gate cloud signals & system sample chapter

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R. K. Kanodia

Ashish Murolia

JHUNJHUNUWALA

SIGNALS & SYSTEMS

Jaipur

GATE CLOUD


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GATE CLOUD Signals & Systems, 1e

R. K. Kanodia, Ashish Murolia

AA1213

Information contained in this book has been obtained by author, from sources believes to be reliable.

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Preface to First Edition

Authors

GATE Question Cloudcaters a versatile collection of Multiple Choice Questions to the students who arepreparing for GATE(Gratitude Aptitude Test in Engineering) examination. This book contains over 1500multiple choice solved problems for the subject of Signals & Systems, which has a significant weightage inthe GATE examinations of EC, EE & IN branches.

which leads to some improvement.

Wishyou all the success in conqueringGATE.

The GATE examination is based on multiple choiceproblems which are tricky, conceptual and tests the basic understanding of the subject. So, the problemsincluded in the book are designed to be as exam-like as possible. The solutions are presented using step bystep methodology whichenhance your problem solving skills.The book is categorized into eleven chapters covering all the topics of syllabus of the examination. Eachchapter contains :

Exercise 1 :Exercise 2 :Exercise 3 :Exercise 4 :Detailed Solutions to Exercise 2, 3 & 4Summary of useful theorems

Although we have put a vigorous effort in preparing this book, some errors may have crept in. We shallappreciate and greatly acknowledge the comments, criticism and suggestion from the users of this book

Theoretical & One line Questions

Level 1

Level 2

Mixed Questions taken form previous examinations of GATE & IES.


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DIGITAL ELECTRONICSR. K . Kanodia & Ashish Murolia

GATE CLOUD

GATE CLOUDis an exclusive series of books which offers a completely solved question bank

to GATE aspirants. The book of this series are featured as

Over 1300 Multiple Choice Questions with full & detailed explanations.Questions are graded in the order of complexity from basic to advanced level.

Contains all previous year GATE and IES exam questions from various

branches.

Each question is designed to GATE exam level.

Circuit Analysis

Analog Circuit and Devices

(For EC, EE & IN branches)

(For EC, EE & IN branches)

(For EC, EE & IN branches)Control Systems

Upcoming titles in this series


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CHAPTER 6

THE Z TRANSFORM


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EXERCISE 6.1

MCQ 6.1.1 The z-transform is used to analyze

(A) discrete time signals and system (B) continuous time signals and system

(C) both (A) and (B) (D) none

MCQ 6.1.2 Which of the following expression is correct for the bilateral z-transform of [ ]x n?

(A) [ ]x n znn 0

3

=/ (B) [ ]x n z nn 03

=/(C) [ ]x n zn

n 3

3

=

/ (D) [ ]x n z nn 3

3

=

/MCQ 6.1.3 The unilateral z-transform of sequence [ ]x nis defined as

(A) [ ]x n zn

n 0

3

=

/ (B) [ ]x n znn 3

3

=

/

(C) [ ]x n z n

n 0

3

=

/ (D) [ ]x n z nn 3

3

=

/MCQ 6.1.4 The z-transform of a causal signal [ ]x nis given by

(A) [ ]x n zn

n 3

3

=

/ (B) [ ]x n znn 0

3

=

/

(C) [ ]x n z n

n 3

3

=

/ (D) [ ]x n z nn 0

3

=

/MCQ 6.1.5 For a signal [ ]x n, its unilateral z-transform is equivalent to the bilateral z-transform of

(A) [ ] [ ]x n r n (B) [ ] [ ]x n n

(C) [ ] [ ]x n u n (D) none of these

MCQ 6.1.6 The ROC of z-transform ( )X z is defined as the range of values of zfor which ( )X z

(A) zero (B) diverges

(C) converges (D) none

MCQ 6.1.7 In the z-plane the ROC of z-transform ( )X z consists of a

(A) strip (B) parabola

(C) rectangle (D) ring

MCQ 6.1.8 If [ ]x nis a right-sided sequence, and if the circle z r0= is in the ROC, then

(A) the values of zfor which z r> 0will also be in the ROC


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Page 484 The Z Transform Chapter 6

(B) the values of zfor which z r< 0will also be in the ROC

(C) both (A) & (B)

(D) none of these

MCQ 6.1.9 The ROC does not contain any

(A) poles (B) 1s

(C) zeros (D) none

MCQ 6.1.10 Let [ ] ( )x n X z Z

be a z-transform pair. If [ ] [ ]x n n= , then the ROC of ( )X z is

(A) z 1< (B) z 1>

(C) entire z-plane (D) none of the above

MCQ 6.1.11 The ROC of z-transform of unit-step sequence [ ]u n, is

(A) entirez-plane (B)

z1

(D) none of the above

MCQ 6.1.12 The ROC of the unilateral z-transform of n is

(A) z > (B) z <

(C) 1z < (D) z 1>

MCQ 6.1.13 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

MCQ 6.1.14 The z-transform of unit step sequence is

(A) 1 (B) 1z1

(C) zz

1 (D) 0

MCQ 6.1.15 The ROC for the z-transform of the sequence [ ] [ ]x n u n = is

(A) z 0> (B) 1z (D) does not exist

MCQ 6.1.16 Let [ ] ( )x n X z Z

, then unilateral z-transform of sequence [ ] [ 1]x n x n 1 = will be

(A) ( ) ( ) [0]X z z X z x 11

= + (B) ( ) ( ) [ ]X z z X z x 11

1=

(C) ( ) ( ) [ 1]X z z X z x 11

= (D) ( ) [ ] [ 1]X z z X z x 1

1= +

MCQ 6.1.17 Let [ ] ( )x n X z Z

, the bilateral z-transform of [ ]x n n0 is given by

(A) ( )zX z (B) ( )z X zn0

(C) ( )z X zn0 (D) ( )z

X z1


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Chapter 6 The Z Transform Page 485

MCQ 6.1.18 If the ROC of z-transform of [ ]x nis Rxthen the ROC of z-transform of [ ]x n is

(A) Rx (B) Rx

(C) /R1 x (D) none of these

MCQ 6.1.19 If ( ) { [ ]}X z x n Z= , then ( ) { [ ]}X z a x n Z n= will be

(A) ( )X az (B) Xaza k

(C) Xzaa k (D) X az1b l

MCQ 6.1.20 If [ ]x nand [ ]y nare two discrete time sequences, then the z-transform of correlation

of the sequences [ ] [ ]andx n y n is

(A) ( ) ( )X z Y z 1 1 (B) ( ) ( )X z Y z 1

(C) ( ) ( )X z Y z * (D) * ( ) * ( )X z Y z 1

MCQ 6.1.21 If ( ) { [ ]}X z x n Z= , then, value of [0]x is equal to

(A) ( )lim zX zz 0"

