1. 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.

2. 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.

3. 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.

4. 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.

5. What is the one-sided z-transform of x(n)=δ(n+k)?

A. z-k
B. 0
C. zk
D. 1

Answer: B

Since the signal x(n)=δ(n+k) is an anti causal signal i.e., it is defined for n<0 and x(n)=1 at z=-k. Since the one-sided z-transform is defined only for causal signal, in this case X+(z)=0.

6. If X+(z) is the one sided z-transform of x(n), then what is the one sided z-transform of x(n-k)?

A. z-k X+(z)

B. zk X+(z-1)

C. z-k \([X^+(z)+\sum_{n=1}^k x(-n)z^n]\); k>0

D. z-k \([X^+(z)+\sum_{n=0}^k x(-n)z^n]\); k>0

Answer: C

From the definition of one sided z-transform we have,

8. If x(n)=an, then what is one sided z-transform of x(n+2)?

A. \(\frac{z^{-2}}{1-az^{-1}}\) + a-1z-1 + a-2

B. \(\frac{z^{-2}}{1-az^{-1}}\) – a-1z-1 + a-2

C. \(\frac{z^2}{1-az^{-1}}\) + a z + z2

D. \(\frac{z^2}{1+az^{-1}}\) – z2 – az

Answer: D

We will apply the time advance theorem with the value of k=2.We obtain,
Z+{x(n+2)}=z2 X+(z)-x(0)z2-x(1)z

=>X1+(z)=\(\frac{z^2}{1+az^{-1}}\) – z2 – az.

9. If X+(z) is the one sided z-transform of the signal x(n), then
\(\lim_{n \rightarrow \infty} x(n)=\lim_{z\rightarrow 1}(z-1) X^+(z)\) is called Final value theorem.

A. True
B. False

Answer: A

In the above theorem, we are calculating the value of x(n) at infinity, so it is called as final value theorem.

10. The impulse response of a relaxed LTI system is h(n)=anu(n), |a|<1. What is the value of the step response of the system as n→∞?

A. \(\frac{1}{1+a}\)

B. \(\frac{1}{1-a}\)

C. \(\frac{a}{1+a}\)

D. \(\frac{a}{1-a}\)

Answer: B

The step response of the system is y(n)=x(n)*h(n) where x(n)=u(n)
On applying z-transform on both sides, we get