“Autocorrelation”的概念、定义、习题解题方法及翻译-数学辞典

单词

Autocorrelation

释义

Autocorrelation

The autocorrelation function is defined by

(1)

where

denotes Convolution and

denotes Cross-Correlation. A finite autocorrelation is given by

			(2)
			(3)

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 FourierTransform known as the Wiener-Khintchine Theorem. Let , and denote the ComplexConjugate of , then the Fourier Transform of the Absolute Square of is given by

(8)

The autocorrelation is a Hermitian Operator since . is Maximum atthe 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.

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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