---
title: "Energy or power: e^(−5t)u(t+2)"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:energy-power-shifted-exponential"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/energy-power-shifted-exponential
site: BCT Engineering
---

# Energy or power: e^(−5t)u(t+2)

Signals and Systems · Chapter 1: Signals

- Asked in: 2077 Chaitra, 3+4 marks
- Video: [15 Example of Energy and power Signals 𝒙(𝒕)=𝒆^(−𝟐𝒕) 𝒖(𝒕) | Signals and Systems | AL AMIN](https://www.youtube.com/watch?v=HTNnRkuD2ho) — AL AMIN
- 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-shifted-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^(−5t)u(t+2) is energy signal or power signal or neither energy nor power signal.

## Short answer

Energy signals have finite non-zero energy and zero power; power signals have finite non-zero power and infinite energy. For x(t)=e^{-5t}u(t+2): E = ∫_{-2}^{∞} e^{-10t} dt = e^{20}/10 ≈ 4.85×10^7 (finite). Hence it is an energy 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. $e^{t}$, $t\,u(t)$).

#### $x(t) = e^{-5t}u(t+2)$

$u(t+2) = 1$ for $t \ge -2$, so the signal starts at $t = -2$ and then decays.

**Energy:**

$$
\begin{aligned}
E &= \int_{-2}^{\infty}e^{-10t}\,dt = \left[\frac{e^{-10t}}{-10}\right]_{-2}^{\infty} \\
&= 0 + \frac{e^{20}}{10} = \frac{e^{20}}{10} \approx 4.85 \times 10^{7}
\end{aligned}
$$

This is a large but **finite** value.

**Power:**

$$
P = \lim_{T\to\infty}\frac{1}{2T}\cdot\frac{e^{20}}{10} = 0
$$

**Answer:** $x(t)$ is an **energy signal** with $E = e^{20}/10 \approx 4.85 \times 10^7$ J (normalised) and $P = 0$.

## Check yourself

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

1. u(t+2) equals 1 for:
   A. t >= -2
   B. t >= 2
   C. t <= -2
   D. t >= 0
   Answer: A. The step is shifted left, switching on at t = -2. (Key point: Signal start)

2. The energy of e^(-5t)u(t+2) is:
   A. 1/10
   B. e^20/10
   C. Infinite
   D. e^(-20)/10
   Answer: B. Integral of e^(-10t) from -2 to infinity is e^(20)/10. (Key point: Energy)

3. Since e^20/10 is about 4.85 x 10^7, the energy is:
   A. Infinite
   B. Finite
   C. Zero
   D. Undefined
   Answer: B. A large number is still finite, so the energy criterion is satisfied. (Key point: Large but finite)

4. The signal is:
   A. Power signal with P = 1/2
   B. Energy signal with P = 0
   C. Neither
   D. Power signal with P = e^20
   Answer: B. Finite E implies P = 0, so it is an energy signal. (Key point: Classification)

## Close variants set in other exams

- [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: 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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