---
title: "Define z-transform; differ from Fourier transform"
subject: "Applied Mathematics"
chapter: 2
key: "semester_4/applied_mathematics:z-transform-vs-fourier"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/z-transform-vs-fourier
site: BCT Engineering
---

# Define z-transform; differ from Fourier transform

Applied Mathematics · Chapter 2: The Z-Transform

- Asked in: 2073 Bhadra, 2 marks
- Video: [Understanding the Z-Transform](https://www.youtube.com/watch?v=XJRW6jamUHk) — MATLAB
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/z-transform-vs-fourier
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-2

## Question

Define z-transform. How does it differ from Fourier transform?

## Short answer

Z-transform uses complex z=re^{iω}; Fourier transform is DTFT on |z|=1 (z=e^{iω}). Z handles growing sequences and transient analysis, DTFT needs absolute summability. X(e^{iω})=X(z) on unit circle if ROC contains it.

## Answer

#### Z-transform
The z-transform of a sequence x(k) is X(z) = Σ (k = −∞ to ∞) x(k) z⁻ᵏ, where z = re^(iω) is a complex variable. It converts a difference equation into an algebraic equation and is widely used for analysing discrete-time systems.

#### Difference from the Fourier transform
| Point | z-transform | Fourier transform (DTFT) |
|---|---|---|
| Variable | Complex z = re^(iω) | Real frequency ω (z = e^(iω)) |
| Definition | Σ x(k) z⁻ᵏ | Σ x(k) e^(−iωk) |
| Convergence | Exists for z in a region of convergence (ROC) | Converges if Σ\|x(k)\| < ∞ (the unit circle must lie in the ROC) |
| Signals handled | Also growing sequences (e.g. 2ᵏ) | Needs absolutely summable sequences |
| Relation | General | A special case: the z-transform on the unit circle |
| Main use | Transient analysis, stability, solving difference equations with initial conditions | Frequency (steady-state) spectrum |

So X(e^(iω)) = X(z) evaluated at |z| = 1, provided the unit circle lies in the ROC.

## Check yourself

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

1. z in the z-transform?
   A. Real frequency
   B. Time
   C. Complex, z = re^{iω}
   D. Integer
   Answer: C. Fourier is r=1. (Key point: Variable)

2. How are they related?
   A. They are unrelated
   B. Z-transform on the real axis
   C. DTFT on |z|=2
   D. DTFT = z-transform on |z|=1 if the unit circle is in the ROC
   Answer: D. X(e^{iω}). (Key point: Relation)

3. Which handles 2^k?
   A. z-transform with ROC |z|>2
   B. Fourier transform
   C. Both
   D. Neither
   Answer: A. DTFT needs absolute summability. (Key point: Growing signals)

## Related in Applied Mathematics

- [Fourier transform vs Fourier series](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Afourier-transform-vs-series) — Applied Mathematics, chapter 4
- [Fourier transform of 1-x²; evaluate integral](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Afourier-transform-1-x2-evaluate) — Applied Mathematics, chapter 4, asked 2×
- [State and prove Fourier convolution theorem](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Aconvolution-theorem-fourier-prove) — Applied Mathematics, chapter 4, asked 3×
- [Sine transform of e^{-x}; ∫x²/(1+x²)²](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Afourier-sine-transform-x2-over-1-x2-sq) — Applied Mathematics, chapter 4, asked 4×
- [Sine transform of e^{-x}; ∫x sin mx/(x²+1)](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Afourier-sine-transform-x-sin-mx) — Applied Mathematics, chapter 4, asked 4×
- [Cosine transform of e^{-x}; Parseval identity](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Afourier-cosine-parseval-e-x) — Applied Mathematics, chapter 4, asked 3×

## Related in other subjects

- [Fourier transform of e^(−a|t|)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-two-sided-exponential) — Signals and Systems, chapter 3, asked 5×
- [Fourier transform of e^(−at)cos ω₀t](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-damped-cosine) — Signals and Systems, chapter 3, asked 4×
- [Fourier transform of e^(−2|t−1|)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-shifted-two-sided-exponential) — Signals and Systems, chapter 3, asked 4×
- [Derive the CT Fourier transform pair](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actft-derivation) — Signals and Systems, chapter 3, asked 3×