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
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 .
See also Autocorrelation, Cross-Correlation Theorem, Fourier Transform
© 1996-9 Eric W. Weisstein
1999-05-26