Signals and Systems

Z-Transform

Practice Sheet-01
DISCUSSION By- SUJAL PATEL SIR
TOPICS TO
01 Question
BE
COVERED
02 Discussion
Q.1 The z-transform of a discrete sequence x[n] is X(z), then the z-
transform of x[2n] will be
A. X ( 2z )

z
B. X 
2
1
C. [ X ( z ) + X ( − z )]
2

D. X ( z)
 =- f: . Odd.._
/ I /c] �

0.,
I<
..::c [l<J
/ [/<.J ==- ..,,z- lfr) + c- I)
� -Z
/{ Oqd, :
k .· e __x [,k) ::::: 0

veh :
_;r [Jc7 � ;;t [Ji_]
Q.2 A discrete time causal signal x[n] has the z-transform
z
X(z ) = , ROC :| z |  0.4
z − 0.4
The ROC for z-transform of the even part of x[n] will be

A. same as ROC of X(z)

B. 0.4  | z |  2.5

C. z  0.2

D. z  0.8
Q.3 The z-transform of the discrete time signal x[n] shown in the
figure is

z −k
A.
1 − z −1
z −k
B.
1 + z −1
1 − z −k
C.
1 − z −1
1 + z −k
D.
1 − z −1
Q.4 Given X(z) is z-transform of a discrete sequence x[n], and is given
𝑧 2 +5𝑧
by 𝑋 𝑧 =
𝑧 2 −2𝑧−3

If ROC of X(z) is 1< |z| < 3, the signal x[n] would be

A. [2(3)𝑛 − (−1)𝑛 ]𝑢[𝑛]

B. [−2(3)𝑛 + (−1)𝑛 ]𝑢[−𝑛 − 1]

C. −2(3)𝑛 𝑢[−𝑛 − 1] − (−1)𝑛 𝑢[𝑛]

D. [2(3)𝑛 + (−1)𝑛 ]𝑢[−𝑛 − 1]

Q.5 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}
List-I (Impulse Response) List-I (Impulse Response)
P. [(−4)2𝑛 + 6(3)𝑛 ]𝑢[𝑛] 1. does not exist
Q. (−4)2𝑛 𝑢[𝑛] + (−6)3𝑛 𝑢[−𝑛 − 1] 2. |𝑧| > 3
R. 4 2𝑛 𝑢 −𝑛 − 1 + −6 3𝑛 𝑢 −𝑛 − 1 3. |𝑧| < 2
S. 4(2)𝑛 𝑢[−𝑛 − 1] + (−6)3𝑛 𝑢[𝑛] 4. 2 < |𝑧| < 3

Codes:
P Q R S
A. 4 1 3 2

B. 2 1 3 4

C. 1 4 2 3

D. 2 4 3 1
Q.6 For a signal 𝑥[𝑛] = [𝛼 𝑛 + 𝛼 −𝑛 ]𝑢[𝑛] , the ROC of its z-transform
would be

 1 
A. | z |  min |  |, 
 |  | 
B. | z |  |  |

 1 
C. | z |  max |  |, 
 |  | 
D. | z |  |  |
Q.7 Consider a discrete-time signal
n n
1 1
x | n | =   u [n] +   u [−n − 1]
3 2
The ROC of its z-transform is

A. 3 < |z| < 2

1
B. |z|
2
1
C. |z| 
3
1 1
D. |z|
3 2
 1 1
The time signal corresponding to , | z |  is
Q.8 1 2
1 − z −2
4

2−n , n even and n  0

A. 
 0, otherwise
2n
1
B.  4  u [n]
 
2−n , n odd, n  0
C. 
 0, n even

D. 2−n u [n]
 10
1 −k
Q.9 The time signal corresponding to  z , | z |  0 is
k =5 k

10
1
A.  k [n + k ]
k =5

10
1
B.  k [n − k ]
k =5
10
1
C.  k [−n + k ]
k =5
10
1
D.  k [−n − k ]
k =5
Q.10 The z-transform of 𝑥[𝑛] = {2,4, 5, 7,0,1}
↑

