Smitry at 06:36, 13 October 2006

2006-10-13T06:36:58Z

← Older revision		Revision as of 23:36, 12 October 2006
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			*[[Signals and systems\|Signals and Systems]]
	Rayleigh's Theorem is derived from the equation for Energy		Rayleigh's Theorem is derived from the equation for Energy
	*<math> W = \int_{-\infty}^{\infty}p(t)\,dt </math>		*<math> W = \int_{-\infty}^{\infty}p(t)\,dt </math>

Sherna at 09:36, 11 October 2006

2006-10-11T09:36:17Z

New page

Rayleigh's Theorem is derived from the equation for Energy
*<math> W = \int_{-\infty}^{\infty}p(t)\,dt </math>
If we assume that the circuit is a Voltage applied over a load then <math> p(t)=\frac{x^2(t)}{R_L}</math>
for matters of simplicity we can assume <math>R_L = 1\, \Omega</math>
<br>
This leaves us with
*<math> W = \int_{-\infty}^{\infty}|x|^2(t)\,dt</math>
This is the same as the dot product so to satisfy the condition for complex numbers it becomes
*<math> W = \int_{-\infty}^{\infty}x(t)\,x^*(t)\,dt</math>
If we substitute <math> x(t) = \int_{-\infty}^{\infty}X(f)\,e^{j2\pi ft}\,df </math> and <math>x^*(t)= \int_{-\infty}^{\infty}X(f')\,e^{-j2\pi f't}\,df'</math>
<br>
<br>Substituting this back into the original equation makes it
*<math>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</math>
*<math>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</math>
The time integral becomes <math> \delta (f-f') \,which \ is\ 0\ except\ for\ when\ f' = f</math>
This simplifies the above equation such that
*<math>W = \int_{-\infty}^{\infty}X(f)\,\int_{-\infty}^{\infty}X^*(f')\left(\delta (f-f') \right)\,df'\,df</math>
*<math>W = \int_{-\infty}^{\infty}X(f)\,X^*(f)\,df</math>
Proving that the energy in the time domain is the same as that in the frequency domain
*<math> W = \int_{-\infty}^{\infty}X(f)\,X^*(f)\,df = \int_{-\infty}^{\infty}x(t)\,x^*(t)\,dt</math>

Rayleigh's Theorem - Revision history

Smitry at 06:36, 13 October 2006

Sherna at 09:36, 11 October 2006