 1] (D) X1 (z) = z-1 X [z] + x [- 1]

MCQ 6.3.17

Let x [n]

Z

Z

(A) zX (z) (C) z-n X (z) 0

X (z), the bilateral z -transform of x [n - n 0] is given by (B) zn X (z) (D) 1 X (z) z 0

MCQ 6.3.18

If the ROC of z -transform of x [n] is Rx then the ROC of z -transform of x [- n] is (A) Rx (B) - Rx (C) 1/Rx (D) none of these

MCQ 6.3.19

If X (z) = Z {x [n]} {, then X (z) = Z {a-n x [n]} { will be (B) X a z k (A) X (az) a (C) X a a k (D) X b 1 l z az Buy Online: shop.nodia.co.in

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GATE STUDY PACKAGE Electronics & Communication 10 Subject-wise books by R. K. Kanodia General Aptitude Engineering Mathematics Networks Electronic Devices Analog Electronics Digital Electronics Signals & Systems Control Systems Communication Systems Electromagnetics Page 540 Chap 6

MCQ 6.3.20

If x [n] and y [n] are two discrete time sequences, then the z -transform of correlation of the sequences x [n] and y [n] is (A) X (z-1) Y (z-1) (B) X (z) Y (z-1) (C) X (z) * Y (z) (D) X * (z) Y * (z-1)

MCQ 6.3.21

If X (z) = Z {x [n]}, then, value of x [0] is equal to (A) lim zX (z) (B) lim (z - 1) X (z) z"0 z"1 (C) lim X (z) (D) lim X (z)

The Z-Transform

z"3

z"0

MCQ 6.3.22

The choice of realization of structure depends on (A) computational complexity (B) memory requirements (C) parallel processing and pipelining (D) all the above

MCQ 6.3.23

Which of the following schemes of system realization uses separate delays for input and output samples ? (A) parallel form (B) cascade form (C) direct form-I (D) direct form-II

MCQ 6.3.24

The direct form-I and II structures of IIR system will be identical in (A) all pole system (B) all zero system (C) both (A) and (B) (D) first order and second order systems

MCQ 6.3.25

i. n

o .c

a i d

o n

The number of memory locations required to realize the system, -2 -3 H (z) = 1 + 3z -2+ 2z-4 is 1 + 2z + z (A) 5 (B) 7 (C) 2 (D) 10

. w w

w

MCQ 6.3.26

The mapping z = esT from s -plane to z -plane, is (A) one to one (B) many to one (C) one to many (D) many to many

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EXERCISE 6.4

Page 541 Chap 6 The Z-Transform

MCQ 6.4.1 IES EC 2007

What is the z -transform of the signal x [n] = an u [n] ? 1 z-1 (C) X (z) = z z-a

1 1-z (D) X (z) = 1 z-a

(A) X (z) =

(B) X (z) =

MCQ 6.4.2

The z -transform of the time function

GATE EC 1998

(A) z - 1 z (C)

3

/ d [n - k] is k=0

z (z - 1) 2

(B)

z z-1

(D)

(z - 1) 2 z

i. n o c . a i d o n . w w w

MCQ 6.4.3

The z -transform F (z) of the function f (nT) = anT is

GATE EC 1999

z z - aT z (C) z - a-T

MCQ 6.4.4

The discrete-time signal x [n] X (z) = n3= 0 2 +3 n z2n , where a transform-pair relationship, is orthogonal to the signal n (A) y1 [n] ) Y1 (z) = n3= 0 ` 2 j z - n ( B 3n 3 - (2n + 1) y2 [n] ) Y2 (z) = n = 0 (5 - n) z

GATE EE 2006

(A)

z z + aT z (D) z + a-T

(B)

Z

/ / (C) y3 [n] ) Y3 (z) = / n3=- 3 2 - n z - n

MCQ 6.4.5 IES E & T 1994

MCQ 6.4.6 IES EC 2002

MCQ 6.4.7 IES EC 2006

/

n

denotes )

(D) y4 [n] ) Y4 (z) = 2z - 4 + 3z - 2 + 1

Which one of the following is the region of convergence (ROC) for the sequence x [n] = bn u [n] + b-n u [- n - 1]; b < 1 ? (A) Region z < 1 (B) Annular strip in the region b > z > 1 b (C) Region z > 1 (D) Annular strip in the region b < z < 1 b Assertion (A) : The signals an u [n] and - an u [- n - 1] have the same z -transform, z/ (z - a). Reason (R) : The Reason of Convergence (ROC) for an u [n] is z > a , whereas the ROC for an u [- n - 1] is z < a . (A) Both A and R are true and R is the correct explanation of A (B) Both A and R are true but R is NOT the correct explanation of A (C) A is true but R is false (D) A is false but R is true Which one of the following is the correct statement ? The region of convergence of z -transform of x [n] consists of the values of z for which x [n] r-n is (A) absolutely integrable (B) absolutely summable (C) unity (D) < 1 Buy Online: shop.nodia.co.in

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GATE STUDY PACKAGE Electronics & Communication 10 Subject-wise books by R. K. Kanodia General Aptitude Engineering Mathematics Networks Electronic Devices Analog Electronics Digital Electronics Signals & Systems Control Systems Communication Systems Electromagnetics Page 542 Chap 6 The Z-Transform

MCQ 6.4.8 GATE EC 2009

MCQ 6.4.9 GATE EC 2005

MCQ 6.4.10 GATE IN 2008

MCQ 6.4.11 GATE EC 2001

MCQ 6.4.12 IES EC 2005

n n The ROC of z -transform of the sequence x [n] = b 1 l u [n] - b 1 l u [- n - 1] is 3 2 (A) 1 < z < 1 (B) z > 1 3 2 2 (C) z < 1 (D) 2 < z < 3 3 The region of convergence of z - transform of the sequence 5 n 6 n b 6 l u [n] - b 5 l u [- n - 1] must be (A) z < 5 (B) z > 5 6 6 (C) 5 < z < 6 (D) 6 < z < 3 6 5 5 The region of convergence of the z -transform of the discrete-time signal x [n] = 2n u [n] will be (A) z > 2 (B) z < 2 (C) z > 1 (D) z < 1 2 2

The region of convergence of the z - transform of a unit step function is (A) z > 1 (B) z < 1 (C) (Real part of z ) > 0 (D) (Real part of z ) < 0

i. n

o .c

Match List I (Discrete Time signal) with List II (Transform) and select the correct answer using the codes given below the lists : List I

a i d

o n

A. Unit step function

. w w

2.

z - cos wT z - 2z cos wT + 1

C. sin wt, t = 0, T, 2T

3.

z z-1

D. cos wt, t = 0, T, 2T, .....

4.

z sin wT z - 2z cos wT + 1

Codes : A (A) 2 (B) 3 (C) 2 (D) 3

IES EC 2006

MCQ 6.4.14 IES E & T 1997

1. 1

B. Unit impulse function

w MCQ 6.4.13

List II

B 4 1 1 4

C 1 4 4 1

2

2

D 3 2 3 2

What is the inverse z -transform of X (z) (A) 1 # X (z) zn - 1 dz (B) 2pj # X (z) zn + 1 dz 2pj (C) 1 # X (z) z1 - n dz (D) 2pj # X (z) z- (n + 1) dz 2pj Which one of the following represents the impulse response of a system defined by H (z) = z-m ? (A) u [n - m] (B) d [n - m] (C) d [m] (D) d [m - n]

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1 with z > 1, then what is the corresponding x [n] ? 1 - z-1 (B) en (D) d (n)

Page 543 Chap 6

MCQ 6.4.15

If X (z) is

IES EC 2008

(A) e-n (C) u [n]

MCQ 6.4.16

The z -transform X (z) of a sequence x [n] is given by X [z] = 1 -0.25z . It is given that the region of convergence of X (z) includes the unit circle. The value of x [0] is (A) - 0.5 (B) 0 (C) 0.25 (D) 05

GATE EC 2007

MCQ 6.4.17 GATE EE 2005

MCQ 6.4.18 IES EC 2002

-1

If u (t) is the unit step and d (t) is the unit impulse function, the inverse z -transform of F (z) = z +1 1 for k > 0 is (A) (- 1) k d (k)

(B) d (k) - (- 1) k

(C) (- 1) k u (k)

(D) u (k) - (- 1) k

5 ^2z - 6 h 1 1 ^z - 2 h^z - 3 h Match List I (The sequences) with List II (The region of convergence ) and select the correct answer using the codes given below the lists :

For a z -transform X (z) =

i. n o c . a i d o n . w w w List I

IES EC 2005

MCQ 6.4.20 GATE EE 2008

List II

A. [(1/2) n + (1/3) n] u [n]

1. (1/3) < z < (1/2)

B. (1/2) n u [n] - (1/3) n u [- n - 1]

2.

z < (1/3)

C. - (1/2) n u [- n - 1] + (1/3) n u [n]

3.

z < 1/3 & z > 1/2

D. - [(1/2) n + (1/3) n] u [- n - 1]

4.

z > 1/2

Codes : A (A) 4 (B) 1 (C) 4 (D) 1 MCQ 6.4.19

B 2 3 3 2

C 1 4 1 4

D 3 2 2 3

Which one of the following is the inverse z -transform of z X (z) = , z a , the residue of X (z) zn - 1 at z = a for 2 ( z a ) n $ 0 will be (A) an - 1 (B) an (D) nan - 1

(C) nan MCQ 6.4.21 GATE IN 2004

The Z-Transform

1 2

1

3 + , a and b < 1 with the ROC specified as 1 - az-1 1 - bz-1 a < z < b , then x [0] of the corresponding sequence is given by (B) 5 (A) 1 3 6 (C) 1 (D) 1 2 6

Given X (z) =

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GATE STUDY PACKAGE Electronics & Communication 10 Subject-wise books by R. K. Kanodia General Aptitude Engineering Mathematics Networks Electronic Devices Analog Electronics Digital Electronics Signals & Systems Control Systems Communication Systems Electromagnetics Page 544 Chap 6

MCQ 6.4.22

The Z-Transform

IES EC 2002

MCQ 6.4.23 GATE EC 2010

MCQ 6.4.24 GATE IN 2003

MCQ 6.4.25 GATE EC 2006

MCQ 6.4.26 IES EC 2000

-3 If X (z) = z + z-1 then x [n] series has z+z (A) alternate 0’s (C) alternate 2’s

Consider the z -transform x (z) = 5z2 + 4z-1 + 3; 0 < z < 3. The inverse z - transform x [n] is (A) 5d [n + 2] + 3d [n] + 4d [n - 1] (B) 5d [n - 2] + 3d [n] + 4d [n + 1] (C) 5u [n + 2] + 3u [n] + 4u [n - 1] (D) 5u [n - 2] + 3u [n] + 4u [n + 1] The sequence x [n] whose z -transform is X (z) = e1/z is (B) 1 u [- n] (A) 1 u [n] n! - n! 1 (C) (- 1) n 1 u [n] (D) u [- n - 1] n! - (n + 1) ! If the region of convergence of x1 [n] + x2 [n] is 1 < z < 2 then the region of 3 3 convergence of x1 [n] - x2 [n] includes (A) 1 < z < 3 (B) 2 < z < 3 3 3 (C) 3 < z < 3 (D) 1 < z < 2 2 3 3 Match List I with List II and select the correct answer using the codes given below the lists :

i. n

o .c

a i d

List I

o n

A. an u [n]

. w w

B. - an u [- n - 1] n

C. - na u [- n - 1]

w

D. nan u [n]

Codes : A (A) 2 (B) 1 (C) 1 (D) 2

MCQ 6.4.27 IES EC 2007

(B) alternate 1’s (D) alternate - 1’s

B 4 3 4 3

C 3 4 3 4

List II

1.

az-1 , ROC : z > a (1 - az-1) 2

2.