(B) ( ) ( )lim z X z1z 1

"

(C) ( )limX zz" 3

(D) ( )limX zz 0"

MCQ 6.1.22 The choice of realization of structure depends on

(A) computational complexity (B) memory requirements

(C) parallel processing and pipelining (D) all the above

MCQ 6.1.23 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 6.1.24 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 6.1.25 The number of memory locations required to realize the system,

( )H zz z

z z1 21 3 2

2 4

2 3

=+ +

+ +

is

(A) 5 (B) 7

(C) 2 (D) 10

MCQ 6.1.26 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 6.2

MCQ 6.2.1 Consider a DT signal which is defined as follows

[ ]x n,

,

n

n

21 0

0 0

(C) ,maxz 1> e o (D) z <

MCQ 6.2.7 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. [ 2]u n 1.( )

, 1z z

z11

1

2

Codes :

P Q R S

(A) 1 4 2 3

(B) 2 4 1 3(C) 4 1 3 2

(D) 4 2 3 1

MCQ 6.2.8 The z-transform of signal [ ] [ ]x n e u n jn= is

(A) , : 1ROCz

z z1

>+

(B) , : 1ROCz j

z z >

(C) , : 1ROCz

zz

1


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MCQ 6.2.14 If ROC of ( )X z is z1 3< < , the signal [ ]x nwould be

(A) [ ( ) ( ) ] [ ]u n2 3 1n n (B) [ ( ) ( ) ] [ ]u n2 3 1 1n n +

(C) ( ) [ ] ( ) [ ]u n u n 2 3 1 1n n (D) [ ( ) ( ) ] [ ]u n2 3 1 1n n+

MCQ 6.2.15 Consider a DT sequence [ ]x n [ ] [ ]x n x n 1 2= + where, [ ]x n1 ( . ) [ ]u n0 7 1n

= and

[ ] ( 0.4) [ 2]x n u n n2 = . The region of convergence of z-transform of [ ]x nis

(A) . .z0 4 0 7< < (B) .z 0 7>

(C) .z 0 4< (D) none of these

MCQ 6.2.16 The z-transform of a DT signal [ ]x nis ( )X z z z

z8 2 12

=

. What will be the z

-transform of [ ]x n 4 ?

(A) ( ) ( )

( )

z z

z

8 4 2 4 1

42+ +

+

(B) z zz

8 2 12

5

(C)z z

z128 8 1

42

(D)

z z z8 21

5 4 3

MCQ 6.2.17 If [ ] [ ]x n u n n= , then the z-transform of [ ] [ ]x n u n 3+ will be

(A)z

z 2

(B)z

z4

(C)z

z3

a k (D)

zz 3

MCQ 6.2.18 Let [ ], [ ]x n x n 1 2 and [ ]x n3 be three discrete time signals and ( ), ( )X z X z 1 2 and ( )X z3

are their z-transform respectively given as

( )X z1 ( )( . )z zz

1 0 5

2

=

,

( )X z2 ( )( . )z zz

1 0 5=

and ( )X z3 ( )( . )z z1 0 51

=

Then [ ], [ ]x n x n 1 2 and [ ]x n3 are related as

(A) [ ] [ ] [ ]x n x n x n 2 11 2 3 = = (B) [ ] [ ] [ ]x n x n x n 2 11 2 3+ = + =

(C) [ ] [ ] [ ]x n x n x n 1 21 2 3= = (D) [ ] [ ] [ ]x n x n x n 1 11 2 3+ = =

MCQ 6.2.19 The inverse z-transform of a function ( )X zz

z 9

=

is

(A) [ ]u n 10n 10 (B) [ ]u n 10n

(C) [ ]u n/n 10 (D) [ ]u n 9n 9


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MCQ 6.2.20 Let [ ] ( )x n X z Z

be a z-transform pair, where ( )X z z

z3

2

=

. The value of [ ]x 5 is

(A) 3 (B) 9

(C) 1 (D) 0

MCQ 6.2.21 The z-transform of the discrete time signal [ ]x nshown in the figure is

(A) zz1

k

1

(B) zz1

k

1+

(C)zz

11 k

1

(D)zz

11 k

1

+

MCQ 6.2.22 Consider the unilateral z-transform pair [ ] ( )x n X z z

z1

Z=

. The z-transform

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

(A)z

z1

2

,

z 11

(B)

z 11

,

zz

1

2

(C)z 1

1

,z

z1 (D)

zz

1,

zz

1

2

MCQ 6.2.23 A discrete time causal signal [ ]x nhas the z-transform

( )X z .

, : 0.4ROCz

zz

0 4 >=

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

(A) same as ROC of ( )X z (B) . .z0 4 2 5< (D) .z 0 8>

MCQ 6.2.24 The z-transform of a discrete time sequence [ ] [ 1] [ ]y n n n u n = + is

(A)( )z

z1

23

2

(B)

( )

( )

z

z z

1

13

+

(C)( )z

z1 2

(D)( )z 1

12


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MCQ 6.2.25 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) List-II (z-transform)

P. ( ) [ ]n u n1 n 1.( )

, :ROCz

z z1

1>1 21

Q. [ ]nu n 1 2.( )

, : 1ROCz

z1

1>1

+

R. ( ) [ ]u n1 n 3.( )

, : 1ROCz

zz

1 1 2

1

+

Codes : P Q R S

(A) 4 1 2 3

(B) 4 3 2 1

(C) 3 1 4 2

(D) 2 4 1 3

MCQ 6.2.26 A signal [ ]x n has the following z-transform ( )X z (1 2 ), :log ROCz z < 21= .

The signal [ ]x nis

(A) [ ]u n21 nb l (B) [ ]n u n1 21

nb l(C) [ ]

n u n

121 1

n

b l (D) [ ]u n21 1n

b lMCQ 6.2.27 A discrete time sequence is defined as [ ]x n ( 2) [ 1]u nn

n1=

. The z-transform

of [ ]x nis

(A) , :log ROCz z21

21


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MCQ 6.2.29 Let ( )X z be the z-transform of a causal signal [ ] [ ]x n a u n n= with :ROC z a>

. Match the discrete sequences , ,S S S1 2 3and S4with ROC of their z-transforms

,R R1 2and R3.

