### Z transform and Properties of Z Transform multiple choice questions

# **Z transform**

1. What is the set of all values of z for which X(z) attains a finite value?

a) Radius of convergence

b) Radius of divergence

c) Feasible solution

d) None of the mentioned**
Answer: Radius of convergence**

2. Is the discrete time LTI system with impulse response h(n)=a^{n}(n) (|a| < 1) BIBO stable?

a) True

b) False**
Answer: True**

3. What is the ROC of a causal infinite length sequence?

a) |z|<r_{1}

b) |z|>r_{1}

c) r_{2}<|z|<r_{1}

d) None of the mentioned**
Answer: |z|>r _{1}**

4. The Z-Transform X(z) of a discrete time signal x(n) is defined as ____________

a) ∑∞k=-∞ x(n)z^{n}

b) ∑∞k=-∞ x(n)z^{-n}

c) ∑∞k=0x(n)z^{n}

d) None of the mentioned**
Answer: ∑∞k=-∞ x(n)z ^{-n}**

5. What is the z-transform of the following finite duration signal?

x(n)={2, 4, 5, 7, 0, 1}?

↑

a) 2 + 4z + 5z^{2} + 7z^{3} + z^{4}

b) 2 + 4z + 5z^{2} + 7z^{3} + z^{5}

c) 2 + 4z^{-1} + 5z^{-2} + 7z^{-3} + z^{-5}

d) 2z^{2} + 4z + 5 +7z^{-1} + z^{-3}**
Answer: 2z ^{2} + 4z + 5 +7z^{-1} + z^{-3}**

6. What is the ROC of the signal x(n)=Î´(n-k), k>0?

a) z=0

b) z=∞

c) Entire z-plane, except at z=0

d) Entire z-plane, except at z=∞**
Answer: Entire z-plane, except at z=0**

7. What is the z-transform of the signal x(n) = -Î±^{n}u(-n-1)?

a) 1/(1−Î±z^−1);ROC |z|<|Î±|

b) −1/(1+Î±z^−1);ROC |z|<|Î±|

c) −1/(1−Î±z^−1);ROC |z|>|Î±|

d) −1/(1−Î±z^−1);ROC |z|<|Î±|**
Answer: −1/(1−Î±z^−1);ROC |z|<|Î±|**

8. What is the z-transform of the signal x(n)=(0.5)^{n}u(n)?

a) 1/(1−0.5z^−1);ROC|z|>0.5

b) 1/(1−0.5z^−1);ROC|z|<0.5

c) 1/(1+0.5z^−1);ROC|z|>0.5

d) 1/(1+0.5z^−1);ROC|z|<0.5**
Answer: 1/(1−0.5z^−1);ROC|z|>0.5**

9. Which of the following series has an ROC as mentioned below?

^{-n}u(n)

b) Î±

^{n}u(n)

c) Î±

^{-n}u(-n)

d) Î±

^{n}u(n)

**Answer: Î±**

^{n}u(n)10. What is the ROC of the z-transform of the signal x(n)= a^{n}u(n)+b^{n}u(-n-1)?

a) |a|<|z|<|b|

b) |a|>|z|>|b|

c) |a|>|z|<|b|

d) |a|<|z|>|b|**
Answer: |a|<|z|<|b|**

11. What is the ROC of z-transform of an two sided infinite sequence?

a) |z|>r_{1}

b) |z|<r_{1}

c) r_{2}<|z|<r_{1}

d) None of the mentioned**
Answer: r _{2}<|z|<r_{1}**

12. The z-transform of a sequence x(n) which is given as X(z)= ∑∞k=-∞ x(n)z^{-n }is known as _____________

a) Uni-lateral Z-transform

b) Bi-lateral Z-transform

c) Tri-lateral Z-transform

d) None of the mentioned**
Answer: Bi-lateral Z-transform**

13. What is the ROC of the system function H(z) if the discrete time LTI system is BIBO stable?

a) Entire z-plane, except at z=0

b) Entire z-plane, except at z=∞

c) Contain unit circle

d) None of the mentioned**
Answer: Contain unit circle**

14. The ROC of z-transform of any signal cannot contain poles.

a) True

b) False

Answer: True

15. What is the ROC of z-transform of finite duration anti-causal sequence?

a) z=0

b) z=∞

c) Entire z-plane, except at z=0

d) Entire z-plane, except at z=∞**
Answer: Entire z-plane, except at z=∞**

## Properties of Z Transform

1. According to Time shifting property of z-transform, if X(z) is the z-transform of x(n) then what is the z-transform of x(n-k)?a) z

^{k}X(z)

b) z

^{-k}X(z)

c) X(z-k)

d) X(z+k)

**Answer: z**

^{-k}X(z)2. Which of the following justifies the linearity property of z-transform?[x(n)↔X(z)].

