---
title: "Properties of the Fourier transform"
subject: "Data Communication"
chapter: 3
key: "semester_5/data_communication:fourier-transform-properties"
source: https://www.bctengineer.com/q/semester_5/data_communication/fourier-transform-properties
site: BCT Engineering
---

# Properties of the Fourier transform

Data Communication · Chapter 3: Signal Analysis

- Asked in: 2076 Asoj, 4 marks; 2073 Chaitra, 6 marks (asked 2 times)
- Video: [ECE300 Lecture 5-3: Fourier Transforms Properties](https://www.youtube.com/watch?v=zij3L1UHjoo) — Rose-Hulman Online
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/data_communication/fourier-transform-properties
- Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-3

## Question

What are the properties of the Fourier transform?

## Short answer

Key properties include linearity, time shift (e^{-jωt0}X(ω)), frequency shift (X(ω-ω0)), scaling (1/|a| X(ω/a)), convolution ↔ multiplication, differentiation ↔ jωX(ω), and Parseval's theorem. For real signals, magnitude is even and phase odd; a real even signal has real even spectrum.

## Answer

Let x(t) ↔ X(ω) denote a Fourier transform pair, with X(ω) = ∫ x(t) e^(-jωt) dt. The main properties are:

| Property | Time domain | Frequency domain |
|---|---|---|
| Linearity | a·x₁(t) + b·x₂(t) | a·X₁(ω) + b·X₂(ω) |
| Time shifting | x(t - t₀) | e^(-jωt₀) X(ω) |
| Frequency shifting (modulation) | e^(jω₀t) x(t) | X(ω - ω₀) |
| Time scaling | x(at) | (1/\|a\|) X(ω/a) |
| Time reversal | x(-t) | X(-ω) |
| Conjugation | x*(t) | X*(-ω) |
| Duality | X(t) | 2π x(-ω) |
| Time differentiation | dx/dt | jω X(ω) |
| Time integration | ∫ x(τ) dτ | X(ω)/(jω) + π X(0) δ(ω) |
| Frequency differentiation | t·x(t) | j dX/dω |
| Convolution in time | x₁(t) * x₂(t) | X₁(ω) X₂(ω) |
| Multiplication in time | x₁(t) x₂(t) | (1/2π) X₁(ω) * X₂(ω) |
| Parseval's theorem | ∫\|x(t)\|² dt | (1/2π) ∫\|X(ω)\|² dω |

Additional points:
- For a real x(t), X(-ω) = X*(ω): the magnitude is even and the phase is odd.
- If x(t) is real and even, X(ω) is real and even; if x(t) is real and odd, X(ω) is purely imaginary and odd.
- Time scaling shows that compression in time gives expansion in frequency (a short pulse has a wide bandwidth).

## Check yourself

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

1. x(t-t0) transforms to
   A. e^(jωt0)X(ω)
   B. X(ω-t0)
   C. X(ω)/t0
   D. e^(-jωt0)X(ω)
   Answer: D. Phase shift. (Key point: Time shift)

2. x(at) transforms to
   A. (1/|a|)X(ω/a)
   B. aX(aω)
   C. X(aω)
   D. |a|X(ω/a)
   Answer: A. Compression widens spectrum. (Key point: Time scaling)

3. Convolution in time gives
   A. Convolution
   B. Product X1X2
   C. Sum
   D. Quotient
   Answer: B. Property. (Key point: Convolution)

4. For real x(t), the magnitude spectrum is
   A. Odd, phase even
   B. Both even
   C. Even, phase odd
   D. Both odd
   Answer: C. X(-ω) = X*(ω). (Key point: Real signals)

## Related in Data Communication

- [Why the Fourier transform is needed](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Awhy-fourier-transform-needed) — Data Communication, chapter 3, asked 2×
- [Fourier transform of e^(-at)u(t) and spectra](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aft-exp-causal-spectra) — Data Communication, chapter 3, asked 2×
- [Fourier transform of a DC signal](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aft-dc-signal) — Data Communication, chapter 3
- [Fourier transform of e^(-a|t|)](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aft-exp-abs) — Data Communication, chapter 3
- [Fourier transform of a ramp pulse](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aft-ramp-signal) — Data Communication, chapter 3
- [Fourier series vs Fourier transform](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Afs-vs-ft) — Data Communication, chapter 3

## Related in other subjects

- [Fourier transform properties; Rayleigh energy theorem](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Afourier-properties-rayleigh) — Communication Systems, chapter 1
- [Hilbert vs Fourier transform; properties of HT](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ahilbert-vs-fourier-properties) — Communication Systems, chapter 2
- [Autocorrelation and ESD as Fourier transform pair](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aautocorrelation-esd-fourier-pair) — Communication Systems, chapter 1
- [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×