---
title: "Fourier transform property: Convolution"
subject: "Communication Systems"
chapter: 1
key: "semester_6/communication_systems:ft-property-convolution"
source: https://www.bctengineer.com/q/semester_6/communication_systems/ft-property-convolution
site: BCT Engineering
---

# Fourier transform property: Convolution

Communication Systems · Chapter 1: Introduction

- Asked in: 2067 Mangsir (CS I), 2 marks
- Video: [Fourier Transform Properties CONVOLUTION](https://www.youtube.com/watch?v=7ApZ8cjNfSI) — Francesco Fedele
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/ft-property-convolution
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1

## Question

State and explain the convolution property of Fourier Transform.

## Short answer

The convolution property states that time-domain convolution corresponds to frequency-domain multiplication: g₁(t)*g₂(t) ↔ G₁(f)G₂(f). Conversely, time-domain multiplication corresponds to frequency-domain convolution: g₁(t)g₂(t) ↔ G₁(f)*G₂(f). This simplifies LTI system analysis since output Y(f)=X(f)H(f), and explains modulation/sampling spectra.

## Answer

**Statement:** if $g_1(t)\leftrightarrow G_1(f)$ and $g_2(t)\leftrightarrow G_2(f)$, then

$$
g_1(t)*g_2(t)\leftrightarrow G_1(f)\,G_2(f),\qquad g_1(t)\,g_2(t)\leftrightarrow G_1(f)*G_2(f)
$$

where $g_1(t)*g_2(t)=\int_{-\infty}^{\infty}g_1(\tau)g_2(t-\tau)d\tau$.

**Explanation:** convolution in the time domain becomes simple multiplication in the frequency domain (time-convolution theorem), and multiplication in time becomes convolution in frequency (multiplication theorem).

**Significance:**
- The output of an LTI system $y(t)=x(t)*h(t)$ is found easily as $Y(f)=X(f)H(f)$; filtering is just multiplying spectra.
- Multiplication of a message by a carrier or by a sampling pulse train corresponds to convolving the spectra, explaining modulation and sampling spectra.

## Check yourself

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

1. g1(t)*g2(t) corresponds to
   A. G1(f)*G2(f)
   B. G1(f)G2(f)
   C. G1(f)+G2(f)
   D. G1(f)/G2(f)
   Answer: B. Convolution in time becomes multiplication in frequency. (Key point: Time convolution theorem)

2. g1(t)g2(t) corresponds to
   A. G1(f)G2(f)
   B. G1(f)*G2(f)
   C. G1(f)-G2(f)
   D. G1(-f)G2(f)
   Answer: B. Multiplication in time becomes convolution in frequency. (Key point: Multiplication theorem)

3. Why is the convolution property useful for LTI systems?
   A. It lets Y(f) = X(f)H(f) replace the convolution integral
   B. It makes all systems causal
   C. It removes noise
   D. It converts H(f) to a delta function
   Answer: A. Filtering becomes simple multiplication of the input spectrum by the transfer function. (Key point: Application to LTI filtering)

## Related in Communication Systems

- [Fourier transform property: Modulation](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aft-property-modulation) — 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
- [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: Scaling](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aft-property-scaling) — Communication Systems, chapter 1
- [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 properties; Rayleigh energy theorem](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Afourier-properties-rayleigh) — Communication Systems, chapter 1

## Related in other subjects

- [Derive convolution property of z-transform](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Az-convolution-property-derive) — Digital Signal Analysis and Processing, chapter 2
- [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×
- [DTFT convolution property](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtft-convolution-property) — Signals and Systems, chapter 3, asked 12×
- [DTFT properties; convolution property](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtft-properties-convolution) — Signals and Systems, chapter 3, asked 2×