--- title: "Time shifting property of CTFT" subject: "Signals and Systems" chapter: 3 key: "semester_4/signals_and_systems:ctft-time-shifting" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/ctft-time-shifting site: BCT Engineering --- # Time shifting property of CTFT Signals and Systems · Chapter 3: Fourier Transform - Asked in: 2075 Bhadra, 2+5 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_4/signals_and_systems/ctft-time-shifting - Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-3 ## Question State and prove time shifting property of continuous time Fourier Transform. ## Short answer If x(t) transforms to X(jω), then x(t - t0) transforms to e^(-jωt0) X(jω). Proof uses the substitution τ = t - t0; a time delay leaves magnitude unchanged and adds linear phase -ωt0. ## Answer **Statement:** If $x(t) \leftrightarrow X(j\omega)$, then a time-shifted signal has the transform $$ x(t - t_0) \;\longleftrightarrow\; e^{-j\omega t_0}\,X(j\omega) $$ A delay of $t_0$ seconds leaves the magnitude spectrum unchanged and adds a linear phase $-\omega t_0$. **Proof:** By definition, $$ \mathcal{F}\{x(t-t_0)\} = \int_{-\infty}^{\infty} x(t-t_0)\,e^{-j\omega t}\,dt $$ Put $\tau = t - t_0$, so $t = \tau + t_0$ and $dt = d\tau$; the limits stay $-\infty$ to $\infty$: $$ \begin{aligned} \mathcal{F}\{x(t-t_0)\} &= \int_{-\infty}^{\infty} x(\tau)\,e^{-j\omega(\tau+t_0)}\,d\tau \\ &= e^{-j\omega t_0}\int_{-\infty}^{\infty} x(\tau)\,e^{-j\omega\tau}\,d\tau \\ &= e^{-j\omega t_0}\,X(j\omega) \end{aligned} $$ Hence proved. Similarly, an advance gives $x(t+t_0) \leftrightarrow e^{j\omega t_0}X(j\omega)$. **Interpretation:** - $|e^{-j\omega t_0}X(j\omega)| = |X(j\omega)|$: shifting does not change which frequencies are present or their strengths. - $\angle = \angle X(j\omega) - \omega t_0$: each frequency component is delayed by the same time $t_0$, which needs a phase shift proportional to frequency (linear phase). This is why distortionless systems need linear phase. **Example:** $e^{-a|t|} \leftrightarrow \frac{2a}{a^2+\omega^2}$, so $e^{-a|t-2|} \leftrightarrow \frac{2a}{a^2+\omega^2}e^{-j2\omega}$. Likewise $\delta(t-t_0) \leftrightarrow e^{-j\omega t_0}$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the FT of x(t - t0)? A. e^(jωt0) X(jω) B. X(j(ω - t0)) C. e^(-jωt0) X(jω) / ω D. e^(-jωt0) X(jω) Answer: D. A delay by t0 multiplies the spectrum by e^(-jωt0). (Key point: Statement) 2. What substitution is used in the proof? A. τ = t - t0, so t = τ + t0 B. τ = t t0 C. τ = t/t0 D. τ = ω t0 Answer: A. With τ = t - t0 the exponent becomes e^(-jω(τ+t0)), letting e^(-jωt0) come out of the integral. (Key point: Proof substitution) 3. What does a time shift do to magnitude and phase? A. Magnitude scaled by t0, phase unchanged B. Magnitude unchanged, phase gets -ωt0 (linear) C. Magnitude unchanged, phase gets a constant shift D. Both unchanged Answer: B. |e^(-jωt0)| = 1, while its angle -ωt0 is linear in ω. (Key point: Effect on spectra) 4. Why does a delay need a phase proportional to frequency? A. So higher frequencies are attenuated more B. So the signal becomes causal C. So every component is delayed by the same time t0 D. So the energy is conserved Answer: C. A phase shift of -ωt0 corresponds to a time delay of t0 for each frequency, which is why distortionless systems need linear phase. (Key point: Why linear phase) ## Close variants set in other exams - [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× - [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× ## Related in Signals and Systems - [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× - [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× - [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× - [FS coefficients; time shifting and conjugation](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Afs-information-shift-conjugation) — Signals and Systems, chapter 2 ## 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