---
title: "Energy or power: x(t) = A cos2πft"
subject: "Communication Systems"
chapter: 1
key: "semester_6/communication_systems:energy-power-acos"
source: https://www.bctengineer.com/q/semester_6/communication_systems/energy-power-acos
site: BCT Engineering
---

# Energy or power: x(t) = A cos2πft

Communication Systems · Chapter 1: Introduction

- Asked in: 2076 Bhadra (CS I), 2+5 marks; 2065 Kartik (CS I), 2+6 marks (asked 2 times)
- Video: [Signal Power and Energy](https://www.youtube.com/watch?v=7I9XEhAup4c) — Iain Explains Signals, Systems, and Digital Comms
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/energy-power-acos
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1

## Question

Define energy and power signal. Find whether the signal x(t) = A cos2πft is energy type or power type signal.

## Short answer

An energy signal has finite nonzero energy and zero average power; a power signal has finite nonzero average power and infinite energy. Since x(t)=A cos(2πft) yields infinite energy but finite average power P = A²/2, it is a power signal.

## Answer

#### Energy and power signals
- **Energy signal:** a signal with finite, non-zero total energy, $0 < E < \infty$, where $E = \int_{-\infty}^{\infty} |x(t)|^2\,dt$. Its average power is zero. Example: a single rectangular pulse, $e^{-at}u(t)$.
- **Power signal:** a signal with finite, non-zero average power, $0 < P < \infty$, where $P = \lim_{T\to\infty}\frac{1}{T}\int_{-T/2}^{T/2} |x(t)|^2\,dt$. Its energy is infinite. Example: periodic signals such as sinusoids, the unit step.

A signal cannot be both; some signals (e.g. $e^{t}$, a ramp $t\,u(t)$) are neither.

#### Is $x(t) = A\cos 2\pi f t$ an energy or power signal?
Let $f = f_0 = 1/T_0$.

**Energy:**

$$
\begin{aligned}
E &= \lim_{T\to\infty}\int_{-T/2}^{T/2} A^2\cos^2(2\pi f_0 t)\,dt \\
&= \lim_{T\to\infty}\int_{-T/2}^{T/2} \frac{A^2}{2}\left[1+\cos(4\pi f_0 t)\right]dt \\
&= \lim_{T\to\infty}\left[\frac{A^2 T}{2} + \frac{A^2\sin(2\pi f_0 T)}{4\pi f_0}\right] = \infty
\end{aligned}
$$

The second term stays bounded while $A^2T/2$ grows without limit, so the energy is infinite. It is **not** an energy signal.

**Power:**

$$
\begin{aligned}
P &= \lim_{T\to\infty}\frac{1}{T}\int_{-T/2}^{T/2} A^2\cos^2(2\pi f_0 t)\,dt \\
&= \lim_{T\to\infty}\frac{1}{T}\left[\frac{A^2 T}{2} + \frac{A^2\sin(2\pi f_0 T)}{4\pi f_0}\right] \\
&= \frac{A^2}{2} + \lim_{T\to\infty}\frac{A^2\sin(2\pi f_0 T)}{4\pi f_0 T} = \frac{A^2}{2}
\end{aligned}
$$

(Same result by averaging over one period $T_0$, since the signal is periodic.)

Since $0 < P = A^2/2 < \infty$ and $E = \infty$:

**Answer:** $x(t) = A\cos 2\pi f t$ is a **power signal** with average power $P = \dfrac{A^2}{2}$ (normalised to a 1 $\Omega$ load); its energy is infinite.

## Check yourself

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

1. Which statement correctly describes an energy signal and a power signal?
   A. Energy signal: finite energy and zero average power; power signal: finite non-zero power and infinite energy
   B. Energy signal: infinite energy; power signal: zero energy
   C. Both can hold for the same signal if it is periodic
   D. Energy signal: finite power and infinite energy; power signal: finite energy
   Answer: A. A signal is energy type if 0 < E < infinity (then P = 0), and power type if 0 < P < infinity (then E is infinite); it cannot be both. (Key point: Energy vs power signal definitions)

2. Why is the energy of x(t) = A cos 2 pi f t infinite?
   A. The cosine is not integrable over any interval
   B. The integral of cos^2 over all time is zero
   C. The term A^2 T/2 grows without bound while the other term stays bounded
   D. Because the amplitude A is unbounded
   Answer: C. cos^2 = (1 + cos 2x)/2 gives A^2 T/2 plus a bounded oscillating term, so E tends to infinity as T grows. (Key point: Energy of A cos(2 pi f t))

3. What is the average power of x(t) = A cos 2 pi f t (1 ohm), and what is its type?
   A. A^2, energy signal
   B. A^2/2, power signal
   C. A^2/2, energy signal
   D. A/2, power signal
   Answer: B. Averaging A^2 cos^2 over a period gives A^2/2, which is finite and non-zero, while E is infinite, so it is a power signal. (Key point: Average power of a cosine)

## Related in Communication Systems

- [Energy and power signal; bandwidth of a system](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aenergy-power-bandwidth-system) — Communication Systems, chapter 1
- [AM numerical: S(t) = 10(1 + 0.2cos2π10³t)cos2π10⁶t](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aam-numerical-10-0-2) — Communication Systems, chapter 2
- [AM numerical: u(t) = [20 + 2cos3000πt + 10cos6000πt]cos2πfct](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aam-20-2cos-10cos-fc-1e5) — Communication Systems, chapter 2
- [Shannon-Hartley theorem; energy/power of e^(−2t)u(t)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ashannon-hartley-energy-e2t) — Communication Systems, chapter 1
- [Unit step in terms of signum; energy or power](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aunit-step-signum-energy-power) — Communication Systems, chapter 1
- [ESD of an energy signal (ideal BPF example)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aesd-energy-signal-ideal-bpf) — Communication Systems, chapter 1

## Related in other subjects

- [Power of energy signal is zero](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aenergy-vs-power-signal-example) — Signals and Systems, chapter 1
- [Energy or power: x(t) = e^(−at)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aenergy-power-one-sided-exponential) — Signals and Systems, chapter 1, asked 6×
- [Energy or power: x(t) = e^(−3t)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aenergy-power-decaying-exponential) — Signals and Systems, chapter 1, asked 5×
- [Energy or power: e^(−5t)u(t+2)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aenergy-power-shifted-exponential) — Signals and Systems, chapter 1, asked 5×