--- title: "Frequency shifting property of CTFT" subject: "Signals and Systems" chapter: 3 key: "semester_4/signals_and_systems:ft-frequency-shifting" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/ft-frequency-shifting site: BCT Engineering --- # Frequency shifting property of CTFT Signals and Systems · Chapter 3: Fourier Transform - Asked in: 2074 Bhadra, 2+3 marks; 2073 Bhadra, 3 marks (asked 2 times) - Video: [Fourier Transform - Frequency Shifting Property](https://www.youtube.com/watch?v=FZBZqTP6T-o) — TutorialsPoint - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/ft-frequency-shifting - Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-3 ## Question State and prove frequency shifting property of the continuous time Fourier transform. ## Short answer Frequency shifting: e^{jω₀t} x(t) ↔ X(ω−ω₀). Proof: ∫ x(t)e^{jω₀t} e^{−jωt} dt = ∫ x(t)e^{−j(ω−ω₀)t} dt = X(ω−ω₀). Modulation: x(t) cos(ω₀t) ↔ ½[X(ω−ω₀)+X(ω+ω₀)]. ## Answer #### Statement If $x(t) \xleftrightarrow{F} X(\omega)$, then $$ e^{j\omega_0 t}x(t) \xleftrightarrow{F} X(\omega - \omega_0) $$ Multiplying a signal by a complex exponential shifts its spectrum by $\omega_0$. #### Proof $$ F\{e^{j\omega_0 t}x(t)\} = \int_{-\infty}^{\infty}x(t)e^{j\omega_0 t}e^{-j\omega t}dt = \int_{-\infty}^{\infty}x(t)e^{-j(\omega - \omega_0)t}dt = X(\omega - \omega_0) $$ The last integral is the definition of $X$, evaluated at $\omega - \omega_0$. #### Application: modulation Since $\cos\omega_0 t = \frac{1}{2}(e^{j\omega_0 t} + e^{-j\omega_0 t})$: $$ x(t)\cos\omega_0 t \xleftrightarrow{F} \frac{1}{2}\left[X(\omega - \omega_0) + X(\omega + \omega_0)\right] $$ This is amplitude modulation: the baseband spectrum is copied to $\pm\omega_0$ with half the height. ```text X(w) FT of x(t) cos(w0 t) 1 1/2 1/2 /\ /\ /\ / \ / \ / \ -+--+-- w -----+----+--0--+----+---- w -W W -w0 w0 ``` #### Examples - With $x(t) = 1$ (so $X(\omega) = 2\pi\delta(\omega)$): $e^{j\omega_0 t} \leftrightarrow 2\pi\delta(\omega - \omega_0)$, and therefore $\cos\omega_0 t \leftrightarrow \pi[\delta(\omega - \omega_0) + \delta(\omega + \omega_0)]$. - With $x(t) = e^{-at}u(t)$: $e^{-at}e^{j\omega_0 t}u(t) \leftrightarrow \frac{1}{a + j(\omega - \omega_0)}$, a spectrum whose peak has moved from $\omega = 0$ to $\omega = \omega_0$. **Dual of time shifting:** a shift in time multiplies the spectrum by $e^{-j\omega t_0}$. A shift in frequency multiplies the signal by $e^{j\omega_0 t}$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. If x(t) <-> X(ω), the transform of e^(jω0 t) x(t) is: A. X(ω + ω0) B. X(ω - ω0) C. e^(-jωω0) X(ω) D. X(ω) e^(jω0) Answer: B. Multiplying by e^(jω0 t) shifts the spectrum to the right by ω0, giving X(ω - ω0). (Key point: Statement) 2. In the proof, how does X(ω - ω0) appear? A. The exponent combines to e^(-j(ω - ω0)t) inside the transform integral B. By differentiating with respect to ω C. By changing the integration limits D. By using the convolution theorem Answer: A. e^(jω0 t)e^(-jωt) = e^(-j(ω-ω0)t), and the integral is the definition of X evaluated at ω - ω0. (Key point: Proof step) 3. What is the transform of x(t)cos(ω0 t)? A. X(ω - ω0) + X(ω + ω0) B. (1/2)[X(ω - ω0) + X(ω + ω0)] C. (1/2)[X(ω - ω0) - X(ω + ω0)] D. X(ω) cos(ω0) Answer: B. cos is half the sum of two exponentials, so the spectrum is copied to ±ω0 with half the height. (Key point: Modulation) ## Close variants set in other exams - [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× ## Related in Signals and Systems - [Frequency shifting property of DTFS](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtfs-frequency-shifting) — Signals and Systems, chapter 2, asked 3× - [CTFT linearity and time shifting](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actft-linearity-time-shifting) — Signals and Systems, chapter 3 - [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× - [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× - [Duality property of CTFT](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actft-duality) — Signals and Systems, chapter 3, asked 3× - [DTFT conjugation and frequency shifting](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtft-conjugate-frequency-shift) — Signals and Systems, chapter 3 ## Related in other subjects - [Fourier transform property: Time shifting](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aft-property-time-shifting) — Communication Systems, chapter 1 - [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 - [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