1 , ROC : z > a (1 - az-1)

3.

1 , ROC : z < | a (1 - az-1)

4.

az-1 , ROC : z < | a (1 - az-1) 2

D 1 2 2 1

Algebraic expression for z -transform of expression for z -transform of {e j w n x [n]} (A) X (z - z 0) (C) X (e jw z) 0

0

MCQ 6.4.28 IES E & T 1997

x [n] is X [z]. What is the algebraic ? (B) X (e-j w z) (D) X (z) e j w z 0

0

Given that F (z) and G (z) are the one-sided z -transforms of discrete time functions f (nT) and g (nT), the z -transform of / f (kT) g (nT - kT) is given by (A) (C)

/ f (nT) g (nT) z-n / f (kT) g (nT - kT) z-n

(B) (D)

/ f (nT) z-n / g (nT) z-n / f (nT - kT) g (nT) z-n

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Sample Chapter of Signals MCQ 6.4.29 IES E & T 1997

and Systems (Vol-7, GATE Study Package)

Match List-I (x [n]) with List-II (X (z)) and select the correct answer using the codes given below the Lists:

Page 545 Chap 6 The Z-Transform

List-I A.

a n u [ n]

1.

az (z - a) 2

B.

an - 2 u [n - 2]

2.

ze-j ze-j - a

C.

e jn an

3.

z z-a

D.

nan u [n]

4.

z-1 z-a

Codes : A (A) 3 (B) 2 (C) 3 (D) 1 MCQ 6.4.30 IES EC 2005

List-II

B 2 3 4 4

C 4 4 2 2

D 1 1 1 3

i. n o c . a i d o n . w w w

The output y [n] of a discrete time LTI system is related to the input x [n] as given below : y [n] =

3

/ x [k ]

k=0

MCQ 6.4.31 IES EC 2010

Which one of the following correctly relates the z -transform of the input and output, denoted by X (z) and Y (z), respectively ? (A) Y (z) = (1 - z-1) X (z) (B) Y (z) = z-1 X (z) X (z) dX (z) (C) Y (z) = (D) Y (z) = -1 dz 1-z Convolution of two sequence x1 [n] and x2 [n] is represented as (A) X1 (z) * X2 (z) (B) X1 (z) X2 (z) (C) X1 (z) + X2 (z) (D) X1 (z) /X2 (z) 1z-1 (1 - z-4) . Its final value is 4 (1 - z-1) 2 (B) zero (D) infinity

MCQ 6.4.32

The z -transform of a signal is given by C (z) =

GATE EC 1999

(A) 1/4 (C) 1.0

MCQ 6.4.33

Consider a system described by the following difference equation: y (n + 3) + 6y (n + 2) + 11y (n + 1) + 6y (n)= r (n + 2) + 9r (n + 1) + 20r (n) where y is the output and r is the input. The transfer function of the system is 2 2 (B) 3 z +29z + 20 (A) 32z + 2z + 20 3z + 2z + z + 6 z + 6z + 6z + 11 3 2 (C) z +26z + 6z + 11 (D) none of the above z + 9z + 20

IES E & T 1996

MCQ 6.4.34 IES E & T 1998

If the function H1 (z) = (1 + 1.5z-1 - z-2) and H2 (z) = z2 + 1.5z - 1, then (A) the poles and zeros of the functions will be the same (B) the poles of the functions will be identical but not zeros (C) the zeros of the functions will be identical but not the poles (D) neither the poles nor the zeros of the two functions will be identical Buy Online: shop.nodia.co.in

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GATE STUDY PACKAGE Electronics & Communication 10 Subject-wise books by R. K. Kanodia General Aptitude Engineering Mathematics Networks Electronic Devices Analog Electronics Digital Electronics Signals & Systems Control Systems Communication Systems Electromagnetics Page 546 Chap 6

MCQ 6.4.35

The state model 0 1 0 x [k + 1] = > x [k ] + > H u [k ] - b - aH 1 x 1 [k ] y [k] = 80 1B> x2 [k]H

IES EC 1999

The Z-Transform

is represented in the difference equation as (A) c [k + 2] + ac [k + 1] + bc [k] = u [k] (B) c [k + 1] + ac [k] + bc [k - 1] = u [k - 1] (C) c [k - 2] + ac [k - 1] + bc [k] = u [k] (D) c [k - 1] + ac [k] + bc [k + 1] = u [k + 1] MCQ 6.4.36 IES EC 2000

The impulse response of a discrete system with a simple pole shown in the figure below. The pole of the system must be located on the

i. n

(A) (B) (C) (D) MCQ 6.4.37 IES EC 2001

o .c

a i d

real axis at z =- 1 real axis between z = 0 and z = 1 imaginary axis at z = j imaginary axis between z = 0 and z = j

. w w

o n

Which one of the following digital filters does have a linear phase response ? (A) y [n] + y [n - 1] = x [n] - x [n - 1] (B) y [n] = 1/6 (3x [n] + 2x [n - 1] + x [n - 2]) (C) y [n] = 1/6 (x [n] + 2x [n - 1] + 3x [n - 2]) (D) y [n] = 1/4 (x [n] + 2x [n - 1] + x [n - 2])

w MCQ 6.4.38 IES EC 2001

MCQ 6.4.39 IES EC 2001

The poles of a digital filter with linear phase response can lie (A) only at z = 0 (B) only on the unit circle (C) only inside the unit circle but not at z = 0 (D) on the left side of Real (z) = 0 line The impulse response of a discrete system with a simple pole is shown in the given figure

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Sample Chapter of Signals The pole must be located (A) on the real axis at z = 1 (C) at the origin of the z-plane MCQ 6.4.40 IES EC 2002

and Systems (Vol-7, GATE Study Package) Page 547 Chap 6

(B) on the real axis at z =- 1 (D) at z = 3

The Z-Transform

The response of a linear, time-invariant discrete-time system to a unit step input u [n] is the unit impulse d [n]. The system response to a ramp input nu [n] would be (A) u [n] (B) u [n - 1] (C) nd [n]

(D)

3

/ k d [n - k] k=0

MCQ 6.4.41

A system can be represented in the form of state equations as

IES EC 2002

s [n + 1] = As [n] + Bx [n] y [n] = Cs [n] + Dx [n] where A, B, C and D are matrices, s [n] is the state vector. x [n] is the input and y [n] is the output. The transfer function of the system H (z) = Y (z) /X (z) is given by (A) A (zI - B) -1 C + D (B) B (zI - C ) -1 D + A (C) C (zI - A) -1 B + D (D) D (zI - A) -1 C + B

MCQ 6.4.42 IES EC 2004

MCQ 6.4.43

i. n o c . a i d o n . w w w

What is the number of roots of the polynomial F (z) = 4z3 - 8z2 - z + 2 , lying outside the unit circle ? (A) 0 (B) 1 (C) 2 (D) 3 y [ n] =

n

/ x [k]

k =- 3

IES EC 2004

Which one of the following systems is inverse of the system given above ? (A) x [n] = y [n] - y [n - 1] (B) x [n] = y [n] (C) x [n] = y [n + 4] (D) x [n] = ny [n]

MCQ 6.4.44

For the system shown, x [n] = kd [n], and y [n] is related to x [n] as = x [n] y [n] - 12 y [n - 1]

IES EC 2006

What is y [n] equal to ? (A) k (C) nk MCQ 6.4.45 IES EC 2010

MCQ 6.4.46 IES EC 2011

(B) (1/2) n k (D) 2n

Unit step response of the system described by the equation y [n] + y [n - 1] = x [n] is z2 z (B) (A) (z + 1) (z - 1) (z + 1) (z - 1) z (z - 1) (C) z + 1 (D) z-1 (z + 1) Unit step response of the system described by the equation y [n] + y [n - 1] = x [n] is z2 z (B) (A) (z + 1) (z - 1) (z + 1) (z - 1) (z + 1) z (z - 1) (C) (D) (z - 1) (z + 1) Buy Online: shop.nodia.co.in

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GATE STUDY PACKAGE Electronics & Communication 10 Subject-wise books by R. K. Kanodia General Aptitude Engineering Mathematics Networks Electronic Devices Analog Electronics Digital Electronics Signals & Systems Control Systems Communication Systems Electromagnetics Page 548 Chap 6

MCQ 6.4.47 IES EC 2011

The Z-Transform

MCQ 6.4.48 GATE EC 2009

MCQ 6.4.49 GATE EC 2004

MCQ 6.4.50 GATE EC 2003

System transformation function H (z) for a discrete time LTI system expressed in state variable form with zero initial conditions is (A) c (zI - A) -1 b + d (B) c (zI - A) -1 (C) (zI - A) -1 z (D) (zI - A) -1 A system with transfer function H (z) has impulse response h (.) defined as h (2) = 1, h (3) =- 1 and h (k) = 0 otherwise. Consider the following statements. S1 : H (z) is a low-pass filter. S2 : H (z) is an FIR filter. Which of the following is correct? (A) Only S2 is true (B) Both S1 and S2 are false (C) Both S1 and S2 are true, and S2 is a reason for S1 (D) Both S1 and S2 are true, but S2 is not a reason for S1 The z -transform of a system is H (z) = impulse response of the system is (A) (0.2) n u [n] (B) (0.2) n u [- n - 1] (C) - (0.2) n u [n] (D) - (0.2) n u [- n - 1]

z z - 0.2

. If the ROC is z < 0.2 , then the

i. n

o .c

a i d

A sequence x (n) with the z -transform X (z) = z 4 + z2 - 2z + 2 - 3z-4 is applied as an input to a linear, time-invariant system with the impulse response h [n] = 2d [n - 3] where 1, n = 0 d [ n] = ) 0, otherwise The output at n = 4 is (A) - 6 (B) zero (C) 2 (D) - 4

o n

. w w

w

MCQ 6.4.51

GATE EE 2009

The z-transform of a signal x [n] is given by 4z - 3 + 3z - 1 + 2 - 6z2 + 2z3 It is applied to a system, with a transfer function H (z) = 3z - 1 - 2 Let the output be y [n]. Which of the following is true ? (A) y [n] is non causal with finite support (B) y [n] is causal with infinite support (C) y [n] = 0; n > 3 (D) Re [Y (z)] z = e =- Re [Y (z)] z = e ji

- ji

Im [Y (z)] z = e = Im [Y (z)] z = e ; - p # q < p ji

MCQ 6.4.52 GATE EE 2008

- ji

H (z) is a transfer function of a real system. When a signal x [n] = (1 + j) n is the input to such a system, the output is zero. Further, the Region of convergence (ROC) of ^1 - 12 z - 1h H(z) is the entire Z-plane (except z = 0 ). It can then be inferred that H (z) can have a minimum of (A) one pole and one zero (B) one pole and two zeros (C) two poles and one zero (D) two poles and two zeros Buy Online: shop.nodia.co.in

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Sample Chapter of Signals MCQ 6.4.53 GATE IN 2004

and Systems (Vol-7, GATE Study Package)

A discrete-time signal, x [n], suffered a distortion modeled by an LTI system with H (z) = (1 - az-1), a is real and a > 1. The impulse response of a stable system that exactly compensates the magnitude of the distortion is n (A) b 1 l u [n] a n (B) -b 1 l u [- n - 1] a

