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

$W=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}P(t)dt$

Energy of a Signal

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

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

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

$W=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}|v{|}^2(t)dt$

This can be written using bra-ket notation as

$<v(t)|v(t)>$ or $<v|v>$

By Rayleigh's Theorem,

$<v|v>=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}|V(f){|}^{2}df$

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

$W=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}|V(f){|}^{2}df$