---
title: "Fourier transform of e^(-a|t|)"
subject: "Data Communication"
chapter: 3
key: "semester_5/data_communication:ft-exp-abs"
source: https://www.bctengineer.com/q/semester_5/data_communication/ft-exp-abs
site: BCT Engineering
---

# Fourier transform of e^(-a|t|)

Data Communication · Chapter 3: Signal Analysis

- Asked in: 2071 Chaitra, 4 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/data_communication/ft-exp-abs
- Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-3

## Question

Find the Fourier transform of the signal x(t) = e^(-a|t|), where (0 < a < ∞) is real-valued and |t| denotes the absolute value of (t).

## Short answer

For x(t)=e^{−a|t|}, the Fourier transform is X(ω)=2a/(a²+ω²). The spectrum is real, even, and positive, peaking at 2/a at ω=0.

## Answer

x(t) = e^(-a|t|), a > 0, which is e^(at) for t < 0 and e^(-at) for t ≥ 0.

X(ω) = ∫_{-∞}^{0} e^(at) e^(-jωt) dt + ∫₀^∞ e^(-at) e^(-jωt) dt

First integral = ∫_{-∞}^0 e^((a - jω)t) dt = 1/(a - jω)

Second integral = ∫₀^∞ e^(-(a + jω)t) dt = 1/(a + jω)

X(ω) = 1/(a - jω) + 1/(a + jω) = [(a + jω) + (a - jω)] / (a² + ω²)

**X(ω) = 2a / (a² + ω²)**

The spectrum is real, even and positive (as the signal is real and even), with a peak 2/a at ω = 0 and falling to zero as ω → ∞.

## Check yourself

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

1. The integral is split at
   A. t = a
   B. t = 1
   C. t = 0
   D. ω = 0
   Answer: C. |t| changes definition. (Key point: Method)

2. FT of e^(-a|t|) is
   A. 1/(a+jω)
   B. a/(a^2+ω^2)
   C. 2/(a+ω)
   D. 2a/(a^2+ω^2)
   Answer: D. Sum of both parts. (Key point: Result)

3. The spectrum is
   A. Real and even
   B. Imaginary and odd
   C. Complex
   D. Zero
   Answer: A. x is real and even. (Key point: Nature)

4. Peak value at ω=0 is
   A. 1/a
   B. 2/a
   C. 2a
   D. a
   Answer: B. 2a/a^2. (Key point: Peak)

## Related in Data Communication

- [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 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
- [Why the Fourier transform is needed](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Awhy-fourier-transform-needed) — Data Communication, chapter 3, asked 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×
- [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×

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- [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×
- [Fourier transform of e^(−2|t−1|)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-shifted-two-sided-exponential) — Signals and Systems, chapter 3, asked 4×