---
title: "Energy or power: e^(-a|t|)u(t)"
subject: "Data Communication"
chapter: 2
key: "semester_5/data_communication:energy-power-exp-abs"
source: https://www.bctengineer.com/q/semester_5/data_communication/energy-power-exp-abs
site: BCT Engineering
---

# Energy or power: e^(-a|t|)u(t)

Data Communication · Chapter 2: Signals and Systems

- Asked in: 2072 Chaitra, 4 marks
- 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_5/data_communication/energy-power-exp-abs
- Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-2

## Question

Justify whether a signal x(t) = e^(-a|t|)·u(t) (a > 0) is energy or power signal.

## Short answer

For t≥0, x(t)=e^{-at}u(t). Energy E = ∫₀^∞ e^{-2at} dt = 1/(2a) J; average power P = 0. Finite energy, zero power → energy signal.

## Answer

x(t) = e^(-a|t|) u(t), a > 0

Since u(t) is zero for t < 0, only t ≥ 0 counts, where |t| = t. So x(t) = e^(-at) u(t).

#### Energy
E = ∫₀^∞ e^(-2at) dt = [-e^(-2at)/(2a)]₀^∞ = **1/(2a)** (finite and non-zero)

#### Power
P = lim (T→∞) (1/2T) ∫₀^T e^(-2at) dt = lim (1/2T) · (1 - e^(-2aT))/(2a) = 0

**Answer: E = 1/(2a) < ∞ and P = 0, so x(t) is an energy signal.**

## Check yourself

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

1. Why does e^(-a|t|)u(t) reduce to e^(-at)u(t)?
   A. Because a > 0
   B. Because |t| = -t
   C. Because the signal is even
   D. u(t) removes t < 0 where |t| = t
   Answer: D. Only t >= 0 counts. (Key point: Reduction)

2. What is the energy?
   A. 1/(2a)
   B. 1/a
   C. 2a
   D. a/2
   Answer: A. Integral of e^(-2at) is 1/(2a). (Key point: Energy)

3. What is the average power?
   A. 1/(2a)
   B. 0
   C. 1
   D. Infinity
   Answer: B. Finite energy over infinite time. (Key point: Power)

4. The signal is
   A. A power signal
   B. Neither
   C. An energy signal
   D. Both
   Answer: C. E finite, P = 0. (Key point: Classification)

## Related in Data Communication

- [Energy or power of DT cos(3πk/16)](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Adt-signal-cos-energy-power) — Data Communication, chapter 2
- [Energy or power: unit step, impulse](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aenergy-power-check-signals) — Data Communication, chapter 2, asked 2×
- [Power and energy of e^(-2t)u(t)](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Apower-energy-e-2t) — Data Communication, chapter 2
- [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
- [AM wave power and bandwidth numerical](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aam-numeric-40-carrier-power) — Data Communication, chapter 5

## 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
- [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
- [Justify: series motor power; shunt motor for traction](https://www.bctengineer.com/question.md?key=semester_7%2Futilization_of_electrical_energy%3Ajustify-series-motor-power-shunt-traction) — Utilization of Electrical Energy, chapter 4
- [Justify higher voltage/more circuits for more power](https://www.bctengineer.com/question.md?key=semester_8%2Ftransmission_and_distribution_system_design%3Ajustify-higher-voltage-more-circuits) — Transmission and Distribution System Design, chapter 2