---
title: "Define Region of Convergence"
subject: "Digital Signal Analysis and Processing"
chapter: 2
key: "semester_7/digital_signal_analysis_and_processing:define-roc"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/define-roc
site: BCT Engineering
---

# Define Region of Convergence

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

- Asked in: 2074 Chaitra, 1 marks; 2073 Shrawan, 1 marks; 2073 Chaitra, 1 marks; 2071 Chaitra, 1 marks; 2070 Asar, 1 marks (asked 5 times)
- Video: [Z Transform Region of Convergence Explained ("the best explanation in the internet!")](https://www.youtube.com/watch?v=uq_qv3Spzbs) — Iain Explains Signals, Systems, and Digital Comms
- 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/define-roc
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-2

## Question

Define Region of Convergence (ROC).

## Short answer

The ROC is the set of complex z for which Σ|x[n]||z|^(-n) < ∞. It is an annular region r1<|z|<r2 bounded by poles and contains no poles; it is needed to uniquely specify the sequence. For (1/2)^n u[n], the ROC is |z|>1/2.

## Answer

The **Region of Convergence (ROC)** of the z-transform X(z) = Σ(n = −∞ to ∞) x[n] z⁻ⁿ is the set of all complex values z for which the series converges (the sum is finite), i.e. Σ |x[n]| |z|⁻ⁿ < ∞.

It is generally a ring r₁ < |z| < r₂ in the z-plane, bounded by poles, and it contains no poles. The ROC is needed to specify X(z) completely: the same expression, for example 1/(1 − az⁻¹), corresponds to aⁿu[n] if |z| > |a| and to −aⁿu[−n−1] if |z| < |a|.

Example: x[n] = (½)ⁿ u[n] gives X(z) = 1/(1 − ½z⁻¹) with ROC |z| > ½.

## Check yourself

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

1. The ROC is the set of z for which
   A. X(z) = 0
   B. the series converges
   C. x[n] = 0
   D. poles are located
   Answer: B. sum |x[n]||z|^-n < infinity. (Key point: Definition)

2. Why is the ROC needed?
   A. the same expression can belong to different sequences
   B. to find zeros
   C. to find x[0]
   D. to decide LTI
   Answer: A. 1/(1-az^-1) gives a^n u[n] or -a^n u[-n-1] depending on ROC. (Key point: Role of ROC)

3. For (1/2)^n u[n] the ROC is
   A. |z| < 1/2
   B. |z| > 2
   C. all z
   D. |z| > 1/2
   Answer: D. Right-sided with pole at 1/2. (Key point: Example)

## Related in Digital Signal Analysis and Processing

- [Properties of ROC of z-transform](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Aroc-properties) — Digital Signal Analysis and Processing, chapter 2, asked 6×

## Related in other subjects

- [Define z-transform and ROC](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Az-transform-definition-roc) — Applied Mathematics, chapter 2
- [ROC of x(k) = −a^k u(−k−1)](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Aroc-left-sided-sequence) — Digital Control System, chapter 2