Sequences ROC

:S1 [ ]x n 2 :R1 z a>

:S2 [ ]x n 2+ :R2 z a3

3

=

1. Non causal but stable

Q.. ( )

( . )

, : 1.2ROCH z

z

zz

1 2

or 1.9P <


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MCQ 6.2.54 Consider three stable LTI systems ,S S1 2and S3whose transfer functions are

S1: ( )H z 2z z

z2

21

163

21

=+

S2: ( )H z

z z z

z 1

32 3 21 2 34=

+ +

+

S3: ( )H z 1 1

1

z z z

z z1

31 1

21 1

21 2

34 1

=

+

^ ^h hWhich of the above systems is/are causal?

(A) S1only (B) S1and S2

(C) S1and S3 (D) ,S S1 2and S3

MCQ 6.2.55 The transfer function for the system realization shown in the figure will be

(A)zz

42 3

+ (B)

zz

24 3

+

(C)z

z2 3

4

+ (D)zz

23

+

MCQ 6.2.56 Consider a cascaded system shown in the figure

where, [ ]h n1 [ ] [ ]n n21 1 = + and [ ]h n2 [ ]u n2

1 n= b l

If an input [ ] ( )cosx n n= is applied, then output [ ]y nequals to

(A) ( )cos n31

(B) ( )cos n65

(C) ( )cos n613

(D) ( )cos n

MCQ 6.2.57 The block diagram of a discrete time system is shown in the figure below

The range of for which the system is BIBO stable, will be

(A) 1> (B) 1 1< (D) 0


25/111

EXERCISE 6.3

MCQ 6.3.1 Let [ ] [ 1] [ 2]x n n n = + + . The unilateral z-transform is

(A) z 2 (B) z2

(C) 2z 2 (D) 2z2

MCQ 6.3.2 The unilateral z-transform of signal [ ] [ 4]x n u n = + is

(A) 1 3z z z z 2 4+ + + + (B)z1

1

(C) 1 z z z z 1 2 3 4+ + + + (D)z1

11

MCQ 6.3.3 The z-transform of [ ], 0n k k > is

(A) , 0z z >k (B) , 0z z >k

(C) , 0z zk ! (D) , 0z zk !

MCQ 6.3.4 The z-transform of [ ], 0n k k > + is

(A) , 0z zk!

(B) , 0z zk!

(C) ,z k all z (D) zk, all z

MCQ 6.3.5 The z-transform of [ ]u nis

(A) , 1z

z1

1>1

(B) , 1z

z1

14

5 5

(B)

( . )

( . ), .

z z

zz

0 25

0 250 5>4

5 5

(C)( . )

( . ), 0.25

z z

zz

0 25

0 25

(B) ,z

zz

4 14

41

(D) ,z

z1 4

141

(B) , 3zz z3

(D) , 3

z z

33

(C)( )

( ),

z z

z z

2 1

1 22

+

0 1z< < (D)

( )

( ),

z z

z z

2 1

1 22

+

1z >

MCQ 6.3.11 The z-transform of {3, 0, 0, 0, 0, , 1, 4}6 -

(A) 3 6 4 , 0z z z z


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MCQ 6.3.15 The time signal corresponding to , 4z

z zz

16

3>2

241

is

(A) ( ) [ ]u n3249 4

3247 4n n +: D (B) [ ]u n3249 4 3247 4n n+: D

(C) ( 4) [ ] 4 [ ]u n u n 3249

3247n n

+ (D) 4 [ ] ( 4) [ ]u n u n 3249

3247n n

+

MCQ 6.3.16 The time signal corresponding to , 1z

z z zz

12 2 2

>2

4 3 2

is

(A) 2 [ 2] [1 ( 1) ] [ 2]n u nn + (B) 2 [ 2] [1 ( 1) ] [ 2]n u nn + + +

(C) 2 [ 2] [( 1) 1] [ 2]n u nn + + + (D) 2 [ 2] [( 1) 1] [ 2]n u nn +

MCQ 6.3.17 The time signal corresponding to 1 2 4 , 0z z z >6 8+ + is

(A) [ ] 2 [ 6] 4 [ 8]n n n + + (B) [ ] 2 [ 6] 4 [ 8]n n n + + + +

(C) [ ] 2 [ 6] 4 [ 8]n n n + + + + (D) [ ] 2 [ 6] 4 [ 8]n n n + +

MCQ 6.3.18 The time signal corresponding to ,k

z z1 0>k

k 5

10

=

/ is

(A) [ ]k

n k1

k 5

10

+=

/ (B) [ ]k

n k1

k 5

10

=

/

(C) [ ]k

n k1

k 5

10

+=

/ (D) [ ]k

n k1

k 5

10

=

/

MCQ 6.3.19 The time signal corresponding to (1 )z 1 3+ , 0z > is

(A) [ ] 3 [ 1] 3 [ 2] [ 3]n n n n + + + (B) [ ] 3 [ 1] 3 [ 2] [ 3]n n n n + + + + + +

(C) [ ] 3 [ 1] 3 [ 2] [ 3]n n n n + + + + + +

(D) [ ] 3 [ 1] 3 [ 2] [ 3]n n n n + + +

MCQ 6.3.20 The time signal corresponding to 3 2 , 0z z z z z >6 2 3 4+ + + + is

(A) [ 6] [ 2] 3 [ ] 2 [ 3] [ 4]n n n n n + + + + + +

(B) [ 6] [ 2] 3 [ ] 2 [ 3] [ 4]n n n n n + + + + + +

(C) [ 6] [ 2] 3 [ ] 2 [ 3] [ 4]n n n n n + + + + + + + +

(D) [ 6] [ 2] 3 [ ] 2 [ 3] [ 4]n n n n n + + + +

MCQ 6.3.21 The time signal corresponding to ,z

z1

121

>41 2

(A)2 , 0

0,

n neven and

otherwise

n$

* (B) [ ]u n41 n2b l

(C)2 , , 0

0,

n n

n

odd

even

>n* (D) 2 [ ]u nn


28/111


MCQ 6.3.22 The time signal corresponding to ,z

z1

121

1+ is

(A)( )

[ ]

k

n k1 k 1

(B)( )

[ ]

k

n k1 k 1

+

(C)( )

[ ]k

n k1 k

(D)( )

