This Demonstration studies the equivalence of linear and circular convolutions. In signal processing, linear convolution (or simply convolution) refers to the convolution between infinitely supported sequences and filters, while circular convolution refers to the convolution between finitely supported and circularly extended sequences and filters (circular extension makes such sequences and filters periodic).

Given a sequence of length

and a filter with an impulse response of length

, linear and circular convolutions are equivalent when the period of the circular convolution,

, satisfies

In this Demonstration, the first graphic shows the sequence

of length

, the second graphic shows the filter with impulse response

of length

, and the third graphic shows the results of linear convolution,

(in black), and circular convolution,

(in red, repeated with period

). For

, linear and circular convolutions are equivalent (black and red stems are identical within one period); for

, linear and circular convolutions are not equivalent (black and red stems are not identical within a single period).