Rayleigh's Theorem

• Signals and Systems

Rayleigh's Theorem is derived from the equation for Energy

• $W=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}p(t)dt$

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 $R_L=1\mathrm{\Omega }$
This leaves us with

• $W=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}|x{|}^2(t)dt$

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

• $W=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(t)x^{*}(t)dt$

If we substitute $x(t)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}X(f)e^{j2\pi ft}df$ and $x^{*}(t)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}X(f^{\prime })e^{-j2\pi f^{\prime }t}d{f}^{\prime }$

Substituting this back into the original equation makes it

• $W=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}(\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}X(f)e^{j2\pi ft}df)(\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{X}^{*}(f^{\prime })e^{-j2\pi f^{\prime }t}d{f}^{\prime })dt$
• $W=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}X(f)\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{X}^{*}(f^{\prime })\left(\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{j2\pi (f-f^{\prime })t}dt\right)d{f}^{\prime }df$

The time integral becomes $$\begin{gathered} \delta (f-f^{\prime })which \\ is \\ 0 \\ except \\ for \\ when \\ {f}^{\prime }=f \end{gathered}$$ This simplifies the above equation such that

• $W=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}X(f)\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{X}^{*}(f^{\prime })(\delta (f-f^{\prime }))d{f}^{\prime }df$
• $W=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}X(f)X^{*}(f)df$

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

• $W=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}X(f)X^{*}(f)df=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(t)x^{*}(t)dt$