# Rayleigh's Theorem

Rayleigh's Theorem is derived from the equation for Energy

• $W = \int_{-\infty}^{\infty}p(t)\,dt$

If we assume that the circuit is a Voltage applied over a load then $p(t)=\frac{x^2(t)}{R_L}$ for matters of simplicity we can assume $R_L = 1\, \Omega$
This leaves us with

• $W = \int_{-\infty}^{\infty}|x|^2(t)\,dt$

This is the same as the dot product so to satisfy the condition for complex numbers it becomes

• $W = \int_{-\infty}^{\infty}x(t)\,x^*(t)\,dt$

If we substitute $x(t) = \int_{-\infty}^{\infty}X(f)\,e^{j2\pi ft}\,df$ and $x^*(t)= \int_{-\infty}^{\infty}X(f')\,e^{-j2\pi f't}\,df'$

Substituting this back into the original equation makes it

• $W = \int_{-\infty}^{\infty}\left(\int_{-\infty}^{\infty}X(f)\,e^{j2\pi ft}\,df\right) \,\left(\int_{-\infty}^{\infty}X^*(f')\,e^{-j2\pi f't}\,df'\right)\,dt$
• $W = \int_{-\infty}^{\infty}X(f)\,\int_{-\infty}^{\infty}X^*(f')\left(\int_{-\infty}^{\infty}e^{j2\pi (f-f')t}\,dt\right)\,df'\,df$

The time integral becomes $\delta (f-f') \,which \ is\ 0\ except\ for\ when\ f' = f$ This simplifies the above equation such that

• $W = \int_{-\infty}^{\infty}X(f)\,\int_{-\infty}^{\infty}X^*(f')\left(\delta (f-f') \right)\,df'\,df$
• $W = \int_{-\infty}^{\infty}X(f)\,X^*(f)\,df$

Proving that the energy in the time domain is the same as that in the frequency domain

• $W = \int_{-\infty}^{\infty}X(f)\,X^*(f)\,df = \int_{-\infty}^{\infty}x(t)\,x^*(t)\,dt$