The autocorrelation function is defined by

(1) |

(2) | |||

(3) |

If is a Real Function,

(4) |

(5) |

(6) |

(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) |

(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) |

**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

1999-05-25