Derive convolution property of z-transform

Statement: If x₁[n] ↔ X₁(z) with ROC R₁ and x₂[n] ↔ X₂(z) with ROC R₂, then

x₁[n] * x₂[n] ↔ X₁(z) X₂(z), with ROC containing R₁ ∩ R₂.

Derivation. Let y[n] = x₁[n] * x₂[n] = Σ(k = −∞ to ∞) x₁[k] x₂[n − k]. Then

Y(z) = Σ(n) y[n] z⁻ⁿ = Σ(n) [ Σ(k) x₁[k] x₂[n − k] ] z⁻ⁿ

Interchange the order of summation (valid inside the common ROC, where both converge absolutely):

Y(z) = Σ(k) x₁[k] [ Σ(n) x₂[n − k] z⁻ⁿ ]

Put m = n − k, so n = m + k and z⁻ⁿ = z⁻ᵐ z⁻ᵏ:

Y(z) = Σ(k) x₁[k] z⁻ᵏ · Σ(m) x₂[m] z⁻ᵐ = X₁(z) X₂(z)

The ROC is at least the overlap R₁ ∩ R₂; it can be larger if a pole of one is cancelled by a zero of the other.

Use: the output of an LTI system is Y(z) = X(z)H(z), so convolution in time becomes simple multiplication in the z-domain.