“Convolution”的概念、定义、习题解题方法及翻译-数学辞典

单词

Convolution

释义

Convolution

A convolution is an integral which expresses the amount of overlap of one function as it is shifted over anotherfunction . It therefore ``blends'' one function with another. For example, in synthesis imaging, the measured Dirty Map is a convolution of the ``true'' CLEAN Map with theDirty Beam (the Fourier Transform of the sampling distribution). The convolution is sometimes also known byits German name, Faltung (``folding''). A convolution over a finite range is givenby

(1)

where the symbol

(occasionally also written as

) denotes convolution of

and

. Convolution is moreoften taken over an infinite range,

(2)

Let

, and

be arbitrary functions and

a constant. Convolution has the following properties:

(3)

(4)

(5)

(6)

The Integral identity

(7)

also gives a convolution. Taking the Derivative of a convolution gives

(8)

The Area under a convolution is the product of areas under the factors,



			(9)

The horizontal Centroids add

(10)

as do the Variances

(11)

where

(12)

See also Autocorrelation, Convolution Theorem, Cross-Correlation,Wiener-Khintchine Theorem

References

Convolution

Bracewell, R. ``Convolution.'' Ch. 3 in The Fourier Transform and Its Applications. New York: McGraw-Hill, pp. 25-50, 1965.

Hirschman, I. I. and Widder, D. V. The Convolution Transform. Princeton, NJ: Princeton University Press, 1955.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 464-465, 1953.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Convolution and Deconvolution Using the FFT.'' §13.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 531-537, 1992.

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