Convolution

y (t) = ò ¥-¥ x(t) h(t - t) dt º x(t) ´ h(t) = ò ¥-¥ h(t) x(t - t) dt y (t) is convolution of function x(t), h(t), "sort of" a sliding weighted average of 1st function [x(s)] with 2nd function providing weights. Pick a t, integrate over all s then plot result at t, choose next t, and repeat.

If you convolve a function with an impulse function - get original function at impulse points. Example: y(t) = ò ¥ -¥ [d (t - T) + d (t + T)] x (t - t) dt but property of direct delta function ò ¥ -¥ d(t - T) x(t) dt = x(T) so y(t) = x(t - T) + x(t + T) Importance of convolution integral ® with respect to FT. Multiplication in one domain = convolution in other domain. eg. h(x)· k(x) Û H(f) ´ K(f) spatial domain frequency (or time) Note: Remember logarithms where multiplication of numbers become addition in "log. domain." So multiplication of function by sampling function in spatial domain = convolution of FT's of each of the functions.

Last changed April 01, 1998, G. Scott Owen, owen@siggraph.org