Commutativity and associativity of convolution Show by applying the convolution masks a) and d) from Exercise 4.2 to a step edge . . . 0 0 0 0 0 1 1 1 1 1 . . . that convolution is commutative and associative.
Convolution masks with even number of coefficients
Also for filters with an even number of coefficients (2R), it is possible to define filters with even and odd symmetry if we imagine the convolution result is put on an intermediate grid. The convolution mask can be written as [h−R, . . . , h−1,h1, . . . , hR].
The reference part (_R11) gives the equations for the transfer functions of these masks.
1. Prove these equations by applying a shift of half a grid distance to the general equation for the transfer function Eq. (4.23).
2. Compute the transfer functions of the two elementary masks [1 1]/2 (mean of two neighboring points) and [1 − 1] (difference of two neighboring points).