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

A. \(\sum_{n=-\infty}^{\infty}x(n)z^n\)

B. \(\sum_{n=-\infty}^{\infty}x(n)z^{-n}\)

C. \(\sum_{n=0}^{\infty}x(n)z^n\)

D. None of the mentioned

Answer: B

The z-transform of a real discrete time sequence x(n) is defined as a power of ‘z’ which is equal to X(z)=\(\sum_{n=-{\infty}}^{\infty} x(n)z^{-n}\), where ‘z’ is a complex variable.

2. 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: A

Since X(z) is an infinite power series, it is defined only at a few values of z. The set of all values of z where X(z) converges to a finite value is called as Radius of Convergence(ROC.)

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

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: D

We know that, for a given signal x(n) the z-transform is defined as

X(z)=\(\sum_{n=-\infty}^{\infty} x(n)z^{-n}\)

Substitute the values of n from -2 to 3 and the corresponding signal values in the above formula

We get, X(z) = 2z2 + 4z + 5 + 7z-1 + z-3.

4. 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: C
We know that, the z-transform of a signal x(n) is

X(z)=\(\sum_{n=-\infty}^{\infty} x(n)z^{-n}\)

Given x(n)=δ(n-k)=1 at n=k

=> X(z)=z-k

From the above equation, X(z) is defined at all values of z except at z=0 for k>0.

So ROC is defined as the Entire z-plane, except at z=0.

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

A. \(\frac{1}{1-0.5z^{-1}};ROC |z|>0.5\)

B. \(\frac{1}{1-0.5z^{-1}};ROC |z|<0.5\)

C. \(\frac{1}{1+0.5z^{-1}};ROC |z|>0.5\)

D. \(\frac{1}{1+0.5z^{-1}};ROC |z|<0.5\)

Answer: A

For a given signal x(n), its z-transform

X(z)=\(\sum_{n=-\infty}^{\infty} x(n)z^{-n}\)

Given x(n)=(0.5)nu(n)=(0.5)n for n≥0

So, X(z)=\(\sum_{n=0}^{\infty} 0.5^n z^{-n}=\sum_{n=0}^{\infty} (0.5z^{-1})^n\)

This is an infinite GP whose sum is given as

X(z)=\(\frac{1}{1-0.5z^{-1}}\) under the condition that |0.5z-1|<1

=> X(z)=\(\frac{1}{1-0.5z^{-1}}\) and ROC is |z|>0.5.

6. Which of the following series has a ROC as mentioned below?

A. α-nu(n)
B. αnu(n)
C. α-nu(-n)
D. αnu(n)

Answer: B

Let x(n)= αnu(n)
The z-transform of the signal x(n) is given as

8. 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

We know that,
ROC of z-transform of anu(n) is |z|>|a|.
ROC of z-transform of bnu(-n-1) is |z|<|b|.
By combining both the ROC we get the ROC of z-transform of the signal x(n) as |a|<|z|<|b|.

9. What is the ROC of the 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: D

Let us an example of an anti causal sequence whose z-transform will be in the form X(z)=1+z+z2 which has a finite value at all values of ‘z’ except at z=∞. So, the ROC of an anti-causal sequence is the entire z-plane except at z=∞.

10. What is the ROC of the z-transform of a two-sided infinite sequence?

A. |z|>r1
B. |z|<r1
C. r2<|z|<r1
D. None of the mentioned

Answer: C

Let us plot the graph of z-transform of any two-sided sequence which looks as follows.

From the above graph, we can state that the ROC of a two-sided sequence will be of the form r2 < |z| < r1.