Page 549 Chap 6 The Z-Transform

(C) an u [n] (D) an u [- n - 1] MCQ 6.4.54 IES E & T 1998

MCQ 6.4.55 IES EC 1999

MCQ 6.4.56 IES EC 1999

MCQ 6.4.57 IES EC 2005

Assertion (A) : A linear time-invariant discrete-time system having the system function H (z) = z 1 is a stable system. z+ 2 Reason (R) : The pole of H (z) is in the left-half plane for a stable system. (A) Both A and R are true and R is the correct explanation of A (B) Both A and R are true but R is NOT a correct explanation of A (C) A is true but R is false (D) A is false but R is true

i. n o c . a i d o n . w w w

Assertion (A) : An LTI discrete system represented by the difference equation y [n + 2] - 5y [n + 1] + 6y [n] = x [n] is unstable. Reason (R) : A system is unstable if the roots of the characteristic equation lie outside the unit circle. (A) Both A and R are true and R is the correct explanation of A. (B) Both A and R are true but R is NOT the correct explanation of A. (C) A is true but R is false. (D) A is false but R is true. Consider the following statements regarding a linear discrete-time system z2 + 1 H (z) = (z + 0.5) (z - 0.5) 1. The system is stable 2. The initial value h (0) of the impulse response is - 4 3. The steady-state output is zero for a sinusoidal discrete time input of frequency equal to one-fourth the sampling frequency. Which of these statements are correct ? (A) 1, 2 and 3 (B) 1 and 2 (C) 1 and 3 (D) 2 and 3 Assertion (A) : The discrete time system described by y [n] = 2x [n] + 4x [n - 1] is unstable, (here y [n] is the output and x [n] the input) Reason (R) : It has an impulse response with a finite number of non-zero samples. (A) Both A and R are true and R is the correct explanation of A (B) Both A and R are true but R is NOT the correct explanation of A (C) A is true but R is false (D) A is false but R is true

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GATE STUDY PACKAGE Electronics & Communication 10 Subject-wise books by R. K. Kanodia General Aptitude Engineering Mathematics Networks Electronic Devices Analog Electronics Digital Electronics Signals & Systems Control Systems Communication Systems Electromagnetics Page 550 Chap 6

MCQ 6.4.58 GATE EC 2002

The Z-Transform

MCQ 6.4.59 GATE IN 2010

MCQ 6.4.60

If the impulse response of discrete - time system is h [n] =- 5n u [- n - 1], then the system function H (z) is equal to (A) - z and the system is stable z-5 (B) z and the system is stable z-5 (C) - z and the system is unstable z-5 (D) z and the system is unstable z-5 H (z) is a discrete rational transfer function. To ensure that both H (z) and its inverse are stable its (A) poles must be inside the unit circle and zeros must be outside the unit circle. (B) poles and zeros must be inside the unit circle. (C) poles and zeros must be outside the unit circle (D) poles must be outside the unit circle and zeros should be inside the unit circle

i. n

Assertion (A) : The stability of the system is assured if the Region of Convergence (ROC) includes the unit circle in the z -plane. Reason (R) : For a causal stable system all the poles should be outside the unit circle in the z -plane. (A) Both A and R are true and R is the correct explanation of A (B) Both A and R are true but R is NOT the correct explanation of A. (C) A is true but R is false (D) A is false but R is true

o .c

IES EC 2002

MCQ 6.4.61 IES EC 2002

a i d

o n

. w w

Assertion (A) : For a rational transfer function H (z) to be causal, stable and causally invertible, both the zeros and the poles should lie within the unit circle in the z -plane. Reason (R) : For a rational system, ROC bounded by poles. (A) Both A and R are true and R is the correct explanation of A (B) Both A and R are true but R is NOT the correct explanation of A (C) A is true but R is false (D) A is false but R is true

w

2 - 34 z-1 1 - 34 z-1 + 18 z-2

MCQ 6.4.62

The transfer function of a discrete time LTI system is H (z) =

GATE EC 2010

Consider the following statements: S1: The system is stable and causal for ROC: z > 1/2 S2: The system is stable but not causal for ROC: z < 1/4 S3: The system is neither stable nor causal for ROC: 1/4 < z < 1/2 Which one of the following statements is valid ? (A) Both S1 and S2 are true (B) Both S2 and S3 are true (C) Both S1 and S3 are true (D) S1, S2 and S3 are all true

MCQ 6.4.63

A causal LTI system is described by the difference equation

GATE EC 2004

2y [n] = ay [n - 2] - 2x [n] + bx [n - 1] The system is stable only if (A) a = 2 , b < 2 (B) a > 2, b > 2 (C) a < 2 , any value of b (D) b < 2 , any value of a Buy Online: shop.nodia.co.in

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Sample Chapter of Signals MCQ 6.4.64 IES EC 2000

and Systems (Vol-7, GATE Study Package)

Two linear time-invariant discrete time systems s1 and s2 are cascaded as shown in the figure below. Each system is modelled by a second order difference equation. The difference equation of the overall cascaded system can be of the order of

Page 551 Chap 6 The Z-Transform

(A) 0, 1, 2, 3 or 4 (B) either 2 or 4 (C) 2 (D) 4 MCQ 6.4.65 IES EC 2000

Consider the compound system shown in the figure below. Its output is equal to the input with a delay of two units. If the transfer function of the first system is given by H1 (z) = z - 0.5 , z - 0.8

i. n o c . a i d o n . w w w

then the transfer function of the second system would be -2 -3 -2 -3 (B) H2 (z) = z - 0.8z-1 (A) H2 (z) = z - 0.2z-1 1 - 0.4z 1 - 0.5z -1 -3 -2 -3 (C) H2 (z) = z - 0.2z-1 (D) H2 (z) = z + 0.8z-1 1 - 0.4z 1 + 0.5z MCQ 6.4.66 GATE EC 2011

Two systems H1 (z ) and H2 (z ) are connected in cascade as shown below. The overall output y [n] is the same as the input x [n] with a one unit delay. The transfer function of the second system H2 (z ) is

1 - 0.6z-1 z-1 (1 - 0.4z-1) z-1 (1 - 0.4z-1) (C) (1 - 0.6z-1) (A)

MCQ 6.4.67 GATE EC 2010

MCQ 6.4.68 GATE EE 2009

(B)

z-1 (1 - 0.6z-1) (1 - 0.4z-1)

(D)

1 - 0.4 z-1 z-1 (1 - 0.6z-1)

Two discrete time system with impulse response h1 [n] = d [n - 1] and h2 [n] = d [n - 2] are connected in cascade. The overall impulse response of the cascaded system is (A) d [n - 1] + d [n - 2] (B) d [n - 4] (C) d [n - 3] (D) d [n - 1] d [n - 2] A cascade of three Linear Time Invariant systems is causal and unstable. From this, we conclude that (A) each system in the cascade is individually causal and unstable (B) at least on system is unstable and at least one system is causal (C) at least one system is causal and all systems are unstable (D) the majority are unstable and the majority are causal

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GATE STUDY PACKAGE Electronics & Communication 10 Subject-wise books by R. K. Kanodia General Aptitude Engineering Mathematics Networks Electronic Devices Analog Electronics Digital Electronics Signals & Systems Control Systems Communication Systems Electromagnetics Page 552 Chap 6

MCQ 6.4.69 IES EC 2001

The Z-Transform

MCQ 6.4.70 GATE IN 2004

The minimum number of delay elements required in realizing a digital filter with the transfer function 1 + az-1 + bz-2 H (z) = 1 + cz-1 + dz-2 + ez-3 (A) 2 (B) 3 (C) 4 (D) 5 1 A direct form implementation of an LTI system with H (z) = 1 - 0.7z-1 + 0.13z-2 is shown in figure. The value of a 0, a1 and a2 are respectively

(A) 1.0, 0.7 and - 0.13 (C) 1.0, - 0.7 and 0.13 MCQ 6.4.71 GATE IN 2010

(B) - 0.13, 0.7 and 1.0 (D) 0.13, - 0.7 and 1.0

i. n

A digital filter having a transfer function H (z) = implemented

o .c

p 0 + p1 z-1 + p 3 z-3 is 1 + d 3 z-3

using Direct Form-I and Direct Form-II realizations of IIR structure. The number of delay units required in Direct Form-I and Direct Form-II realizations are, respectively (A) 6 and 6 (B) 6 and 3 (C) 3 and 3 (D) 3 and 2 MCQ 6.4.72 GATE EE 2007

a i d

o n

. w w

Consider the discrete-time system shown in the figure where the impulse response of G (z) is g (0) = 0, g (1) = g (2) = 1, g (3) = g (4) = g = 0

w

This system is stable for range of values of K (A) [- 1, 12 ] (B) [- 1, 1] (C) [- 12 , 1] (D) [- 12 , 2] MCQ 6.4.73 GATE IN 2006

In the IIR filter shown below, a is a variable gain. For which of the following cases, the system will transit from stable to unstable condition ?

(A) 0.1 < a < 0.5 (B) 0.5 < a < 1.5 (C) 1.5 < a < 2.5 (D) 2 < a < 3 Buy Online: shop.nodia.co.in

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Sample Chapter of Signals MCQ 6.4.74 IES EC 2002

MCQ 6.4.75 IES EC 2007

MCQ 6.4.76 IES EC 2008

and Systems (Vol-7, GATE Study Package)

The poles of an analog system are related to the corresponding poles of the digital system by the relation z = est . Consider the following statements. 1. Analog system poles in the left half of s -plane map onto digital system poles inside the circle z = 1. 2. Analog system zeros in the left half of s -plane map onto digital system zeros inside the circle z = 1. 3. Analog system poles on the imaginary axis of s -plane map onto digital system zeros on the unit circle z = 1. 4. Analog system zeros on the imaginary axis of s -plane map onto digital system zeros on the unit circle z = 1. Which of these statements are correct ? (A) 1 and 2 (B) 1 and 3 (C) 3 and 4 (D) 2 and 4

i. n o c . a i d o n . w w w

Assertion (A) : The z -transform of the output of an ideal sampler is given by Z [f (t)] = K 0 + K1 + K22 + .... + Knn z z z Reason (R) : The relationship is the result of application of z = e-sT , where T stands for the time gap between the samples. (A) Both A and R are true and R is the correct explanation of A (B) Both A and R are true but R is NOT the correct explanation of A (C) A is true but R is false (D) A is false but R is true z and Laplace transform are related by

IES EC 2010

(A) s = ln z (C) s = z

IES EC 2010

MCQ 6.4.79 GATE IN 2004

The Z-Transform

Which one of the following rules determines the mapping of s -plane to z -plane ? (A) Right half of the s -plane maps into outside of the unit circle in z -plane (B) Left half of the s -plane maps into inside of the unit circle (C) Imaginary axis in s -plane maps into the circumference of the unit circle (D) All of the above

MCQ 6.4.77

MCQ 6.4.78

Page 553 Chap 6

(B) s = ln z T (D) T ln z

Frequency scaling [relationship between discrete time frequency (W) and continuous time frequency (w)] is defined as (A) w = 2 W (B) w = 2TS /W (C) W = 2w/TS (D) W = wTS A casual, analog system has a transfer function H (s) = s +a a . Assuming a sampling time of T seconds, the poles of the transfer function H (z) for an equivalent digital system obtained using impulse in variance method are at (B) a j a , - j a k (A) (eaT , e-aT ) T T 2

(C) (e jaT , e-jaT )

2

(D) (eaT/2, e-aT/2) ***********

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GATE STUDY PACKAGE Electronics & Communication 10 Subject-wise books by R. K. Kanodia General Aptitude Engineering Mathematics Networks Electronic Devices Analog Electronics Digital Electronics Signals & Systems Control Systems Communication Systems Electromagnetics

