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

# Fourier transform property: Scaling

Communication Systems · Chapter 1: Introduction

- Asked in: 2067 Mangsir (CS I), 2 marks
- 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-scaling
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1

## Question

State and explain the scaling property of Fourier Transform.

## Short answer

The scaling property states that if g(t) ↔ G(f), then g(at) ↔ (1/|a|)G(f/a). Time compression (|a|>1) expands the spectrum in frequency with reduced amplitude, while time stretching (|a|<1) shrinks the spectrum. The 1/|a| factor preserves energy. For a=-1, time reversal gives g(-t) ↔ G(-f).

## Answer

**Statement:** if $g(t)\leftrightarrow G(f)$, then for a real constant $a$

$$
g(at)\leftrightarrow\frac{1}{\lvert a\rvert}G\left(\frac{f}{a}\right)
$$

**Explanation:** if $\lvert a\rvert>1$ the signal is **compressed** in time, and its spectrum is **expanded** in frequency (with reduced height); if $\lvert a\rvert<1$ the signal is stretched and its spectrum shrinks. The factor $1/\lvert a\rvert$ keeps the energy relation consistent. For $a=-1$, $g(-t)\leftrightarrow G(-f)$ (time reversal).

**Example:** a rectangular pulse of width $T$ has spectrum $T\,\text{sinc}(fT)$ with first null at $1/T$. Halving the width to $T/2$ moves the first null to $2/T$, doubling the bandwidth.

**Significance:** this is the inverse relation between pulse duration and bandwidth: faster data (shorter pulses) needs more bandwidth. Playing a tape at double speed raises all pitches by an octave.

## Check yourself

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

1. g(at) has Fourier transform
   A. a G(f/a)
   B. (1/|a|) G(f/a)
   C. G(af)
   D. (1/|a|) G(af)
   Answer: B. Time scaling by a scales frequency by 1/a with amplitude factor 1/|a|. (Key point: Scaling statement)

2. If a signal is compressed in time (|a| > 1), its spectrum
   A. Is compressed
   B. Is expanded with reduced height
   C. Is unchanged
   D. Is shifted in frequency
   Answer: B. Narrower pulses have wider spectra: duration and bandwidth are inversely related. (Key point: Compression and bandwidth)

3. A rectangular pulse of width T has first null at 1/T. If the width is halved to T/2, the first null moves to
   A. 1/(2T)
   B. 1/T
   C. 2/T
   D. 4/T
   Answer: C. Halving the width doubles the bandwidth, so the first null is at 2/T. (Key point: Rect pulse example)

## 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: 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

## Related in other subjects

- [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×
- [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×
- [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
- [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