---
title: "Energy or power: x(t) = e^(−3t)"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:energy-power-decaying-exponential"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/energy-power-decaying-exponential
site: BCT Engineering
---

# Energy or power: x(t) = e^(−3t)

Signals and Systems · Chapter 1: Signals

- Asked in: 2081 Chaitra, 3+3 marks; 2080 Asoj, 4 marks; 2073 Magh, 3+4 marks (asked 3 times)
- Video: [Energy and Power Signals | Solved Problems / Examples](https://www.youtube.com/watch?v=bZgQD_7d5VM) — Shrenik Jain
- 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-decaying-exponential
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

Define energy signal and power signal. Determine whether the signal x(t) = e^(−3t) is a power signal or energy signal or neither energy nor power signal.

## Short answer

An energy signal has finite nonzero total energy and zero average power; a power signal has finite nonzero average power and infinite energy. The two-sided signal e^{-3t} has infinite energy and infinite power, so it is neither an energy nor a power signal.

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

#### Checking $x(t) = e^{-3t}$ (defined for all $t$)

$|x(t)|^2 = e^{-6t}$.

**Energy:**

$$
\begin{aligned}
E &= \lim_{T\to\infty}\int_{-T}^{T} e^{-6t}\,dt = \lim_{T\to\infty}\frac{e^{6T}-e^{-6T}}{6} = \infty
\end{aligned}
$$

**Power:**

$$
\begin{aligned}
P &= \lim_{T\to\infty}\frac{1}{2T}\cdot\frac{e^{6T}-e^{-6T}}{6} = \lim_{T\to\infty}\frac{e^{6T}}{12T} = \infty
\end{aligned}
$$

(The exponential grows faster than $T$, so the limit is infinite.)

Since both $E$ and $P$ are infinite, **$x(t) = e^{-3t}$ is neither an energy signal nor a power signal.** The signal blows up as $t \to -\infty$.

**Note:** if the signal is taken as one-sided, $x(t) = e^{-3t}u(t)$, then

$$
E = \int_0^{\infty} e^{-6t}\,dt = \frac{1}{6}\ \text{J}, \qquad P = 0
$$

and it is an **energy signal**.

**Answer:** $e^{-3t}$ for all $t$: neither ($E = \infty$, $P = \infty$); $e^{-3t}u(t)$: energy signal with $E = 1/6$.

## Check yourself

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

1. Which condition defines an energy signal?
   A. 0 < P < infinity and E = infinity
   B. 0 < E < infinity, so average power P = 0
   C. E = infinity and P = infinity
   D. E = 0 and P = 0
   Answer: B. An energy signal has finite non-zero total energy, which forces its average power to zero. (Key point: Energy signal definition)

2. A power signal has which pair of properties?
   A. Finite non-zero energy and zero power
   B. Infinite energy and infinite power
   C. Finite non-zero average power and infinite total energy
   D. Zero energy and finite power
   Answer: C. A power signal has 0 < P < infinity, and since power persists forever, its total energy is infinite. (Key point: Power signal definition)

3. For x(t) = e^(-3t) defined for all t, what is E = integral of e^(-6t) dt from -T to T as T goes to infinity?
   A. 1/6
   B. Zero
   C. Infinite, because e^(6T) grows without bound
   D. Finite but large
   Answer: C. The integral equals (e^(6T) - e^(-6T))/6, which blows up as T grows because the signal explodes as t goes to -infinity. (Key point: Energy of e^(-3t) for all t)

4. Why is e^(-3t) (all t) classified as neither an energy nor a power signal?
   A. Because it is periodic
   B. Because both E and P are infinite
   C. Because P = 0 and E is infinite
   D. Because E is finite but P is infinite
   Answer: B. e^(6T)/(12T) tends to infinity, so power is infinite as well as energy; neither finite-energy nor finite-power condition holds. (Key point: Classification as neither)

## Close variants set in other exams

- [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^(−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×

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