Z-Transform of Delta Function

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

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.