---
title: "CTFT conjugation and conjugate symmetry"
subject: "Signals and Systems"
chapter: 3
key: "semester_4/signals_and_systems:ctft-conjugation-symmetry"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/ctft-conjugation-symmetry
site: BCT Engineering
---

# CTFT conjugation and conjugate symmetry

Signals and Systems · Chapter 3: Fourier Transform

- Asked in: 2079 Jestha, 4+4 marks; 2072 Asoj, 4+2 marks (asked 2 times)
- Video: [Properties of Fourier Transform: Conjugation & Conjugate Symmetry with example](https://www.youtube.com/watch?v=yJh-m_yek7M) — PMRF Teachings
- 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-conjugation-symmetry
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-3

## Question

State and prove the complex conjugation property for continuous time aperiodic signals. Also show that if the CT signal is purely real, its Fourier Transform is conjugate symmetric.

## Short answer

Conjugation property: x*(t) ↔ X*(−ω). Proof: conjugate definition to get X*(ω) = ∫ x*(t)e^{jωt} dt, replace ω by −ω. For real x(t), X(−ω) = X*(ω), i.e., conjugate symmetric: real part even, imaginary part odd, magnitude even, phase odd.

## Answer

#### Conjugation property
**Statement:** if $x(t) \xleftrightarrow{F} X(\omega)$, then
$$
x^*(t) \xleftrightarrow{F} X^*(-\omega)
$$
**Proof:** start from the definition and take the conjugate:
$$
X(\omega) = \int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt \;\Rightarrow\; X^*(\omega) = \int_{-\infty}^{\infty}x^*(t)e^{j\omega t}dt
$$
Replace $\omega$ by $-\omega$:
$$
X^*(-\omega) = \int_{-\infty}^{\infty}x^*(t)e^{-j\omega t}dt = F\{x^*(t)\}
$$
So the Fourier transform of $x^*(t)$ is $X^*(-\omega)$.

#### Conjugate symmetry for real signals
**Statement:** if $x(t)$ is real, then
$$
X(-\omega) = X^*(\omega)
$$
**Proof:** a real signal satisfies $x(t) = x^*(t)$. Taking the Fourier transform of both sides and using the conjugation property:
$$
X(\omega) = X^*(-\omega)
$$
Replacing $\omega$ by $-\omega$ gives $X(-\omega) = X^*(\omega)$.

The same result follows directly:
$$
X(-\omega) = \int x(t)e^{j\omega t}dt = \left[\int x(t)e^{-j\omega t}dt\right]^* = X^*(\omega) \quad (x \text{ real})
$$

#### Consequences
Write $X(\omega) = \mathrm{Re}\{X(\omega)\} + j\,\mathrm{Im}\{X(\omega)\} = |X(\omega)|e^{j\angle X(\omega)}$. Then for real $x(t)$:

| Quantity | Symmetry |
|---|---|
| $\mathrm{Re}\{X(\omega)\}$ | even in $\omega$ |
| $\mathrm{Im}\{X(\omega)\}$ | odd in $\omega$ |
| $\lvert X(\omega)\rvert$ | even in $\omega$ |
| $\angle X(\omega)$ | odd in $\omega$ |

Further, if $x(t)$ is real and even, $X(\omega)$ is real and even. If $x(t)$ is real and odd, $X(\omega)$ is purely imaginary and odd. Because of this symmetry, only the positive frequencies need to be plotted for real signals.

#### Example
$x(t) = e^{-at}u(t)$ ($a > 0$) is real, and $X(\omega) = \frac{1}{a + j\omega}$:
$$
X(-\omega) = \frac{1}{a - j\omega} = \left[\frac{1}{a + j\omega}\right]^* = X^*(\omega)
$$
$|X(\omega)| = 1/\sqrt{a^2 + \omega^2}$ is even, and $\angle X(\omega) = -\tan^{-1}(\omega/a)$ is odd.

## Check yourself

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

1. If x(t) <-> X(ω), the Fourier transform of x*(t) is:
   A. X*(ω)
   B. X*(-ω)
   C. X(-ω)
   D. -X*(ω)
   Answer: B. Conjugating the definition gives X*(ω) with e^(+jωt); replacing ω by -ω gives the transform of x*(t) as X*(-ω). (Key point: Conjugation property)

2. For a real signal x(t), which relation holds?
   A. X(-ω) = -X(ω)
   B. X(-ω) = X*(ω)
   C. X(-ω) = X(ω) only
   D. X*(ω) = -X(-ω)
   Answer: B. Real means x = x*, so X(ω) = X*(-ω), i.e. X(-ω) = X*(ω): conjugate symmetry. (Key point: Real signal symmetry)

3. For real x(t), which pair of symmetries is correct?
   A. Re X even, Im X even
   B. Re X odd, Im X even
   C. Re X even, Im X odd
   D. |X| odd, phase even
   Answer: C. Conjugate symmetry makes the real part and magnitude even, and the imaginary part and phase odd. (Key point: Even and odd parts)

4. If x(t) is real and odd, X(ω) is:
   A. Real and even
   B. Purely imaginary and odd
   C. Real and odd
   D. Purely imaginary and even
   Answer: B. A real odd signal has only the sine part in the transform, giving a purely imaginary odd spectrum. (Key point: Real odd signal)

## Related in Signals and Systems

- [Fourier transform of CT periodic signals](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-periodic-signal-derivation) — Signals and Systems, chapter 3, asked 3×
- [Parseval's theorem for CT aperiodic signals](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Act-parseval-aperiodic) — Signals and Systems, chapter 3, asked 5×
- [Fourier transform of a CT signal](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-unspecified-ct-signal) — Signals and Systems, chapter 3
- [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×
- [Harmonic exponentials; CTFS conjugate symmetry](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aharmonic-exponentials-conjugation) — Signals and Systems, chapter 2
- [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×

## Related in other subjects

- [Fourier transform property: Time shifting](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aft-property-time-shifting) — Communication Systems, chapter 1
- [Fourier transform property: Duality](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aft-property-duality) — Communication Systems, chapter 1
- [Z-transform of ramp-and-hold continuous signal](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Awaveform-ramp-to-1-at-6) — Digital Control System, chapter 2
- [Derive time shifting property of z-transform](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Az-time-shifting-derive) — Digital Signal Analysis and Processing, chapter 2