Z-Transform Examples and Solutions
1. Z-Transform of an
Given: x[n] = an
Definition of Z-transform:
X(z) = ∑ n=0 to ∞ x[n]z⁻ⁿ
Substitute x[n] = an
X(z) = ∑ aⁿ z⁻ⁿ = ∑ (az⁻¹)ⁿ
Using geometric series:
∑ rⁿ = 1 / (1 - r)
X(z) = 1/(1 − az⁻¹) = z/(z − a)
ROC: |z| > |a|
2. Z-Transform of Unit Step u[n]
Given: x[n] = u[n]
X(z) = ∑ z⁻ⁿ
This is a geometric series.
X(z) = 1 / (1 − z⁻¹) = z / (z − 1)
ROC: |z| > 1
3. Z-Transform of n
Given: x[n] = n
X(z) = ∑ n z⁻ⁿ
X(z) = z / (z − 1)²
ROC: |z| > 1
4. Z-Transform of n an
Given: x[n] = n an
Using differentiation property of Z-transform.
X(z) = az / (z − a)²
ROC: |z| > |a|
5. Z-Transform of anu[n]
Given: x[n] = anu[n]
X(z) = ∑ aⁿ z⁻ⁿ
X(z) = 1 / (1 − az⁻¹)
ROC: |z| > |a|
Summary Table
| Sequence x[n] | Z-Transform X(z) | ROC |
|---|---|---|
| an | z / (z − a) | |z| > |a| |
| u[n] | z / (z − 1) | |z| > 1 |
| n | z / (z − 1)2 | |z| > 1 |
| n an | az / (z − a)2 | |z| > |a| |