SOLUTIONS 6.1

Page 554 Chap 6 The Z-Transform

SOL 6.1.1

Option (C) is correct. The z -transform is 3

/ x [ n] z

X (z) =

-n

=

n =- 3 -1

-1

-b 1 l z-n 2 n =- 3 n

/

-1

=-

1 -1 n ^2z h-n b 2 z l =n =- 3 n =- 3

=-

/ (2z)

/

/

3

m

Let - n = m so,

m=1

The above series converges if 2z < 1 or z < 1 2 X (z) =- 2z = 2z , z 1 2z 2

SOL 6.1.3

ROC is intersection of both, therefore ROC : 1 < z < 2 2 Option (D) is correct. X (z) = = = =

3

/ x [ n] z n =- 3 3

/

n =- 3 -1

/

n =- 3 3

-n

1 n -n b 2 l z u [ - n - 1] + 1 -n b2l z +

/ (2z)

n=1

n

n

+

3

3

/ n =- 3

1 n -n b 3 l z u [n]

/ b 31z l

n

n=0

3

/ b 31z l

n

=

n=0

Series I converges, when 2z < 1 or z < 1 2 1 Series II converges, when < 1 or z > 1 3z 3

2z + 1 1 - 2z 1 - 13 z-1 S 14243 I II

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SOL 6.1.4

and Systems (Vol-7, GATE Study Package)

So ROC of X (z) is intersection of both ROC: 1 < z < 1 3 2 Option (C) is correct. z -transform of x [n] X (z) = = =

3

Page 555 Chap 6 The Z-Transform

/ x [ n] z

-n

n =- 3 3

/

3

/a

an z-n u [n] +

n =- 3 3

-n -n

z u [ n]

n =- 3

/ (az

-1 n

) +

n=0

3

/ (az)

-n

=

n=0

1 1 + 1 - az-1 1 - (az) -1 14 2 4 3 1 44 2 44 3 I II

-1

Series I converges, if az < 1 or z > a Series II converges, if (az) -1 < 1 or az > 1 or z > 1 a So ROC is interaction of both ROC : z > max e a , 1 o a SOL 6.1.5

i. n o c . a i d o n . w w w

Option (C) is correct. (P " 4)

x1 [n] = u [n - 2]

X1 (z) =

3

/ u [n - 2] z

-n

=-

z , 1 - z-1

z >1

3

/ u [- n - 3] z

-n

n =- 3 -3

/z

-n

3

/z

=-

m

3

/ u [n + 4] z

-n

3

-n

n =- 4

= z -1 = -4 1 -1 , 1-z z (1 - z ) n x 4 [n] = (1) u [- n]

=

z 1

/ u [ - n] z n =- 3 0

-n

/z

-n

=

n =- 3

SOL 6.1.6

=

n=2

n =- 3 3

(S " 2)

-n

-2

x2 [n] =- u [- n - 3]

X2 [z] =-

(R " 3)

/z

=

n =- 3

(Q " 1)

3

3

/z

m

=

m=0

1 = - z-1 , 1 - z 1 - z-1

z 1 1 - e jp z-1 = z jp = z z+1 z-e Option (D) is correct. We can write, transfer function Az2 H (z) = (z - 2) (z - 3) =

SOL 6.1.7

a e jp =- 1

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GATE STUDY PACKAGE Electronics & Communication 10 Subject-wise books by R. K. Kanodia General Aptitude Engineering Mathematics Networks Electronic Devices Analog Electronics Digital Electronics Signals & Systems Control Systems Communication Systems Electromagnetics A = 1 or A = 2 (- 1) (- 2) 2z2 so, H (z) = (z - 2) (z - 3) H (z) 2z = z (z - 2) (z - 3) From partial fraction H (z) = - 4z + 6z (z - 2) (z - 3) We can see that for ROC : z > 3 , the system is causal and unstable because ROC is exterior of the circle passing through outermost pole and does not include unit circle. so, h [n] = [(- 4) 2n + (6) (3) n] u [n], z > 3 (P " 2) For ROC 2 < z < 3 , The sequence corresponding to pole at z = 2 corresponds to right-sided sequence while the sequence corresponds to pole at z = 3 corresponds to left sided sequence

Page 556 Chap 6

H (1) =

The Z-Transform

h [n] = (- 4) 2n u [n] + (- 6) 3n u [- n - 1] (Q " 4) For ROC : z < 2 , ROC is interior to circle passing through inner most pole, hence the system is non causal.

i. n

h [n] = (4) 2n u [- n - 1] + (- 6) 3n u [- n - 1]

o .c

For the response n

h [n] = 4 (2) u [- n - 1] + (- 6) 3n u [n] ROC : z < 2 and z > 3 which does not exist SOL 6.1.8

a i d

Option (A) is correct.

(R " 3)

(S " 1)

X (z) = ez + e1/z

o n

. w w

SOL 6.1.9

2 3 X (z) = c1 + z + z + z + .....m + b1 + 1 + 1 12 + .....l z 2! z 2! 3!

2 3 -2 = c1 + z + z + z + ....m + b1 + z-1 + z + ....l 2! 3! 2! Taking inverse z -transform x [n] = d [ n] + 1 n! Option (A) is correct. 2 z (z + 5) X (z) = 2 z + 5z = (z - 3) (z + 1) z - 2z - 3 X (z) z+5 By partial fraction = 2 - 1 = z z-3 z+1 (z - 3) (z + 1) Thus X (z) = 2z - z z-3 z+1 Poles are at z = 3 and z =- 1

w

ROC : z < 1, which is not exterior of circle outside the outermost pole z = 3 . So, x [n] is anticausal given as Buy Online: shop.nodia.co.in

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Sample Chapter of Signals

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x [n] = [- 2 (3) n + (- 1) n] u [- n - 1] SOL 6.1.10

Page 557 Chap 6

Option (A) is correct. X (z) = 2z - z z-3 z+1 If z > 3 , ROC is exterior of a circle outside the outer most pole, x [n] is causal.

The Z-Transform

x [n] = [2 (3) n - (- 1) n] u [n] SOL 6.1.11

i. n o c . a i d o n . w w w

Option (C) is correct.

X (z) = 2z - z z-3 z+1

If ROC is 1 < z < 3 , x [n] is two sided with anticausal part 2z , z < 3 z-3 z , z >1 and causal part z+1

x [n] =- 2 (3) n u [- n - 1] - (- 1) n u [n]

SOL 6.1.12

Option (D) is correct. X1 (z) =

3

/ (0.7)

n -n

z u [ n - 1] =

n =- 3

3

/ (0.7z

-1 n

)

n=1

0.7z-1 1 - 0.7z-1 < 1 or z > 0.7 =

ROC : 0.7z-1

X2 (z) = = =

3

/

(- 0.4) n z-n u [- n - 2] =

n =- 3 3

/ (- 0.4)

m=2 3

/

/ (- 0.4)

n -n

z

n =- 3 -m m

Let n =- m

z

[(- 0.4) -1 z ] m =

m=2

-2

- (0.4) -1 z 1 + (0.4) -1 z

ROC : (0.4) -1 z < 1 or z < 0.4 The ROC of z -transform of x [n] is intersection of both which does not exist. SOL 6.1.13

Option (D) is correct. Buy Online: shop.nodia.co.in

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GATE STUDY PACKAGE Electronics & Communication 10 Subject-wise books by R. K. Kanodia General Aptitude Engineering Mathematics Networks Electronic Devices Analog Electronics Digital Electronics Signals & Systems Control Systems Communication Systems Electromagnetics Page 558 Chap 6

Z

If x [n] X (z) From time shifting property

The Z-Transform

x [n - n 0] So SOL 6.1.14

0

z (x [n - 4]) = z-4 X (z) =

1 8z5 - 2z 4 - z3

X1 (z) = z1 X2 (z) = z2 X 3 (z) z-2 X1 (z) = z-1 X2 (z) = X 3 (z) x1 [n - 2] = x2 [n - 1] = x 3 [n]

Option (C) is correct. x [n] can be written in terms of unit sequence as x [n] = u [ n] - u [n - k ] -k X (z) = z - z-k z = 1 - z-1 z-1 z-1 1-z

so SOL 6.1.16

z-n X (z)

Option (A) is correct. We can see that or So

SOL 6.1.15

Z

Option (C) is correct. For positive shift then, So

i. n

o .c

x [n]

Z

X (z)

x [n - n 0]

Z

x [ n - 1]

Z

z-n X (z), n 0 $ 0 z-1 a z k = 1 z-1 z-1

If,

For negative shift

a i d

o n

x [n + n 0]

. w w

Z

0

zn e X (z) 0

n0 - 1

/ x [n] z

-m

o, n0 > 0

m=0

x [ n + 1] z ^X (z) - x [0]h We know that x [n] = u [n] so x [0] = 1

w

and

SOL 6.1.17

Z

x [ n + 1]

Z

z ^X (z) - 1h = z a z - 1k = z z-1 z-1

Option (B) is correct. xe [n] = 1 (x [n] + x [- n]) 2 z - transform of xe [n], Xe (z) = 1 ;X (z) + X b 1 lE 2 z

Even part of x [n],

a x [- n]

Z

Xb 1 l z

1/z = 1a z k + 1e 2 z - 0.4 2 1/z - 0.4 o 1 44 2 44 3 1 444 2 4 44 3 I II

Region of convergence for I series is z > 0.4 and for II series it is z < 2.5 . Therefore, Xe (z) has ROC 0.4 < z < 2.5 SOL 6.1.18

Option (B) is correct. (P " 4) We know that

If then,

y [n] = n (- 1) n u [n]

(- 1) n u [n]

Z

x [n]

Z

nx [n]

Z

1 , 1 + z-1 X (z) dX (z) -z dz

z >1

(z -domain differentiation)

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Sample Chapter of Signals n (- 1) n u [n]

so,

and Systems (Vol-7, GATE Study Package)

- z d ; 1 -1 E, ROC : z > 1 dz 1 + z

Z

Page 559 Chap 6

- z-1 , ROC : z > 1 (1 + z-1) 2 y [n] =- nu [- n - 1]

The Z-Transform

Y (z) = (Q " 3) We know that,

- 1 , ROC : z < 1 (1 - z-1) Again applying z -domain differentiation property Z - nu [- n - 1] z d : - 1-1 D, ROC : z < 1 dz 1 - z z-1 , ROC : z < 1 Y (z) = (1 - z-1) 2 (R " 2) y [n] = (- 1) n u [n] u [- n - 1]

Z

Y (z) =

3

/ (- 1)

3

/ (- z

n -n

z u [n] =

n =- 3

-1 n

)

n=0

1 , ROC : z > 1 1 + z-1 = nu [n] Z 1 , ROC : z > 1 1 - z-1 Z z d b 1 -1 l, ROC : z > 1 dz 1 - z -1 z , ROC : z > 1 = (1 - z-1) 2

i. n o c . a i d o n . w w w =

(S " 1)

y [n]

We know that

u [n]

so,

nu [n] Y (z)

SOL 6.1.19

Option (A) is correct. Its difficult to obtain z -transform of x [n] directly due to the term 1/n . Let y [n] = nx [n] = (- 2) -n u [- n - 1] So z -transform of y [n] Y (z) = - z 1 , ROC : z < 1 2 z+ 2 y [n] = nx [n] dX (z) so, (Differentiation in z -domain) Y (z) =- z dz dX (z) -z = - z1 dz z+ 2 dX (z) = 11 dz z+ 2 or X (z) = log bz + 1 l, ROC : z < 1 2 2 Option (A) is correct. Since

SOL 6.1.20

X (z) = Y (z) = = SOL 6.1.21

3

/ x [ n] z n =- 3 3

/

-n

, ROC : Rx

y [n] z-n =

n =- 3 3

3

/ a x [n] z n

-n

n =- 3

x [n] a z k a n =- 3

/

-n

= X a z k, ROC : aRx a

Option (B) is correct. Using time shifting property of z -transform If,

x [n]

