Energy in a signal

From Class Wiki
Jump to: navigation, search

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