---
title: "Energy or power: 3cos(2πt), cos(n/6)"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:energy-power-cos-signals"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/energy-power-cos-signals
site: BCT Engineering
---

# Energy or power: 3cos(2πt), cos(n/6)

Signals and Systems · Chapter 1: Signals

- Asked in: 2070 Bhadra, 4 marks
- Video: [Energy and Power Signals- Numerical 2](https://www.youtube.com/watch?v=w5UqbfMy6nU) — Prof. Brinal Colaco
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/energy-power-cos-signals
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

Determine whether the following signals are energy or power signals. a) f(t) = 3cos(2πt) b) x[n] = cos(n/6)

## Short answer

f(t)=3cos(2πt) has infinite energy and average power 4.5 W, thus it is a power signal. x[n]=cos(n/6) also has infinite energy and average power 0.5 W, so it is a power signal.

## Answer

A signal is an energy signal if $0 < E < \infty$ ($P = 0$) and a power signal if $0 < P < \infty$ ($E = \infty$).

#### (a) $f(t) = 3\cos(2\pi t)$
This is periodic with $T = 1$ s, so its energy over all time is infinite. Average power over one period:

$$
\begin{aligned}
P &= \frac{1}{T}\int_0^T 9\cos^2(2\pi t)\,dt = \int_0^1 \frac{9}{2}\left[1 + \cos(4\pi t)\right]dt \\
&= \frac{9}{2} + 0 = 4.5\ \text{W}
\end{aligned}
$$

$E = \infty$, $P = 4.5$ W, so $f(t)$ is a **power signal** (in general, $A\cos\omega_0 t$ has $P = A^2/2$).

#### (b) $x[n] = \cos(n/6)$
Here $\omega_0 = 1/6$ and $\omega_0/2\pi = 1/(12\pi)$ is irrational, so $x[n]$ is **not periodic**. Its power must be found with the limit:

$$
\begin{aligned}
P &= \lim_{N\to\infty}\frac{1}{2N+1}\sum_{n=-N}^{N}\cos^2\frac{n}{6} \\
&= \lim_{N\to\infty}\frac{1}{2N+1}\sum_{n=-N}^{N}\frac{1}{2}\left[1 + \cos\frac{n}{3}\right] = \frac{1}{2}
\end{aligned}
$$

The sum of $\cos(n/3)$ stays bounded, so divided by $2N+1$ it goes to 0. The energy $\sum \cos^2(n/6)$ grows without limit, so $E = \infty$.

**Answer:** $E = \infty$, $P = 0.5$ W, so $x[n]$ is a **power signal** (even though it is aperiodic).

## Check yourself

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

1. What is the average power of 3 cos(2πt)?
   A. 9 W
   B. 3 W
   C. 4.5 W
   D. 1.5 W
   Answer: C. P = A^2/2 = 9/2 = 4.5 W. (Key point: Power of 3cos(2πt))

2. How is 3 cos(2πt) classified?
   A. Energy signal
   B. Neither
   C. Aperiodic signal
   D. Power signal
   Answer: D. It is periodic, so E is infinite and P is finite. (Key point: Classification of (a))

3. Is x[n] = cos(n/6) periodic?
   A. No, since ω0/2π = 1/(12π) is irrational
   B. Yes, with N = 6
   C. Yes, with N = 12
   D. Yes, with N = 12π
   Answer: A. No integer period exists. (Key point: Periodicity of cos(n/6))

4. How is cos(n/6) classified and why?
   A. Energy signal, because it is aperiodic
   B. Power signal with P = 0.5, because the cos(n/3) sum stays bounded
   C. Neither, because it is aperiodic
   D. Power signal with P = 1
   Answer: B. Aperiodic does not mean energy signal: E = ∞ and P = 1/2. (Key point: Aperiodic power signal)

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