---
title: "Duality property of CTFT"
subject: "Signals and Systems"
chapter: 3
key: "semester_4/signals_and_systems:ctft-duality"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/ctft-duality
site: BCT Engineering
---

# Duality property of CTFT

Signals and Systems · Chapter 3: Fourier Transform

- Asked in: 2081 Asoj, 1.5+3.5 marks; 2077 Chaitra, 5 marks (asked 2 times)
- Video: [Duality Property of Fourier Transform](https://www.youtube.com/watch?v=9OK_i-n8gN8) — 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/ctft-duality
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-3

## Question

State and prove (explain with derivation) the duality property of continuous time Fourier transform.

## Short answer

If x(t) has Fourier transform X(ω), then X(t) has transform 2π x(−ω); for even x(t), X(t) ↔ 2π x(ω). Proof: from the inverse transform, 2π x(−t) = ∫ X(ω) e^{−jωt} dω; exchanging variable names t and ω gives 2π x(−ω) = ∫ X(t) e^{−jωt} dt.

## Answer

#### Statement
If $x(t) \xleftrightarrow{F} X(\omega)$, then a time function with the same shape as $X$ has a transform with the shape of $x$:
$$
X(t) \xleftrightarrow{F} 2\pi\,x(-\omega)
$$
For an even $x(t)$ this simplifies to $X(t) \leftrightarrow 2\pi x(\omega)$.

#### Proof
Start from the inverse transform:
$$
x(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty}X(\omega)e^{j\omega t}d\omega \;\Rightarrow\; 2\pi x(t) = \int_{-\infty}^{\infty}X(\omega)e^{j\omega t}d\omega
$$
Replace $t$ by $-t$:
$$
2\pi x(-t) = \int_{-\infty}^{\infty}X(\omega)e^{-j\omega t}d\omega
$$
Now exchange the names of the variables $t$ and $\omega$:
$$
2\pi x(-\omega) = \int_{-\infty}^{\infty}X(t)e^{-j\omega t}dt = F\{X(t)\}
$$
This proves $X(t) \leftrightarrow 2\pi x(-\omega)$.

#### Example
A rectangular pulse $x(t) = 1$ for $|t| < T_1$ has $X(\omega) = \frac{2\sin\omega T_1}{\omega}$. By duality, with $x$ even:
$$
\frac{2\sin T_1 t}{t} \xleftrightarrow{F} 2\pi\,x(\omega) = \begin{cases}2\pi, & |\omega| < T_1\\ 0, & |\omega| > T_1\end{cases}
$$
Dividing by $2\pi$ and writing $W$ for $T_1$:
$$
\frac{\sin Wt}{\pi t} \xleftrightarrow{F} \begin{cases}1, & |\omega| < W\\ 0, & |\omega| > W\end{cases}
$$
A sinc pulse in time is an ideal low-pass spectrum. Similarly, $\delta(t) \leftrightarrow 1$ gives $1 \leftrightarrow 2\pi\delta(\omega)$.

## Check yourself

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

1. If x(t) <-> X(ω), the duality property states that X(t) <-> ?
   A. x(ω)
   B. 2π x(-ω)
   C. x(-ω)/2π
   D. 2π x(ω)
   Answer: B. A time function shaped like X has a transform shaped like x, reversed and scaled by 2π: 2π x(-ω). (Key point: Duality statement)

2. What is the starting point of the duality proof?
   A. The Laplace transform of x(t)
   B. The inverse Fourier transform expression 2π x(t) = ∫X(ω)e^(jωt)dω
   C. The convolution property
   D. Parseval's relation
   Answer: B. The inverse transform is rewritten, replace t by -t, then swap the variable names t and ω to get the forward transform of X(t). (Key point: Starting point of proof)

3. After writing 2π x(-t) = ∫X(ω)e^(-jωt)dω, what step completes the proof?
   A. Differentiating both sides
   B. Exchanging the names of t and ω to get 2π x(-ω) = ∫X(t)e^(-jωt)dt
   C. Taking the complex conjugate
   D. Scaling time by 2π
   Answer: B. Interchanging the symbols turns the right side into the Fourier transform integral of X(t). (Key point: Role of variable swap)

4. The rectangular pulse (|t| < T1) has X(ω) = 2 sin(ωT1)/ω. By duality, what is the transform of sin(Wt)/(πt)?
   A. A impulse at ω = W
   B. A rectangle equal to 1 for |ω| < W and 0 elsewhere
   C. A triangle of width 2W
   D. A constant 2π
   Answer: B. A sinc pulse in time maps to an ideal low-pass rectangular spectrum of cutoff W. (Key point: Duality example)

## Close variants set in other exams

- [Time shifting property of CTFT](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actft-time-shifting) — Signals and Systems, chapter 3, asked 5×

## Related in Signals and Systems

- [CTFT convolution property](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actft-convolution-property) — Signals and Systems, chapter 3, asked 12×
- [Frequency shifting property of CTFT](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-frequency-shifting) — Signals and Systems, chapter 3, asked 3×
- [FT of rectangular pulse and constant; duality](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-rect-pulse-constant-duality) — Signals and Systems, chapter 3
- [Time-scaling property of CTFS](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actfs-time-scaling) — Signals and Systems, chapter 2, asked 4×
- [CTFT linearity and time shifting](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actft-linearity-time-shifting) — Signals and Systems, chapter 3
- [Time-scaling property of DTFS](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtfs-time-scaling) — Signals and Systems, chapter 2, asked 6×

## Related in other subjects

- [Fourier transform property: Duality](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aft-property-duality) — Communication Systems, chapter 1
- [Fourier transform property: Time shifting](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aft-property-time-shifting) — Communication Systems, chapter 1
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- [Fourier transform property: Scaling](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aft-property-scaling) — Communication Systems, chapter 1