## Wiener-Khintchine Theorem

Recall the definition of the Autocorrelation function of a function ,

 (1)

Also recall that the Fourier Transform of is defined by
 (2)

giving a Complex Conjugate of
 (3)

Plugging and into the Autocorrelation function therefore gives

 (4)

so, amazingly, the Autocorrelation is simply given by the Fourier Transform of the Absolute Square of ,
 (5)

The Wiener-Khintchine theorem is a special case of the Cross-Correlation Theorem with .