ListConvolve

Usage

• ListConvolve[ker, list] forms the convolution of the kernel ker with list.
• ListConvolve[ker, list, k] forms the cyclic convolution in which the k element of ker is aligned with each element in list.
• ListConvolve[ker, list, {, }] forms the cyclic convolution whose first element contains list[[1]] ker[[]] and whose last element contains list[[-1]] ker[[]].
• ListConvolve[ker, list, klist, p] forms the convolution in which list is padded at each end with repetitions of the element p.
• ListConvolve[ker, list, klist, {, , ... }] forms the convolution in which list is padded at each end with cyclic repetitions of the .
• ListConvolve[ker, list, klist, padding, g, h] forms a generalized convolution in which g is used in place of Times and h in place of Plus.
• ListConvolve[ker, list, klist, padding, g, h, lev] forms a convolution using elements at level lev in ker and list.

Notes

• With kernel and list , ListConvolve[ker, list] computes , where the limits of the sum are such that the kernel never overhangs either end of the list.

• Example: ListConvolve[x,y, a,b,c].

• The values of and in ListConvolve[ker, list, {, }] determine the amount of overhang to allow at each end of list.

• Common settings for {, } are:

• Examples: ListConvolve[x,y, a,b,c, 1,1].

• ListConvolve[x,y, a,b,c, 1,-1].

• ListConvolve[ker, list, {, }, padlist] effectively lays down repeated copies of padlist, then superimposes one copy of list on them and forms a convolution of the result.

p	pad with repetitions of a single element
{, , ... }	pad with cyclic repetitions of a sequence of elements
list	pad by treating list as cyclic (default)
{}	do no padding

• ListConvolve[ker, list, {{, , ... }, {, , ... }}] forms the cyclic convolution whose {1,1, ... } element contains ker[[, , ... ]] list[[1,1, ... ]] and whose {-1,-1, ... } element contains ker[[, , ... ]] list[[-1,-1, ... ]].

• {, } is taken to be equivalent to {{, , ... }, {, , ... }}.

Further Examples