Definition of Z Transform MCQ Quiz - Objective Question with Answer for Definition of Z Transform - Download Free PDF

Last updated on Apr 23, 2025

Latest Definition of Z Transform MCQ Objective Questions

Definition of Z Transform Question 1:

A sequence x(n) with the z-transform X(z) = z4 + z2 – 2z + 2 – 3z-4 is applied as an input to a linear, time-invariant system with the impulse response h(n) = 2δ (n - 3) where

δ(n)={1n=00otherwise

The output at n = 4 is

Answer (Detailed Solution Below) 0

Definition of Z Transform Question 1 Detailed Solution

h(n) = 2δ (n – 3)

H(z) = 2z-3

x(z) = z4 + z2 – 2z + 2 – 3z-4

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

= 2(z + z-1 – 2z-2 + 2z-3 – 3z-7)

By applying inverse z-transform,

y(n) = 2[δ (n + 1) + δ (n - 1) -2δ (n - 2) + 2δ (n - 3) – 3δ (n - 7)]

At n = 4, y (4) = 0

Definition of Z Transform Question 2:

If the z-transform of a sequence x[n] = {1, 1, -1, -1} is X[z], then the value of X(1/2) is

  1. -9
  2. 1.875
  3. -1.125
  4. 15

Answer (Detailed Solution Below)

Option 1 : -9

Definition of Z Transform Question 2 Detailed Solution

X(z)=n=x[n]zn 

=n=03x[n]zn 

= z-0 + z-1 – z-2 – z-3

=1+1z1z21z3 

x(12)=1+248=9 

Definition of Z Transform Question 3:

A sequence x(n) with the z-transform X(z) = z4 + z2 – 2z + 2 – 3z-4 is applied as an input to a linear, time-invariant system with the impulse response h(n) = 2δ (n - 3) where

δ(n)={1n=00otherwise

The output at n = 4 is

  1. - 6
  2. zero
  3. 2
  4. -4

Answer (Detailed Solution Below)

Option 2 : zero

Definition of Z Transform Question 3 Detailed Solution

h(n) = 2δ (n – 3)

H(z) = 2z-3

x(z) = z4 + z2 – 2z + 2 – 3z-4

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

= 2(z + z-1 – 2z-2 + 2z-3 – 3z-7)

By applying inverse z-transform,

y(n) = 2[δ (n + 1) + δ (n - 1) -2δ (n - 2) + 2δ (n - 3) – 3δ (n - 7)]

At n = 4, y (4) = 0

Definition of Z Transform Question 4:

The value of n=0n(12)n is _________.

Answer (Detailed Solution Below) 2

Definition of Z Transform Question 4 Detailed Solution

Concept:

z-transform of a discrete signal can be written as;

X(z)=n=+x[n]zn      …1)

Calculation:

We have to calculate:

n=0(n)(12)n=n=+nu[n].(2)n      …2)

On comparing equation (1) & (2) we find that;

n=0(n)(2)n=X(z)=X(2)

Where X(z) is z-transform of nu[n] which is:

X(z)=z1(1z1)2

So, X(z)=z1(1z1)2 

X(2)=21(121)2=Y2Y4=2

Hence our required summation is

n=0(n)(2)n=X(2)=2

Definition of Z Transform Question 5:

The z-transform of a discrete-time signal x(n) is X(z)=z+1z(z1). Then, x(0) + x(1) + x(2) is _____.

Answer (Detailed Solution Below) 3

Definition of Z Transform Question 5 Detailed Solution

Concept:

The z-transform of a signal αn u(n) is:

αnu(n)11αz1

Also, the z transform of δ(n) ↔ 1

If, x(n) ↔ X(z)

Then, x(n - n0) ↔ X(z).z-n0

Calculation:

Given,

X(z)=z+1z(z1)

Doing partial fraction;

z+1z(z1)=Az+Bz1

⇒ z + 1 = A(z - 1) + B(z)

Putting, z = 0

1 = -A

A = -1

Putting z = 1

B = 2

So, x(z)=z+1z(z1)=1z+2z1

=z1+2z11z1

δ(n) ↔ 1

δ(n - 1) ↔ z-1

Also, u(n)11z1

u(nn0)zno1z1

u(n1)z11z1

So, the inverse z-transform of the above given will be x(n) = -δ(n - 1) + 2 u(n - 1)

Now, x(0) = -δ(-1) + 2u(0 - 1) = 0 + 0 = 0

x(1) = -δ(1 - 1) + 2(1 - 1) = -1 + 2 = 1

x(2) = -δ(2 - 1) + 2u(2 - 1) = 0 + 2 = 2

Now, x(0) + x(1) + x(2) = 0 + 1 + 2 = 3

Top Definition of Z Transform MCQ Objective Questions

The value of n=0n(12)n is _________.

