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

# CTFT linearity and time shifting

Signals and Systems · Chapter 3: Fourier Transform

- Asked in: 2070 Bhadra, 4 marks
- Video: [Properties of Fourier Transform (Part 1)](https://www.youtube.com/watch?v=0USI-48ovJI) — Neso Academy
- 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-linearity-time-shifting
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-3

## Question

Explain the linearity and time shifting properties of continuous time Fourier transform.

## Short answer

Linearity: a x₁(t) + b x₂(t) ↔ a X₁(jω) + b X₂(jω). Time shifting: x(t – t₀) ↔ e^{-jω t₀} X(jω); delay adds linear phase –ω t₀ but does not change the magnitude spectrum.

## Answer

#### Linearity property

If $x_1(t) \leftrightarrow X_1(j\omega)$ and $x_2(t) \leftrightarrow X_2(j\omega)$, then for any constants $a$ and $b$:

$$
a\,x_1(t) + b\,x_2(t) \;\longleftrightarrow\; a\,X_1(j\omega) + b\,X_2(j\omega)
$$

**Proof:** $\int[a x_1(t) + b x_2(t)]e^{-j\omega t}dt = a\int x_1e^{-j\omega t}dt + b\int x_2e^{-j\omega t}dt = aX_1 + bX_2$, because integration is linear.

**Meaning:** The FT of a sum is the sum of the FTs; superposition holds. Example: $\cos\omega_0t = \frac{1}{2}(e^{j\omega_0t} + e^{-j\omega_0t}) \leftrightarrow \pi[\delta(\omega-\omega_0) + \delta(\omega+\omega_0)]$.

#### Time shifting property

$$
x(t - t_0) \;\longleftrightarrow\; e^{-j\omega t_0}\,X(j\omega)
$$

**Proof:** Put $\tau = t - t_0$:

$$
\int_{-\infty}^{\infty}x(t-t_0)e^{-j\omega t}dt = \int_{-\infty}^{\infty}x(\tau)e^{-j\omega(\tau+t_0)}d\tau = e^{-j\omega t_0}X(j\omega)
$$

**Meaning:** A delay does not change the magnitude spectrum, $|X(j\omega)|$; it only adds a linear phase $-\omega t_0$. Example: $\delta(t) \leftrightarrow 1$, so $\delta(t-2) \leftrightarrow e^{-j2\omega}$ (magnitude 1, phase $-2\omega$).

## Check yourself

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

1. What does linearity of the FT state?
   A. x1 x2 <-> X1 X2
   B. a x1 + b x2 <-> X1 X2
   C. a x1 + b x2 <-> a b (X1 + X2)
   D. a x1 + b x2 <-> a X1 + b X2
   Answer: D. The integral is linear, so superposition holds. (Key point: Linearity)

2. What is the FT of x(t - t0)?
   A. e^(-jωt0) X(jω)
   B. e^(jωt0) X(jω)
   C. X(ω - t0)
   D. t0 X(jω)
   Answer: A. A delay multiplies by e^(-jωt0). (Key point: Time shift)

3. What does a delay change?
   A. Magnitude only
   B. Only phase, by -ωt0
   C. Both magnitude and phase
   D. Neither
   Answer: B. |e^(-jωt0)| = 1. (Key point: Effect on spectrum)

## Related in Signals and Systems

- [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×
- [DTFT linearity and time shifting](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtft-linearity-time-shift) — Signals and Systems, chapter 3
- [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×
- [Properties of continuous time systems](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Act-system-properties) — Signals and Systems, chapter 5, asked 2×
- [CTFS shifting properties; coefficients of A cos ω₀t](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actfs-shifting-cosine-coefficients) — Signals and Systems, chapter 2
- [CTFS time shifting and time scaling](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actfs-shift-scale-properties) — Signals and Systems, chapter 2, asked 2×

## Related in other subjects

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- [Properties of continuous-time Fourier series](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Afourier-series-properties) — Data Communication, chapter 3
- [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
- [Properties of the Fourier transform](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Afourier-transform-properties) — Data Communication, chapter 3, asked 2×