---
title: "Properties of ROC of z-transform"
subject: "Digital Signal Analysis and Processing"
chapter: 2
key: "semester_7/digital_signal_analysis_and_processing:roc-properties"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/roc-properties
site: BCT Engineering
---

# Properties of ROC of z-transform

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

- Asked in: 2072 Chaitra, 6 marks; 2074 Asoj, 3 marks; 2074 Chaitra, 3 marks; 2072 Kartik, 3 marks; 2069 Chaitra, 3 marks; 2070 Chaitra, 2 marks (asked 6 times)
- Video: [ROC and its Properties](https://www.youtube.com/watch?v=LyOQVvPVIuE) — Neso Academy
- 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/roc-properties
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-2

## Question

State and explain the properties of Region of Convergence (ROC) of the z-transform, with examples.

## Short answer

The ROC is a connected annulus r1<|z|<r2 with no poles; finite sequences have ROC all z except possibly 0/infinity, right-sided sequences have |z|>r1, left-sided |z|<r2, two-sided a ring. It includes the unit circle iff the sequence is absolutely summable; for example a^n u[n] has ROC |z|>|a|, and a causal stable system has all poles inside the unit circle.

## Answer

The **region of convergence (ROC)** of X(z) = Σ x[n] z⁻ⁿ is the set of z for which the sum converges absolutely, Σ |x[n]| |z|⁻ⁿ < ∞.

#### Properties
1. **The ROC is a ring (annulus) centred at the origin:** r₁ < |z| < r₂, where 0 ≤ r₁ and r₂ ≤ ∞. Convergence depends only on |z|, not on the angle.
2. **The ROC contains no poles**, since X(z) is infinite at a pole. It is bounded by poles (or extends to 0 or ∞).
3. **Finite-duration sequence:** the ROC is the entire z-plane, except possibly z = 0 and/or z = ∞. Example: x[n] = {1, 2, 3} for n = 0, 1, 2 gives X(z) = 1 + 2z⁻¹ + 3z⁻², ROC all z except z = 0.
4. **Right-sided sequence** (x[n] = 0 for n < N₁): the ROC is the exterior of a circle, |z| > r₁ (the largest pole magnitude); it includes z = ∞ if N₁ ≥ 0 (causal). Example: aⁿu[n] ↔ 1/(1 − az⁻¹), |z| > |a|.
5. **Left-sided sequence** (x[n] = 0 for n > N₂): the ROC is the interior of a circle, |z| < r₂ (the smallest pole magnitude); it includes z = 0 if N₂ ≤ 0. Example: −aⁿu[−n−1] ↔ 1/(1 − az⁻¹), |z| < |a|.
6. **Two-sided sequence:** the ROC is a ring r₁ < |z| < r₂, or empty if the right-sided and left-sided parts do not overlap. Example: (½)ⁿu[n] − 3ⁿu[−n−1] has ROC ½ < |z| < 3.
7. **The ROC is a connected region** (one ring). For a rational X(z) with several poles, the ROC is one of the rings between consecutive pole magnitudes, so different ROCs give different sequences for the same X(z).
8. **Stability:** x[n] is absolutely summable (its DTFT exists) if and only if the ROC includes the unit circle |z| = 1.
9. **Causality of a system:** for a causal LTI system the ROC of H(z) lies outside the outermost pole; for a stable and causal rational system, all poles are inside the unit circle.

**Example:** X(z) = 1/((1 − ½z⁻¹)(1 − 2z⁻¹)) has poles ½ and 2, so three possible ROCs: |z| < ½ (left-sided), ½ < |z| < 2 (two-sided, includes the unit circle, stable), |z| > 2 (right-sided, causal but unstable).

## Check yourself

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

1. The ROC of a z-transform is
   A. a half plane
   B. a ring centred at origin
   C. a line
   D. a single point
   Answer: B. Convergence depends only on |z|. (Key point: ROC shape)

2. The ROC
   A. contains no poles
   B. contains all poles
   C. is bounded by zeros
   D. always includes z = 0
   Answer: A. X(z) is infinite at poles. (Key point: Poles)

3. The ROC of a right-sided sequence is
   A. interior of a circle
   B. a ring
   C. the whole plane
   D. exterior of a circle
   Answer: D. |z| > largest pole magnitude. (Key point: Right-sided)

4. x[n] has a DTFT iff the ROC
   A. excludes the unit circle
   B. includes the origin
   C. includes the unit circle
   D. is empty
   Answer: C. Absolute summability corresponds to the unit circle in ROC. (Key point: Stability)

## Related in Digital Signal Analysis and Processing

- [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×
- [Z-transform and ROC of two-sided exponential](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Az-transform-two-sided-exp) — Digital Signal Analysis and Processing, chapter 2
- [Inverse z-transform with repeated pole](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Ainverse-z-double-pole) — Digital Signal Analysis and Processing, chapter 2, asked 5×
- [Inverse z-transform for three ROCs](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Ainverse-z-three-rocs) — Digital Signal Analysis and Processing, chapter 2, asked 5×

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