Z

X (z), ROC : Rx

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GATE STUDY PACKAGE Electronics & Communication 10 Subject-wise books by R. K. Kanodia General Aptitude Engineering Mathematics Networks Electronic Devices Analog Electronics Digital Electronics Signals & Systems Control Systems Communication Systems Electromagnetics Page 560 Chap 6

x [n - n 0] z-n X (z) with same ROC except the possible deletion or addition of z = 0 or z = 3. So, ROC for x [n - 2] is Rx (S1, R1) Similarly for x [n + 2], ROC : Rx (S2, R1) Using time-reversal property of z -transform Z

then,

The Z-Transform

If,

x [n]

Z

If,

x [n]

Z

an x [n]

Z

0

X (z), ROC : Rx Z then, x [- n] X b 1 l, ROC : 1 z Rx Z For S 3 , x [- n] X b 1 l, z Because z is replaced by 1/z , so ROC would be z < 1 (S 3, R 3) a S 4 :(- 1) n x [n] Using the property of scaling in z -domain, we have then,

X (z), ROC : Rx Xa z k a

i. n

z is replaced by z/a so ROC will be Rx a Z n Here (- 1) x [n] X a z k, a = 1 1

o .c

ROC : z > a

so, SOL 6.1.22

Option (D) is correct. Time scaling property : If,

o n

. w w

then, Time shifting property : If,

a i d

(S 4, R1)

x [n]

Z

X (z)

x [n/2]

Z

X (z2)

x [n]

Z

X (z)

(P " 2)

then, x [ n - 2] u [ n - 2 ] z-2 X (z) For x [n + 2] u [n] we can not apply time shifting property directly. Let, y [n] = x [n + 2] u [n]

w

so,

Z

= an + 2 u [n + 2] u [n] = an + 2 u [n] Y (z) =

3

/ y [n] z

-n

n =- 3 3 2

=a Let,

=

3

/a

n + 2 -n

z

n=0

/ (az

-1 n

) = a2 X (z)

(R " 4)

n=0 2n

g [ n] = b x [n] G (z) = =

3

/

3

/b

g [n] z-n =

n =- 3 3 n

2n

an z-n u [n]

n =- 3 n

/ a b bz

2

n=0

SOL 6.1.23

(Q " 1)

z l = Xb b 2 l

(S " 3)

Option (C) is correct. Let, y [n] = x [2n] Y (z) = =

3

/ x [2n] z n =- 3 3

-n

/ x [ k] z

-k/2

k =- 3

Put 2n = k or n = k , k is even 2

Since k is even, so we can write Buy Online: shop.nodia.co.in

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Sample Chapter of Signals

and Systems (Vol-7, GATE Study Package)

x [k] + (- 1) k x [k] -k/2 Y (z) = ; Ez 2 k =- 3 3

Page 561 Chap 6

/

3

The Z-Transform

3

=1 x [k] z-k/2 + 1 x [k] (- z1/2) k 2 k =- 3 2 k =- 3 = 1 8X ( z ) + X (- z )B 2

/

SOL 6.1.24

/

Option (A) is correct. From the accumulation property we know that x [n]

Z

/ x [k ]

Z

If, then,

n

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

k =- 3

Here,

y [n] =

n

/ x [k ] k=0

Y (z) = SOL 6.1.25

z X (z) = 4z2 (z - 1) (z - 1) (8z2 - 2z - 1)

Option (B) is correct. By taking z -transform of both the sequences

i. n o c . a i d o n . w w w

X (z) = (- 1 + 2z-1 + 0 + 3z-3) H (z) = 2z2 + 3 Convolution of sequences x [n] and h [n] is given as

y [n] x [n ] * h [n] Applying convolution property of z -transform, we have Y (z) = X (z) H (z) or, SOL 6.1.26

= (- 1 + 2z-1 + 3z-3) (2z2 + 3) =- 2z2 + 4z - 3 + 12z-1 + 9z-3 y [n] = {- 2, 4, - 3, 12, 0, 9} -

Option (C) is correct. The z -transform of signal x)[n] is given as follows Z {x)[n]} =

3

/x

n =- 3

3

3

) -n

@

)

...(1)

n =- 3

Let z -transform of x [n] is X (z) X (z) =

/ 6x [n] (z )

[n] z-n =

)

/ x [ n] z

-n

n =- 3

Taking complex conjugate on both sides of above equation X)(z) =

3

/ [ x [ n] z

-n )

]

n =- 3

Replacing z " z), we will get X)(z)) =

3

/

) -n 6x [n] (z ) @

)

...(2)

n =- 3

Comparing equation (1) and (2) Z {x)[n]} = X)(z)) SOL 6.1.27

Option (B) is correct. By taking z -transform on both sides of given difference equation Y (z) - 1 z-1 6Y (z) + y [- 1] z @ = X (z) 2 Let impulse response is H (z), so the impulse input is X (z) = 1 H (z) - 1 z-1 6H (z) + 3z @ = 1 2 Buy Online: shop.nodia.co.in

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GATE STUDY PACKAGE Electronics & Communication 10 Subject-wise books by R. K. Kanodia General Aptitude Engineering Mathematics Networks Electronic Devices Analog Electronics Digital Electronics Signals & Systems Control Systems Communication Systems Electromagnetics H (z) [1 - 1 z-1] = 5 2 2

Page 562 Chap 6

5/2 = 5b z 1 l 2 z- 2 1 - 12 z-1 n h [ n] = 5 b 1 l , n $ 0 2 2

The Z-Transform

H (z) =

SOL 6.1.28

Option (B) is correct. h [n] = (2) n u [n] Taking z -transform Y (z) H (z) = z = z - 2 X (z) so, (z - 2) Y (z) = zX (z) or, (1 - 2z-1) Y (z) = X (z) Taking inverse z -transform y [n] - 2y [n - 1] = x [n]

SOL 6.1.29

Option (B) is correct. h [ n] = d [ n] - b - 1 l u [n] 2 z -transform of h [n]

i. n

n

o .c

1 z = 2 = Y (z) X (z) z + 12 z + 12 1 1 bz + 2 l Y (z) = 2 X (z) 1 -1 1 -1 b1 + 2 z l Y (z) = 2 z X (z) Taking inverse z -transform y [n] + 1 y [n - 1] = 1 x [n - 1] 2 2

H (z) = 1 -

a i d

o n

SOL 6.1.30

. w w

w

y [n] + 0.5y [n - 1] = 0.5x [n - 1]

Option (C) is correct. We have y [n] - 0.4y [n - 1] = (0.4) n u [n] Zero state response refers to the response of system with zero initial condition. So, by taking z -transform z Y (z) - 0.4z-1 Y (z) = z - 0.4 z2 (z - 0.4) 2 Taking inverse z -transform Y (z) =

y [n] = (n + 1) (0.4) n u [n] SOL 6.1.31

Option (B) is correct. Zero state response refers to response of the system with zero initial conditions. Taking z -transform Y (z) - 1 z-1 Y (z) = X (z) 2 Y (z) = a z k X (z) z - 0.5 For an input x [n] = u [n], X (z) = z z-1 Buy Online: shop.nodia.co.in

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Sample Chapter of Signals so,

Y (z) =

and Systems (Vol-7, GATE Study Package)

z2 z z = (z - 1) (z - 0.5) (z - 0.5) (z - 1)

Y (z) z = z (z - 1) (z - 0.5) = 2 - 1 z - 1 z - 0.5 Thus Y (z) = 2z - z z - 1 z - 0.5 Taking inverse z -transform

Page 563 Chap 6 The Z-Transform

By partial fraction

y [n] = 2u [n] - (0.5) n u [n] SOL 6.1.32

Option (C) is correct. Input, x [n] = 2d [n] + d [n + 1] By taking z -transform X (z) = 2 + z Y (z) = H (z), X (z)

Y (z) is z -transform of output y [n]

i. n o c . a i d o n . w w w

Y (z) = H (z) X (z) 2z (z - 1) = (z + 2) (z + 2) 2 2z (z - 1) = = 2z - 6z z+2 (z + 2) Taking inverse z -transform

y [n] = 2d [n + 1] - 6 (- 2) n u [n]

SOL 6.1.33

Option (B) is correct. Poles of the system function are at z = ! j ROC is shown in the figure.

Causality : We know that a discrete time LTI system with transfer function H (z) is causal if and only if ROC is the exterior of a circle outside the outer most pole. For the given system ROC is exterior to the circle outside the outer most pole (z = ! j). The system is causal. Stability : A discrete time LTI system is stable if and only if ROC of its transfer function H (z) includes the unit circle z = 1. The given system is unstable because ROC does not include the unit circle. Impulse Response : H (z) = 2 z z +1 We know that Buy Online: shop.nodia.co.in

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GATE STUDY PACKAGE Electronics & Communication 10 Subject-wise books by R. K. Kanodia General Aptitude Engineering Mathematics Networks Electronic Devices Analog Electronics Digital Electronics Signals & Systems Control Systems Communication Systems Electromagnetics Page 564 Chap 6

z sin W0 , z >1 z - 2z cos W0 + 1 z2 + 1 = z2 - 2z cos W0 + 1 2z cos W0 = 0 or W0 = p 2

sin (W0 n) u [n]

The Z-Transform

Here So

Z

2

Taking the inverse Laplace transform of H (z) h [n] = sin a p n k u [n] 2 SOL 6.1.34

Option (D) is correct. Statement (A), (B) and (C) are true.

SOL 6.1.35

Option (D) is correct. First we obtain transfer function (z -transform of h [n]) for all the systems and then check for stability 1 z (A) H (z) = 3 1 2 ^z - 3 h Stable because all poles lies inside unit circle. (B) h [ n] = 1 d [ n] 3 1 (absolutely summable) h [n] = 3 Thus this is also stable. n (C) h [ n] = d [ n] - b - 1 l u [n] 3 H (z) = 1 - z 1 z+ 3 Pole is inside the unit circle, so the system is stable. (D) h [n] = [(2) n - (3) n] u [n] H (z) = z - z z-2 z-3 Poles are outside the unit circle, so it is unstable.

i. n

/

o .c

a i d

o n

SOL 6.1.36

. w w

w

Option (B) is correct. By taking z -transform (1 + 3z-1 + 2z-2) Y (z) = (2 + 3z-1) X (z) So, transfer function Y (z) (2 + 3z-1) 2z2 + 3z H (z) = = = X (z) (1 + 3z-1 + 2z-2) z2 + 3z + 2 H (z) or By partial fraction = 2 2z + 3 = 1 + 1 z z+2 z+1 z + 3z + 2 Thus H (z) = z + z z+2 z+1 Both the poles lie outside the unit circle, so the system is unstable.

SOL 6.1.37

Option (B) is correct. y [ n ] = x [ n ] + y [ n - 1] Put x [n] = d [n] to obtain impulse response h [n] h [ n ] = d [ n ] + h [ n - 1] For n = 0 , h [0] = d [0] + h [- 1] ( h [- 1] = 0 , for causal system) h [0] = 1 n = 1, h [1] = d [1] + h [0]

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Sample Chapter of Signals

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h [1] = 1 h [2] = d [2] + h [1]

n = 2,

Page 565 Chap 6 The Z-Transform

h [2] = 1 In general form h [ n] = u [ n] Let

x [n]

Z

h [ n]

Z

Thus, statement 1 is true.

