--- title: "Energy or power: x(t) = e^(−at)" subject: "Signals and Systems" chapter: 1 key: "semester_4/signals_and_systems:energy-power-one-sided-exponential" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/energy-power-one-sided-exponential site: BCT Engineering --- # Energy or power: x(t) = e^(−at) Signals and Systems · Chapter 1: Signals - Asked in: 2072 Asoj, 3+3 marks - Video: [Signals and system/15/ Energy Signal and Power Signals, x (t)= e-2t u(t), x(t)= cost](https://www.youtube.com/watch?v=ubRE6kT5l-c) — SHERYL RADLEY- ECE Easy Engineering - 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-one-sided-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 is a power signal or energy signal. x(t) = e^(−at) for a > 0 ## Short answer Energy signal: finite nonzero energy, zero power (e.g., e^(-2t)u(t)). Power signal: finite nonzero power, infinite energy (e.g., cosine). For x(t)=e^(-at)u(t) with a>0, E=1/(2a), P=0, so 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. $e^{t}$, $t\,u(t)$). #### $x(t) = e^{-at}$, $a > 0$ The exponential is usually taken as one-sided, $x(t) = e^{-at}u(t)$, i.e. it starts at $t = 0$. **Energy:** $$ \begin{aligned} E &= \int_{0}^{\infty}e^{-2at}\,dt = \left[\frac{e^{-2at}}{-2a}\right]_0^{\infty} = \frac{1}{2a} \end{aligned} $$ Finite, since $a > 0$. **Power:** $$ P = \lim_{T\to\infty}\frac{1}{2T}\int_0^{T}e^{-2at}\,dt = \lim_{T\to\infty}\frac{1-e^{-2aT}}{4aT} = 0 $$ **Answer:** $e^{-at}u(t)$ is an **energy signal** with $E = \frac{1}{2a}$ and $P = 0$. **Note:** if $e^{-at}$ is taken for **all** $t$ ($-\infty < t < \infty$), it grows without bound as $t \to -\infty$: $E = \infty$ and $P = \lim_{T\to\infty} \frac{e^{2aT}}{4aT} = \infty$, so it is **neither** an energy nor a power signal. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which defines an energy signal? A. 0 < E < ∞, and hence P = 0 B. 0 < P < ∞, and hence E = ∞ C. E = ∞ and P = ∞ D. E = 0 Answer: A. Energy signals have finite non-zero energy and zero average power. (Key point: Energy signal definition) 2. What is E for x(t) = e^(-at)u(t), a > 0? A. 1/a B. 1/(2a) C. a/2 D. Infinite Answer: B. E = ∫0∞ e^(-2at) dt = 1/(2a). (Key point: Energy value) 3. How is e^(-at)u(t) classified? A. Power signal with P = 1/2a B. Neither C. Energy signal with P = 0 D. Power signal with infinite energy Answer: C. E is finite and non-zero, so power tends to zero. (Key point: Classification) 4. If e^(-at) is taken for all t, how is it classified? A. Energy signal B. Power signal with P = 1 C. Periodic D. Neither, because it blows up as t → -∞ Answer: D. For t → -∞, e^(-at) grows, giving E = ∞ and P = ∞. 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