---
title: "Power of energy signal is zero"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:energy-vs-power-signal-example"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/energy-vs-power-signal-example
site: BCT Engineering
---

# Power of energy signal is zero

Signals and Systems · Chapter 1: Signals

- Asked in: 2074 Bhadra, 2+5 marks
- 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/energy-vs-power-signal-example
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

Differentiate between energy signal and power signal. Show by giving suitable example that power of the energy signal is zero and energy of the power signal is infinite.

## Short answer

An energy signal has finite total energy E (0<E<∞) and zero average power; example: e^(-at)u(t) with a>0 gives E=1/(2a), P=0. A power signal has finite nonzero power P and infinite energy; example: A cos(ω₀t) gives P=A²/2, E→∞. Thus power of an energy signal is zero and energy of a power signal is infinite.

## Answer

| Point | Energy signal | Power signal |
|---|---|---|
| Energy | Finite, $0<E<\infty$ | Infinite |
| Average power | Zero | Finite, $0<P<\infty$ |
| Duration | Usually finite or decaying | Infinite duration |
| Typical type | Aperiodic, transient | Periodic or random |
| Examples | Pulse, $e^{-at}u(t)$ | $A\cos\omega t$, $u(t)$ |
| Spectrum | Energy spectral density | Power spectral density |

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

#### Power of an energy signal is zero
Take $x(t) = e^{-at}u(t)$, $a > 0$.

$$
E = \int_0^{\infty}e^{-2at}dt = \frac{1}{2a} \quad\text{(finite)}
$$

$$
P = \lim_{T\to\infty}\frac{1}{2T}\int_0^{T}e^{-2at}dt = \lim_{T\to\infty}\frac{1-e^{-2aT}}{4aT} = 0
$$

In general, $P = \lim_{T\to\infty} E/2T$; a finite $E$ divided by $2T \to \infty$ gives zero.

#### Energy of a power signal is infinite
Take $x(t) = A\cos\omega_0 t$.

$$
P = \frac{1}{T_0}\int_0^{T_0}A^2\cos^2\omega_0 t\,dt = \frac{A^2}{2} \quad\text{(finite)}
$$

$$
E = \lim_{T\to\infty}\int_{-T}^{T}A^2\cos^2\omega_0 t\,dt \approx \lim_{T\to\infty}\frac{A^2}{2}(2T) = \infty
$$

In general, $E = \lim_{T\to\infty} P\cdot 2T$; a non-zero $P$ multiplied by $2T \to \infty$ gives infinity. Hence a signal cannot be both an energy and a power signal.

## Check yourself

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

1. Why is the average power of an energy signal zero?
   A. Because the signal is zero everywhere
   B. P = lim E/2T, and finite E divided by 2T → ∞ gives 0
   C. Because energy equals power
   D. Because E is infinite
   Answer: B. With finite E, dividing by an ever-growing 2T makes P vanish. (Key point: Energy signal power)

2. What is the energy of x(t) = e^(-at)u(t), a > 0?
   A. 1/a
   B. 2a
   C. 1/(2a)
   D. Infinite
   Answer: C. E = ∫0∞ e^(-2at) dt = 1/(2a). (Key point: Energy of e^(-at)u(t))

3. Why is the energy of a power signal infinite?
   A. Because its power is zero
   B. Because it decays
   C. Because it is aperiodic
   D. E = lim P·2T, and a non-zero P times 2T → ∞ diverges
   Answer: D. For A cos ω0 t, P = A^2/2, which grows without limit when accumulated over infinite time. (Key point: Power signal energy)

4. Which statement is correct?
   A. A signal cannot be both an energy and a power signal
   B. Every energy signal is a power signal
   C. Every periodic signal is an energy signal
   D. Power signals have finite energy
   Answer: A. Energy signals have P = 0 and power signals have E = ∞, so the classes are disjoint. (Key point: Mutual exclusivity)

## Related in Signals and Systems

- [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×
- [Energy or power: x(t) = e^(−a|t|)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aenergy-power-two-sided-exponential) — Signals and Systems, chapter 1, asked 2×
- [Energy and power: 4sin(2πt), cos(3n/4)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aenergy-power-4sin-cos) — Signals and Systems, chapter 1, asked 2×
- [Energy or power: 3cos(4πt), sin(n/6)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aenergy-power-3cos-sin) — Signals and Systems, chapter 1, asked 2×

## Related in other subjects

- [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
- [Energy of particle in infinite well](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aparticle-infinite-well-energy-eigenvalue) — Engineering Physics, chapter 10, asked 7×
- [Infinite potential well: quantized energy and wave function](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Ainfinite-well-energy-quantized) — Electrical Engineering Material, chapter 1, asked 4×
- [Power transfer and pf of motor on infinite bus](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machines_ii%3Amotor-infinite-bus-power-transfer) — Electric Machines II, chapter 2