X (z) X (z) = z z - 0.5 H (z) =

H (z)

z z-1

Y (z) = H (z) X (z) = a z ka z k = 2z - z By partial fraction z - 1 z - 0.5 z - 1 z - 0.5 Inverse z -transform Statement 3 is also true. y [n] = 2u [n] - (0.5) n u [n] H (z) = z z-1 Output

i. n o c . a i d o n . w w w

System pole lies at unit circle z = 1, so the system is not BIBO stable. SOL 6.1.38

Option (C) is correct. (P " 3) ROC is exterior to the circle passing through outer most pole at z = 1.2 , so it is causal. ROC does not include unit circle, therefore it is unstable. (Q " 1) ROC is not exterior to the circle passing through outer most pole at z = 1.2 , so it is non causal. But ROC includes unit circle, so it is stable. (R " 2), ROC is not exterior to circle passing through outermost pole z = 0.8 , so it is not causal. ROC does not include the unit circle, so it is unstable also. (S " 4), ROC contains unit circle and is exterior to circle passing through outermost pole, so it is both causal and stable.

SOL 6.1.39

Option (D) is correct.

P (z - 0.9) z - 0.9 + Pz P (z - 0.9) z - 0.9 = P = 1 + P (1 + P) z - 0.9 f z - 0.9 p 1+P z = 0. 9 1+P

H (z) =

Pole at

For stability pole lies inside the unit circle, so z - 0.1 or P < - 1.9

SOL 6.1.40

Option (A) is correct. For a system to be causal and stable, H (z) must not have any pole outside the unit circle z = 1. z- 1 z - 12 S1 : H (z) = 2 1 2 3 = 1 3 z + 2 z - 16 ^z - 4 h^z + 4 h Buy Online: shop.nodia.co.in

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GATE STUDY PACKAGE Electronics & Communication 10 Subject-wise books by R. K. Kanodia General Aptitude Engineering Mathematics Networks Electronic Devices Analog Electronics Digital Electronics Signals & Systems Control Systems Communication Systems Electromagnetics Poles are at z = 1/4 and z =- 3/4 , so it is causal. z+1 H (z) = S2 : 4 1 -3 ^z + 3 h^1 - 2 z h one pole is at z =- 4/3 , which is outside the unit circle, so it is not causal. S 3 : one pole is at z = 3, so it is also non-causal.

Page 566 Chap 6 The Z-Transform

SOL 6.1.41

Option (D) is correct. 3

3

n =- 3

n =- 3

/ x [n] z-n = / d [n - k] z-n = z-k, z ! 0

X (z) =

ROC : We can find that X (z) converges for all values of z except z = 0 , because at z = 0 X (z) " 3. SOL 6.1.42

Option (D) is correct. 3

3

n =- 3

n =- 3

/ x [n] z-n = / d [n + k] z-n = zk , all z

X (z) =

ROC : We can see that above summation converges for all values of z . SOL 6.1.43

Option (A) is correct. 3

=

3

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

X (z) =

n =- 3 3 -n

/z

=

1 1 - z-1

o .c

S I ROC : Summation I converges if n=0

3

/z

-n

SOL 6.1.44

Option (D) is correct.

o n

. w w

w

X (z) = = =

z > 1, because when

a i d

" 3.

n=0

3

i. n

n =- 3

z < 1, then

/ x [n] z

n =- 3 3

/

n =- 3 4

-n

1 n -n b 4 l (u [n] - u [n - 5]) z

/ b 14 z-1l

n

u [n] - u [n - 5] = 1, for 0 # n # 4

n=0

1 44 2 44 3 I

1 - ^ 14 z-1h5 z5 - (0.25) 5 , all z = = 4 1 - ^ 14 z-1h z (z - 0.5) ROC : Summation I converges for all values of z because n has only four value.

SOL 6.1.45

Option (D) is correct. X (z) =

3

/ x [n] z

-n

3

/

=

n =- 3 0

n =- 3

1 n -n b 4 l u [ - n] z

0 n = / b 1 z-1 l = / ^4z h-n 4 n =- 3 n =- 3

=

/ (4z) m = 1 -1 4z ,

z 4 has a radius greater that the pole at z =- 4 and z = 4 , therefore both the terms of X (z) corresponds to right sided sequences. Taking inverse z - transform we have x [n] = : 49 (- 4) n + 47 4nD u [n] 32 32 SOL 6.1.54

i. n o c . a i d o n . w w w

Option (C) is correct. Using partial fraction expansion X (z) can be simplified as 4 3 2 X (z) = 2z - 22z - 2z z -1 -1 2z-2 z2 = ;2 - 2z E 1 - z-2 = ;2 + 1 -1 + - 1-1 E z2 1+z 1-z Poles are at z =- 1 and z = 1. Location of poles and ROC is shown in the following figure

ROC : z > 1 has radius grater than both the poles at z =- 1 and z = 1, therefore both the terms in X (z) corresponds to right sided sequences. Z 1 (right-sided) (- 1) n u [n] 1 + z-1 Z 1 (right-sided) u [ n] -1 1-z Now, using time shifting property the complete inverse z -transform of X (z) is -1

-1

x [n] = 2d [n + 2] + ((- 1) n - 1) u [n + 2]

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GATE STUDY PACKAGE Electronics & Communication 10 Subject-wise books by R. K. Kanodia General Aptitude Engineering Mathematics Networks Electronic Devices Analog Electronics Digital Electronics Signals & Systems Control Systems Communication Systems Electromagnetics Page 570 Chap 6

SOL 6.1.55

The Z-Transform

Option (A) is correct. We have, X (z) = 1 + 2z-6 + 4z-8, Taking inverse z -transform we get

z >0

x [n] = d [n] + 2d [n - 6] + 4d [n - 8] SOL 6.1.56

z-n

0

Z-1

d [n - n 0]

z-n

0

Z-1

d [n - n 0]

Option (B) is correct. Since x [n] is right sided, 10

/ k1 d [n - k]

x [n] =

k=5

SOL 6.1.57

Option (D) is correct. We have , X (z) = (1 + z-1) 3 = 1 + 3z-1 + 3z-2 + z-3 , z >0 Since x [n] is right sided signal, taking inverse z -transform we have z-n

x [n] = d [n] + 3d [n - 1] + 3d [n - 2] + d [n - 3] SOL 6.1.58

0

Z-1

d [n - n 0]

Option (A) is correct. We have, X (z) = z6 + z2 + 3 + 2z-3 + z-4, z > 0 Taking inverse z transform we get

i. n

x [n] = d [n + 6] + d [n + 2] + 3d [n] + 2d [n - 3] + d [n - 4] SOL 6.1.59

o .c

Option (A) is correct. 1

X (z) =

1 -2 4

z >1 2

,

1- z The power series expansion of X (z) with z >

a i d

-2 2

-2

1 2

or

1 -2 4

z

< 1 is written as

-2 3

X (z) = 1 + z + b z l + b z l + ....... 4 4 4

o n

. w w

=

SOL 6.1.60

-2

k=0 1 -2 4 3

3

k

l =

/ b 14 l z k

-2k

k=0

< 1 or z > 12 . Taking inverse z -transform we get k Z z-2k d [n - 2k] x [n] = / b 1 l d [n - 2k] 4 k=0

Series converges for

w

3

/ b 14 z

z

-1

1 , = *b 4 l 0, 2-n, =* 0, n 2

n even and n $ 0 n odd n even and n $ 0 n odd

Option (C) is correct. 1 , z b Summation II converges, if bz < 1 or z < 1 b since b < 1 so from the above two conditions ROC : z < 1. SOL 6.4.6

Option (B) is correct. z -transform of signal an u [n] is Buy Online: shop.nodia.co.in

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GATE STUDY PACKAGE Electronics & Communication 10 Subject-wise books by R. K. Kanodia General Aptitude Engineering Mathematics Networks Electronic Devices Analog Electronics Digital Electronics Signals & Systems Control Systems Communication Systems Electromagnetics Page 588 Chap 6

3

/ an u [n] z-n

X (z) =

=

u [n] = 1, n $ 0

n=0

n =- 3

The Z-Transform

3

/ (az-1) n 1 44 2 44 3 I

1 = z 1 - az-1 z - a Similarly, z -transform of signal an u [- n - 1] is =

3

/ - an u [- n - 1] z-n

X (z) =

n =- 3 -1

/ an z-n

=-

a u [- n - 1] = 1, n #- 1

n =- 3

Let n =- m , then

3

3

X (z) =- / a-m zm = - / (a-1 z) m m=1 m=1 1 44 42 44 3 -1 II z a z == z-a 1 - a-1 z z -transform of both the signal is same. (A) is true ROC : To obtain ROC we find the condition for convergences of X (z) for both the transform. Summation I converges if a-1 z < 1 or z > a , so ROC for an u [n] is z > a Summation II converges if a-1 z < 1 or z < a , so ROC for - an u [- n - 1] is z < a . (R) is true, but (R) is NOT the correct explanation of (A).

i. n

SOL 6.4.7

o .c

a i d

Option (B) is correct. z - transform of x [n] is defined as

o n

. w w

X (z) = =

3

/ x [n] z-n

n =- 3 3

/ x [n] r-n e-jWn

Putting z = re jW

n =- 3

z -transform exists if X (z) < 3 3

/

w

n =- 3

x [n] r-n e-jWn < 3 3

/

or

n =- 3

x [n] r-n < 3

Thus, z -transform exists if x [n] r-n is absolutely summable. SOL 6.4.8

Option (A) is correct. n n x [n] = b 1 l u [n] - b 1 l u [- n - 1] 3 2

Taking z transform we have X (z) = =

n=3

n =- 1

n

/ b 13 l z-n - /

1 n -n b2l z

/ b 13 z-1l

1 -1 n b2z l

n=0 n=3 n=0

n

-

n =- 3 n =- 1

/

n =- 3

First term gives 1 z-1 < 1 " 1 < z 3 3 Second term gives 1 z-1 > 1 " 1 > z 2 2 Thus its ROC is the common ROC of both terms. that is

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Sample Chapter of Signals

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1< z 2 =

SOL 6.4.11

Option (A) is correct. We have h [ n] = u [ n] 3

/ x [ n] z

H (z) =

-n

n =- 3

H (z) is convergent if

=

3

/ 1z

-n

n=0

=

3

/ (z

-1 n

)

n=0

3

/ (z-1) n < 3

n=0

and this is possible when z-1 < 1. Thus ROC is z-1 < 1 or z > 1 SOL 6.4.12

Option (B) is correct. (Please refer to table 6.1 of the book Gate Guide signals & Systems by same authors) Z z (A) u [n] (A " 3) z-1 d [ n]

(B) (C)

sin wt

(D)

cos wt

Z Z

z sin wT z - 2z cos wT + 1 z - cos wT z2 - 2z cos wt + 1 2

t = nT Z t = nT

1

SOL 6.4.13

Option (A) is correct. Inverse z -transform of X (z) is given as x [n] = 1 # X (z) zn - 1 dz 2pj

SOL 6.4.14

Option (B) is correct. so

(B " 1) (C " 4) (D " 2)

H (z) = z-m h [n] = d [n - m]

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GATE STUDY PACKAGE Electronics & Communication 10 Subject-wise books by R. K. Kanodia General Aptitude Engineering Mathematics Networks Electronic Devices Analog Electronics Digital Electronics Signals & Systems Control Systems Communication Systems Electromagnetics Page 590 Chap 6

SOL 6.4.15

Option (C) is correct. We know that

The Z-Transform

an u [n]

Z

u [n]

Z

For a = 1, SOL 6.4.16

1 1 - az-1 1 1 - z-1

Option (B) is correct. 0.5 1 - 2z-1 Since ROC includes unit circle, it is left handed system X (z) =

x [n] =- (0.5) (2) -n u [- n - 1] x (0) = 0 If we apply initial value theorem x (0) = lim X (z) = lim 0.5 -1 = 0.5 z"3 z " 31 - 2z That is wrong because here initial value theorem is not applicable because signal x [n] is defined for n < 0 . SOL 6.4.17

i. n

Option (B) is correct. F (z) =

z -transform

f (k) = d (k) - (- 1) k Z 1 Thus (- 1) k 1 + z- 1 Option (C) is correct. z ^2z - 56 h X (z) = 1 1 ^z - 2 h^z - 3 h 5 ^2z - 6 h X (z) = = 11 + 11 1 1 z ^z - 2 h^z - 3 h ^z - 2 h ^z - 3 h or ...(1) X (z) = z 1 + z 1 ^z - 2 h ^z - 3 h S S term I term II 1 Poles of X (z) are at z = and z = 1 2 3 1 ROC : z > 2 Since ROC is outside to the outer most pole so both the terms in equation (1) corresponds to right sided sequence. so,

SOL 6.4.18

o .c

1 = 1- z = 1- 1 z+1 z+1 1 + z- 1

a i d

o n

. w w

w

So,

n n x [n] = b 1 l u [n] + b 1 l u [n] 2 3

(A " 4)

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Sample Chapter of Signals

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ROC : z < 13 :Since ROC is inside to the innermost pole so both the terms in equation (1) corresponds to left sided signals.

