---
title: "Inverse z-transform of factored X(z)"
subject: "Digital Signal Analysis and Processing"
chapter: 2
key: "semester_7/digital_signal_analysis_and_processing:inverse-z-factored-form"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/inverse-z-factored-form
site: BCT Engineering
---

# Inverse z-transform of factored X(z)

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

- Asked in: 2074 Asoj, 3 marks
- 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/inverse-z-factored-form
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-2

## Question

Find the inverse z-transform of X(z) = z²[1 − (3/2)z⁻¹](1 + z⁻¹)(1 − z⁻¹).

## Short answer

Expand the polynomial: X(z)=z^2 -1.5z -1 +1.5z^{-1}. Taking termwise inverse gives x[n]=δ[n+2] -1.5δ[n+1] -δ[n] +1.5δ[n-1], a finite-length sequence with ROC entire z-plane except 0 and ∞.

## Answer

X(z) = z²(1 − (3/2)z⁻¹)(1 + z⁻¹)(1 − z⁻¹) is a polynomial in z, so expand it; each term cz⁻ᵏ is simply a delayed impulse cδ[n − k].

(1 + z⁻¹)(1 − z⁻¹) = 1 − z⁻²

(1 − 1.5z⁻¹)(1 − z⁻²) = 1 − 1.5z⁻¹ − z⁻² + 1.5z⁻³

Multiply by z²:

X(z) = z² − 1.5z − 1 + 1.5z⁻¹

Use δ[n − k] ↔ z⁻ᵏ (k negative means an advance):

- z² ↔ δ[n + 2]
- −1.5z ↔ −1.5δ[n + 1]
- −1 ↔ −δ[n]
- 1.5z⁻¹ ↔ 1.5δ[n − 1]

**Answer:** x[n] = δ[n + 2] − 1.5δ[n + 1] − δ[n] + 1.5δ[n − 1],

i.e. x[n] = {1, −1.5, −1, 1.5} for n = −2, −1, 0, 1 (zero elsewhere). It is a finite-length sequence, so the ROC is the whole z-plane except z = 0 and z = ∞.

## Check yourself

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

1. (1+z^-1)(1-z^-1) =
   A. 1 - z^-2
   B. 1 + z^-2
   C. 1 - z^-1
   D. z^-2
   Answer: A. Difference of squares. (Key point: Expand)

2. cz^-k inverts to
   A. c delta[n+k]
   B. c u[n-k]
   C. c k
   D. c delta[n-k]
   Answer: D. Delayed impulse. (Key point: Term cz^-k)

3. x[n] =
   A. {1,-1.5,-1,1.5} for n=0..3
   B. {1.5,-1,-1.5,1} for n=-2..1
   C. {1,-1.5,-1,1.5} for n=-2..1
   D. {1,1,1,1}
   Answer: C. z^2 gives delta[n+2]. (Key point: Final answer)

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