A convolution is an integral which expresses the amount of overlap of one function as it is shifted over another
function . 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 the
Dirty Beam (the Fourier Transform of the sampling distribution). The convolution is sometimes also known by
its German name, Faltung (``folding''). A convolution over a finite range is given
by
|
(1) |
where the symbol (occasionally also written as ) denotes convolution of and . Convolution is more
often 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,
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.
© 1996-9 Eric W. Weisstein
1999-05-25