---
title: "Fourier transform of e^(−at)cos ω₀t"
subject: "Signals and Systems"
chapter: 3
key: "semester_4/signals_and_systems:ft-damped-cosine"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/ft-damped-cosine
site: BCT Engineering
---

# Fourier transform of e^(−at)cos ω₀t

Signals and Systems · Chapter 3: Fourier Transform

- Asked in: 2081 Asoj, 5 marks
- 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-damped-cosine
- 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) cos ω₀t, a > 0.

## Short answer

For x(t)=e^{−at} cos(ω₀t) u(t), a>0, the transform is X(ω) = (a+jω) / [(a+jω)²+ω₀²]. Obtained by expanding cos as sum of exponentials, using εe^{−at}u(t)ι ↔⊗1/(a+jω)Ιand frequency shifting.

## Answer

For the Fourier transform to exist, the signal is taken as causal: $x(t) = e^{-at}\cos(\omega_0 t)\,u(t)$ with $a > 0$. Without $u(t)$ the signal grows without bound as $t \to -\infty$, and it has no transform.

#### Method: known pair plus frequency shifting
The known pair is
$$
e^{-at}u(t) \xleftrightarrow{F} \frac{1}{a + j\omega}
$$
Write the cosine using Euler's formula:
$$
x(t) = \frac{1}{2}e^{-at}u(t)e^{j\omega_0 t} + \frac{1}{2}e^{-at}u(t)e^{-j\omega_0 t}
$$
The frequency-shift property, $e^{\pm j\omega_0 t}g(t) \leftrightarrow G(\omega \mp \omega_0)$, gives
$$
\begin{aligned}
X(\omega) &= \frac{1}{2}\left[\frac{1}{a + j(\omega - \omega_0)} + \frac{1}{a + j(\omega + \omega_0)}\right] \\
&= \frac{1}{2}\cdot\frac{2(a + j\omega)}{(a + j\omega - j\omega_0)(a + j\omega + j\omega_0)} \\
&= \frac{a + j\omega}{(a + j\omega)^2 + \omega_0^2}
\end{aligned}
$$
The second line uses a common denominator: the numerators add to $2a + 2j\omega$. The last step uses $(p - jq)(p + jq) = p^2 + q^2$ with $p = a + j\omega$ and $q = \omega_0$.

$$
\boxed{X(\omega) = \frac{a + j\omega}{(a + j\omega)^2 + \omega_0^2}}
$$

#### Check by direct integration
$$
\begin{aligned}
X(\omega) &= \int_0^{\infty}e^{-at}\frac{e^{j\omega_0 t} + e^{-j\omega_0 t}}{2}e^{-j\omega t}dt \\
&= \frac{1}{2}\left[\frac{1}{a + j(\omega - \omega_0)} + \frac{1}{a + j(\omega + \omega_0)}\right]
\end{aligned}
$$
This is the same expression. A numerical check with $a = 1.3$, $\omega_0 = 2$ and $\omega = 0.9$ gives $0.288 + j0.046$ from both the integral and the formula.

#### Spectrum
$|X(\omega)|$ has peaks near $\omega = \pm\omega_0$, of height about $\frac{1}{2a}$ when $a \ll \omega_0$. The width of each peak is set by $a$. This is the spectrum of a damped oscillation, like the impulse response of a second-order underdamped system.

## Check yourself

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

1. Why is the signal taken as e^(-at)cos(ω0 t)u(t)?
   A. Without u(t) it grows without bound as t -> -infinity, so no transform exists
   B. To make it periodic
   C. To make it real
   D. Because cos is odd
   Answer: A. e^(-at) with a>0 blows up for negative t, so the two-sided signal has no Fourier transform. (Key point: Causality assumption)

2. Which known pair is used with frequency shifting?
   A. u(t) <-> 1/(jω)
   B. e^(-at)u(t) <-> 1/(a + jω)
   C. e^(-a|t|) <-> 2a/(a^2 + ω^2)
   D. δ(t) <-> 1
   Answer: B. Writing cos with Euler's formula gives two shifted copies of the transform of e^(-at)u(t). (Key point: Known pair)

3. The Fourier transform of e^(-at)cos(ω0 t)u(t) is:
   A. (a + jω)/((a + jω)^2 + ω0^2)
   B. ω0/((a + jω)^2 + ω0^2)
   C. (a + jω)/((a + jω)^2 - ω0^2)
   D. 1/(a + jω)
   Answer: A. Adding (1/2)[1/(a+j(ω-ω0)) + 1/(a+j(ω+ω0))] over a common denominator gives (a+jω)/((a+jω)^2 + ω0^2). (Key point: Result)

4. For a << ω0, where does |X(ω)| peak and what sets the peak width?
   A. At ω = 0, width set by ω0
   B. Near ±ω0, width set by a
   C. At ω = a, width set by ω0
   D. It has no peaks
   Answer: B. The two shifted terms peak near ±ω0 with height about 1/(2a), and the decay rate a controls width. (Key point: Spectrum shape)

## Close variants set in other exams

- [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×
- [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^(−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×

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