---
title: "Define z-transform and ROC"
subject: "Applied Mathematics"
chapter: 2
key: "semester_4/applied_mathematics:z-transform-definition-roc"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/z-transform-definition-roc
site: BCT Engineering
---

# Define z-transform and ROC

Applied Mathematics · Chapter 2: The Z-Transform

- Asked in: 2073 Magh, 2 marks
- Video: [Region of Convergence of Z-Transform](https://www.youtube.com/watch?v=a05UJmw5wak) — TutorialsPoint
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/z-transform-definition-roc
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-2

## Question

Define Z-transform and its region of convergence.

## Short answer

Definition as above. ROC is set of z where sum converges. Right-sided (causal): |z|>R; left-sided: |z|<R; two-sided: ring. For a^k u(k), ROC is |z|>|a|; ROC never contains poles.

## Answer

#### Z-transform
The z-transform of a sequence x(k) is

X(z) = Z[x(k)] = Σ (k = −∞ to ∞) x(k) z⁻ᵏ

For a causal sequence this is the one-sided transform Σ (k = 0 to ∞) x(k) z⁻ᵏ, where z is a complex variable.

#### Region of convergence (ROC)
The ROC is the set of values of z in the complex plane for which the series Σ x(k) z⁻ᵏ converges (to a finite value).

- Right-sided (causal) sequence: ROC is |z| > R, outside a circle.
- Left-sided sequence: ROC is |z| < R, inside a circle.
- Two-sided sequence: ROC is a ring R₁ < |z| < R₂ (possibly empty).
- Finite-length sequence: ROC is the whole z-plane except possibly z = 0 or ∞.

**Example:** x(k) = aᵏ for k ≥ 0 gives X(z) = Σ (a/z)ᵏ = z/(z − a), converging only when |a/z| < 1, i.e. ROC: |z| > |a|. The ROC never contains a pole of X(z).

## Check yourself

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

1. What is ROC?
   A. Set of zeros
   B. Set of z where Σ x(k) z⁻ᵏ converges
   C. Set of poles
   D. |z|=1
   Answer: B. Convergence region. (Key point: ROC)

2. Causal sequence ROC?
   A. |z| < R
   B. R1<|z|<R2
   C. |z| > R
   D. Whole plane
   Answer: C. Outside a circle. (Key point: Causal ROC)

3. ROC and poles?
   A. ROC contains all poles
   B. ROC is the poles
   C. Poles on the boundary of the region are always included
   D. ROC never contains a pole
   Answer: D. X→∞ at a pole. (Key point: Poles)

4. ROC of a^k, k≥0?
   A. |z|>|a|
   B. |z|<|a|
   C. |z|>1
   D. All z
   Answer: A. |a/z|<1. (Key point: Example)

## Related in other subjects

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- [Define Region of Convergence](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Adefine-roc) — Digital Signal Analysis and Processing, chapter 2, asked 5×
- [Inverse z-transform, ROC |z| < 0.4](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Ainverse-z-roc-less-0-4) — Digital Signal Analysis and Processing, chapter 2, asked 5×
- [Inverse z-transform, ROC |z| < 1](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Ainverse-z-z-minus-1-z-minus-2) — Digital Signal Analysis and Processing, chapter 2, asked 5×