Difference between revisions of "Energy in a signal"

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(Definition of Energy)
 
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*[[Signals and systems|Signals and Systems]]
 
===Definition of Energy===
 
===Definition of Energy===
Enegry is the ability or pentential for something to create change. Scientifically enegry is defined as total work done force. Work can be mathematically calculated as the line integral of force per infinatesimal unit distance,
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Energy is the ability or potential for something to create change. Scientifically energy is defined as total work done by a force. Work can be mathematically calculated as the line integral of force per infinatesimal unit distance,
: <math> '''W''' = \int \mathbf{F} \cdot \mathrm{d}\mathbf{s}</math>
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: <math> W = \int \mathbf{F} \cdot \mathrm{d}\mathbf{s}</math>
   
===Energy of a signal===
 
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Power represents a change in energy.
From circiut analysis we know that the energy of a voltage source is,
 
 
: <math> P(t) = \frac{dW}{dt} </math>
: <math>W = {V^2 \over R}</math>
 
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This means we can also write energy as
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: <math> W = \int_{-\infty}^{\infty} P(t)\,dt</math>
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===Energy of a Signal===
 
From circuit analysis we know that the power generated by a voltage source is,
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: <math>P(t) = {v^2(t) \over R}</math>
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Assuming that R is 1 then the total energy is just,
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: <math>W = \int_{-\infty}^\infty |v|^2(t) \, dt</math>
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This can be written using [http://en.wikipedia.org/wiki/Bra-ket_notation bra-ket] notation as
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: <math> <v(t) | v(t)> \!</math> or <math> <v|v> \!</math>
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By [[Rayleigh's Theorem]],
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: <math> <v|v> = \int_{-\infty}^{\infty} |V(f)|^2\,df </math>
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This implies that the energy of a signal can be found by the fourier transform of the signal,
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: <math> W = \int_{-\infty}^{\infty} |V(f)|^2\,df </math>

Latest revision as of 02:38, 11 October 2006

Definition of Energy

Energy is the ability or potential for something to create change. Scientifically energy is defined as total work done by a force. Work can be mathematically calculated as the line integral of force per infinatesimal unit distance,

 W = \int \mathbf{F} \cdot \mathrm{d}\mathbf{s}

Power represents a change in energy.

 P(t) = \frac{dW}{dt}

This means we can also write energy as

 W = \int_{-\infty}^{\infty} P(t)\,dt

Energy of a Signal

From circuit analysis we know that the power generated by a voltage source is,

P(t) = {v^2(t) \over R}

Assuming that R is 1 then the total energy is just,

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

This can be written using bra-ket notation as

 <v(t) | v(t)> \! or  <v|v> \!

By Rayleigh's Theorem,

 <v|v> = \int_{-\infty}^{\infty} |V(f)|^2\,df

This implies that the energy of a signal can be found by the fourier transform of the signal,

 W = \int_{-\infty}^{\infty} |V(f)|^2\,df