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)=an(n) (|a| < 1) BIBO stable?
a) True
b) False
Answer: True

3. What is the ROC of a causal infinite length sequence?
a) |z|<r1
b) |z|>r1
c) r2<|z|<r1
d) None of the mentioned
Answer: |z|>r1

4. The Z-Transform X(z) of a discrete time signal x(n) is defined as ____________
a)  ∑k=-∞ x(n)zn
b)  ∑k=-∞ x(n)z-n
c)  ∑k=0x(n)zn
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 + 5z2 + 7z3 + z4
b) 2 + 4z + 5z2 + 7z3 + z5
c) 2 + 4z-1 + 5z-2 + 7z-3 + z-5
d) 2z2 + 4z + 5 +7z-1 + z-3
Answer: 2z2 + 4z + 5 +7z-1 + z-3

6. What is the ROC of the signal x(n)=δ(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) = -αnu(-n-1)?
a) 1/(1−αz^−1);ROC |z|<|α|
b) −1/(1+αz^−1);ROC |z|<|α|
c) −1/(1−αz^−1);ROC |z|>|α|
d) −1/(1−αz^−1);ROC |z|<|α|
Answer: −1/(1−αz^−1);ROC |z|<|α|

8. What is the z-transform of the signal x(n)=(0.5)nu(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?





a) α-nu(n)
b) αnu(n)
c) α-nu(-n)
d) αnu(n)
Answer: αnu(n)

10. What is the ROC of the z-transform of the signal x(n)= anu(n)+bnu(-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|>r1
b) |z|<r1
c) r2<|z|<r1
d) None of the mentioned
Answer: r2<|z|<r1

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) zkX(z)
b) z-kX(z)
c) X(z-k)
d) X(z+k)
Answer: z-kX(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 anx(n)?
a) X(az)
b) X(az-1)
c) X(a-1z)
d) None of the mentioned
Answer: X(a-1z)

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)=an(sinω0n)u(n)?
a) [az-1sinω0]/[1+2az-1cosω0+a2z-2 ]
b) [az-1sinω0]/[1−2az-1cosω0−a2z-2 ]
c) [(az)-1cosω0]/[1−2az-1cosω0+a2z-2 ]
d) [az-1sinω0]/[1−2az-1cosω0+a2z-2 ]
Answer: [az-1sinω0]/[1−2az-1cosω0+a2z-2 ]

6. If the ROC of X(z) is r1<|z|<r2, then what is the ROC of X(a-1z)?
a) |a|r1<|z|<|a|r2
b) |a|r1>|z|>|a|r2
c) |a|r1<|z|>|a|r2
d) |a|r1>|z|<|a|r2
Answer: |a|r1<|z|<|a|r2

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)=nanu(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(jω0n)u(n)?
a) (z-1sinω0)/1+2z-1cosω0+z-2
b) (z-1sinω0)/1−2z-1cosω0−z-2
c) (z-1cosω0)/1−2z-1cosω0+z-2
d) (z-1sinω0)/1−2z-1cosω0+z-2
Answer: (z-1sinω0)/1−2z-1cosω0+z-2

11. What is the z-transform of the signal defined as x(n)=u(n)-u(n-N)?
a) 1+zN/1+z-1
b) 1−zN/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 x1(n)={1,-2,1} and x2(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 x1(n)={1,2,3} and x2(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{x1(n)}=X1(z) and Z{x2(n)}=X2(z) then Z{x1(n)*x2(n)}=?
a) X1(z).X2(z)
b) X1(z)+X2(z)
c) X1(z)*X2(z)
d) None of the mentioned
Answer: X1(z).X2(z)

16. What is the z-transform of the signal defined as x(n)=u(n)-u(n+N)?
a) 1+zN/1+z-1
b) 1−zN/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(3n)-9(7n)]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{x1(n)}=X1(z) and Z{x2(n)}=X2(z) then what is the z-transform of correlation between the two signals?
a) X1(z).X2(z-1)
b) X1(z).X2(z-1)
c) X1(z).X2(z)
d) X1(z).X2(-z)
Answer: X1(z).X2(z-1)

20. What is the z-transform of the signal x(n)=δ(n-n0)?
a) zn0
b) z-n0
c) zn-n0
d) zn+n0
Answer: z-n0

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)= x1(n).x2*(n)?
a) 1/2πj ∮X1(v)X2(zv)v-1dv
b) 1/2πj ∮X1(v)X*2(z*/v*)v-1dv
c) 1/2πj ∮X1(v)X*2(z/v)v-1dv
d) None of the mentioned
Answer: 1/2πj ∮X1(v)X*2(z*/v*)v-1dv

23. What is the signal whose z-transform is given as X(z)=1/2πj ∮X1(v)X2(zv)v-1dv?
a) x1(n)*x2(n)
b) x1(n)*x2(-n)
c) x1(n).x2(n)
d) x1(n)*x2*(n)
Answer: x1(n).x2(n)

24. What is the signal x(n) whose z-transform X(z)=log(1+az-1);|z|>|a|?
a) (−1)n.an/n.u(n−1)
b) (−1)n.an/n.u(n+1)
c) (−1)n-1.an/n.u(n−1)
d) (−1)n-1.an/n.u(n+1)
Answer: (−1)n-1.an/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(jω0n)u(n)?
a) [z-1sinω0]/[1+2z-1cosω0+z-2 ]
b) [z-1sinω0]/[1−2z-1cosω0−z-2 ]
c) [z-1cosω0]/[1−2z-1cosω0+z-2 ]
d) [z-1sinω0]/[1−2z-1cosω0+z-2 ]
Answer: [z-1cosω0]/[1−2z-1cosω0+z-2 ]

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