Page 591 Chap 6 The Z-Transform

n n So, x [n] =-b 1 l u [- n - 1] - b 1 l u [- n - 1] (D " 2) 2 3 1 1 1 ROC : 3 < z < 2 : ROC is outside to the pole z = 3 , so the second term of equation (1) corresponds to a causal signal. ROC is inside to the pole at z = 12 , so First term of equation (1) corresponds to anticausal signal.

So,

i. n o c . a i d o n . w w w n n x [n] =-b 1 l u [- n - 1] + b 1 l u [n] 2 3

(C " 1)

ROC : z < 13 & z > 12 : ROC : z < 13 is inside the pole at z = 13 so second term of equation (1) corresponds to anticausal signal. On the other hand, ROC : z > 12 is outside to the pole at z = 12 , so the first term in equation (1) corresponds to a causal signal.

So,

n n x [n] = b 1 l u [n] - b 1 l u [- n - 1] 2 3

(B " 3)

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GATE STUDY PACKAGE Electronics & Communication 10 Subject-wise books by R. K. Kanodia General Aptitude Engineering Mathematics Networks Electronic Devices Analog Electronics Digital Electronics Signals & Systems Control Systems Communication Systems Electromagnetics Page 592 Chap 6

SOL 6.4.19

Option (A) is correct. X (z) =

Given,

The Z-Transform

z , z a Given that X (z) = (z - a) 2 Residue of X (z) zn - 1 at z = a is = d (z - a) 2 X (z) zn - 1 z = a dz z zn - 1 = d (z - a) 2 dz (z - a) 2 z=a d n n-1 = z = nz = nan - 1 z=a dz z = a

. w w

w SOL 6.4.21

Option (C) is correct. X (z) =

1 2

1

3 , + 1 - az-1 1 - bz-1 Poles of the system are z = a , z = b ROC : a < z < b

ROC : a < z < b

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Sample Chapter of Signals

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Since ROC is outside to the pole at z = a , therefore the first term in X (z) corresponds to a causal signal. 1 2

1 (a) n u [n] 2 1 - az ROC is inside to the pole at z = b , so the second term in X (z) corresponds to a anticausal signal. 1 3

Z-1

Page 593 Chap 6 The Z-Transform

-1

- 1 (b) n u [- n - 1] 3 x [n] = 1 (a) n u [n] - 1 (b) n u [- n - 1] 2 3 x [0] = 1 u [0] - 1 u [- 1] = 1 2 3 2 Z-1

-1

1 - bz

SOL 6.4.22

Option (A) is correct. (z + z-3) z (1 + z-4) = z (1 + z-2) (z + z-1) = (1 + z-4) (1 + z-2) -1 Writing binomial expansion of (1 + z-2) -1 , we have X (z) = (1 + z-4) (1 - z-2 + z-4 - z-6 + ....) = 1 - z-2 + 2z-4 - 2z-6 + ... For a sequence x [n], its z -transform is X (z) =

i. n o c . a i d o n . w w w X (z) =

Comparing above two

3

/ x [n] z-n

n =- 3

x [n] = d [n] - d [n - 2] + 2d [n - 4] - 2d [n - 6] + ... = "1, 0, - 1, 0, 2, 0, - 2, ...., x [n] has alternate zeros. SOL 6.4.23

Option (A) is correct.

Z-1

ad [n ! a] aZ ! a 2 X (z) = 5z + 4z-1 + 3 x [n] = 5d [n + 2] + 4d [n - 1] + 3d [n]

We know that

Given that Inverse z -transform SOL 6.4.24

Option (A) is correct. X (z) = e1/z X (z) = e1/z = 1 + 1 + 1 2 + 1 3 + .... z 2z 3z z -transform of x [n] is given by 3 x [ 1] x [ 2] x [ 3] X (z) = / x [n] z-n = x [0] + + 2 + 3 + .... z z z n=0 Comparing above two 1 1 1 "x [0], x [1], x [2], x [3] ...., = '1, , , , ....1 1 2 3 x [n ] = 1 u [ n] n

SOL 6.4.25

Option (D) is correct. The ROC of addition or subtraction of two functions x1 [n] and x2 [n] is R1 + R2 . We have been given ROC of addition of two function and has been asked ROC of subtraction of two function. It will be same.

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GATE STUDY PACKAGE Electronics & Communication 10 Subject-wise books by R. K. Kanodia General Aptitude Engineering Mathematics Networks Electronic Devices Analog Electronics Digital Electronics Signals & Systems Control Systems Communication Systems Electromagnetics Page 594 Chap 6

SOL 6.4.26

The Z-Transform

Option (D) is correct. (A) x [n] = an u [n] 3

/ x [n] z-n

X (z) = =

n =- 3 3

=

3

/ an z-n

u [n] = 1, n $ 0

n=0

/ (az-1) n

n=0

1 , az-1 < 1 or z > a 1 - az-1 x [n] =- an u [- n - 1] =

(B)

3

/ an u [- n - 1] z-n

X (z) ==-

n =- 3 -1

/ an z-n

u [- n - 1] = 1, n #- 1

n =- 3 3 -m m

X (z) =- / a

Let n =- m ,

3

z =- / (a-1 z) m

m=1 -1

m=1

= - a -z1 , a-1 z < 1 or z < a 1-a z 1 , z < a = (1 - az-1) x [n] =- nan u [- n - 1]

i. n

(C) We have,

(A " 2)

(B " 3)

o .c

a i d

1 , z < a (1 - az-1) From the property of differentiation in z -domain Z z < a - z d : 1 -1 D, - nan u [- n - 1] dz 1 - az Z az-1 , z < a (1 - az-1) 2 (D) x [n] = nan u [n] Z 1 We have, , z > a an u [n] (1 - az-1) From the property of differentiation in z -domain Z 1 , z > a -z d ; nan u [n] dz (1 - az-1)E Z az-1 , z > a (1 - az-1) 2 - an u [- n - 1]

Z

o n

. w w

(C " 4)

w SOL 6.4.27

(D " 1)

Given that, z transform of x [n] is 3

/ x [n] z-n

X (z) = z -transform of {x [n] e

n =- 3 jw0 n

}

3

/ x [n] e jw n z-n

Y (z) =

0

n =- 3

so,

=

3

/ x [n] (ze-jw ) -n 0

n =- 3

= X (zl) zl = ze

-jw0

Y (z) = X (ze-jw ) 0

SOL 6.4.28

Option (A) is correct.

SOL 6.4.29

Option (C) is correct. We know that, z z-a From time shifting property a n u [ n]

Z

(A " 3)

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Sample Chapter of Signals

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z z-a From the property of scaling in z -domain an - 2 u [n - 2]

Z

x [n]

Z

If, then,

an x [n]

so

(e j ) n an

From the property of a n u [ n]

If, then,

nan u [n]

z-2

X (z) Z Xa z k a z -j ae j k Z = ze j z ze - a ae j - a k differentiation in z -domain Z z z-a Z d z = az dz a z - a k (z - a) 2

(B " 4)

Page 595 Chap 6 The Z-Transform

(C " 2)

(D " 1)

SOL 6.4.30 SOL 6.4.31

Option (C) is correct. The convolution of a signal x [n] with unit step function u [n] is given by

i. n o c . a i d o n . w w w y [n] = x [n] * u [n] =

3

/ x [k ]

k=0

Taking z -transform

Y (z) = X (z)

SOL 6.4.32

Option (B) is correct. From the property of z -transform. x 1 [ n] * x 2 [ n]

SOL 6.4.33

1 1 - z-1

Z

X1 (z) X2 (z)

Option (C) is correct. Given z transform

z-1 (1 - z-4) 4 (1 - z-1) 2 Applying final value theorem C (z) =

lim f (n) = lim (z - 1) f (z)

n"3

z"1

z-1 (1 - z-4) z"1 z"1 4 (1 - z-1) 2 z-1 (1 - z-4) (z - 1) = lim z"1 4 (1 - z-1) 2 z-1 z-4 (z 4 - 1) (z - 1) = lim z"1 4z-2 (z - 1) 2 -3 (z - 1) (z + 1) (z2 + 1) (z - 1) = lim z z"1 4 (z - 1) 2 -3 = lim z (z + 1) (z2 + 1) = 1 z"1 4 Option (C) is correct. lim (z - 1) F (z) = lim (z - 1)

SOL 6.4.34

H1 (z) = 1 + 1.5z-1 - z-2 2 = 1 + 3 - 12 = 2z + 32z - 2 2z z 2z Poles z2 = 0 & z = 0 zeros (2z2 + 3z - 2) = 0 & bz - 1 l (z + 2) = 0 & z = 1 , z =- 2 2 2 zeros of the two systems are identical. Buy Online: shop.nodia.co.in

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GATE STUDY PACKAGE Electronics & Communication 10 Subject-wise books by R. K. Kanodia General Aptitude Engineering Mathematics Networks Electronic Devices Analog Electronics Digital Electronics Signals & Systems Control Systems Communication Systems Electromagnetics Page 596 Chap 6

SOL 6.4.35

The Z-Transform

Option (D) is correct. Taking z -transform on both sides of given equation. z3 Y (z) + 6z2 Y (z) + 11zY (z) + 6Y (z) = z2 R (z) + 9zR (z) + 20R (z) Transfer function 2 Y (z) = 3 z +29z + 20 R (z) z + 6z + 11z + 6

SOL 6.4.36

Option (A) is correct. Characteristic equation of the system zI - A = 0 0 1 z -1 z 0 zI - A = > H - > => H b -a b z + aH 0 z zI - A = z (z + a) + b = 0 z + za + b = 0 In the given options, only option (A) satisfies this characteristic equation. 2

c [k + 2] + ac [k + 1] + bc [k] = u [k] z2 + za + b = 0 SOL 6.4.37

i. n

Option (B) is correct. We can see that the given impulse response is decaying exponential, i.e. h [n] = an u [n], 0 < a < 1 H (z) = z z -transform of h [n] z-a Pole of the transfer function is at z = a , which is on real axis between 0 < a < 1.

o .c

SOL 6.4.38

a i d

o n

Option (A) is correct.

. w w

y [ n ] + y [ n - 1] = x [ n ] - x [ n - 1] Taking z -transform

SOL 6.4.39

w

Y (z) + z-1 Y (z) = X (z) - z-1 X (z) (1 - z-1) Y (z) = (1 + z-1)

which has a linear phase response.

