---
title: "Energy or power: u(t) and δ(t)"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:energy-power-step-impulse"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/energy-power-step-impulse
site: BCT Engineering
---

# Energy or power: u(t) and δ(t)

Signals and Systems · Chapter 1: Signals

- Asked in: 2079 Chaitra, 2+4 marks
- Video: [Energy and Power of Continuous Time Signals](https://www.youtube.com/watch?v=0Kmqp-XX_F0) — Neso Academy
- 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-step-impulse
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

Define Power and Energy Type continuous time signal with suitable examples. Check whether these signals are Energy or Power Type with required calculation. a) x(t) = u(t) b) x(t) = δ(t)

## Short answer

A signal with finite non-zero energy (0<E<∞, P=0) is an energy signal; one with finite non-zero power (0<P<∞, E=∞) is a power signal. For u(t): E=∞, P=1/2 → power signal. For δ(t): E=∞, P=0 → neither energy nor power.

## 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. $e^{t}$, $t\,u(t)$).

#### (a) $x(t) = u(t)$

$$
E = \lim_{T\to\infty}\int_{-T}^{T}u^2(t)\,dt = \lim_{T\to\infty}\int_0^{T}1\,dt = \lim_{T\to\infty}T = \infty
$$

$$
P = \lim_{T\to\infty}\frac{1}{2T}\int_0^{T}1\,dt = \lim_{T\to\infty}\frac{T}{2T} = \frac{1}{2}
$$

$u(t)$ is a **power signal**, $P = 1/2$.

#### (b) $x(t) = \delta(t)$

Model $\delta(t)$ as a pulse $p_\Delta(t)$ of width $\Delta$ and height $1/\Delta$ (area 1), and let $\Delta \to 0$:

$$
E = \lim_{\Delta\to 0}\int_0^{\Delta}\left(\frac{1}{\Delta}\right)^2dt = \lim_{\Delta\to 0}\frac{1}{\Delta} = \infty
$$

The energy is infinite, so $\delta(t)$ is **not** an energy signal. For any fixed pulse width, the energy is finite and the average power is

$$
P = \lim_{T\to\infty}\frac{1}{2T}\cdot\frac{1}{\Delta} = 0
$$

So the power is not a finite non-zero value either. Hence **$\delta(t)$ is neither an energy nor a power signal** (it is not square-integrable; note that its **area** $\int\delta(t)dt = 1$ is not its energy).

**Answer:** (a) $u(t)$: power signal, $P = 0.5$, $E = \infty$. (b) $\delta(t)$: neither ($E = \infty$).

## Check yourself

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

1. For x(t) = u(t), P equals:
   A. 1
   B. 1/2
   C. 0
   D. Infinity
   Answer: B. P = lim T/(2T) = 1/2, while E = infinity, so it is a power signal. (Key point: u(t) power)

2. Modelling delta(t) as a pulse of width Delta and height 1/Delta, E equals:
   A. 1
   B. Delta
   C. 1/Delta, which tends to infinity
   D. 0
   Answer: C. E = (1/Delta)^2 x Delta = 1/Delta, infinite as Delta -> 0. (Key point: Energy of impulse)

3. For fixed Delta, the power (1/2T)(1/Delta) as T -> infinity is:
   A. Infinite
   B. 1/2
   C. 0
   D. 1
   Answer: C. Finite energy averaged over infinite time gives zero. (Key point: Power of fixed-width pulse)

4. delta(t) is classified as:
   A. Energy signal
   B. Power signal
   C. Neither
   D. Both
   Answer: C. Its energy is infinite and power is not finite non-zero; area 1 is not its energy. (Key point: Classification of impulse)

## Related in Signals and Systems

- [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
- [Continuous-time vs discrete-time signals](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Act-vs-dt-signals) — Signals and Systems, chapter 1, asked 3×
- [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
- [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
- [Define energy and power signals](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adefine-energy-power-signals) — Signals and Systems, chapter 1, asked 2×
- [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×

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