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

# Fourier transform property: Modulation

Communication Systems · Chapter 1: Introduction

- Asked in: 2067 Mangsir (CS I), 2 marks
- Video: [Fourier Transform Modulation Property | Modulation Property and its Mathematical Proof](https://www.youtube.com/watch?v=8kBKFBhBKBQ) — Muhammed Mustaqim
- 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-modulation
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1

## Question

State and explain the modulation property of Fourier Transform.

## Short answer

Modulation property: g(t)cos(2πfct) ↔ ½[G(f-fc)+G(f+fc)]. Cosine expressed as sum of exponentials shifts spectrum by ±fc with half amplitude. This is the basis of amplitude modulation, moving baseband message to carrier band for transmission and back at receiver.

## Answer

**Statement:** if $g(t)\leftrightarrow G(f)$, then multiplying $g(t)$ by a sinusoidal carrier shifts its spectrum to $\pm f_c$:

$$
g(t)\cos(2\pi f_ct)\leftrightarrow\frac{1}{2}\left[G(f-f_c)+G(f+f_c)\right]
$$

**Explanation:** write $\cos2\pi f_ct=\frac{1}{2}(e^{j2\pi f_ct}+e^{-j2\pi f_ct})$. By the frequency-shift property, $g(t)e^{j2\pi f_ct}\leftrightarrow G(f-f_c)$, so each exponential term moves the whole spectrum by $\pm f_c$, and the factor $\frac{1}{2}$ halves its height.

**Significance:** this is the basis of amplitude modulation (DSB-SC). A baseband message of bandwidth $W$ is moved to a band $f_c-W$ to $f_c+W$; multiplying again by the carrier at the receiver moves it back (coherent demodulation).

## Check yourself

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

1. g(t)cos(2πfct) has Fourier transform
   A. G(f - fc) + G(f + fc)
   B. (1/2)[G(f - fc) + G(f + fc)]
   C. G(f)cos(2πfc f)
   D. (1/2)G(f - fc)
   Answer: B. The cosine splits into two exponentials, each shifting the spectrum by ±fc with half the height. (Key point: Modulation property statement)

2. The spectrum shift follows from
   A. Writing cos as a sum of complex exponentials and using the frequency-shift property
   B. Time scaling
   C. Differentiation in time
   D. Duality
   Answer: A. cos = (e^{j..}+e^{-j..})/2 and g(t)e^{j2πfct} gives G(f - fc). (Key point: Why the spectrum shifts)

3. A baseband message of bandwidth W multiplied by a carrier occupies
   A. 0 to W
   B. fc - W to fc + W
   C. 0 to fc
   D. W to 2W
   Answer: B. The message spectrum is moved to a band around the carrier, the basis of DSB-SC AM. (Key point: Significance)

## Related in Communication Systems

- [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
- [Fourier transform property: Convolution](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aft-property-convolution) — 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

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