---
title: "Z-transform and ROC of two-sided exponential"
subject: "Digital Signal Analysis and Processing"
chapter: 2
key: "semester_7/digital_signal_analysis_and_processing:z-transform-two-sided-exp"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/z-transform-two-sided-exp
site: BCT Engineering
---

# Z-transform and ROC of two-sided exponential

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

- Asked in: 2070 Chaitra, 4 marks
- Video: [ZT & ROC of Infinite Duration Double - Sided Sequence](https://www.youtube.com/watch?v=HER7oDITVIs) — Neeharika 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/z-transform-two-sided-exp
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-2

## Question

Find the z-transform and locate the ROC of the signal x[n] = (−1/3)^n u[n] − (1/3)^n u[−n−1].

## Short answer

Term (-1/3)^n u[n] gives 1/(1+(1/3)z^{-1}) with ROC |z|>1/3. Term -(1/3)^n u[-n-1] gives 1/(1-(1/3)z^{-1}) with ROC |z|<1/3. Their ROCs are disjoint, so the signal has no convergent z-transform.

## Answer

x[n] = (−⅓)ⁿ u[n] − (⅓)ⁿ u[−n−1]. Take the two terms separately.

**First term** (right-sided): x₁[n] = (−⅓)ⁿ u[n]

X₁(z) = Σ(n=0 to ∞) (−⅓ z⁻¹)ⁿ = 1/(1 + ⅓z⁻¹), ROC: |z| > ⅓

**Second term** (left-sided): x₂[n] = −(⅓)ⁿ u[−n−1]

X₂(z) = −Σ(n = −∞ to −1) (⅓ z⁻¹)ⁿ = 1/(1 − ⅓z⁻¹), ROC: |z| < ⅓

**Combined:** x[n] = x₁[n] + x₂[n], so

X(z) = 1/(1 + ⅓z⁻¹) + 1/(1 − ⅓z⁻¹) = 2/(1 − (1/9)z⁻²) = 2z²/(z² − 1/9)

The poles are at z = ±⅓.

**ROC:** the ROC of the sum must lie in both |z| > ⅓ and |z| < ⅓. These two regions do not overlap.

```text
 Im
  |      ROC1: |z| > 1/3  (outside)
  |      ROC2: |z| < 1/3  (inside)
 -x---o---x-- Re       no common ring
```

**Answer:** The algebraic form is X(z) = 2z²/(z² − 1/9), but the intersection of the ROCs is empty, so **the z-transform of this x[n] does not converge for any z**: the ROC does not exist (the signal has no z-transform, and no DTFT). (X(z) = 2z²/(z²−1/9) would correspond to the signal with ROC |z| > ⅓ only, i.e. to a different sequence.)

## Check yourself

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

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

2. -(1/3)^n u[-n-1] has ROC
   A. |z| > 1/3
   B. |z| > 3
   C. all z
   D. |z| < 1/3
   Answer: D. Left-sided gives interior ROC. (Key point: Left-sided term)

3. The overall ROC is
   A. 1/3 < |z| < 1/3 ring
   B. |z| > 1/3
   C. empty
   D. all z
   Answer: C. The two regions do not overlap. (Key point: Combined ROC)

4. Therefore the z-transform of x[n]
   A. is 2z^2/(z^2-1/9) with |z|>1/3
   B. does not converge for any z
   C. is stable
   D. is causal
   Answer: B. No ROC means no z-transform. (Key point: Conclusion)

## Related in Digital Signal Analysis and Processing

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