21. What is the inverse z-transform of
X(z)=\(\frac{1}{1-1.5z^{-1}+0.5z^{-2}}\) 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
Answer: A
The partial fraction expansion for the given X(z) is
\(X(z)= \frac{2z}{z-1}-\frac{z}{z-0.5}\)
In case when ROC is |z|>1, the signal x(n) is causal and both the terms in the above equation are causal terms. Thus, when we apply inverse z-transform to the above equation, we get
x(n)=2(1)nu(n)-(0.5)nu(n)=(2-0.5n)u(n).
22. What is the inverse z-transform of
X(z)=\(\frac{1}{1-1.5z^{-1}+0.5z^{-2}}\) 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)
Answer: C
The partial fraction expansion for the given X(z) is
\(X(z)= \frac{2z}{z-1}-\frac{z}{z-0.5}\)
In case when ROC is |z|<0.5, the signal is anti causal. Thus both the terms in the above equation are anti causal terms. So, if we apply inverse z-transform to the above equation we get
x(n)= [-2+0.5n]u(-n-1).
23. What is the inverse z-transform of
X(z)=\(\frac{1}{1-1.5z^{-1}+0.5z^{-2}}\) 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)
Answer: B
The partial fraction expansion of the given X(z) is
\(X(z)= \frac{2z}{z-1}-\frac{z}{z-0.5}\)
In this case, ROC is 0.5<|z|<1 is a ring, which implies that the signal is two-sided. Thus one of the signals corresponds to a causal signal and the other corresponds to an anti causal signal.
Obviously, the ROC given is the overlapping of the regions |z|>0.5 and |z|<1. Hence the pole p2=0.5 provides the causal part and the pole p1=1 provides the anti causal part. SO, if we apply the inverse z-transform we get
x(n)= -2u(-n-1)-(0.5)nu(n).
24. What is the causal signal x(n) having the z-transform
X(z)=\(\frac{1}{(1+z^{-1})(1-z^{-1})^2}\)?
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)
When we apply the inverse z-transform for the above equation, we get
x(n)=[1/4(-1)n+3/4+n/2]u(n).
25. The z-transform of a signal x(n) whose definition is given by \(X(z)=\sum_{n=0}^{\infty} x(n)z^{-n}\) is known as ________
A. Unilateral z-transform
B. Bilateral z-transform
C. Rational z-transform
D. None of the mentioned
Answer: A
The z-transform of the x(n) whose definition exists in the range n=-∞ to +∞ is known as a bilateral or two-sided z-transform. But in the given question the value of n=0 to +∞. So, such a z-transform is known as a Unilateral or one-sided z-transform.
26. For what kind of signals is one-sided z-transform unique?
A. All signals
B. Anti-causal signal
C. Causal signal
D. None of the mentioned
Answer: C
One-sided z-transform is unique only for causal signals because only these signals are zero for n<0.
27. What is the one sided z-transform X+(z) of the signal x(n)={1,2,5,7,0,1}?
A. z2+2z+5+7z-1+z-3
B. 5+7z+z3
C. z-2+2z-1+5+7z+z3
D. 5+7z-1+z-3
Answer: D
Since the one sided z-transform is valid only for n>=0, the z-transform of the given signal will be X+(z)= 5+7z-1+z-3.
28. What is the one-sided z-transform of x(n)=δ(n-k)?
A. z-k
B. zk
C. 0
D. 1
Answer: A
Since the signal x(n)= δ(n-k) is a causal signal i.e., it is defined for n>0 and x(n)=1 at z=k
So, from the definition of one-sided z-transform X+(z)=z-k.