Fundamentals of Statistics contains material of various lectures and courses of H. Lohninger on statistics, data analysis and here for more.

Convolution - Calculation Rules

The following table gives a survey on some mathematical rules concerning the convolution operator:

Commutativity fg = gf
The order of the operands (functions) is exchangeable.
Associativity f(gh) = (fg)h = fgh
Convolution operations can be combined in arbitrary ways (the order in which the operations are performed does not matter).
Distributivity f(g+h) = fg + fh
The distributive law corresponds to the distributive law of the arithmetic multiplication.
Scalar Multiplication a(fg) = (af)g = f(ag)
If a function is multiplied by a scalar, the results of the convolution has to be multiplied by this scalar as well.
Convolution Theorem F (gh) = F (g)F (h)
The Fourier transform F() of a convolution product is proportional to the product of the Fourier transforms of the individual functions g and h.
Derivation Rule D(gh) = D(g)h = gD(h)
The first derivative D() of a convolution product is equal to the convolution of one of the original functions and the first derivative of the other function. Please note that for discrete sequences the first derivative is the difference between two consecutive elements of the sequence: D(gn) = gn+1 - gn

Last Update: 2012-10-08