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

# Fourier transform property: Time shifting

Communication Systems · Chapter 1: Introduction

- Asked in: 2067 Mangsir (CS I), 2 marks
- Video: [Fourier Transform - Time Shifting Property](https://www.youtube.com/watch?v=ITrHCSmQ4iw) — TutorialsPoint
- 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-time-shifting
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1

## Question

State and explain the time shifting property of Fourier Transform.

## Short answer

Time shifting property: g(t-t₀) ↔ G(f)e^{-j2πft₀}. Proof by substitution shows amplitude spectrum unchanged, only linear phase shift added. Higher frequencies get more phase shift so all components delay equally. Distortionless transmission requires channel with linear phase response.

## Answer

**Statement:** if $g(t)\leftrightarrow G(f)$, then delaying the signal by $t_0$ gives

$$
g(t-t_0)\leftrightarrow G(f)\,e^{-j2\pi ft_0}
$$

**Proof:** put $\lambda=t-t_0$ in $\int g(t-t_0)e^{-j2\pi ft}dt=\int g(\lambda)e^{-j2\pi f(\lambda+t_0)}d\lambda=e^{-j2\pi ft_0}G(f)$.

**Explanation:** a time shift does not change the amplitude spectrum, $\lvert G(f)e^{-j2\pi ft_0}\rvert=\lvert G(f)\rvert$; it only adds a phase $-2\pi ft_0$ that is linear in frequency. Higher frequencies get more phase shift so that all components are delayed by the same time.

**Significance:** this gives the distortionless transmission condition: a channel with $H(f)=Ke^{-j2\pi ft_d}$ (linear phase) only delays the signal without changing its shape.

## Check yourself

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

1. g(t - t0) has Fourier transform
   A. G(f - t0)
   B. G(f)e^{-j2πf t0}
   C. G(f)e^{+j2πf t0}
   D. G(f/t0)
   Answer: B. Delay multiplies the spectrum by e^{-j2πf t0}. (Key point: Time shifting statement)

2. A time shift changes the spectrum by
   A. Scaling its magnitude by t0
   B. Adding a phase linear in frequency while |G(f)| is unchanged
   C. Shifting it to a new frequency
   D. Removing high frequencies
   Answer: B. |e^{-j2πf t0}| = 1, so only a linear phase -2πf t0 is added. (Key point: Effect on amplitude spectrum)

3. Which channel response only delays a signal by td without changing shape?
   A. H(f) = K
   B. H(f) = K e^{-j2πf td}
   C. H(f) = K f
   D. H(f) = K e^{-f^2}
   Answer: B. Constant gain with linear phase delays all components by the same time. (Key point: Distortionless transmission)

## 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: 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

## Related in other subjects

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