[ ]k

n k1 k

+

MCQ 6.3.24 If z-transform is given by ( ) ( ), 0cosX z z z >3= , the value of [12]x is

(A)241

(B)241

(C)61

(D)61

MCQ 6.3.25 [ ]X zof a system is specified by a pole zero pattern as following :

Consider three different solution of [ ]x n

[ ]x n1 [ ]u n2 31n n= b l; E

[ ]x n2 2 [ 1] [ ]u n u n 31n

n=

[ ]x n3 2 [ 1] [ 1]u n u n 31n

n= +

Correct solution is

(A) [ ]x n1 (B) [ ]x n2

(C) [ ]x n3 (D) All three


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MCQ 6.3.26 Consider three different signal

[ ]x n1 [ ]u n2 21n n

= b l; E [ ]x n2 2 [ 1] [ 1]u n u n

2

1nn= +

[ ]x n3 2 [ 1] [ ]u n u n 21n

n=

Following figure shows the three different region. Choose the correct for the ROC

of signal

R1 R2 R3(A) [ ]x n1 [ ]x n2 [ ]x n3(B) [ ]x n2 [ ]x n3 [ ]x n1(C) [ ]x n1 [ ]x n3 [ ]x n2(D) [ ]x n3 [ ]x n2 [ ]x n1

MCQ 6.3.27 Given the z-transform

( )X z z z

z

1 1

1

21 1

31 1

67 1

=

+

+

For three different ROC consider there different solution of signal [ ]x n:

(a) , [ ] [ ]z x n u n 21

21

31

> n

n

1=

b l; E(b) , [ ] [ 1]z x n u n

31

21

31

< n

n

1=

+

+ b l; E

(c) , [ ] [ 1] [ ]z x n u n u n 31

21

21

31

< < n

n

1=

b lCorrect solution are(A) (a) and (b) (B) (a) and (c)

(C) (b) and (c) (D) (a), (b), (c)

MCQ 6.3.28 The ( )X z has poles at z 21= and 1z = . If [1] 1, [ 1] 1x x= = , and the ROC

includes the point z 43= . The time signal [ ]x nis

(A) [ ] ( 1) [ 1]u n u n 21

nn

1 (B) [ ] ( 1) [ 1]u n u n 21

nn


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31/111


MCQ 6.3.34 The z-transform of the signal [ ]nx nis

(A)( )z

z16

322 2

2

(B)

( )zz16

322 2

2

(C) ( )z

z

16

322 2 (D) ( )z

z

16

322 2

MCQ 6.3.35 The z-transform of the signal [ 1] [ 1]x n x n + + is

(A)( )

( )

( )

( )

z

z

z

z

1 16

1

1 16

12

2

2

2

+

++

(B)

( 1)

z

z z

162

2

+

(C)( )

z

z z

16

12

2

+ (D) None of the above

MCQ 6.3.36 The z-transform of the signal [ ] [ 3]x n x n * is

(A)

( )z

z

16

2 2

3

(B)

( )z

z

16

2 2

7

(C)( )z

z162 25

(D)

( )zz162 2

Statement for Q. 37-41 :

Given the z-transform pair 3 [ ] ( )n u n X z n z2

MCQ 6.3.37 The time signal corresponding to (2 )X z is

(A) 3 [2 ]n u nn2 (B) [ ]n u n23 n 2

b l(C) [ ]n u n23

n2b l (D) 6 [ ]n u nn 2

MCQ 6.3.38 The time signal corresponding to ( )X z 1 is

(A) 3 [ ]n u nn2 (B) 3 [ ]n u nn2

(C) 3 [ ]n

u n1

n2

1 (D) 3 [ ]

n u n

1n

2

1

MCQ 6.3.39 The time signal corresponding to ( )dzd

X z is

(A) ( 1) 3 [ 1]n u nn3 1 (B) 3 [ 1]n u nn3

(C) (1 ) 3 [ 1]n u nn3 1 (D) ( 1) 3 [ ]n u nn3 1

MCQ 6.3.40 The time signal corresponding to2

( )z z X z2 2

b l is

(A) ( [ 2] [ 2])x n x n 21

+ (B) [ 2] [ 2]x n x n +

(C) [ 2] [ 2])x n x n 21

+ (D) [ 2] [ 2]x n x n +

MCQ 6.3.41 The time signal corresponding to { ( )}X z 2is


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(A) [ [ ]]x n 2 (B) [ ] [ ]x n x n *

(C) ( ) [ ]x n x n * (D) [ ] [ ]x n x n *

MCQ 6.3.42 A causal system has

Input, [ ]x n [ ] [ 1] [ 2]n n n41

81 = + and

Output, [ ]y n [ ] [ 1]n n43

=

The impulse response of this system is

(A) [ ]u n31 5

21 2

41n n

b bl l; E (B) [ ]u n31 5 21 2 41n n+ b bl l; E(C) [ ]u n

31 5

21 2

41n n

b bl l; E (D) [ ]u n31 5 21 2 41n n+b bl l; E

MCQ 6.3.43 A causal system has input [ ] ( 3) [ ]x n u n n= and output [ ]y n ( ) ( ) [ ]u n4 2 n n21=

6 @.

The impulse response of this system is

(A) [ ]u n721 10

21n n

b bl l; E (B) ( ) [ ]u n7 2 10 21n n b l; E(C) ( ) [ ]u n10

21 7 2 n

2

b l; E (D) ( ) [ ]u n10 2 7 21n nb l; EMCQ 6.3.44 A system has impulse response [ ] ( ) [ ]h n u n n2

1= . The output [ ]y nto the input [ ]x n

is given by [ ] 2 [ 4]y n n= . The input [ ]x nis

(A) 2 [ 4] [ 5]n n (B) 2 [ 4] [ 5]n n + +

(C) 2 [ 4] [ 5]n n + + (D) 2 [ 4] [ 5]n n

MCQ 6.3.45 A system is described by the difference equation

[ ]y n [ ] [ 2] [ 4] [ 6]x n x n x n x n = +

The impulse response of system is

(A) [ ] 2 [ 2] 4 [ 4] 6 [ 6]n n n n + + + +

(B) [ ] 2 [ 2] 4 [ 4] 6 [ 6]n n n n + +

(C) [ ] [ 2] [ 4] [ 6]n n n n +

(D) [ ] [ 2] [ 4] [ 6]n n n n + + + +

MCQ 6.3.46 The impulse response of a system is given by [ ]h n [ 1]u n43

n= . The differenceequation representation for this system is

(A) 4 [ ] [ 1] 3 [ 1]y n y n x n = (B) 4 [ ] [ 1] 3 [ 1]y n y n x n + = +

(C) 4 [ ] [ 1] 3 [ 1]y n y n x n + = (D) 4 [ ] [ 1] 3 [ 1]y n y n x n + + = +

MCQ 6.3.47 The impulse response of a system is given by [ ]h n [ ] [ 5]n n = . The difference

equation representation for this system is

(A) [ ] [ ] [ 5]y n x n x n = (B) [ ] [ ] [ 5]y n x n x n = +

(C) [ ] [ ] 5 [ 5]y n x n x n = + (D) [ ] [ ] 5 [ 5]y n x n x n = +


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MCQ 6.3.48 Consider the following three systems