a) x(n)+y(n) ↔ X(z)Y(z)

b) x(n)+y(n) ↔ X(z)+Y(z)

c) x(n)y(n) ↔ X(z)+Y(z)

d) x(n)y(n) ↔ X(z)Y(z)**
Answer: x(n)+y(n) ↔ X(z)+Y(z)**

3. If X(z) is the z-transform of the signal x(n) then what is the z-transform of a^{n}x(n)?

a) X(az)

b) X(az^{-1})

c) X(a^{-1}z)

d) None of the mentioned**
Answer: X(a ^{-1}z)**

4. What is the z-transform of the signal x(n)=[3(2n)-4(3n)]u(n)?

a) (3/1− 2z^{-1})−(4/1−3z^{-1})

b) (3/1−2z^{-1})−(4/1+3z^{-1})

c) (3/1−2z)−(4/1−3z)

d) None of the mentioned**
Answer: (3/1− 2z ^{-1})−(4/1−3z^{-1})**

5. What is the z-transform of the signal x(n)=a^{n}(sinÏ‰_{0}n)u(n)?

a) [az^{-1}sinÏ‰0]/[1+2az^{-1}cosÏ‰0+a^{2}z^{-2} ]

b) [az^{-1}sinÏ‰0]/[1−2az^{-1}cosÏ‰0−a^{2}z^{-2} ]

c) [(az)^{-1}cosÏ‰0]/[1−2az^{-1}cosÏ‰0+a^{2}z^{-2} ]

d) [az^{-1}sinÏ‰0]/[1−2az^{-1}cosÏ‰0+a^{2}z^{-2} ]**
Answer: [az ^{-1}sinÏ‰0]/[1−2az^{-1}cosÏ‰0+a^{2}z^{-2} ]**

6. If the ROC of X(z) is r_{1}<|z|<r_{2}, then what is the ROC of X(a^{-1}z)?

a) |a|r_{1}<|z|<|a|r_{2}

b) |a|r_{1}>|z|>|a|r_{2}

c) |a|r_{1}<|z|>|a|r_{2}

d) |a|r_{1}>|z|<|a|r_{2}**
Answer: |a|r _{1}<|z|<|a|r_{2}**

7. If X(z) is the z-transform of the signal x(n), then what is the z-transform of the signal x(-n)?

a) X(-z)

b) X(z^{-1})

c) X^{-1}(z)

d) None of the mentioned**
Answer: X(z ^{-1})**

8. What is the z-transform of the signal x(n)=na^{n}u(n)?

a) (az)^{-1}/(1−(az)^{-1})^{2}

b) (az)^{-1}/(1−(az)^{-1})^{2}

c) (az)^{-1}/(1−az^{-1})^{2}

d) (az)^{-1}/(1+az^{-1})^{2}**
Answer: (az) ^{-1}/(1−az^{-1})^{2}**

9. X(z) is the z-transform of the signal x(n), then what is the z-transform of the signal nx(n)?

a) −z{dX(z)/dz}

b) z{dX(z)/dz}

c) -z^{-1}{dX(z)/dz}

d) z^{-1}{dX(z)/dz}**
Answer: −z{dX(z)/dz}**

10. What is the z-transform of the signal x(n)=sin(jÏ‰_{0}n)u(n)?

a) (z^{-1}sinÏ‰0)/1+2z^{-1}cosÏ‰0+z^{-2}

b) (z^{-1}sinÏ‰0)/1−2z^{-1}cosÏ‰0−z^{-2}

c) (z-1cosÏ‰0)/1−2z^{-1}cosÏ‰0+z^{-2}

d) (z^{-1}sinÏ‰0)/1−2z^{-1}cosÏ‰0+z^{-2}**
Answer: (z ^{-1}sinÏ‰0)/1−2z^{-1}cosÏ‰0+z^{-2}**

11. What is the z-transform of the signal defined as x(n)=u(n)-u(n-N)?

a) 1+z^{N}/1+z^{-1}

b) 1−z^{N}/1+z^{-1}

c) 1+z^{-N}/1+z^{-1}

d) 1−z^{-N}/1−z^{-1}**
Answer: 1−z ^{-N}/1−z^{-1}**

12. If Z{x(n)}=X(z) and the poles of X(z) are all inside the unit circle, then the final value of x(n) as n→∞ is given by i.e., lim n→∞x(n)=?

a) lim z→1[(z−1)X(z)]

b) lim z→0[(z−1)X(z)]

c) lim z→−1[(z−1)X(z)]

d) lim z→1[(z+1)X(z)]**
Answer: lim z→1[(z−1)X(z)]**

13. What is the convolution x(n) of the signals x_{1}(n)={1,-2,1} and x_{2}(n)={1,1,1,1,1,1}?

a) {1,1,0,0,0,0,1,1}

b) {-1,-1,0,0,0,0,-1,-1}

c) {-1,1,0,0,0,0,1,-1}

d) {1,-1,0,0,0,0,-1,1}**
Answer: {1,-1,0,0,0,0,-1,1}**

14. If x_{1}(n)={1,2,3} and x_{2}(n)={1,1,1}, then what is the convolution sequence of the given two signals?

a) {1,2,3,1,1}

b) {1,2,3,4,5}

c) {1,3,5,6,2}

d) {1,2,6,5,3}**
Answer: {1,2,6,5,3}**

15. If Z{x_{1}(n)}=X_{1}(z) and Z{x_{2}(n)}=X_{2}(z) then Z{x_{1}(n)*x_{2}(n)}=?

a) X_{1}(z).X_{2}(z)

