Convolution

Z←L f⍡g R returns the convolution (moving window inner product) of L vs. R.

L and R are vectors; scalars are promoted to one-element vectors.

f and g are functions.

Z is a vector whose shape is (⍴L)+(⍴R)-1.

The result is obtained by dragging the reverse of the shorter argument through all positions of the longer argument (as in a moving window) and performing an f.g inner product between the two, including leading and trailing prefixes. For example,

      L←1 3 2 1 ⋄ R←2 1 ⋄ L+⍡×R

2	1 2	1 2	1 2	1	multiply the rows together
1 3 2 1	1 3 2 1	1 3 2 1	1 3 2 1	1 3 2 1
2	1 6	3 4	2 2	1	add the products

L+⍡×R 2 7 7 4 1

Interestingly, this algorithm solves a diverse set of problems such as weighted moving average (used to remove pixelization from a digital image), polynomial multiplication, and overlapping string searching.

Polynomial multiplication is illustrated in the above example using the functions + and ×. Overlapping string searching uses the functions ∧ and = as in

L←'abababc' ⋄ R←'aba'

(0⌈¯1+(⍴L)⌊⍴R)↓L∧⍡=⌽R

1 0 1 0 0 0 0

where (0⌈¯1+(⍴L)⌊⍴R)↓ is needed to remove the leading prefix comparisons.