[ ]y n1 0.2 [ 1] [ ] 0.3 [ 1] 0.02 [ 2]y n x n x n x n = + +

[ ]y n2 [ ] 0.1 [ 1]x n x n =

[ ]y n

3 0.5 [ 1] 0.4 [ ] 0.3 [ 1]y n x n x n = +

The equivalent system are

(A) [ ]y n1 and [ ]y n2 (B) [ ]y n2 and [ ]y n3

(C) [ ]y n3 and [ ]y n1 (D) all

MCQ 6.3.49 The z-transform function of a stable system is ( )( )( )

H zz z

z

1 2 1

21

21 1

23 1

= +

. The

impulse response [ ]h nis

(A) 2 [ 1] [ ]u n u n 21n n

+ b l (B) 2 [ 1] [ ]u n u n 21n n

+ b l

(C) 2 [ 1] [ ]u n u n 21n

n

b l (D) 2 [ ] [ ]u n u n 21n n

b lMCQ 6.3.50 The z-transform of a anti causal system is ( )X z

z zz

3 7 1212 21

2= +

. The value of [0]x is

(A)47

(B) 0

(C) 4 (D) Does not exist

MCQ 6.3.51 The transfer function of a causal system is ( )H z z z

z6

52

2

=

. The impulse response is

(A) (3 ( 1) 2 ) [ ]u nn n n 1+ + (B) (3 2( 2) ) [ ]u nn n1 + +

(C) (3 ( 1) 2 ) [ ]u nn n n1 1+ + (D) (3 ( 2) ) [ ]u nn n1 1 +

MCQ 6.3.52 The transfer function of a system is given by ( )H z ( )

z z

z z3 22

41=

. The system is

(A) causal and stable (B) causal, stable and minimum phase

(C) minimum phase (D) none of the above

MCQ 6.3.53 The z-transform of a signal [ ]x nis ( )X z z z13

310 1 2= +

. If ( )X z converges on the

unit circle, [ ]x nis

(A)( )

[ ] [ 1]u n u n 3 8

1 83nn

1

3

+

(B)( )

[ ]( )

[ ]u n u n 3 8

18

3nn

1

3

+

(C)( )

[ ]( )

[ ]u n u n 3 8

18

3n

n

1

3

+

(D)( )

[ ]( )

[ ]u n u n 3 8

18

3n

n

1

3

+

MCQ 6.3.54 The transfer function of a system is ( )H z ,z

zz

1

441

>

41 1 2

1

=

^ h . The [ ]h nis(A) stable (B) causal

(C) stable and causal (D) none of the above


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MCQ 6.3.55 The transfer function of a system is given as

( )H z z z

z2

21

31

21

=

+

^ ^^h h

hConsider the two statements

Statement (i) : System is causal and stable.

Statement (ii) : Inverse system is causal and stable.

The correct option is

(A) (i) is true (B) (ii) is true

(C) Both (i) and (ii) are true (D) Both are false

MCQ 6.3.56 The system [ ]y n [ 1] 0.12 [ 2] [ 1] [ 2]cy n y n x n x n = + + is stable if

(A) 1.12c < (B) 1.12c >

(C) 1.12c < (D) 1.12c >

MCQ 6.3.57 The impulse response of the system shown below is

(A) 2 (1 ( 1) ) [ ] [ ]u n n21n2n

2 + +^ h (B) (1 ( 1) ) [ ] [ ]u n n22

21n n + +

(C) 2 (1 ( 1) ) [ ] [ ]u n n21n2n

2

+

^ h (D) [1 ( 1) ] [ ] [ ]u n n22

21n n

+

MCQ 6.3.58 The system diagram for the transfer function ( )H z z z

z12

=+ +

. is shown below.

The system diagram is a

(A) Correct solution (B) Not correct solution

(C) Correct and unique solution (D) Correct but not unique solution

***********


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EXERCISE 6.4

MCQ 6.4.1 What is the z-transform of the signal [ ] [ ]x n u n n= ?

(A) ( )X zz 1

1=

(B) ( )X z

z11

=

(C) ( )X zz

z

=

(D) ( )X zz

1

=

MCQ 6.4.2 The z-transform of the time function [ ]n kk 0

3

=

/ is

(A)z

z 1 (B)

zz

1

(C)( )z

z1 2

(D)( )

zz 1 2

MCQ 6.4.3 The z-transform ( )F z of the function ( )f nT anT= is

(A) z az

T (B) z az

T+

(C)z a

zT

(D)z a

zT

+

MCQ 6.4.4 The discrete-time signal [ ] ( )x n X z z 23

nn

n

0

2Z n=

3

+=/ , where denotes a

transform-pair relationship, is orthogonal to the signal

(A) [ ] ( )y n Y z z 32 n

n

n1 1

0) =

3

=

-` j/ (B) [ ] ( ) ( )y n Y z n z 5 ( )nnn

2 20

2 1) =

3

=

- +/

(C) [ ] ( )y n Y z z 2 nn

n3 3) =

3

3 -

=-

-/ (D) [ ] ( )y n Y z z z 2 3 14 4 4 2) = + +- -

MCQ 6.4.5 Which one of the following is the region of convergence (ROC) for the sequence

[ ]x n [ ] [ 1]; 1b u n b u n b >

(C) Region z 1>

(D) Annular strip in the region b zb1

< , whereas

the ROC for [ 1]a u nn 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

MCQ 6.4.7 Which one of the following is the correct statement ?

The region of convergence of z-transform of [ ]x nconsists of the values of z for

which [ ]x n r n is(A) absolutely integrable (B) absolutely summable

(C) unity (D) 1

(C) z3

1< (D) z2 3<

(C) z65

56

< < (D) z56

< < 3

MCQ 6.4.10 The region of convergence of the z-transform of the discrete-time signal [ ] 2 [ ]x n u n n=

will be

(A) 2z > (B) 2z (D) z21

(B) z 1 (D) (Real part of z) 0 , then what is the corresponding [ ]x n?

