---
title: "Derive convolution property of z-transform"
subject: "Digital Signal Analysis and Processing"
chapter: 2
key: "semester_7/digital_signal_analysis_and_processing:z-convolution-property-derive"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/z-convolution-property-derive
site: BCT Engineering
---

# Derive convolution property of z-transform

Digital Signal Analysis and Processing · Chapter 2: Z-transform

- Asked in: 2072 Kartik, 3 marks
- Video: [Convolution Property of Z-transform](https://www.youtube.com/watch?v=Wa2snZZlcZ8) — Engineers choice tutor
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/z-convolution-property-derive
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-2

## Question

Derive the convolution property of z-transform.

## Short answer

If x1[n]↔X1(z) with ROC R1 and x2[n]↔X2(z) with ROC R2, then y[n]=x1[n]*x2[n] has transform Y(z)=X1(z)X2(z) with ROC at least R1∩R2. Proof: Σ_n (Σ_k x1[k]x2[n-k]) z^(-n) = Σ_k x1[k]z^(-k) Σ_m x2[m]z^(-m).

## Answer

**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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. x1[n]*x2[n] has z-transform
   A. X1(z)+X2(z)
   B. X1(z)*X2(z) convolved
   C. X1(z)X2(z)
   D. X1(z)/X2(z)
   Answer: C. Convolution becomes multiplication. (Key point: Statement)

2. Derivation uses
   A. differentiating
   B. interchanging sums and substituting m = n-k
   C. integrating
   D. Parseval
   Answer: B. Then the double sum separates. (Key point: Key step)

3. The ROC of the product contains
   A. R1 intersect R2
   B. R1 union R2
   C. R1 only
   D. empty set always
   Answer: A. It can be larger if pole-zero cancellation occurs. (Key point: ROC)

## Related in Digital Signal Analysis and Processing

- [Derive time shifting property of z-transform](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Az-time-shifting-derive) — Digital Signal Analysis and Processing, chapter 2
- [State and prove circular convolution property](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Acircular-conv-property-prove) — Digital Signal Analysis and Processing, chapter 7

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