Energy in a signal

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

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

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

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

This page is far from complete please feel free to pick up where it has been left off.