(A) e n (B) en

(C) [ ]u n (D) ( )n

MCQ 6.4.16 The ztransform ( )X z of a sequence [ ]x nis given by [ ]X z.

z1 2

0 51

=

. It is given thatthe 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

MCQ 6.4.17 If ( )u t is the unit step and ( )t is the unit impulse function, the inverse z-transform

of ( )F z 1z1= + for k 0> is

(A) ( ) ( )k1 k (B) ( ) ( )k 1 k

(C) ( ) ( )u k1 k (D) ( ) ( )u k 1 k

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MCQ 6.4.18 For a z-transform ( )X z z z

z2

21

31

65

=

^ ^^ h hhMatch List I (The sequences) with List II (The region of convergence ) and select

the correct answer using the codes given below the lists :List I List II

A. [(1/2) (1/3) ] [ ]u nn n+ 1. ( / ) ( / )z1 3 1 2<

Codes :

A B C D

(A) 4 2 1 3(B) 1 3 4 2

(C) 4 3 1 2

(D) 1 2 4 3

MCQ 6.4.19 Which one of the following is the inverse z-transform of

( )X z ( )( )

, 2z z

zz

2 3 , the residue of ( )X z zn 1 at z a= for n 0$ will

be

(A) an 1 (B) an

(C) nan (D) nan 1-

MCQ 6.4.21 Given ( ) ,X zaz bz 1 1121

131

=

+

a and 1b < with the ROC specified as

a z b< < , then [ ]x 0 of the corresponding sequence is given by

(A)31 (B)

65

(C)21 (D)

61

MCQ 6.4.22 If ( )X zz zz z

1

3

=+

+

then [ ]x nseries has

(A) alternate 0s (B) alternate 1s

(C) alternate 2s (D) alternate 1 s

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MCQ 6.4.23 Consider the z-transform ( ) 5 4 3; 0x z z z z <

B. [ 1]u nn 2.( )z1

11

, ROC : z >

C. [ 1]n u nn 3.( )z1

11

, ROC : |z <

D. [ ]n u nn

4. ( )zz1 1 21

, ROC : |z <

Codes :

A B C D

(A) 2 4 3 1

(B) 1 3 4 2

(C) 1 4 3 2

(D) 2 3 4 1

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MCQ 6.4.27 Match List-I ( [ ])x n with List-II ( ( ))X z and select the correct answer using the codes

given below the Lists:

List-I List-II

A. [ ]a u nn 1.( )z a

az2

B. [ ]a u n 2n 2 2.ze a

zej

j

C. e ajn n 3.z a

z

D. [ ]na u nn 4.z a

z 1

Codes :

A B C D(A) 3 2 4 1

(B) 2 3 4 1

(C) 3 4 2 1

(D) 1 4 2 3

MCQ 6.4.28 Algebraic expression for z-transform of [ ]x nis [ ]X z. What is the algebraic expression

for z-transform of { [ ]}e x nj n0 ?

(A) ( )X z z0 (B) ( )X e zj 0

(C) ( )X e zj 0 (D) ( )X z ej z0

MCQ 6.4.29 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) ( ) ( )f nT g nT z n/ (B) ( ) ( )f nT z g nT zn n //(C) ( ) ( )f kT g nT kT z n / (D) ( ) ( )f nT kT g nT z n /

MCQ 6.4.30 The output [ ]y nof a discrete time LTI system is related to the input [ ]x nas given

below :

[ ]y n [ ]x kk 0

=3

=

/

Which one of the following correctly relates the z-transform of the input andoutput, denoted by ( )X z and ( )Y z , respectively ?

(A) ( ) ( ) ( )Y z z X z 1 1= (B) ( ) ( )Y z z X z 1=

(C) ( )( )

Y zz

X z

1 1=

(D) ( )( )

Y zdz

dX z=

MCQ 6.4.31 Convolution of two sequence [ ]x n1 and [ ]x n2 is represented as

(A) ( ) ( )X z X z 1 2* (B) ( ) ( )X z X z 1 2

(C) ( ) ( )X z X z 1 2+ (D) ( )/ ( )X z X z 1 2

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MCQ 6.4.32 The z-transform of a signal is given by ( )C z ( )

( )

z

z z

4 1

1 11 2

1 4

=

. Its final value is

(A) 1/4 (B) zero

(C) 1.0 (D) infinity

MCQ 6.4.33 Consider a system described by the following difference equation:

( ) ( ) ( ) ( )y n y n y n y n 3 6 2 11 1 6+ + + + + + ( ) ( ) ( )r n r n r n 2 9 1 20= + + + +

Where yis the output and ris the input. The transfer function of the system will

be

(A)3z z z

z z2 6

2 203 2

2

+ + +

+ + (B)z z z

z z6 6 11

9 203 2

2

+ + +

+ +

(C)z z

z z z9 20

6 6 112

3 2

+ +

+ + + (D) none of the above

MCQ 6.4.34 If the function ( ) ( . )H z z z 1 1 511 2

= + and ( ) .H z z z 1 5 12

2= + , 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

MCQ 6.4.35 The state model

[ 1]x k+ [ ] [ ]x k u k 0 1 0

1 =

+> >H H

[ ]y k[ ]

[ ]

x k

x k0 1

1

2= 8 >B H

is represented in the difference equation as

(A) [ 2] [ 1] [ ] [ ]c k c k c k u k + + + + =

(B) [ 1] [ ] [ 1] [ 1]c k c k c k u k + + + =

(C) [ 2] [ 1] [ ] [ ]c k c k c k u k + + =

(D) [ 1] [ ] [ 1] [ 1]c k c k c k u k + + + = +

MCQ 6.4.36 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

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(A) real axis at z 1=

(B) real axis between z 0= and z 1=

(C) imaginary axis at z j=

(D) imaginary axis between z 0= and z j=

MCQ 6.4.37 Which one of the following digital filters does have a linear phase response ?

