linguistic:translation, information:filtering
Circular discrete convolution

When a function g N {\displaystyle g_{_{N}}} is periodic, with period N, {\displaystyle N,} then for functions, f, {\displaystyle f,} such that f ∗ g N {\displaystyle f*g_{_{N}}} exists, the convolution is also periodic and identical to:

The summation on k {\displaystyle k} is called a periodic summation of the function f. {\displaystyle f.}

If g N {\displaystyle g_{_{N}}} is a periodic summation of another function, g, {\displaystyle g,} then f ∗ g N {\displaystyle f*g_{_{N}}} is known as a circular convolution of f {\displaystyle f} and g. {\displaystyle g.}

When the non-zero durations of both f {\displaystyle f} and g {\displaystyle g} are limited to the interval, {\displaystyle,} f ∗ g N {\displaystyle f*g_{_{N}}} reduces to these common forms:

The notation f ∗ N g {\displaystyle f*_{N}g} for cyclic convolution denotes convolution over the cyclic group of integers modulo N.

Circular convolution arises most often in the context of fast convolution with a fast Fourier transform (FFT) algorithm.