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

# Fourier transform of e^(-at)u(t) and spectra

Data Communication · Chapter 3: Signal Analysis

- Asked in: 2072 Kartik, 6 marks; 2071 Shrawan, 5+3 marks (asked 2 times)
- 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-causal-spectra
- Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-3

## Question

Find the Fourier transform of the single sided exponential function x(t) = e^(-at)u(t) (a>0). Plot its magnitude and phase spectra.

## Answer

x(t) = e^(-at) u(t), a > 0

X(ω) = ∫ x(t) e^(-jωt) dt = ∫₀^∞ e^(-at) e^(-jωt) dt = ∫₀^∞ e^(-(a + jω)t) dt

= [ -e^(-(a + jω)t) / (a + jω) ]₀^∞ = 1/(a + jω)

The integral converges since a > 0 (the signal is absolutely integrable).

**X(ω) = 1/(a + jω)**

#### Magnitude and phase
|X(ω)| = 1 / √(a² + ω²)

φ(ω) = ∠X(ω) = -tan⁻¹(ω/a)

| ω | \|X\| | φ |
|---|---|---|
| 0 | 1/a | 0 |
| a | 1/(a√2) | -45° |
| ∞ | 0 | -90° |
| -a | 1/(a√2) | +45° |

```text
Magnitude spectrum |X(w)|
            1/a
          .-'-.
        .'  |  '.      1/(a*sqrt2) at w = -a and w = +a
     _.'    |    '._
 ---'-------+-------'---> w
      -a    0    a

Phase spectrum (degrees)
 +90 -._
        '-._           +45 at w = -a
   0 -------'-._---------> w
                '-._   -45 at w = +a
 -90                '-
      -a    0    a
```
The magnitude spectrum is even and bell-shaped, with the peak 1/a at ω = 0 and a half-power (3 dB) point at ω = a. The phase spectrum is odd: it falls from +90° (as ω → -∞) through 0 at ω = 0 to -90° (as ω → +∞). This is the response of a first-order RC low-pass filter with a = 1/RC.

## Check yourself

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

1. FT of e^(-at)u(t) is
   A. 1/(a-jω)
   B. a/(a^2+ω^2)
   C. 2a/(a^2+ω^2)
   D. 1/(a+jω)
   Answer: D. Integral of e^(-(a+jω)t). (Key point: Result)

2. |X(ω)| =
   A. 1/sqrt(a^2+ω^2)
   B. 1/(a^2+ω^2)
   C. a/ω
   D. sqrt(a^2+ω^2)
   Answer: A. Modulus. (Key point: Magnitude)

3. Phase is
   A. tan^-1(ω/a)
   B. -tan^-1(ω/a)
   C. -tan^-1(a/ω)
   D. 0
   Answer: B. Angle of 1/(a+jω). (Key point: Phase)

4. At ω = a the phase is
   A. +45°
   B. -90°
   C. -45°
   D. 0°
   Answer: C. -tan^-1(1). (Key point: Phase at ω=a)

## Related in Data Communication

- [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 a DC signal](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aft-dc-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×
- [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

## Related in other subjects

- [Z-transform and ROC of two-sided exponential](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Az-transform-two-sided-exp) — Digital Signal Analysis and Processing, chapter 2
- [Magnitude and phase of DTFT of aⁿu[n]](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtft-geometric-magnitude-phase) — Signals and Systems, chapter 3
- [Single-phase PWM inverter rms and Fourier series](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Asingle-phase-pwm-inverter-fourier) — Power Electronics, chapter 5
- [Fourier transform of a function](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-transform-of-a-function) — Mathematics III, chapter 4