(A) [ ] [ 1] [ ] [ 1]y n y n x n x n + =

(B) [ ] 1/6(3 [ ] 2 [ 1] [ 2])y n x n x n x n = + +

(C) [ ] 1/6( [ ] 2 [ 1] 3 [ 2])y n x n x n x n = + +

(D) [ ] 1/4( [ ] 2 [ 1] [ 2])y n x n x n x n = + +

MCQ 6.4.38 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 ( ) 0zReal = line

MCQ 6.4.39 The impulse response of a discrete system with a simple pole is shown in the given

figure

The pole must be located

(A) on the real axis at z 1= (B) on the real axis at z 1=

(C) at the origin of the z-plane (D) at z 3=

MCQ 6.4.40 The response of a linear, time-invariant discrete-time system to a unit step input[ ]u nis the unit impulse [ ]n . The system response to a ramp input [ ]nu nwould be

(A) [ ]u n (B) [ 1]u n

(C) [ ]n n (D) [ ]k n kk 0

3

=

/

MCQ 6.4.41 A system can be represented in the form of state equations as

[ 1]s n+ [ ] [ ]As n Bx n = +

[ ]y n [ ] [ ]Cs n Dx n = +

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where , ,A B Cand Dare matrices, [ ]s nis the state vector. [ ]x nis 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 C D 1 + (B) ( )B zI C D A1 +

(C) ( )C zI A B D 1

+

(D) ( )D zI A C B 1

+

MCQ 6.4.42 What is the number of roots of the polynomial ( ) 4 2F z z z z 83 2= + , lying

outside the unit circle ?

(A) 0 (B) 1

(C) 2 (D) 3

MCQ 6.4.43 [ ] [ ]y n x k k

n

=3=

/

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 k n = , and [ ]y nis related to [ ]x nas [ ] [ 1]y n y n 21

[ ]x n=

What is [ ]y nequal to ?

(A) k (B) ( / ) k1 2 n

(C) nk (D) 2n

MCQ 6.4.45 Unit step response of the system described by the equation [ ] [ 1] [ ]y n y n x n + = is

(A)( )( )z z

z1 1

2

+ (B)

( )( )z zz

1 1+

(C)zz

11

+ (D)

( )( )z

z z

11

+

MCQ 6.4.46 Unit step response of the system described by the equation [ ] [ 1] [ ]y n y n x n + = is

(A)( 1)( 1)z z

z2

+ (B)

( )( )z zz

1 1+

(C)( )( )

z

z

11

+ (D)

( )( )z

z z

11

+

MCQ 6.4.47 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 b d 1 + (B) ( )c zI A 1

(C) ( )zI A z 1 (D) ( )zI A 1

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MCQ 6.4.48 A system with transfer function ( )H z has impulse response (.)h defined as

( ) , ( )h h2 1 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

MCQ 6.4.49 The z-transform of a system is ( )H z .zz0 2= . If the ROC is .z 0 2< , then the

impulse response of the system is

(A) ( . ) [ ]u n0 2 n (B) ( . ) [ ]u n0 2 1n

(C) ( . ) [ ]u n0 2 n (D) ( . ) [ ]u n0 2 1n

MCQ 6.4.50 A sequence ( )x n with the ztransform ( ) 2 2 3X z z z z z 4 2 4= + + is applied as an

input to a linear, time-invariant system with the impulse response [ ] 2 [ 3]h n n=

where

[ ]n ,

,

n1 0

0 otherwise=

=)The output at n 4= is

(A) 6 (B) zero

(C) 2 (D) 4

MCQ 6.4.51 The z-transform of a signal [ ]x nis given by z z z z 4 3 2 6 23 1 2 3+ + +- -

It is applied to a system, with a transfer function ( )H z z3 21= -

Let the output be [ ]y n. Which of the following is true ?

(A) [ ]y nis non causal with finite support

(B) [ ]y nis causal with infinite support

(C) [ ] ;y n n0 3>=

(D) [ ( )] [ ( )]

[ ( )] [ ( )] ;

Re Re

Im Im

Y z Y z

Y z Y z . The impulse response of a stable system

that exactly compensates the magnitude of the distortion is

(A) [ ]a u n1 n

b l (B) [ 1]a u n1 n

b l(C) [ ]a u nn (D) [ 1]a u nn

MCQ 6.4.55 Assertion (A) :A linear time-invariant discrete-time system having the system

function

( )H z z

z

21= +is a stable system.

Reason (R):The pole of ( )H z is in the left-half plane for a stable system.


(B) Both A and R are true but R is NOT a correct explanation of A(C) A is true but R is false


MCQ 6.4.56 Assertion (A) :An LTI discrete system represented by the difference equation

[ 2] 5 [ 1] 6 [ ] [ ]y n y n y 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.

MCQ 6.4.57 Consider the following statements regarding a linear discrete-time system

( )H z ( . )( . )z z

z0 5 0 5

12=

+

+

1. The system is stable

2. The initial value ( )h 0 of the impulse response is 4

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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

MCQ 6.4.58 Assertion (A) :The discrete time system described by [ ] [ ] [ ]y n x n x n 2 4 1= + is

unstable, (here [ ]y nis the output and [ ]x nthe input)

Reason (R) :It has an impulse response with a finite number of non-zero samples.





MCQ 6.4.59 If the impulse response of discrete - time system is [ ] [ ]h n u n 5 1n= , then the

system function ( )H z is equal to

(A)z

z5

and the system is stable (B)z

z5

and the system is stable

(C)z

z5

and the system is unstable (D)z

z5

and the system is unstable

MCQ 6.4.60 ( )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

MCQ 6.4.61 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.


(B) Both A and R are true but R is NOT the correct explanation of A.



MCQ 6.4.62 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.

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IES EC 2002

IES EC 2002


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Reason (R) :For a rational system, ROC bounded by poles.





MCQ 6.4.63 The transfer function of a discrete time LTI system is given by

( )H z 1 z z

z2

43 1

81 2

43 1

= +

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: 1/z 4 , 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 [( ) ( )( ) ] [ ]u n4 2 6 3n n= + , 3z > ( 2)P "


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For ROC z2 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

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

n n= +

( 4)Q"

For : 2ROC z < , ROC is interior to circle passing through inner most pole, hence

the system is non causal.

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

For the response

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

: 2ROC z < and z 3> which does not exist ( 1)S "

SOL 6.2.10 Option (B) is correct.

( )X z

( )z z

z

1

1=

+

2z z z

zz

z11

2 11

1= +

= +

a k By partial fraction

Taking inverse z-transform

[ ]x n [ ] [ ]n u n1 2 1= +

[ ]x 0 0 0 0= + =

[ ]x 1 1 2 1= + =

[ ]x 2 0 2 2= + =

SOL 6.2.11 Option (A) is correct.

