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

# Define the z-transform

Applied Mathematics · Chapter 2: The Z-Transform

- Asked in: 2075 Baisakh, 1 marks
- Video: [Introduction to Z-Transform](https://www.youtube.com/watch?v=mJgVOV9jRZU) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/define-z-transform
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-2

## Question

Define z-transform.

## Short answer

The z-transform of a sequence x(k) is X(z)=Σ_{k=−∞}^{∞} x(k)z^{−k}, with complex z. For causal sequences, one-sided sum from k=0. For sampled signals, x(k)=x(kT). Example: a^k u(k) ↔ z/(z−a).

## Answer

The **z-transform** of a discrete-time sequence x(k) (or of the samples x(kT) of a signal x(t)) is the function

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

where z is a complex variable. For a causal sequence (x(k) = 0 for k < 0), the one-sided transform is X(z) = Σ (k = 0 to ∞) x(k) z⁻ᵏ. For a sampled signal, x(k) is replaced by x(kT), with T the sampling period.

Example: for x(k) = aᵏ, k ≥ 0, X(z) = Σ (a/z)ᵏ = z/(z − a), |z| > |a|.

## Check yourself

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

1. Z-transform of x(k)?
   A. Σ x(k) zᵏ
   B. Σ x(k) e^{−ωk}
   C. ∫ x(t) e^{−st} dt
   D. Σ x(k) z⁻ᵏ
   Answer: D. Powers of z⁻¹. (Key point: Definition)

2. One-sided for a causal sequence?
   A. Sum from k=0 to ∞
   B. Sum from −∞ to 0
   C. Sum from k=1 only
   D. Sum of x(k)²
   Answer: A. x(k)=0 for k<0. (Key point: One-sided)

3. Z[a^k] = ?
   A. 1/(z−a)
   B. z/(z−a), |z|>|a|
   C. z/(z+a)
   D. z/(1−a)
   Answer: B. Geometric series. (Key point: Example)