Correlation Theorem

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

$$\zbox {x\star y \leftrightarrow \overline{X}\cdot Y}$$

where the correlation operation `$\star$' was defined in §7.2.4.

Proof:

$$\begin{eqnarray*} (x\star y)_n &\isdef & \sum_{m=0}^{N-1}\overline{x(m)}y(n+m)... ...{x})\ast y\right)_n \\ &\leftrightarrow & \overline{X} \cdot Y \end{eqnarray*}$$

The last step follows from the convolution theorem and the result FLIP$(\overline{x}) \leftrightarrow \overline{X}$ from §7.4.2. Also, the summation range in the second line is equivalent to the range $[N-1,0]$ because all indexing is modulo $N$.