stylistic:style_transfer
Discrete convolution

For complex-valued functions f {\displaystyle f} and g {\displaystyle g} defined on the set Z {\displaystyle \mathbb {Z} } of integers, the discrete convolution of f {\displaystyle f} and g {\displaystyle g} is given by:

or equivalently (see commutativity) by:

The convolution of two finite sequences is defined by extending the sequences to finitely supported functions on the set of integers. When the sequences are the coefficients of two polynomials, then the coefficients of the ordinary product of the two polynomials are the convolution of the original two sequences. This is known as the Cauchy product of the coefficients of the sequences.

Thus when g has finite support in the set { − M, − M + 1, …, M − 1, M } {\displaystyle \{-M,-M+1,\dots,M-1,M\}} (representing, for instance, a finite impulse response), a finite summation may be used: