# 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,

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

Power represents a change in energy.

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

This means we can also write energy as

${\displaystyle W=\int _{-\infty }^{\infty }P(t)\,dt}$

### Energy of a Signal

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

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

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

${\displaystyle W=\int _{-\infty }^{\infty }|v|^{2}(t)\,dt}$

This can be written using bra-ket notation as

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

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

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