Rayleigh's Theorem

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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