--- title: "FT of e^(−at)u(t); amplitude and phase" subject: "Signals and Systems" chapter: 3 key: "semester_4/signals_and_systems:ft-one-sided-exponential" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/ft-one-sided-exponential site: BCT Engineering --- # FT of e^(−at)u(t); amplitude and phase Signals and Systems · Chapter 3: Fourier Transform - Asked in: 2076 Bhadra, 6+2 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-one-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^(−at)u(t), where a is real constant. Also, draw its amplitude and phase spectra. ## Short answer For a > 0, the Fourier transform is 1/(a + jω), with magnitude 1/sqrt(a² + ω²) and phase -arctan(ω/a). The amplitude spectrum is even, the phase spectrum is odd, and the 3 dB bandwidth is ω = a. ## Answer For the transform to exist, $a > 0$ (otherwise $e^{-at}u(t)$ grows and is not absolutely integrable). #### Fourier transform $$ \begin{aligned} X(j\omega) &= \int_{-\infty}^{\infty} e^{-at}u(t)\,e^{-j\omega t}\,dt = \int_{0}^{\infty} e^{-(a+j\omega)t}\,dt \\ &= \left[\frac{e^{-(a+j\omega)t}}{-(a+j\omega)}\right]_{0}^{\infty} \\ &= 0 - \frac{1}{-(a+j\omega)} = \frac{1}{a+j\omega}, \quad a > 0 \end{aligned} $$ The upper limit is zero because $|e^{-(a+j\omega)t}| = e^{-at} \to 0$ as $t \to \infty$. **Magnitude and phase:** $$ |X(j\omega)| = \frac{1}{\sqrt{a^2+\omega^2}}, \qquad \angle X(j\omega) = -\tan^{-1}\left(\frac{\omega}{a}\right) $$ | $\omega$ | $\lvert X\rvert$ | Phase | |---|---|---| | $0$ | $1/a$ | $0$ | | $\pm a$ | $\frac{1}{\sqrt2\,a} = 0.707/a$ | $\mp 45^\circ$ | | $\to \pm\infty$ | $\to 0$ | $\to \mp 90^\circ$ | #### Amplitude and phase spectra ```text |X(jw)| 1/a * * | * 0.707/a *.........* (half-power points, w = +/- a) * | * * | * --------------+--------------> w -a 0 a phase +90 | ...... +45 | . 0 +--------.---------> w | 0 . -45 | . -90 | ...... ``` - The amplitude spectrum is even and the phase spectrum is odd, as expected for a real signal. - The spectrum is low-pass, with 3 dB bandwidth $\omega = a$ rad/s. Faster decay (larger $a$) gives a wider bandwidth. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why must a > 0 for the FT of e^(-at)u(t)? A. Otherwise the signal is non-causal B. Otherwise u(t) is undefined C. Otherwise the FT is purely imaginary D. Otherwise the signal grows and is not absolutely integrable Answer: D. For a ≤ 0 the signal does not decay, so the Fourier integral does not converge. (Key point: Existence) 2. What is X(jω)? A. 1/(a + jω) B. 1/(a - jω) C. a/(a^2 + ω^2) D. 1/(a^2 + ω^2) Answer: A. ∫0^∞ e^(-(a+jω)t) dt = 1/(a+jω), the upper limit vanishing because e^(-at) → 0. (Key point: Transform) 3. What are |X| and the phase at ω = a? A. 1/a and 0° B. 0.707/a and -45° C. 0.5/a and -90° D. 1/(2a) and +45° Answer: B. |X| = 1/sqrt(a^2 + a^2) = 0.707/a and phase = -tan^-1(1) = -45°. (Key point: Spectra values) 4. Which symmetry holds for this real signal's spectra? A. Magnitude odd, phase even B. Both even C. Magnitude even, phase odd D. Both odd Answer: C. For any real x(t), |X(jω)| is even and ∠X(jω) is odd. (Key point: Symmetry) ## 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 - [FT of rectangular pulse and constant; duality](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-rect-pulse-constant-duality) — Signals and Systems, chapter 3 - [FT of periodic signals; rect pulse and constant](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-periodic-rect-constant) — Signals and Systems, chapter 3 - [Fourier transform of a trapezoidal pulse](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-trapezoidal-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 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× ## Related in other subjects - [Fourier transform of e^(-at)u(t) and spectra](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aft-exp-causal-spectra) — Data Communication, chapter 3, asked 2× - [Fourier transform of a DC signal](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aft-dc-signal) — Data Communication, chapter 3 - [Magnitude and phase spectra of a sum of sinusoids](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Amag-phase-5-sin12t) — 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