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The autocorrelation function is defined by
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(1) |
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(2) |
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(3) |
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(4) |
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(5) |
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(6) |
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(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
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(8) |
The autocorrelation is a Hermitian Operator since
.
is Maximum at
the Origin. In other words,
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(9) |
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(10) |
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(11) |
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(12) |
Define
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(13) |
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(14) |
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(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.
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© 1996-9 Eric W. Weisstein