Z Transform Inversion

Digital Communications Systems

This set of Digital Signal Processing Multiple Choice Questions & Answers (MCQs) focuses on “Inversion of Z Transform”.

1. Which of the following method is used to find the inverse z-transform of a signal?
a) Counter integration
b) Expansion into a series of terms
c) Partial fraction expansion
d) All of the mentioned

2. What is the inverse z-transform of X(z)=11−1.5z−1+0.5z−2 if ROC is |z|>1?
a) {1,3/2,7/4,15/8,31/16,….}
b) {1,2/3,4/7,8/15,16/31,….}
c) {1/2,3/4,7/8,15/16,31/32,….}
d) None of the mentioned

3. What is the inverse z-transform of X(z)=11−1.5z−1+0.5z−2 if ROC is |z| < 0.5?
a) {….62,30,14,6,2}
b) {…..62,30,14,6,2,0,0}
c) {0,0,2,6,14,30,62…..}
d) {2,6,14,30,62…..}


Thus X(z)=11−1.5z−1+0.5z−2=2z2+6z3+14z4+30z5+62z6+… In this case x(n)=0 for n≥0.Thus we obtain x(n)= {…..62,30,14,6,2,0,0}

4. What is the inverse z-transform of X(z)=log(1+az-1) |z|>|a|?
a) x(n)=(-1)n+1 a−nn, n≥1; x(n)=0, n≤0
b) x(n)=(-1)n-1 a−nn, n≥1; x(n)=0, n≤0
c) x(n)=(-1)n+1 a−nn, n≥1; x(n)=0, n≤0
d) None of the mentioned

5. What is the proper fraction and polynomial form of the improper rational transform
X(z)=1+3z−1+116z−2+13z−31+56z−1+16z−2?
a) 1+2z-1+16z−11+56z−1+16z−2
b) 1-2z-1+16z−11+56z−1+16z−2
c) 1+2z-1+13z−11+56z−1+16z−2
d) 1+2z-1–16z−11+56z−1+16z−2

6. What is the partial fraction expansion of the proper function X(z)=11−1.5z−1+0.5z−2?
a) 2zz−1−zz+0.5
b) 2zz−1+zz−0.5
c) 2zz−1+zz+0.5
d) 2zz−1−zz−0.5

7. What is the partial fraction expansion of X(z)=1+z−11−z−1+0.5z−2?
a) z(0.5−1.5j)z−0.5−0.5j–z(0.5+1.5j)z−0.5+0.5j
b) z(0.5−1.5j)z−0.5−0.5j+z(0.5+1.5j)z−0.5+0.5j
c) z(0.5+1.5j)z−0.5−0.5j–z(0.5−1.5j)z−0.5+0.5j
d) z(0.5+1.5j)z−0.5−0.5j+z(0.5−1.5j)z−0.5+0.5j

8. What is the partial fraction expansion of X(z)=1(1+z−1)(1−z−1)2?
a) z4(z+1)+3z4(z−1)+z2(z+1)2
b) z4(z+1)+3z4(z−1)–z2(z+1)2
c) z4(z+1)+3z4(z−1)+z2(z−1)2
d) z4(z+1)+z4(z−1)+z2(z+1)2

9. What is the inverse z-transform of X(z)=11−1.5z−1+0.5z−2 if ROC is |z|>1?
a) (2-0.5n)u(n)
b) (2+0.5n)u(n)
c) (2n-0.5n)u(n)
d) None of the mentioned

10. What is the inverse z-transform of X(z)=11−1.5z−1+0.5z−2 if ROC is |z|<0.5?
a) [-2-0.5n]u(n)
b) [-2+0.5n]u(n)
c) [-2+0.5n]u(-n-1)
d) [-2-0.5n]u(-n-1)

11. What is the inverse z-transform of X(z)=11−1.5z−1+0.5z−2 if ROC is 0.5<|z|<1?
a) -2u(-n-1)+(0.5)nu(n)
b) -2u(-n-1)-(0.5)nu(n)
c) -2u(-n-1)+(0.5)nu(-n-1)
d) 2u(n)+(0.5)nu(-n-1)

12. What is the causal signal x(n) having the z-transform X(z)=1(1+z−1)(1−z−1)2?
a) [1/4(-1)n+3/4-n/2]u(n)
b) [1/4(-1)n+3/4-n/2]u(-n-1)
c) [1/4+3/4(-1)n-n/2]u(n)
d) [1/4(-1)n+3/4+n/2]u(n)

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