Rayleigh energy theorem

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.