# Z Transform MCQ

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

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?

C. Feasible solution
D. None of the mentioned

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

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 = ∞

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. $$\frac{1}{1+0.5z^{-1}};ROC |z|<0.5$$

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)

Let x(n) = αnu(n)

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

X(z) = X(z) = $$\sum_{n = 0}^{\infty} \alpha^n z^{-n} = \sum_{n = 0}^{\infty}\alpha z^{-1}n$$

= > X(z) = $$\frac{1}{1-\alpha z^{-1}}$$ and ROC is |z|>α which is as given in the question.

7. What is the z-transform of the signal x(n) = -αnu(-n-1)?

A. $$\frac{1}{1-\alpha z^{-1}}$$;ROC |z|<|α|

B. $$-\frac{1}{1+\alpha z^{-1}}$$;ROC |z|<|α|

C. $$-\frac{1}{1-\alpha z^{-1}}$$;ROC |z|>|α|

D. $$-\frac{1}{1-\alpha z^{-1}}$$;ROC |z|<|α|

Given

x(n) = -αnu(-n-1) = 0 for n≥0

= -αn for n≤-1

From the definition of z-transform, we have

X(z) = $$\sum_{n = -\infty}^{-1}(-α^n)z^{-n} = -\sum_{n = -\infty}^{-1}(\alpha^{-1} z)^{-n} = -\frac{1}{1-\alpha z^{-1}}$$

and |α-1z|<1 = > |z|<|α|.

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|

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 = ∞

Let us take an example of an anti-causal sequence whose z-transform will be in 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