--- title: "Fourier transform of e^(−2|t−1|)" subject: "Signals and Systems" chapter: 3 key: "semester_4/signals_and_systems:ft-shifted-two-sided-exponential" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/ft-shifted-two-sided-exponential site: BCT Engineering --- # Fourier transform of e^(−2|t−1|) Signals and Systems · Chapter 3: Fourier Transform - Asked in: 2072 Magh, 5 marks - Video: [Fourier Transform of Basic Signals (Exponential Signals)](https://www.youtube.com/watch?v=WjOlJ2lPZOg) — 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/ft-shifted-two-sided-exponential - Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-3 ## Question Find the Fourier transform of the signal x(t) = e^(−2|t−1|) ## Short answer First find FT of e^{-2|t|} = 4/(4+ω^2). By time-shift property, x(t)=e^{-2|t−1|} ↔ X(jω) = 4 e^{-jω}/(4+ω^2). ## Answer **Method:** Find the FT of $e^{-2|t|}$, then use the time-shifting property. **Step 1: FT of $x_1(t) = e^{-2|t|}$.** $$ \begin{aligned} X_1(j\omega) &= \int_{-\infty}^{0} e^{2t}e^{-j\omega t}dt + \int_{0}^{\infty} e^{-2t}e^{-j\omega t}dt \\ &= \frac{1}{2 - j\omega} + \frac{1}{2 + j\omega} \\ &= \frac{(2+j\omega) + (2-j\omega)}{4 + \omega^2} = \frac{4}{4 + \omega^2} \end{aligned} $$ **Step 2: time shift.** $x(t) = x_1(t-1)$, and $x_1(t - t_0) \leftrightarrow e^{-j\omega t_0}X_1(j\omega)$ with $t_0 = 1$: $$ X(j\omega) = \frac{4}{4 + \omega^2}\,e^{-j\omega} $$ **Answer:** $$ e^{-2|t-1|} \;\longleftrightarrow\; \frac{4\,e^{-j\omega}}{4 + \omega^2} $$ - Magnitude: $|X(j\omega)| = \frac{4}{4+\omega^2}$ (same as the unshifted signal; peak 1 at $\omega=0$, equal to the area under $x(t)$). - Phase: $\angle X(j\omega) = -\omega$ (linear phase due to a delay of 1 s). ```text x(t) |X(jw)| 1 ^ . 1 ^ . | . . | . . | . . | . . ..+' '.. ...|' '... ---+---+---+---> t ---+---+---+---> w 0 1 2 -2 0 2 ``` ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the method to find the FT of e^(-2|t-1|)? A. Apply frequency shifting with ω0 = 1 B. Apply time scaling by 2 C. Differentiate in time D. Find FT of e^(-2|t|), then apply time shifting with t0 = 1 Answer: D. x(t) = x1(t - 1) where x1 = e^(-2|t|). (Key point: Method) 2. What is the FT of e^(-2|t|)? A. 4/(4 + ω^2) B. 2/(4 + ω^2) C. 1/(2 + jω) D. 4/(2 + ω^2) Answer: A. 1/(2-jω) + 1/(2+jω) = 4/(4+ω^2). (Key point: Base transform) 3. What is X(jω)? A. 4 e^(jω)/(4 + ω^2) B. 4 e^(-jω)/(4 + ω^2) C. 4 e^(-j2ω)/(4 + ω^2) D. 4/(4+ω^2) shifted to ω = 1 Answer: B. A delay of 1 s multiplies by e^(-jω). (Key point: Final answer) 4. How do magnitude and phase compare with the unshifted signal? A. Magnitude halves, phase zero B. Magnitude same, phase +ω C. Magnitude same, phase is -ω D. Magnitude same, phase -2ω Answer: C. Delay by 1 s leaves |X| unchanged and adds -ω. (Key point: Phase and magnitude) ## Close variants set in other exams - [DTFT of aⁿu[n]](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtft-geometric-sequence) — Signals and Systems, chapter 3, asked 4× - [Fourier transform of e^(−at)cos ω₀t](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-damped-cosine) — Signals and Systems, chapter 3, asked 4× - [DTFT of cos ω₀n](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtft-cosine) — Signals and Systems, chapter 3, asked 4× ## Related in Signals and Systems - [Fourier transform of a CT signal](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-unspecified-ct-signal) — Signals and Systems, chapter 3 - [Fourier transform of e^(−a|t|)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-two-sided-exponential) — Signals and Systems, chapter 3, asked 5× - [Fourier transform of unit step](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-unit-step) — Signals and Systems, chapter 3, asked 3× - [Fourier transform of CT periodic signals](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-periodic-signal-derivation) — Signals and Systems, chapter 3, asked 3× - [Inverse Fourier transform of δ(ω)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Ainverse-ft-delta-omega) — Signals and Systems, chapter 3 - [Fourier series vs transform; DTFT of aⁿ](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Afs-vs-ft-dtft-geometric) — Signals and Systems, chapter 3 ## Related in other subjects - [Fourier transform of a DC signal](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aft-dc-signal) — Data Communication, chapter 3 - [Fourier transform of e^(-a|t|)](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aft-exp-abs) — Data Communication, chapter 3 - [Fourier transform of a ramp pulse](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aft-ramp-signal) — Data Communication, chapter 3 - [Fourier transform of e^(-a|t|)](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aft-exp-abs-ct) — Data Communication, chapter 3