## 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.