A convolution is an integral that 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").
Convolution is implemented in Mathematica as Convolve[f, g, x, y] and DiscreteConvolve[f,
g, n, m].
Abstractly, a convolution is defined as a product of functions and that are objects
in the algebra of Schwartz functions
in . Convolution of two functions and over a finite range
is given by
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(1)
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where the symbol denotes convolution of and .
Convolution is more often taken over an infinite range,
(Bracewell 1965, p. 25) with the variable (in this case ) implied, and also
occasionally written as .
The animations above graphically illustrate the convolution of two boxcar functions (left) and two Gaussians (right). In the plots, the green curve shows the
convolution of the blue and red curves as a function of , the position indicated
by the vertical green line. The gray region indicates the product as
a function of , so its area as a function of is precisely the
convolution. One feature to emphasize and which is not conveyed by these illustrations
(since they both exclusively involve symmetric functions) is that the function must be mirrored before lagging it across and integrating.
The convolution of two boxcar functions and
has the particularly simple form
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(4)
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where is the Heaviside step function. Even more amazingly, the convolution
of two Gaussians
is another Gaussian
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(7)
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Let , , and be arbitrary functions
and a constant. Convolution satisfies the properties
(Bracewell 1965, p. 27), as well as
(Bracewell 1965, p. 49).
Taking the derivative of a convolution
gives
(Bracewell 1965, p. 119).
The area under a convolution is the product
of areas under the factors,
The horizontal function centroids
of a convolution add
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(18)
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and provided that either or has its function centroids at its origin, the variances do as well
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(19)
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(Bracewell 1965, p. 142), where
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(20)
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There is also a definition of the convolution which arises in probability theory and is given by
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(21)
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where is a Stieltjes integral.
Bracewell, R. "Convolution" and "Two-Dimensional Convolution." Ch. 3 in The Fourier Transform and Its Applications. New York: McGraw-Hill,
pp. 25-50 and 243-244, 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.
Weisstein, E. W. "Books about Convolution." http://www.ericweisstein.com/encyclopedias/books/Convolution.html.
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