Power Theorem

Theorem: For all $x,y\in\mathbb{C}^N$ ,

$$\zbox {\left<x,y\right> = \frac{1}{N}\left<X,Y\right>.}$$

Proof:

$$\begin{eqnarray*} \left<x,y\right> &\isdef & \sum_{n=0}^{N-1}x(n)\overline{y(n)} = (y\star x)_0 = \hbox{\sc DFT}_0^{-1}(\overline{Y}\cdot X) \\ &=& \frac{1}{N} \sum_{k=0}^{N-1}X(k)\overline{Y(k)} \isdef \frac{1}{N} \left<X,Y\right>. \end{eqnarray*}$$

As mentioned in §5.8, physical power is energy per unit time.^7.20 For example, when a force produces a motion, the power delivered is given by the force times the velocity of the motion. Therefore, if $x(n)$ and $y(n)$ are in physical units of force and velocity (or any analogous quantities such as voltage and current, etc.), then their product $x(n)y(n)\isdeftext f(n)v(n)$ is proportional to the power per sample at time $n$ , and $\left<f,v\right>$ becomes proportional to the total energy supplied (or absorbed) by the driving force. By the power theorem, ${F(k)}\overline{V(k)}/N$ can be interpreted as the energy per bin in the DFT, or spectral power, i.e., the energy associated with a spectral band of width $2\pi/N$ .^7.21