Option (A) is correct. For the linear phase response output is the delayed version of input multiplied by a constant. y [n] = kx [n - n 0] Y (z) = kz-n X (z) = 0

kX (z) zn 0

Pole lies at z = 0 SOL 6.4.40

Option (B) is correct. Given impulse response can be expressed in mathematical form as h [n] = d [n] - d [n - 1] + d [n - 2] - d [n - 3] + .... By taking z -transform H (z) = 1 - z-1 + z-2 - z-3 + z-4 - z-5 + .... = (1 + z-2 + z-4 + ....) - (z-1 + z-3 + z-5 + ....) -1 2 = 1 -2 - z -2 = 2 z - 2 z 1-z 1-z z -1 z -1 2 (z - z) z (z - 1) Pole at z =- 1 = z = 2 = z+1 1 + 1 ( z ) ( z ) (z - 1) Buy Online: shop.nodia.co.in

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Sample Chapter of Signals SOL 6.4.41

and Systems (Vol-7, GATE Study Package)

Option (B) is correct. Z Let Impulse response of system h [n] H (z) First consider the case when input is unit step. Input, x1 [n] = u [n] or X1 (z) = z (z - 1)

Page 597 Chap 6 The Z-Transform

y1 [n] = d [n] or Y1 (z) = 1 Y1 (z) = X1 (z) H (z) 1 = z H (z) (z - 1) (z - 1) Transfer function, H (z) = z Now input is ramp function Output, so,

x2 [n] = nu [n] z X2 (z) = (z - 1) 2 Y2 (z) = X2 (z) H (z) (z - 1) = : z 2 D= = 1 (z) G (z - 1) (z - 1)

Output,

i. n o c . a i d o n . w w w Y2 (z) 1 (z - 1)

SOL 6.4.42

Z-1 Z

-1

y 2 [n]

u [n - 1]

Option (C) is correct. Given state equations

s [n + 1] = As [n] + Bx [n] y [n] = Cs [n] + Dx [n] Taking z -transform of equation (1)

zS (z) = AS (z) + BX (z) S (z) 6zI - A@ = BX (z) S (z) = (zI - A) -1 BX (z) Now, taking z -transform of equation (2)

...(1) ...(2)

I " unit matrix ...(3)

Y (z) = CS (z) + DX (z) Substituting S (z) from equation (3), we get Y (z) = C (zI - A) -1 BX (z) + DX (z) Transfer function Y (z) H (z) = = C (zI - A) -1 B + D X (z) SOL 6.4.43

Option (B) is correct. F (z) = 4z3 - 8z2 - z + 2 F (z) = 4z2 (z - 2) - z (z - 2) = (4z2 - 2) (z - 2) 4z2 - 2 = 0 and (z - 2) = 0 z = ! 1 and z = 2 2 Only one root lies outside the unit circle.

SOL 6.4.44

Option (A) is correct. We know that convolution of x [n] with unit step function u [n] is given by x [n ] * u [n ] =

n

/ x [k ] k =- 3

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GATE STUDY PACKAGE Electronics & Communication 10 Subject-wise books by R. K. Kanodia General Aptitude Engineering Mathematics Networks Electronic Devices Analog Electronics Digital Electronics Signals & Systems Control Systems Communication Systems Electromagnetics y [ n] = x [ n] * u [ n] Taking z -transform on both sides 1 Y (z) = X (z) z = X (z) (z - 1) (1 - z-1)

so

Page 598 Chap 6 The Z-Transform

Transfer function, H (z) =

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

Now consider the inverse system of H (z), let impulse response of the inverse system is given by H1 (z), then we can write H (z) H1 (z) = 1 X (z) H1 (z) = = 1 - z-1 Y (z) (1 - z-1) Y (z) = X (z) Y (z) - z-1 Y (z) = X (z) Taking inverse z -transform y [n] - y [n - 1] = x [n] SOL 6.4.45

i. n

Option (B) is correct.

o .c

y [ n ] - 1 y [ n - 1] = x [ n ] 2 Taking z -transform on both sides Y (z) - 1 z-1 Y (z) = X (z) 2 Y (z) 1 Transfer function H (z) = = X (z) 1 - 12 z-1 Now, for input x [n] = kd [n]Output is

o n

a i d

. w w

w

Y (z) = H (z) X (z) k = -1 ^1 - 12 z h

Taking inverse z -transform n n y [n] = k b 1 l u [n] = k b 1 l , 2 2

X (z) = k

n$0

SOL 6.4.46

Option (A) is correct.

SOL 6.4.47

y [ n ] + y [ n - 1] = x [ n ] For unit step response, x [n] = u [n] y [ n ] + y [ n - 1] = u [ n ] Taking z -transform Y (z) + z-1 Y (z) = z z-1 (1 + z-1) Y (z) = z (z - 1) (1 + z) Y (z) = z z (z - 1) z2 Y (z) = (z + 1) (z - 1) Option (A) is correct.

SOL 6.4.48

Option (A) is correct.

SOL 6.4.49

Option (A) is correct. We have h (2) = 1, h (3) =- 1 otherwise h [k] = 0 . The diagram of response Buy Online: shop.nodia.co.in

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is as follows :

Page 599 Chap 6 The Z-Transform

It has the finite magnitude values. So it is a finite impulse response filter. Thus S2 is true but it is not a low pass filter. So S1 is false. SOL 6.4.50

Option (D) is correct. H (z) = We know that

z z - 0.2

z < 0.2

1 1 - az-1 h [n] =- (0.2) n u [- n - 1]

- an u [- n - 1] *

i. n o c . a i d o n . w w w

Thus SOL 6.4.51

z a but in the inverse system z < a , for stability of H1 (z). so h1 [n] =- an u [- n - 1] H1 (z) =

SOL 6.4.55

Option (C) is correct.

o .c

a i d

o n

z z + 12 Pole, z =- 1 2 The system is stable if pole lies inside the unit circle. Thus (A) is true, (R) is false.

SOL 6.4.56

. w w

H (z) =

w

Option (A) is correct. Difference equation of the system. y [n + 2] - 5y [n + 1] + 6y [n] = x [n] Taking z -transform on both sides of above equation.

z2 Y (z) - 5zY (z) + 6Y (z) = X (z) (z2 - 5z + 6) y (z) = X (z) Transfer function, Y (z) 1 H (z) = = 2 1 = X (z) (z - 5z + 6) (z - 3) (z - 2) Roots of the characteristic equation are z = 2 and z = 3 We know that an LTI system is unstable if poles of its transfer function (roots of characteristic equation) lies outside the unit circle. Since, for the given system the roots of characteristic equation lies outside the unit circle (z = 2, z = 3) so the system is unstable. SOL 6.4.57

Option (C) is correct. z2 + 1 (z + 0.5) (z - 0.5) Poles of the system lies at z = 0.5, z =- 0.5 . Since, poles are within the unit System function,

H (z) =

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circle, therefore the system is stable. From the initial value theorem

Page 601 Chap 6

(z2 + 1) z"3 z " 3(z + 0.5) (z - 0.5) 1 b1 + z2 l = lim =1 0.5 0.5 z"3 b1 + z lb1 - z l

The Z-Transform

h [0] = lim H (z) = lim

SOL 6.4.58

Option (D) is correct. y [n] = 2x [n] + 4x [n - 1] Taking z -transform on both sides Y (z) = 2X (z) + 4z-1 X (z) Transfer Function, Y (z) H (z) = = 2 + 4z-1 = 2z + 4 z X (z) Pole of H (z), z = 0 Since Pole of H (z) lies inside the unit circle so the system is stable. (A) is not True.

i. n o c . a i d o n . w w w

H (z) = 2 + 4z-1 Taking inverse z -transform

h [n] = 2d [n] + 4d [n - 1] = "2, 4, Impulse response has finite number of non-zero samples. (R) is true. SOL 6.4.59

Option (B) is correct. For left sided sequence we have

1 where z < a 1 - az-1 Z 1 Thus where z < 5 - 5n u [- n - 1] 1 - 5z-1 Z z or where z < 5 - 5n u [- n - 1] z-5 Since ROC is z < 5 and it include unit circle, system is stable. - an u [- n - 1]

Z

ALTERNATIVE METHOD :

h [n] =- 5n u [- n - 1] H (z) = Let n =- m, then

3

/ h [n] z

-n

=

n =- 3

H (z) =-

-3

-1

/-5 z

n -n

=-

n =- 3

/ (5z

-1 -m

)

= 1-

n =- 1

1 = 1, 1 - 5-1 z = 1- 5 = z 5-z z-5

-1

/ (5z

-1 n

)

n =- 3 3

/ (5

-1

z) -m

m=0

5-1 z < 1 or z < 5

SOL 6.4.60

Option (B) is correct. For a system to be stable poles of its transfer function H (z) must lie inside the unit circle. In inverse system poles will appear as zeros, so zeros must be inside the unit circle.

SOL 6.4.61

Option (C) is correct. An LTI discrete system is said to be BIBO stable if its impulse response h [n] Buy Online: shop.nodia.co.in

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Page 602 Chap 6

3

/ h [ n] < 3

The Z-Transform

n =- 3

z -transform of h [n] is given as

H (z) =

3

/ h [n] z-n n =- 3

Let z = e jW (which describes a unit circle in the z -plane), then 3

/ h [n] e-jWn

H (z) = = =

n =- 3 3

/

n =- 3 3

/

n =- 3

h [n] e-jWn h [ n] < 3

which is the condition of stability. So LTI system is stable if ROC of its system function includes the unit circle z = 1. (A) is true. We know that for a causal system, the ROC is outside the outermost pole. For the system to be stable ROC should include the unit circle z = 1. Thus, for a system to be causal & stable these two conditions are satisfied if all the poles are within the unit circle in z -plane. (R) is false.

i. n

SOL 6.4.62

o .c

Option (B) is correct. We know that for a causal system, the ROC is outside the outermost pole. For the system to be stable ROC should include the unit circle z = 1. Thus, for a system to be causal & stable these two conditions are satisfied if all the poles are within the unit circle in z -plane. (A) is true. If the z -transform X (z) of x [n] is rational then its ROC is bounded by poles because at poles X (z) tends to infinity. (R) is true but (R) is not correct explanation of (A).

a i d

o n

SOL 6.4.63

. w w

w

Option (C) is correct. We have, 2 - 34 z - 1 H (z) = 1 - 34 z - 1 + 18 z - 2 1 1 By partial fraction = -1 1 -1 + ^1 - 2 z h ^1 - 14 z h For ROC : z > 1/2 n n 1 h [n] = b 1 l u [n] + b 1 l u [n], n > 0 = an u [n], z > a 2 4 1 - z -1 Thus, system is causal. Since ROC of H (z ) includes unit circle, so it is stable also. Hence S1 is True For ROC : z < 1 4 n n h [n] =-b 1 l u [- n - 1] + b 1 l u [n], z > 1, z < 1 2 2 4 4 System is not causal. ROC of H (z ) does not include unity circle, so it is not stable and S 3 is True.

SOL 6.4.64

Option (C) is correct. We have 2y [n] = ay [n - 2] - 2x [n] + bx [n - 1] Buy Online: shop.nodia.co.in

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Taking z transform we get 2Y (z) = aY (z) z-2 - 2X (z) + bX (z) z-1 Y (z) bz-1 - 2 or ...(1) =c X (z) 2 - az-2 m z ( b - z) or H (z) = 22 a (z - 2 ) It has poles at ! a/2 and zero at 0 and b/2 . For a stable system poles must lie inside the unit circle of z plane. Thus a