Energy in a signal: Difference between revisions

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: <math> W = \int_{-\infty}^{\infty} P(t)\,dt</math>
: <math> W = \int_{-\infty}^{\infty} P(t)\,dt</math>


===Energy of a signal===
===Energy of a Signal===
From circiut analysis we know that the power of a voltage source is,
From circuit analysis we know that the power generated by a voltage source is,
: <math>P(t) = {\mathbf{V}^2(t) \over R}</math>
: <math>P(t) = {v^2(t) \over R}</math>
Assuming that R is 1 then the total energy is just,
Assuming that R is 1 then the total energy is just,
: <math>W = \int_{-\infty}^\infty |\mathbf{V}|^2(t) \mathrm{d}\mathbf{t}</math>
: <math>W = \int_{-\infty}^\infty |v|^2(t) \, dt</math>
This can be written using [http://en.wikipedia.org/wiki/Bra-ket_notation bra-ket] notation as

: <math> <v(t) | v(t)> \!</math> or <math> <v|v> \!</math>
This page is far from complete please feel free to pick up where it has been left off.
By [[Rayleigh's Theorem]],
: <math> <v|v> = \int_{-\infty}^{\infty} |V(f)|^2\,df </math>
This implies that the energy of a signal can be found by the fourier transform of the signal,
: <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,

Power represents a change in energy.

This means we can also write energy as

Energy of a Signal

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

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

This can be written using bra-ket notation as

or

By Rayleigh's Theorem,

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