( )X z e e/z z1

= +

( )X z ! !

.....2!

.....z z zz z

12 3

1 1 1 12 3

2= + + + + + + + +c bm l

2! !....

2!....z z z z z1

31

2 31

2

= + + + + + + + +

c bm lTaking inverse z-transform

[ ]x n [ ]!

nn1

= +


( )X z z z

z z2 3

522=

+

( )( )( )

z zz z

3 15=

++

( )z

X z

( )( )z zz3 1

5=

+

+

z z3

21

1=

+ By partial fraction

Thus ( )X z z

zz

z3

21

=

+

Poles are at z 3= and z 1=


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ROC : z 1< , which is not exterior of circle outside the outermost pole z 3= . So,

[ ]x nis anticausal given as

[ ]x n [ ( ) ( ) ] [ ]u n2 3 1 1n n= +


( )X z z

z

z

z

3

2

1=

+

If z 3> , ROC is exterior of a circle outside the outer most pole, [ ]x nis causal.

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

SOL 6.2.14 Option (C) is correct.

( )X z z

zz

z3

21

=

+

If ROC is z1 3< < , [ ]x nis two sided with anticausal partz

z3

2

, z 3< and

causal partz

z1+

, 1z >

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


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SOL 6.2.15 Option (D) is correct.

( )X z1 ( . ) [ ]z u n0 7 1n n

n

= 3

3

=

/ ( . )z0 7 nn

1

1

=3

=

/

..

z

z

1 0 70 7

1

1

=

ROC : . z0 7 1

( )X z2 ( . ) [ ]z u n0 4 2n n

n

= 3

3

=

/ ( . ) z0 4 n nn

2

= 3

=

/

( . ) z0 4 m m

m 2

= 3

=

/ Let n m=

[( . ) ]z0 4 m

m

1

2

= 3

=

/ ( . )

( . )

z

z

1 0 4

0 41

1

=+

ROC : ( . ) z0 4 1


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We know that [ ]a u nn z a

zZ

or [ ]u n3n z

z

3

Z

[ ]u n3 3n 3 zz

z3

3Z

a k

So [ ]x n [ ]u n3 3n 3=

[ ]x 5 [ ]u3 2 92= =


[ ]x ncan be written in terms of unit sequence as

[ ]x n [ ] [ ]u n u n k =

so ( )X z z

z zz

z1 1

k=

zz

11 k

1=


For positive shift

If, [ ]x n ( )X zZ

then, [ ]x n n0 ( )z X znZ 0 , n 00 $

So [ ]x n 1 zz

zz1 1

11Z

=

a kFor negative shift

[ ]x n n0+ ( ) [ ]z X z x n z n m

m

n

0

1Z

0

0

=

e o/ , n 0>0 [ ]x n 1+ ( ) [ ]z X z x 0

Z

^ hWe know that [ ] [ ]x n u n = so [ ]x 0 1=and [ ]x n 1+ ( )z X z z

zz1

11

Z =

^ ah k z z 1=


Even part of [ ]x n, [ ]x ne ( [ ] [ ])x n x n 21

= +

z- transform of [ ]x ne , ( )X ze ( )X z X z21 1

= + b l; E [ ]x n X z1Za b l

./ .

/z

zz

z21

0 4 21

1 0 4

1

III

=

+

a ek o1 2 344 44 1 2 3444 444Region of convergence for I series is .z 0 4> and for II series it is .z 2 5< .

Therefore, ( )X ze has ROC . .z0 4 2 5< 1

Z

so, [ ]nu n , : 1ROCzdzd

z z

11

>1Z

b l

( )Y z ( )

, : 1ROCz

zz

1 >1 2

1=


Given that ( )X z (1 2 )log z= , z21

ROC is exterior to the circle passing through right most pole so both the term in

equation (i) corresponds to right sided sequences

[ ]x n1 [ ] ( ) [ ]u n u n 31 2

nn = +b l

ROC : 2z31

< 21 gives :R1 2z >

2. Since [ ]x n2 is left-sided signal, so ROC is the region inside a circle having radius

equal to magnitude of smallest pole. So, 2z < and z < 21 gives :R2 z < 2

1

3. Since [ ]x n3 is double sided signal, So ROC is the region in z-plane such as

z > 21 and 2z < which gives R3: 2z< 21 is exterior to the cicle which passes through outtermost pole, so

both the terms in equation (i) contributes to right sided sequences.

[ ]x n [ ] [ ]u n u n 22

31

n

n

= b l


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2. ROC z < 31 is interior to the circle passing through left most poles, so both

the terms in equation (i) corresponds to left sided sequences.

[ ]x n [ 1]u n22

31

n

n

=

+

b l; E3. ROC z<


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SOL 6.3.30 Option (D) is correct.

( )X z z z21

41n n

nn

1 11

0

= +3

3

=

=

b bl l// ( )z z2

14

I II

n

n

nm

m

1

0 1= +

3

= =b l1 2 344 44 1 2 344 44/ / z z1 211

1411

1 1=

ROC :Summation I converges if 1 orz z< >21 1

21 and summation II converges

if 1 orz z4 < < 41 . ROC would be intersection of both which does not exist.


[ ]x nZ

z

z162

2

, ROC 4z 21 Since ROC is outside to the outer most pole so both the terms in

equation (i) corresponds to right sided sequence.

So, [ ]x n [ ] [ ]u n u n 21

31n n

= +b bl l ( 4)A "ROC : z < 3

1 :Since ROC is inside to the innermost pole so both the terms in

equation (i) corresponds to left sided signals.

So, [ ]x n [ ] [ ]u n u n 21 1

31 1

n n

= b bl l ( 2)D "


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ROC : z<


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Poles of ( )X z are 2z = and z 3=

ROC : z 2

Residue of ( )X z zn 1 at z a= is

( ) ( )dzd

z a X z z nz a

2 1=

=

( )( )dz

d z az a

z zn

z a

22

1=

=

dzd

zn

z a

==

nznz a

1=

= nan 1=


( )X z az bz 1 1121

131

=

+

, ROC : a z b< > >H H H

zI A ( )z z 0 = + + =

z z2 + + 0=

In the given options, only option (A) satisfies this characteristic equation.

[ 2] [ 1] [ ]c k c k c k + + + + [ ]u k=

z z2 + + 0=


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We can see that the given impulse response is decaying exponential, i.e.

[ ]h n [ ]a u nn= , a0 1

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