Definition
Let Σ be an alphabet, # a symbol not in Σ.
Let x1x2... x|x|, y1y2... y|y|, z1z2... z|z|, ... be n words (i.e. finite sequences) of elements of Σ. Let denote the length of the longest word, i.e. the maximum of |x|, |y|, |z|, ... .
The convolution of these words is a finite sequence of n-tuples of elements of (Σ ∪ {#}), i.e. an element of :
- ,
where for any index i > |w|, the wi is #.
The convolution of x, y, z, ... is denoted conv( x, y, z, ...), zip( x, y, z, ...) or x ⋆ y ⋆ z ⋆ ...
The inverse to convolution is sometimes denoted unzip.
A variation of the convolution operation is defined by:
where is the minimum length of the input words. It avoids the use of an adjoined element, but destroys information about elements of the input sequences beyond .
Read more about this topic: Convolution (computer Science)
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