---
title: "Rayleigh energy theorem"
subject: "Communication Systems"
chapter: 1
key: "semester_6/communication_systems:rayleigh-energy-theorem-prove"
source: https://www.bctengineer.com/q/semester_6/communication_systems/rayleigh-energy-theorem-prove
site: BCT Engineering
---

# Rayleigh energy theorem

Communication Systems · Chapter 1: Introduction

- Asked in: 2075 Baisakh (CS I), 2+6 marks
- Video: [Parseval's Energy Theorem](https://www.youtube.com/watch?v=IrKSx4mMK5M) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/rayleigh-energy-theorem-prove
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1

## Question

State and prove Rayleigh energy theorem for a given energy signal x(t).

## Short answer

Rayleigh energy theorem states total energy E equals integral of |x(t)|^2 dt and also integral of |X(f)|^2 df. Proof: substitute the inverse Fourier transform for x*(t), interchange integration order, and identify the inner integral as X(f), giving integral |X(f)|^2 df.

## Answer

#### Statement
**Rayleigh's energy theorem** (Parseval's theorem for energy signals) states that the total energy of a signal $x(t)$ is the same whether calculated in the time domain or in the frequency domain:

$$
E = \int_{-\infty}^{\infty} |x(t)|^2\,dt = \int_{-\infty}^{\infty} |X(f)|^2\,df
$$

where $X(f)$ is the Fourier transform of $x(t)$. $|X(f)|^2$ is the energy spectral density.

#### Proof
Write $|x(t)|^2 = x(t)\,x^*(t)$ and express $x^*(t)$ through the inverse Fourier transform:

$$
x(t) = \int_{-\infty}^{\infty} X(f)e^{j2\pi ft}\,df \;\Rightarrow\; x^*(t) = \int_{-\infty}^{\infty} X^*(f)e^{-j2\pi ft}\,df
$$

Then

$$
\begin{aligned}
E &= \int_{-\infty}^{\infty} x(t)\,x^*(t)\,dt \\
&= \int_{-\infty}^{\infty} x(t)\left[\int_{-\infty}^{\infty} X^*(f)e^{-j2\pi ft}\,df\right]dt
\end{aligned}
$$

Interchange the order of integration:

$$
\begin{aligned}
E &= \int_{-\infty}^{\infty} X^*(f)\left[\int_{-\infty}^{\infty} x(t)e^{-j2\pi ft}\,dt\right]df \\
&= \int_{-\infty}^{\infty} X^*(f)\,X(f)\,df \\
&= \int_{-\infty}^{\infty} |X(f)|^2\,df
\end{aligned}
$$

since the bracket is the definition of $X(f)$. Hence proved.

#### Example
For $x(t) = e^{-at}u(t)$, $a > 0$:
- Time domain: $E = \int_0^\infty e^{-2at}dt = \frac{1}{2a}$.
- Frequency domain: $X(f) = \frac{1}{a + j2\pi f}$, $|X(f)|^2 = \frac{1}{a^2 + 4\pi^2f^2}$, and $\int_{-\infty}^{\infty}\frac{df}{a^2+4\pi^2f^2} = \frac{1}{2\pi a}\cdot\pi = \frac{1}{2a}$.

Both give the same energy.

**Use:** the theorem lets us find energy in a band of frequencies, e.g. the bandwidth that contains 90% or 99% of the signal energy.

## Check yourself

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

1. Rayleigh's energy theorem states:
   A. Energy in time is twice that in frequency
   B. Energy equals the peak of X(f)
   C. Energy in time domain equals integral of |X(f)|^2 over frequency
   D. Power equals integral of x(t)
   Answer: C. E = integral |x|^2 dt = integral |X(f)|^2 df. (Key point: Statement)

2. The proof begins by writing |x|^2 = x x* and:
   A. Differentiating x(t)
   B. Assuming x is periodic
   C. Expressing x*(t) through the inverse Fourier transform of X*(f)
   D. Using the convolution theorem on x
   Answer: C. x*(t) = integral X*(f) e^{-j2 pi f t} df. (Key point: Proof starting step)

3. After interchanging the order of integration, the inner integral over t of x(t)e^{-j2 pi f t} is:
   A. X*(f)
   B. x(f)
   C. X(f)
   D. delta(f)
   Answer: C. That is the definition of X(f), leaving integral X*(f)X(f) df. (Key point: Interchange of integrals)

4. For x(t) = e^{-at}u(t), the energy is:
   A. 1/(2a)
   B. a/2
   C. 1/a
   D. 2a
   Answer: A. Integral of e^{-2at} from 0 to infinity = 1/(2a), matching the frequency-domain result. (Key point: Example check)

## Related in Communication Systems

- [Rayleigh energy theorem; power from Fourier series](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Arayleigh-periodic-power-fourier) — Communication Systems, chapter 1
- [Fourier transform properties; Rayleigh energy theorem](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Afourier-properties-rayleigh) — Communication Systems, chapter 1
- [State and prove Nyquist sampling theorem](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anyquist-sampling-prove) — Communication Systems, chapter 4
- [Prove sampling theorem; aliasing and aperture effect](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asampling-prove-aliasing-aperture) — Communication Systems, chapter 4
- [Shannon-Hartley theorem; energy/power of e^(−2t)u(t)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ashannon-hartley-energy-e2t) — Communication Systems, chapter 1
- [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

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