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

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

Data Communication · Chapter 3: Signal Analysis

- Asked in: 2069 Chaitra, 5 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-ct
- Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-3

## Question

Find the Fourier Transform of the continuous time signal x(t) = e^(-a|t|), for a > 0.

## Short answer

The continuous‑time signal x(t)=e^{−a|t|} has Fourier transform X(ω)=2a/(a²+ω²). Splitting the integral yields the sum of two terms, giving a real, even spectrum with maximum 2/a at zero frequency.

## Answer

x(t) = e^(-a|t|), a > 0.

X(ω) = ∫_{-∞}^{∞} e^(-a|t|) e^(-jωt) dt

Split at t = 0:

X(ω) = ∫_{-∞}^{0} e^(at) e^(-jωt) dt + ∫₀^∞ e^(-at) e^(-jωt) dt
= [ e^((a - jω)t) / (a - jω) ]_{-∞}^{0} + [ -e^(-(a + jω)t) / (a + jω) ]₀^∞
= 1/(a - jω) + 1/(a + jω)
= 2a / (a² + ω²)

**X(ω) = 2a / (a² + ω²)**

The result is real and even, because x(t) is real and even. Its peak is 2/a at ω = 0, and the half-peak value occurs at ω = ±a. The area under X(ω) is 2π x(0) = 2π.

## Check yourself

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

1. The integral splits at
   A. t = 1
   B. ω = 0
   C. t = 0
   D. t = a
   Answer: C. Absolute value. (Key point: Split)

2. FT =
   A. 1/(a+jω)
   B. a/(a^2+ω^2)
   C. 2/(a^2+ω^2)
   D. 2a/(a^2+ω^2)
   Answer: D. Sum. (Key point: Result)

3. Result is real because
   A. x(t) is real and even
   B. x is causal
   C. a > 0 only
   D. x is odd
   Answer: A. Symmetry. (Key point: Parity)

4. Half peak occurs at
   A. ω = 0
   B. ω = ±a
   C. ω = ±2a
   D. ω = ±1
   Answer: B. 2a/(2a^2). (Key point: Half 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
- [Properties of continuous-time Fourier series](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Afourier-series-properties) — 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
- [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×

## Related in other subjects

- [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
- [Z-transform of ramp-and-hold continuous signal](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Awaveform-ramp-to-1-at-6) — Digital Control System, chapter 2
- [Fourier transform property: Time shifting](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aft-property-time-shifting) — Communication Systems, chapter 1
- [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×