---
title: "Define energy and power signals"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:define-energy-power-signals"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/define-energy-power-signals
site: BCT Engineering
---

# Define energy and power signals

Signals and Systems · Chapter 1: Signals

- Asked in: 2079 Jestha, 4 marks; 2078 Chaitra, 2 marks (asked 2 times)
- Video: [Energy Signals & Power Signals](https://www.youtube.com/watch?v=w1HU1-kGLjs) — TutorialsPoint
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/define-energy-power-signals
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

Define energy and power signals with examples.

## Short answer

An energy signal has finite nonzero energy (E<∞) and zero average power, e.g., e^{-2t}u(t) with E=1/4. A power signal has finite nonzero power (P<∞) and infinite energy, e.g., A cos(ω0t) with P=A²/2.

## Answer

- **Energy signal:** a signal whose total energy is finite and non-zero, $0 < E < \infty$. Its average power is then zero. Example: a single pulse, $e^{-2t}u(t)$.
- **Power signal:** a signal whose average power is finite and non-zero, $0 < P < \infty$. Its total energy is then infinite. Example: $\cos\omega_0 t$, $u(t)$, any periodic signal.

For a continuous-time signal:

$$
E = \lim_{T\to\infty}\int_{-T}^{T}|x(t)|^2\,dt, \qquad P = \lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}|x(t)|^2\,dt
$$

For a discrete-time signal:

$$
E = \sum_{n=-\infty}^{\infty}|x[n]|^2, \qquad P = \lim_{N\to\infty}\frac{1}{2N+1}\sum_{n=-N}^{N}|x[n]|^2
$$

A signal that has neither finite energy nor finite non-zero power is **neither** an energy nor a power signal (e.g. $e^{t}$, $t\,u(t)$).

#### Examples

1. **Energy signal:** $x(t) = e^{-2t}u(t)$

$$
E = \int_0^\infty e^{-4t}\,dt = \frac{1}{4}, \qquad P = 0
$$

2. **Power signal:** $x(t) = A\cos\omega_0 t$

$$
P = \frac{1}{T_0}\int_0^{T_0} A^2\cos^2\omega_0 t\,dt = \frac{A^2}{2}, \qquad E = \infty
$$

3. **DT energy signal:** $x[n] = (0.5)^n u[n]$, $E = \sum_{n=0}^\infty 0.25^n = \frac{1}{1-0.25} = \frac{4}{3}$.

4. **DT power signal:** $x[n] = u[n]$, $P = \lim_{N\to\infty}\frac{N+1}{2N+1} = \frac{1}{2}$.

In general, time-limited (finite-duration) bounded signals are energy signals, and periodic signals are power signals.

## Check yourself

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

1. What are E and P for x(t) = e^(-2t)u(t)?
   A. E = 1/2, P = 0
   B. E = 1/4, P = 0
   C. E = 1/4, P = 1/2
   D. E = infinity, P = 1/4
   Answer: B. E = integral of e^(-4t) from 0 to infinity = 1/4, and P = 0 since E is finite. (Key point: Energy of e^(-2t)u(t))

2. What is the average power of A cos(w0 t)?
   A. A^2
   B. A^2/2
   C. A/2
   D. A^2/4
   Answer: B. Averaging A^2 cos^2 over a period gives A^2/2; the energy is infinite. (Key point: Power of A cos(w0 t))

3. For x[n] = (0.5)^n u[n], E equals:
   A. 2
   B. 1/2
   C. 4/3
   D. 3/4
   Answer: C. E = sum of 0.25^n = 1/(1 - 0.25) = 4/3. (Key point: DT energy sum)

4. The unit step u[n] in DT has average power:
   A. 0
   B. 1/2
   C. 1
   D. Infinite
   Answer: B. P = lim (N+1)/(2N+1) = 1/2, and E is infinite, so it is a power signal. (Key point: Power of unit step)

## Related in Signals and Systems

- [Short note: Energy and power signals](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Ashort-note-energy-power) — Signals and Systems, chapter 1, asked 2×
- [Even/odd signals; energy or power check](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aeven-odd-energy-power-check) — Signals and Systems, chapter 1
- [Energy/power signals; shifting and scaling](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aenergy-power-shifting-scaling) — Signals and Systems, chapter 1
- [Energy density spectrum of CT signals](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aenergy-density-spectrum) — Signals and Systems, chapter 3, asked 3×
- [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×

## Related in other subjects

- [Energy versus power signals](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aenergy-power-distinguish) — Data Communication, chapter 2, asked 6×
- [Causality, stability; energy and power signals](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Acausality-stability-energy-power) — Data Communication, chapter 2
- [Energy/power signals; periodicity of complex exponential](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Aenergy-power-periodic-1) — Digital Signal Analysis and Processing, chapter 1
- [Energy/power signals; periodicity of cos + sin](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Aenergy-power-periodic-2) — Digital Signal Analysis and Processing, chapter 1