---
title: "Fourier transform of cos ω₀t"
subject: "Signals and Systems"
chapter: 3
key: "semester_4/signals_and_systems:ft-everlasting-sinusoid"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/ft-everlasting-sinusoid
site: BCT Engineering
---

# Fourier transform of cos ω₀t

Signals and Systems · Chapter 3: Fourier Transform

- Asked in: 2070 Bhadra, 4 marks
- Video: [Fourier Transform of e^(jω₀t),  e^-(jω₀t),  cos(ω₀t), sin(ω₀t) || Signals and Systems ](https://www.youtube.com/watch?v=aviDdUCOr9A) — Learning with Chirag
- 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-everlasting-sinusoid
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-3

## Question

Find the Fourier transform of everlasting sinusoid x(t) = cos ω₀t.

## Short answer

cos(ω₀t) ↔ π[δ(ω–ω₀) + δ(ω+ω₀)], two impulses of area π at ±ω₀, indicating all power at that single frequency.

## Answer

**Step 1: write the cosine with exponentials (Euler).**

$$
x(t) = \cos\omega_0 t = \frac{1}{2}\left(e^{j\omega_0t} + e^{-j\omega_0t}\right)
$$

**Step 2: FT of a complex exponential.** The inverse FT of $2\pi\delta(\omega - \omega_0)$ is

$$
\frac{1}{2\pi}\int_{-\infty}^{\infty}2\pi\,\delta(\omega-\omega_0)e^{j\omega t}d\omega = e^{j\omega_0t}
$$

so $e^{j\omega_0t} \leftrightarrow 2\pi\delta(\omega-\omega_0)$ and $e^{-j\omega_0t} \leftrightarrow 2\pi\delta(\omega+\omega_0)$.

**Step 3: linearity.**

$$
\begin{aligned}
X(j\omega) &= \frac{1}{2}\left[2\pi\delta(\omega-\omega_0) + 2\pi\delta(\omega+\omega_0)\right] \\
&= \pi\left[\delta(\omega-\omega_0) + \delta(\omega+\omega_0)\right]
\end{aligned}
$$

**Answer:** $\cos\omega_0t \leftrightarrow \pi[\delta(\omega-\omega_0) + \delta(\omega+\omega_0)]$

```text
          X(jw)
   pi ^           ^ pi
      |           |
------+-----+-----+------> w
    -w0     0     w0
```

**Remarks:**

- The everlasting sinusoid is not absolutely integrable, so its FT exists only in the generalised sense, with impulses.
- All its power is at the single frequency $\omega_0$ (and $-\omega_0$ in the two-sided spectrum).
- In terms of $f$: $\cos 2\pi f_0t \leftrightarrow \frac{1}{2}[\delta(f-f_0) + \delta(f+f_0)]$. Similarly $\sin\omega_0t \leftrightarrow \frac{\pi}{j}[\delta(\omega-\omega_0) - \delta(\omega+\omega_0)]$.

## Check yourself

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

1. How is cos ω0 t written to find its FT?
   A. (1/2j)(e^{jω0t} - e^{-jω0t})
   B. e^{jω0t} + e^{-jω0t}
   C. (1/2)(e^{jω0t} + e^{-jω0t})
   D. (1/2)(e^{jω0t} - e^{-jω0t})
   Answer: C. Euler's formula for cosine. (Key point: Euler)

2. What is the FT of e^{jω0t}?
   A. δ(ω - ω0)
   B. 2π δ(ω + ω0)
   C. π δ(ω - ω0)
   D. 2π δ(ω - ω0)
   Answer: D. Inverse FT of 2πδ(ω-ω0) gives e^{jω0t}. (Key point: Exponential FT)

3. What is the FT of cos ω0 t?
   A. π[δ(ω-ω0) + δ(ω+ω0)]
   B. 2π[δ(ω-ω0) + δ(ω+ω0)]
   C. π[δ(ω-ω0) - δ(ω+ω0)]
   D. (1/2)[δ(ω-ω0) + δ(ω+ω0)]
   Answer: A. Half of 2π at each of ±ω0. (Key point: Result)

4. Why is the FT only in the generalised sense?
   A. It is complex
   B. The signal is not absolutely integrable
   C. It is non-causal
   D. It is discrete
   Answer: B. An everlasting sinusoid has infinite energy. (Key point: Existence)

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