Answer (Detailed Solution Below) 2

Definition of Z Transform Question 6 Detailed Solution

Download Solution PDF

Concept:

z-transform of a discrete signal can be written as;

X(z)=n=+x[n]zn      …1)

Calculation:

We have to calculate:

n=0(n)(12)n=n=+nu[n].(2)n      …2)

On comparing equation (1) & (2) we find that;

n=0(n)(2)n=X(z)=X(2)

Where X(z) is z-transform of nu[n] which is:

X(z)=z1(1z1)2

So, X(z)=z1(1z1)2 

X(2)=21(121)2=Y2Y4=2

Hence our required summation is

n=0(n)(2)n=X(2)=2

Definition of Z Transform Question 7:

A sequence x(n) with the z-transform X(z) = z4 + z2 – 2z + 2 – 3z-4 is applied as an input to a linear, time-invariant system with the impulse response h(n) = 2δ (n - 3) where

δ(n)={1n=00otherwise

The output at n = 4 is

Answer (Detailed Solution Below) 0

Definition of Z Transform Question 7 Detailed Solution

h(n) = 2δ (n – 3)

H(z) = 2z-3

x(z) = z4 + z2 – 2z + 2 – 3z-4

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

= 2(z + z-1 – 2z-2 + 2z-3 – 3z-7)

By applying inverse z-transform,

y(n) = 2[δ (n + 1) + δ (n - 1) -2δ (n - 2) + 2δ (n - 3) – 3δ (n - 7)]

At n = 4, y (4) = 0

Definition of Z Transform Question 8:

Z-transform of  x(n)=(13)|n|is:

  1. 83z11103z1+z2      13<|z|<3

  2. Does not exists    

  3. 83z11103z1+z2       13<|z|<3

  4. 83z11103z1+z2          |z|<13,|z|>3        

Answer (Detailed Solution Below)

Option 3 :

83z11103z1+z2       13<|z|<3

Definition of Z Transform Question 8 Detailed Solution

 X(z)=n=1(13)nzn+0(13)nzn

We have,

n=1(13)nzn=n=1(3)nzn=ZT{3nu[n1]}

From the relation, Math input error

Thus, Math input error

Also, Math input error

Thus, Math input error

X(z)=13z11+13z1(113z1)(13z1)         |13|<|z|<|3|

X(z)=83z1(1103z1+z2)         |13|<|z|<|3|

Definition of Z Transform Question 9:

The value of n=0n(12)n is _________.

Answer (Detailed Solution Below) 2

Definition of Z Transform Question 9 Detailed Solution

Concept:

z-transform of a discrete signal can be written as;

X(z)=n=+x[n]zn      …1)

Calculation:

We have to calculate:

n=0(n)(12)n=n=+nu[n].(2)n      …2)

On comparing equation (1) & (2) we find that;

n=0(n)(2)n=X(z)=X(2)

Where X(z) is z-transform of nu[n] which is:

X(z)=z1(1z1)2

So, X(z)=z1(1z1)2 

X(2)=21(121)2=Y2Y4=2

Hence our required summation is

n=0(n)(2)n=X(2)=2

Definition of Z Transform Question 10:

The z-transform of a discrete-time signal x(n) is X(z)=z+1z(z1). Then, x(0) + x(1) + x(2) is _____.

Answer (Detailed Solution Below) 3

Definition of Z Transform Question 10 Detailed Solution

Concept:

The z-transform of a signal αn u(n) is:

αnu(n)11αz1

Also, the z transform of δ(n) ↔ 1

If, x(n) ↔ X(z)

Then, x(n - n0) ↔ X(z).z-n0

Calculation:

Given,

X(z)=z+1z(z1)

Doing partial fraction;

z+1z(z1)=Az+Bz1

⇒ z + 1 = A(z - 1) + B(z)

Putting, z = 0

1 = -A

A = -1

Putting z = 1

B = 2

So, x(z)=z+1z(z1)=1z+2z1

=z1+2z11z1

δ(n) ↔ 1

δ(n - 1) ↔ z-1

Also, u(n)11z1

u(nn0)zno1z1

u(n1)z11z1

So, the inverse z-transform of the above given will be x(n) = -δ(n - 1) + 2 u(n - 1)

