Rayleigh's Energy Theorem

Energy of a Signal

The energy of a signal x(t) is defined as the area under the curve of square of magnitude of that signal, i.e.,

$$\mathrm{E \:=\: \int_{-\infty}^{\infty}\: |x(t)|^2 \: dt}$$

The energy signal exists only of the energy (E) of the signal is finite, i.e., only if 0 < E < ∞.

Rayleigh's Energy Theorem

Statement - The Rayleigh's energy theorem states that the integral of the square of magnitude of a function (i.e., energy of the function) is equal to the integral of the square of magnitude of its Fourier transform, i.e.,

$$\mathrm{E \:=\: \int_{-\infty}^{\infty}\: |x(t)|^2 \: dt \:=\: \frac{1}{2\pi} \int_{-\infty}^{\infty}\: |X(\omega)|^2 \: d\omega}$$

Proof

Consider a function x(t) such that its Fourier transform pair is,

$$\mathrm{x(t)\:\overset{LT}\longleftrightarrow\:X(\omega)}$$

Assume x^*(t) is the conjugate of the function x(t) and its Fourier transform pair is,

$$\mathrm{x^*(t)\:\overset{LT}\longleftrightarrow\:X^*(−\omega)}$$

Then, the energy of the signal x(t) is given by,

$$\mathrm{E \:=\: \int_{-\infty}^{\infty}\:|x(t)|^2\:dt}$$

$$\mathrm{because\:|x(t)|^2 \:=\: x(t)x^*(t)}$$

$$\mathrm{\therefore \:E \:=\: \int_{-\infty}^{\infty}\: x(t) x^*(t) \: dt \:=\: \int_{-\infty}^{\infty}\: x^*(t) x(t) \: dt}$$

Now, by replacing the function x(t) by its inverse Fourier transform, we get,

$$\mathrm{E\:=\:\int_{-\infty}^{\infty}\:x^{*}(t)\left[\frac{1}{2\pi}\int_{-\infty}^{\infty}\:X(\omega)e^{j\omega t}\:d\omega\right]\:dt}$$

By interchanging the order of integration in above equation, we get,

$$\mathrm{E = \frac{1}{2\pi} \int_{-\infty}^{\infty} X(\omega) \left[ \int_{-\infty}^{\infty} x^*(t) e^{j\omega t} dt \right] d\omega }$$

$$\mathrm{\Rightarrow\: E \:=\: \frac{1}{2\pi} \int_{-\infty}^{\infty}\: X(\omega) X^*(-\omega)\: d\omega }$$

$$\mathrm{\therefore\: X(\omega)\: X^*(-\omega) \:=\: |X(\omega)|^2}$$

Therefore, we have,

$$\mathrm{E \:=\: \frac{1}{2\pi} \int_{-\infty}^{\infty}\: |X(\omega)|^2 \:d\omega \:=\: \int_{-\infty}^{\infty}\: |x(t)|^2\: dt}$$

This is called the Rayleigh's energy theorem. The Rayleigh's energy theorem is called the Parseval's theorem for energy signals.

The above expression proves that the integral of the square of a signal is equal to the integral of square of its Fourier transform.