The autocorrelation function is defined by
(1) |
(2) | |||
(3) |
(4) |
(5) |
(6) |
(7) |
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) |
(10) |
(11) |
(12) |
Define
(13) | |||
(14) |
(15) |
See also Convolution, Cross-Correlation, Quantization Efficiency, Wiener-Khintchine Theorem
References
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.
© 1996-9 Eric W. Weisstein