---
title: "Fourier transform of unit step"
subject: "Signals and Systems"
chapter: 3
key: "semester_4/signals_and_systems:ft-unit-step"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/ft-unit-step
site: BCT Engineering
---

# Fourier transform of unit step

Signals and Systems · Chapter 3: Fourier Transform

- Asked in: 2075 Bhadra, 5 marks; 2073 Magh, 4 marks; 2071 Bhadra, 5 marks (asked 3 times)
- Video: [Fourier Transform of Basic Signals (Unit Step Signal)](https://www.youtube.com/watch?v=blf9rfmxGvo) — 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/ft-unit-step
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-3

## Question

Find the Fourier transform of continuous time unit step signal.

## Short answer

Write u(t) = 1/2 + (1/2)sgn(t). The FT of 1/2 is pi delta(w), and the FT of sgn(t) is 2/(jw). Thus U(w) = pi delta(w) + 1/(jw).

## Answer

The unit step $u(t)$ is not absolutely integrable, so its Fourier transform is found as a limit, using the signum function.

#### Step 1: Write u(t) using the signum function
$$
u(t) = \frac{1}{2} + \frac{1}{2}\mathrm{sgn}(t), \qquad \mathrm{sgn}(t) = \begin{cases} 1, & t > 0 \\ -1, & t < 0 \end{cases}
$$

#### Step 2: FT of the constant 1/2
The inverse Fourier transform of $\delta(\omega)$ is $\frac{1}{2\pi}$, so $1 \xleftrightarrow{F} 2\pi\delta(\omega)$ and
$$
\frac{1}{2} \xleftrightarrow{F} \pi\delta(\omega)
$$

#### Step 3: FT of sgn(t)
Write $\mathrm{sgn}(t) = \lim_{a\to 0}\left[e^{-at}u(t) - e^{at}u(-t)\right]$ with $a > 0$:
$$
\begin{aligned}
F\{\mathrm{sgn}(t)\} &= \lim_{a\to 0}\left[\int_0^{\infty}e^{-(a+j\omega)t}dt - \int_{-\infty}^{0}e^{(a-j\omega)t}dt\right] \\
&= \lim_{a\to 0}\left[\frac{1}{a+j\omega} - \frac{1}{a-j\omega}\right] = \lim_{a\to 0}\frac{-2j\omega}{a^2+\omega^2} = \frac{2}{j\omega}
\end{aligned}
$$
So $\frac{1}{2}\mathrm{sgn}(t) \xleftrightarrow{F} \frac{1}{j\omega}$.

#### Step 4: Combine
$$
\boxed{U(\omega) = \pi\delta(\omega) + \frac{1}{j\omega}}
$$

#### Spectrum
- The impulse $\pi\delta(\omega)$ represents the DC component (average value 1/2) of $u(t)$.
- $|U(\omega)| = \frac{1}{|\omega|}$ for $\omega \ne 0$.
- $\angle U(\omega) = -\pi/2$ for $\omega > 0$ and $+\pi/2$ for $\omega < 0$.

**Check:** $u(t) = \int_{-\infty}^{t}\delta(\tau)d\tau$, and the integration property gives $\frac{1}{j\omega}\cdot 1 + \pi\cdot 1\cdot\delta(\omega)$, which is the same result.

## Check yourself

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

1. How is u(t) written for finding its Fourier transform?
   A. u(t) = 1/2 + (1/2) sgn(t)
   B. u(t) = sgn(t)
   C. u(t) = 1 - sgn(t)
   D. u(t) = e^{-at} as a → 0 only
   Answer: A. The constant 1/2 plus half the signum function equals the step. (Key point: Decomposition)

2. The Fourier transform of the constant 1/2 is:
   A. δ(ω)/2
   B. π δ(ω)
   C. 2π δ(ω)
   D. 1/(jω)
   Answer: B. 1 ↔ 2π δ(ω), so 1/2 ↔ π δ(ω). (Key point: FT of constant)

3. The Fourier transform of sgn(t) is:
   A. 1/(jω)
   B. 2/(jω)
   C. 2jω
   D. 2/ω
   Answer: B. The limit of -2jω/(a^2 + ω^2) as a → 0 is 2/(jω). (Key point: FT of sgn)

4. The Fourier transform of u(t) is:
   A. 1/(jω)
   B. π δ(ω) + 1/(jω)
   C. 2π δ(ω) + 1/(jω)
   D. π δ(ω) - 1/(jω)
   Answer: B. π δ(ω) from the 1/2 part plus 1/(jω) from (1/2) sgn(t). (Key point: Final result)

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