---
title: "Inverse Fourier transform of δ(ω)"
subject: "Signals and Systems"
chapter: 3
key: "semester_4/signals_and_systems:inverse-ft-delta-omega"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/inverse-ft-delta-omega
site: BCT Engineering
---

# Inverse Fourier transform of δ(ω)

Signals and Systems · Chapter 3: Fourier Transform

- Asked in: 2081 Asoj, 2 marks
- Video: [Fourier Transform of Basic Signals (Complex Exponential Signal)](https://www.youtube.com/watch?v=fDrChXcplcA) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/inverse-ft-delta-omega
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-3

## Question

Find inverse Fourier Transform of CT unit impulse signal δ(ω)

## Short answer

Inverse FT of δ(ω): x(t)= (1/2π) ∫ δ(ω) e^{jωt} dω = 1/(2π) by sifting. Hence δ(ω) ↔ 1/(2π), a constant DC signal, meaning all spectral content concentrated at zero frequency.

## Answer

Use the inverse Fourier transform with $X(\omega) = \delta(\omega)$:
$$
x(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty}\delta(\omega)e^{j\omega t}d\omega
$$
By the sifting property, $\int\delta(\omega)f(\omega)d\omega = f(0)$, and here $f(0) = e^{j0\cdot t} = 1$:
$$
x(t) = \frac{1}{2\pi}\cdot 1 = \frac{1}{2\pi}
$$
**Answer:** $\delta(\omega) \xleftrightarrow{F^{-1}} \frac{1}{2\pi}$, a constant (pure DC) signal. Equivalently, $1 \leftrightarrow 2\pi\delta(\omega)$: a constant signal has all its content (its power) at zero frequency.

## Check yourself

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

1. Which property is used to evaluate (1/2π)∫δ(ω)e^(jωt)dω?
   A. Time shifting
   B. Sifting property of the impulse
   C. Parseval's relation
   D. Convolution
   Answer: B. The sifting property picks the integrand value at ω = 0, which is e^0 = 1. (Key point: Sifting property)

2. The inverse Fourier transform of δ(ω) is:
   A. 1
   B. δ(t)
   C. 1/(2π)
   D. 2π
   Answer: C. The integral gives 1, multiplied by the 1/(2π) factor, so x(t) = 1/(2π). (Key point: Result)

3. What does the result imply about a constant signal?
   A. 1 <-> δ(ω)
   B. 1 <-> 2π δ(ω)
   C. 1 <-> 1/(jω)
   D. 1 <-> π δ(ω)
   Answer: B. Multiplying by 2π, a constant 1 has transform 2π δ(ω): all content is at zero frequency. (Key point: Meaning)

## Related in Signals and Systems

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