Convolution

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

$$f(t)*g(t) \equiv \int_0^t f(\tau)g(t-\tau)\,d\tau,$$ (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,

$$f(t)*g(t) \equiv \int_{-\infty}^\infty f(\tau)g(t-\tau)\,d\tau = \int_{-\infty}^\infty g(\tau)f(t-\tau)\,d\tau.$$ (2)

Let $f$, $g$, and $h$ be arbitrary functions and $a$ a constant. Convolution has the following properties:

$$f*g = g*f$$ (3)

$$f*(g*h) = (f*g)*h$$ (4)

$$f*(g+h) = (f*g)+(f*h)$$ (5)

$$a(f*g) = (af)*g = f*(ag).$$ (6)

The Integral identity

$$\int^x_a \int^x_a f(t)\,dt\,dx = \int^x_a (x-t)f(t)\,dt$$ (7)

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

$${d\over dx} (f*g)={df\over dx}*g = f*{dg\over dx}.$$ (8)

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

$$\int_{-\infty}^\infty (f*g)\,dx = \int_{-\infty}^\infty\left[{\int_{-\infty}^\infty f(u)g(x-u)\,du}\right]\,dx$$

$$= \int_{-\infty}^\infty f(u)\left[{\int_{-\infty}^\infty g(x-u)\,dx}\right]\,du$$

$$= \left[{\int_{-\infty}^\infty f(u)\,du}\right]\left[{\int_{-\infty}^\infty g(x)\,dx}\right].$$ (9)

The horizontal Centroids add

$$\int_{-\infty}^\infty \left\langle{x(f*g)}\right\rangle{}\, ... ...eft\langle{xf}\right\rangle{}+\left\langle{xg}\right\rangle{},$$ (10)

as do the Variances

$$\int_{-\infty}^\infty \left\langle{x^2(f*g)}\right\rangle{}\... ...langle{x^2f}\right\rangle{}+\left\langle{x^2g}\right\rangle{},$$ (11)

where

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