b) X_{1}(z)+X_{2}(z)

c) X_{1}(z)*X_{2}(z)

d) None of the mentioned**
Answer: X _{1}(z).X_{2}(z)**

16. What is the z-transform of the signal defined as x(n)=u(n)-u(n+N)?

a) 1+z^{N}/1+z^{-1}

b) 1−z^{N}/1−z^{-1}

c) 1+z^{-N}/1+z^{-1}

d) 1−z^{-N}/1−z^{-1}**
Answer: 1−z ^{-N}/1−z^{-1}**

17. If x(n) is an imaginary sequence, then the z-transform of the real part of the sequence is?

a) (1/2)[X(z)+X**(z**)]

b) (1/2)[X(z)-X**(z**)]

c) (1/2)[X(-z)-X**(z**)]

d) (1/2)[X(-z)+X**(z**)]**
Answer: (1/2)[X(z)+X* (z*)]**

18. What is the z-transform of the signal x(n)=[5(3^{n})-9(7^{n})]u(n)?

a)[ 5/(1−3z^{-1})]−[9/(1−7z^{-1})]

b) [5/1+3z^{-1}]−[9/(1+7z^{-1})]

c) [5/1−3z]−[9/1−7z]

d) None of the mentioned**
Answer: [ 5/(1−3z ^{-1})]−[9/(1−7z^{-1})]**

19. If Z{x_{1}(n)}=X_{1}(z) and Z{x_{2}(n)}=X_{2}(z) then what is the z-transform of correlation between the two signals?

a) X_{1}(z).X_{2}(z^{-1})

b) X_{1}(z).X_{2}(z^{-1})

c) X_{1}(z).X_{2}(z)

d) X_{1}(z).X_{2}(-z)**
Answer: X _{1}(z).X_{2}(z^{-1})**

20. What is the z-transform of the signal x(n)=Î´(n-n_{0})?

a) z^{n}_{0}

b) z^{-n}_{0}

c) z^{n-n}_{0}

d) z^{n+n}_{0}**
Answer: z ^{-n}_{0}**

21. If X(z) is the z-transform of the signal x(n), then what is the z-transform of x*(n)?

a) X(z*)

b) X*(z)

c) X*(-z)

d) X**(z**)**
Answer: X* (z*)**

22. What is the z-transform of the signal x(n)= x_{1}(n).x_{2}*(n)?

a) 1/2Ï€j ∮X_{1}(v)X_{2}(zv)v^{-1}dv

b) 1/2Ï€j ∮X_{1}(v)X*_{2}*(z**/v*)v^{-1}dv

c) 1/2Ï€j ∮X_{1}(v)X*_{2}(z/v)v^{-1}dv

d) None of the mentioned**
Answer: 1/2Ï€j ∮X _{1}(v)X*_{2}(z*/v*)v^{-1}dv**

23. What is the signal whose z-transform is given as X(z)=1/2Ï€j ∮X_{1}(v)X_{2}(zv)v^{-1}dv?

a) x_{1}(n)*x_{2}(n)

b) x_{1}(n)*x_{2}(-n)

c) x_{1}(n).x_{2}(n)

d) x_{1}(n)**x*_{2}*(n)**
Answer: x _{1}(n).x_{2}(n)**

24. What is the signal x(n) whose z-transform X(z)=log(1+az^{-1});|z|>|a|?

a) (−1)^{n}.a^{n}/n.u(n−1)

b) (−1)^{n}.a^{n}/n.u(n+1)

c) (−1)^{n}^{-1}.a^{n}/n.u(n−1)

d) (−1)^{n}^{-1}.a^{n}/n.u(n+1)**
Answer: (−1) ^{n}^{-1}.a^{n}/n.u(n−1)**

25. If x(n) is causal, then lim z→∞ X(z)=?

a) x(-1)

b) x(1)

c) x(0)

d) Cannot be determined**
Answer: x(0)**

26. What is the z-transform of the signal x(n)=cos(jÏ‰_{0}n)u(n)?

a) [z^{-1}sinÏ‰0]/[1+2z^{-1}cosÏ‰0+z^{-2} ]

b) [z^{-1}sinÏ‰0]/[1−2z^{-1}cosÏ‰0−z^{-2} ]

c) [z^{-1}cosÏ‰0]/[1−2z^{-1}cosÏ‰0+z^{-2} ]

d) [z^{-1}sinÏ‰0]/[1−2z^{-1}cosÏ‰0+z^{-2} ]**
Answer: [z ^{-1}cosÏ‰0]/[1−2z^{-1}cosÏ‰0+z^{-2} ]**

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