A. 2z2 + 4z + 5 + 7z + z3, z≠∞

B. 2z–2 + 4z–1 + 5 + 7z + z3, z ≠ ∞

C. 2z–2 + 4z–1 + 5 + 7z + z3, 0 < |z| < ∞

D. 2z2 + 4z + 5 + 7z–1 + z–3, 0 < |z| < ∞

Q.11 The z-transform of
2 𝑛
is
3

−5z 3 2
, − | z | −
A.
(2z − 3)(3z − 2) 2 3
−5z 2 3
, | z |
(2z − 3)(3z − 2) 3
B.
2
5z 2 2
, − | z |
(2z − 3)(3z − 2) 3
C.
3
5z 3 2
, − | z | −
D.
(2z − 3)(3z − 2) 2 3
Q.12 The input-output relationship of a system is given as
𝑦 𝑛 − 0.4𝑦 𝑛 − 1 = 𝑥 𝑛 where, x[n] and y[n] are the input
and output respectively. The zero state response of the
system for an input x[n] = (0.4)n u[n] is

A. 𝑛 0.4 𝑛 𝑢[𝑛]

𝑛2 0.4 𝑛 𝑢[𝑛]
B.

C. 𝑛 + 1 0.4 𝑛 𝑢[𝑛]

1
D. ( ) un
0.4
n
n
Q.13 𝑍
If 𝑥 𝑛 ՜ 𝑋 𝑧 be a z-transform pair, then which of the
following is true?

x * n ⎯⎯
→ X * (− z)
Z
A.

x * n ⎯⎯
→ X * (z)
Z
B.

x * n ⎯⎯
→ X * (z *)
Z
C.

x * n ⎯⎯
→ X * (− z *)
D. Z
Q.14 X[z] of a system is specified by a pole zero pattern as
following :
Consider three different solution of x[n]
 n  1 n 
x1 n = 2 −    un
  3  
1
x2 n = −2n un − 1 − n un
3
1
x3 n = −2 un − 1 + n u−n − 1
n
3
Correct solution is

A. x1 n

B. x2 n

C. x3 n

D. All three
Q.15 Given the z-transform
7
1 + z −1
X (z) = 6
 1 −1  1 −1 
1 − 2 z 1 + 3 z 
  
For three different ROC consider there different solution of signal
x[n] :
1  1 −
 1
n
A. z  , x n =  n−1 −    un
2 2  3  
𝑛
1 1 −1
B. 𝑧 < , 𝑥 𝑛 = − 𝑛−1 + 𝑢 −𝑛 + 1
3 2 3
1 1 1 −1 𝑛
C. < 𝑧 < ,𝑥 𝑛 = − 𝑢 −𝑛 − 1 − 𝑢𝑛
3 2 2𝑛−1 3
Correct solution are

A. (A) and (B)

B. (A) and (C)

C. (B) and (C)

D. (A), (B), (C)

Q.16 The final value theorem is

A. lim x (k ) = lim (z − 1) X + (z)

k → z →1

B. lim x (k) = lim X + (z)

k → z →1

C.
k → z →0
( )
lim x (k) = lim z −1 X + (z)

D. lim x (k) = lim (z − 1)

k → z →0
−1
( )
X + z −1
Q.17 The z-transform corresponding to the Laplace transform
function
10
G (s) = is
s (s + 5)

2ze −5z 2 1 − 𝑒 −5𝑇 𝑧

(z − 1) (z − e )
A. B. 𝑧 − 1 𝑧 − 𝑒 −5𝑇
−T

e −5T e −T
C.
(z − 1) 2 D.
(
z z − e −3T )
Q.18 Which one of the following is the inverse z-transform of
z
X (z) = ,| z | 2
(z − 2)(z − 3)

A. 2n − 3n u (−n − 1)

 
3n − 2n u (−n − 1)
B.  

C. 2n − 3n u (n + 1)
 

D. 2n − 3n u (n)

 
Q.19 Algebraic expression for z-transform of x[n] is X[z]. What is
the algebraic expression for z-transform of 𝑒 𝑗𝜔0 𝑛 x[n]?

A. X (z − z0 )

B. (
X e − j0 z )
C. (
X e j0 z )
D. X (z) e j0 z
Q.20 The output y[n] of a discrete time LTl system is related to the
input x[n] as given below :
∞

𝑦 𝑛 = ෍𝑥 𝑘
𝑘=0

Which one of the following correctly relates the z-transforms

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
C. 𝑌 𝑧 = 𝑑𝑋 𝑧
1 − 𝑧 −1 D. 𝑌 𝑧 =
𝑑𝑧
THANK YOU GW
SOLDIERS !