Energy in a signal

• Signals and Systems

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,

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