Z-Transform of the Delta Function
Definition of Delta Function
The discrete-time delta function is written as:
δ[n]
It is defined as:
- δ[0] = 1
- δ[n] = 0 for n ≠ 0
The delta function represents a unit impulse that occurs only at n = 0.
Z-Transform Definition
X(z) = Σ x[n] z⁻ⁿ (n = 0 to ∞)
Substitute x[n] = δ[n]
X(z) = Σ δ[n] z⁻ⁿ
Since δ[n] is non-zero only when n = 0:
X(z) = 1 × z⁰
Final Result
Z{ δ[n] } = 1
Region of Convergence (ROC)
ROC = Entire z-plane
The delta function acts like a sampling impulse, so its Z-transform becomes 1.
Z-Transform of Shifted Delta Functions
1. Z-Transform of δ[n-1]
X(z) = Σ δ[n-1] z⁻ⁿ
δ[n-1] is non-zero only when n = 1.
X(z) = z⁻¹
Z{ δ[n-1] } = z⁻¹
2. Z-Transform of δ[n-k]
X(z) = Σ δ[n-k] z⁻ⁿ
δ[n-k] is non-zero only when n = k.
X(z) = z⁻ᵏ
Z{ δ[n-k] } = z⁻ᵏ
Important Rule:
Z{ δ[n-k] } = z⁻ᵏ
Z{ δ[n-k] } = z⁻ᵏ
Summary Table
| Sequence | Z-Transform |
|---|---|
| δ[n] | 1 |
| δ[n-1] | z⁻¹ |
| δ[n-k] | z⁻ᵏ |
Summary
If the delta function shifts by k, the Z-transform becomes z⁻ᵏ.