Now, x(0) = -δ(-1) + 2u(0 - 1) = 0 + 0 = 0

x(1) = -δ(1 - 1) + 2(1 - 1) = -1 + 2 = 1

x(2) = -δ(2 - 1) + 2u(2 - 1) = 0 + 2 = 2

Now, x(0) + x(1) + x(2) = 0 + 1 + 2 = 3

Definition of Z Transform Question 11:

If the z-transform of a sequence x[n] = {1, 1, -1, -1} is X[z], then the value of X(1/2) is

  1. -9
  2. 1.875
  3. -1.125
  4. 15

Answer (Detailed Solution Below)

Option 1 : -9

Definition of Z Transform Question 11 Detailed Solution

X(z)=n=x[n]zn 

=n=03x[n]zn 

= z-0 + z-1 – z-2 – z-3

=1+1z1z21z3 

x(12)=1+248=9 

Definition of Z Transform Question 12:

A sequence x(n) with the z-transform X(z) = z4 + z2 – 2z + 2 – 3z-4 is applied as an input to a linear, time-invariant system with the impulse response h(n) = 2δ (n - 3) where

δ(n)={1n=00otherwise

The output at n = 4 is

  1. - 6
  2. zero
  3. 2
  4. -4

Answer (Detailed Solution Below)

Option 2 : zero

Definition of Z Transform Question 12 Detailed Solution

h(n) = 2δ (n – 3)

H(z) = 2z-3

x(z) = z4 + z2 – 2z + 2 – 3z-4

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

= 2(z + z-1 – 2z-2 + 2z-3 – 3z-7)

By applying inverse z-transform,

y(n) = 2[δ (n + 1) + δ (n - 1) -2δ (n - 2) + 2δ (n - 3) – 3δ (n - 7)]

At n = 4, y (4) = 0

Definition of Z Transform Question 13:

The convolution of two sequences x1[n], x2[n] given below is

x1[n]={12103}x2[n]=n[u[n]u[n3]]

  1. {0, 1, 4, 3, -2, 3, 6}
  2. {0, 2, -3, 2, 4, 6}
  3. {1, 4, -2, 3, -4, 6}
  4. {0, 1, -2, -1, -2, 4}

Answer (Detailed Solution Below)

Option 1 : {0, 1, 4, 3, -2, 3, 6}

Definition of Z Transform Question 13 Detailed Solution

X1[n] = {1, 2, -1, 0, 3}; x2[n] = {0, 1, 2}

X1(z) = 1 + 2z-1 – z-2 + 0z-3 + 3z-4

= 1 + 2z-1 – z-2 + 3z-4

X2(z) = 0 × z0 + 1 × z-1 + 2 × z-2 = z-1 + 2z-2

x1[n] * x2[n] = Z-1[X1(z) . X2(z)]

= (z-1 + 2z-2) (1 + 2z-1 – z-2 + 3z-4)

= z-1 + 2z-2 – z-3 + 3z-5 + 2z-2 + 4z-3 – 2z-4 + 6z-6

= z-1 + 4z-2 + 3z-3 – 2z-4 + 3z-5 + 6z-6

∴ x1[n] * x2[n] = {0, 1, 4, 3, -2, 3, 6}

Definition of Z Transform Question 14:

A discrete time signal, x(n) suffered a distortion due to LTI system with H(z) = 1 – az-1, a is real and a > 1. The impulse response of a stable system that compensates the distortion is

  1. (1a)nu(n)
  2. (1a)nu(n1)
  3. an u [n]
  4. -an u[-n-1]

Answer (Detailed Solution Below)

Option 4 : -an u[-n-1]

Definition of Z Transform Question 14 Detailed Solution

The distortion is H(z) = 1 – az-1 To compensate the distortion impulse response of compensator system.

H(z)1=1H(z)=11az1

The stable system impulse response is

11az1Zanu[n1]

NOTE : as a > 1, an u[n] will be an unstable system hence option c is incorrect

Definition of Z Transform Question 15:

The time signal corresponding to the given Z transform  1114z2,|z|>14 is

  1. {2n,nevenandn00,otherwise
  2. (12)2nu[n]
  3. (14)2nu[n]
  4. {2n,noddandn>00,otherwise

Answer (Detailed Solution Below)

Option 1 : {2n,nevenandn00,otherwise

Definition of Z Transform Question 15 Detailed Solution

x(z)=1114z2=(1z24)1=1+z24+(z44)2+=k=0(14z2)kx[n]=k=0(14)kδ[n2k]={(14)n2nevenandn00nodd
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