---
title: "Energy vs power spectral density"
subject: "Signals and Systems"
chapter: 3
key: "semester_4/signals_and_systems:energy-vs-power-spectral-density"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/energy-vs-power-spectral-density
site: BCT Engineering
---

# Energy vs power spectral density

Signals and Systems · Chapter 3: Fourier Transform

- Asked in: 2080 Chaitra, 4 marks
- Video: [What is Power Spectral Density (PSD)?](https://www.youtube.com/watch?v=DoSLMEEo1Y0) — Iain Explains Signals, Systems, and Digital Comms
- 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-vs-power-spectral-density
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-3

## Question

Differentiate between energy and power spectral density.

## Short answer

Energy spectral density Ψ(ω)=|X(jω)|^2 applies to energy signals; total energy E=(1/2π)∫Ψ dω, unit J/Hz. Power spectral density S(ω)=lim_{T→∞}|X_T(jω)|^2/T applies to power signals; total power P=(1/2π)∫S dω, unit W/Hz. Both are real, non‑negative and even for real signals.

## Answer

Energy spectral density (ESD) shows how the **energy** of an energy signal is distributed over frequency, while power spectral density (PSD) shows how the **average power** of a power signal is distributed over frequency.

| Point | Energy spectral density | Power spectral density |
|---|---|---|
| Used for | Energy signals ($0 < E < \infty$, $P = 0$) | Power signals ($0 < P < \infty$, $E = \infty$) |
| Definition | $\Psi(\omega) = \lvert X(j\omega)\rvert^2$ | $S(\omega) = \lim_{T\to\infty}\frac{\lvert X_T(j\omega)\rvert^2}{T}$ |
| Total quantity | $E = \frac{1}{2\pi}\int \Psi(\omega)d\omega$ | $P = \frac{1}{2\pi}\int S(\omega)d\omega$ |
| Unit | J/Hz | W/Hz |
| FT pair with | Energy autocorrelation $R(\tau)=\int x(t)x(t+\tau)dt$ | Power autocorrelation $R(\tau)=\lim\frac{1}{T}\int_{-T/2}^{T/2} x(t)x(t+\tau)dt$ |
| Typical signals | Pulses, decaying exponentials | Periodic signals, DC, random noise |
| Periodic signal case | Not defined | Impulses: $2\pi\sum \lvert a_k\rvert^2\delta(\omega-k\omega_0)$ |

Both are real, non-negative and even in $\omega$ for real signals, and both lose phase information.

**Example:** $x(t) = e^{-t}u(t)$ is an energy signal with ESD $\frac{1}{1+\omega^2}$; $x(t) = \cos\omega_0 t$ is a power signal with PSD $\frac{\pi}{2}[\delta(\omega-\omega_0)+\delta(\omega+\omega_0)]$.

## Check yourself

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

1. Which signal type is each density used for?
   A. ESD for power signals, PSD for energy signals
   B. ESD for energy signals (finite E), PSD for power signals (finite P)
   C. Both for energy signals
   D. Both for periodic signals only
   Answer: B. Energy signals have 0<E<∞ and P=0; power signals have 0<P<∞ and E=∞. (Key point: Signal types)

2. The units of ESD and PSD are respectively:
   A. W/Hz and J/Hz
   B. J/Hz and W/Hz
   C. J and W
   D. V and A
   Answer: B. ESD gives energy per hertz, PSD gives power per hertz. (Key point: Units)

3. Which signal has an ESD and which has a PSD?
   A. cos(ω0 t) has ESD; e^(-t)u(t) has PSD
   B. e^(-t)u(t) has ESD 1/(1+ω^2); cos(ω0 t) has impulsive PSD
   C. Both have ESD
   D. Both have PSD
   Answer: B. The decaying exponential is an energy signal; the cosine is a power signal with impulses at ±ω0. (Key point: Example signals)

4. For real signals both densities are:
   A. Real, non-negative and even, with phase lost
   B. Odd and complex
   C. Negative and even
   D. Phase-preserving
   Answer: A. Both depend on magnitude squared, so they are real, non-negative, even, and phase information is lost. (Key point: Common properties)

## Related in Signals and Systems

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- [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×
- [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×

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