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A convolution is an integral which expresses the amount of overlap of one function $g(t)$ as it is shifted over another function $f(t)$. 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 $[0, t]$ is given by

f(t)*g(t) \equiv \int_0^t f(\tau)g(t-\tau)\,d\tau,
\end{displaymath} (1)

where the symbol $f*g$ (occasionally also written as $f\otimes g$) denotes convolution of $f$ and $g$. Convolution is more often taken over an infinite range,
f(t)*g(t) \equiv \int_{-\infty}^\infty f(\tau)g(t-\tau)\,d\tau = \int_{-\infty}^\infty g(\tau)f(t-\tau)\,d\tau.
\end{displaymath} (2)

Let $f$, $g$, and $h$ be arbitrary functions and $a$ a constant. Convolution has the following properties:
f*g = g*f
\end{displaymath} (3)

f*(g*h) = (f*g)*h
\end{displaymath} (4)

f*(g+h) = (f*g)+(f*h)
\end{displaymath} (5)

a(f*g) = (af)*g = f*(ag).
\end{displaymath} (6)

The Integral identity
\int^x_a \int^x_a f(t)\,dt\,dx = \int^x_a (x-t)f(t)\,dt
\end{displaymath} (7)

also gives a convolution. Taking the Derivative of a convolution gives
{d\over dx} (f*g)={df\over dx}*g = f*{dg\over dx}.
\end{displaymath} (8)

The Area under a convolution is the product of areas under the factors,
$\displaystyle \int_{-\infty}^\infty (f*g)\,dx$ $\textstyle =$ $\displaystyle \int_{-\infty}^\infty\left[{\int_{-\infty}^\infty f(u)g(x-u)\,du}\right]\,dx$  
  $\textstyle =$ $\displaystyle \int_{-\infty}^\infty f(u)\left[{\int_{-\infty}^\infty g(x-u)\,dx}\right]\,du$  
  $\textstyle =$ $\displaystyle \left[{\int_{-\infty}^\infty f(u)\,du}\right]\left[{\int_{-\infty}^\infty g(x)\,dx}\right].$ (9)

The horizontal Centroids add
\int_{-\infty}^\infty \left\langle{x(f*g)}\right\rangle{}\, ...
\end{displaymath} (10)

as do the Variances
\int_{-\infty}^\infty \left\langle{x^2(f*g)}\right\rangle{}\...
\end{displaymath} (11)

\left\langle{x^n f}\right\rangle{}\equiv {\int_{-\infty}^\infty x^n f(x)\,dx\over \int_{-\infty}^\infty f(x)\,dx}.
\end{displaymath} (12)

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



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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