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

# Fourier transform properties; Rayleigh energy theorem

Communication Systems · Chapter 1: Introduction

- Asked in: 2071 Magh (old course), 4+4 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/fourier-properties-rayleigh
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1

## Question

State any four properties of the Fourier Transform. Derive the expression for the Rayleigh Energy Theorem.

## Short answer

Four properties: linearity, time shift, time scaling, and modulation. Rayleigh's energy theorem states that total signal energy equals ∫|g(t)|² dt = ∫|G(f)|² df, derived by substituting g*(t) with the inverse transform of G*(f) and swapping integrals.

## Answer

The Fourier transform pair is $G(f)=\int_{-\infty}^{\infty} g(t)e^{-j2\pi ft}dt$ and $g(t)=\int_{-\infty}^{\infty} G(f)e^{j2\pi ft}df$, written $g(t)\leftrightarrow G(f)$.

#### Four properties of the Fourier transform

| Property | Time domain | Frequency domain |
|---|---|---|
| Linearity | $a g_1(t)+b g_2(t)$ | $aG_1(f)+bG_2(f)$ |
| Time shifting | $g(t-t_0)$ | $G(f)e^{-j2\pi f t_0}$ |
| Time scaling | $g(at)$ | $\frac{1}{\lvert a\rvert}G\left(\frac{f}{a}\right)$ |
| Modulation (frequency shift) | $g(t)\cos 2\pi f_c t$ | $\frac{1}{2}[G(f-f_c)+G(f+f_c)]$ |
| Convolution | $g_1(t)*g_2(t)$ | $G_1(f)G_2(f)$ |
| Duality | $G(t)$ | $g(-f)$ |

- **Linearity:** the transform of a weighted sum is the weighted sum of the transforms.
- **Time shifting:** a delay changes only the phase spectrum (linear phase), not the amplitude spectrum.
- **Scaling:** compressing a signal in time expands its spectrum, and vice versa.
- **Modulation:** multiplying by a carrier shifts the spectrum to $\pm f_c$; this is the basis of AM.

#### Rayleigh energy theorem

**Statement:** the total energy of an energy signal is the same whether calculated in the time domain or in the frequency domain:

$$
E=\int_{-\infty}^{\infty}\lvert g(t)\rvert^2dt=\int_{-\infty}^{\infty}\lvert G(f)\rvert^2df
$$

**Derivation:** write $\lvert g(t)\rvert^2=g(t)g^*(t)$ and express $g^*(t)$ by the inverse transform:

$$
g^*(t)=\left[\int_{-\infty}^{\infty}G(f)e^{j2\pi ft}df\right]^*=\int_{-\infty}^{\infty}G^*(f)e^{-j2\pi ft}df
$$

Substitute into the energy integral and change the order of integration:

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

The inner bracket in the second line is just $G(f)$ by definition. Hence the theorem is proved.

**Meaning:** $\lvert G(f)\rvert^2$ is the **energy spectral density** $\Psi_g(f)$ (unit J/Hz); integrating it over all frequencies gives the total energy. Example: for $g(t)=A\,\text{rect}(t/T)$, time-domain energy is $A^2T$, and integrating $A^2T^2\text{sinc}^2(fT)$ over $f$ also gives $A^2T$.

## Check yourself

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

1. If a signal is compressed in time, its spectrum:
   A. Is compressed too
   B. Is unchanged
   C. Expands in frequency
   D. Shifts to a carrier
   Answer: C. g(at) <-> (1/|a|)G(f/a): compression in time expands the spectrum. (Key point: Time scaling)

2. What does delaying a signal by t0 do to its spectrum?
   A. Changes amplitude spectrum
   B. Multiplies G(f) by e^(-j2 pi f t0), changing only phase
   C. Shifts it in frequency by t0
   D. Scales it by t0
   Answer: B. The amplitude spectrum is unchanged; only linear phase is added. (Key point: Time shift)

3. g(t)cos(2 pi fc t) has spectrum:
   A. G(f)cos(2 pi fc)
   B. G(f-fc)G(f+fc)
   C. (1/2)[G(f-fc)+G(f+fc)]
   D. G(fc)
   Answer: C. Multiplying by a carrier shifts the spectrum to +-fc with half amplitude; this is AM. (Key point: Modulation property)

4. In the Rayleigh derivation, the inner integral of g(t)e^(-j2 pi f t) dt equals:
   A. G*(f)
   B. g(f)
   C. |G(f)|^2
   D. G(f)
   Answer: D. By definition it is G(f), leaving the integral of G*G = |G|^2. (Key point: Rayleigh derivation)

## 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
- [Rayleigh energy theorem](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Arayleigh-energy-theorem-prove) — Communication Systems, chapter 1
- [Hilbert vs Fourier transform; properties of HT](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ahilbert-vs-fourier-properties) — Communication Systems, chapter 2
- [Hilbert transform properties; energy and power of unit step](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ahilbert-properties-unit-step-energy) — Communication Systems, chapter 2
- [Autocorrelation and ESD as Fourier transform pair](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aautocorrelation-esd-fourier-pair) — Communication Systems, chapter 1
- [Hilbert transform properties; distortionless transmission](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ahilbert-properties-distortionless) — Communication Systems, chapter 2, asked 3×

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