## Autocorrelation

The autocorrelation function is defined by

 (1)

where denotes Convolution and denotes Cross-Correlation. A finite autocorrelation is given by
 (2) (3)

If is a Real Function,
 (4)

and an Even Function so that
 (5)

then
 (6)

But let , so , then
 (7)

The autocorrelation discards phase information, returning only the Power. It is therefore not reversible.

There is also a somewhat surprising and extremely important relationship between the autocorrelation and the Fourier Transform known as the Wiener-Khintchine Theorem. Let , and denote the Complex Conjugate of , then the Fourier Transform of the Absolute Square of is given by

 (8)

The autocorrelation is a Hermitian Operator since . is Maximum at the Origin. In other words,

 (9)

To see this, let be a Real Number. Then
 (10)

 (11)

 (12)

Define

 (13) (14)

Then plugging into above, we have . This Quadratic Equation does not have any Real Root, so , i.e., . It follows that
 (15)

with the equality at . This proves that is Maximum at the Origin.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Correlation and Autocorrelation Using the FFT.'' §13.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 538-539, 1992.