---
title: "Rayleigh energy theorem; power from Fourier series"
subject: "Communication Systems"
chapter: 1
key: "semester_6/communication_systems:rayleigh-periodic-power-fourier"
source: https://www.bctengineer.com/q/semester_6/communication_systems/rayleigh-periodic-power-fourier
site: BCT Engineering
---

# Rayleigh energy theorem; power from Fourier series

Communication Systems · Chapter 1: Introduction

- Asked in: 2072 Magh (CS I), 4+2 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-periodic-power-fourier
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1

## Question

Derive Rayleigh energy theorem. How can you calculate the power of a periodic signal when its Fourier-series coefficients Cn are known?

## Short answer

Rayleigh's theorem states energy E = ∫|x(t)|² dt = ∫|X(f)|² df, derived by substituting the inverse transform and interchanging integrals. For a periodic signal x(t)=∑ C_n e^{j2π n f0 t}, Parseval’s power theorem gives average power P = ∑ |C_n|². For a real signal, P = C₀² + 2∑_{n=1}∞ |C_n|².

## Answer

#### Rayleigh energy theorem
For an energy signal $x(t)$ with Fourier transform $X(f)$:

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

**Derivation:** substitute $x^*(t) = \int X^*(f)e^{-j2\pi ft}df$ into the energy integral and interchange the order of integration:

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

So energy can be found from the spectrum; $|X(f)|^2$ is the energy spectral density.

#### Power of a periodic signal from $C_n$
A periodic signal with period $T_0$ ($f_0 = 1/T_0$) can be written $x(t) = \sum_n C_ne^{j2\pi nf_0t}$. Its average power is

$$
\begin{aligned}
P &= \frac{1}{T_0}\int_{T_0}x(t)x^*(t)\,dt = \sum_n\sum_m C_nC_m^*\frac{1}{T_0}\int_{T_0}e^{j2\pi(n-m)f_0t}dt \\
&= \sum_{n=-\infty}^{\infty}|C_n|^2
\end{aligned}
$$

since the integral is 1 for $m = n$ and 0 otherwise. This is **Parseval's power theorem**: the power is the sum of the squared magnitudes of the Fourier coefficients. For a real signal, $P = C_0^2 + 2\sum_{n=1}^{\infty}|C_n|^2$.

**Example:** $x(t) = 2 + 4\cos 2\pi f_0t$ has $C_0 = 2$, $C_{\pm1} = 2$, so $P = 4 + 4 + 4 = 12$ W.

## Check yourself

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

1. What does Rayleigh's energy theorem state?
   A. Energy in the time domain equals the integral of |X(f)|^2 over frequency
   B. Energy is the integral of X(f) over frequency
   C. Energy equals the peak of |x(t)|
   D. Energy is zero for all periodic signals
   Answer: A. E = integral |x|^2 dt = integral |X(f)|^2 df. (Key point: Rayleigh theorem)

2. What is the key step in deriving the theorem?
   A. Differentiate x(t) and integrate by parts
   B. Write x*(t) as an inverse transform of X*(f), then swap the order of integration
   C. Use the sampling theorem
   D. Expand x(t) in a Taylor series
   Answer: B. Substituting x*(t)=integral X*(f)e^(-j2 pi f t) df and interchanging integrals gives integral X*X df. (Key point: Derivation step)

3. How is power of a periodic signal found from C_n?
   A. P = |C_0| only
   B. P = sum of C_n
   C. P = sum of |C_n|^2
   D. P = (sum of |C_n|)^2
   Answer: C. Parseval's power theorem: P is the sum of squared magnitudes of the coefficients. (Key point: Parseval's power)

4. For x(t)=2+4cos(2 pi f0 t), what is the power?
   A. 12 W
   B. 20 W
   C. 18 W
   D. 6 W
   Answer: A. C_0=2, C_{+-1}=2, so P = 4+4+4 = 12 W. (Key point: Worked example)

## Related in Communication Systems

- [Fourier transform properties; Rayleigh energy theorem](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Afourier-properties-rayleigh) — Communication Systems, chapter 1
- [Rayleigh energy theorem](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Arayleigh-energy-theorem-prove) — Communication Systems, chapter 1
- [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
- [ESD of an energy signal (ideal BPF example)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aesd-energy-signal-ideal-bpf) — Communication Systems, chapter 1
- [PSD of a power signal and its interpretation](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Apsdf-power-signal-interpretation) — Communication Systems, chapter 1, asked 2×

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