Power Theorem

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

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

Proof:

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

Note that the power theorem would be more elegant ( $\left<x,y\right> = \left<X,Y\right>$) if the DFT were defined as the coefficient of projection onto the normalized DFT sinusoid ${\tilde s}_k(n) \isdeftext s_k(n)/\sqrt{N}$.

The power theorem is also sometimes called Parseval's theorem [42].