Circular Convolution

The circular convolution, also known as cyclic convolution, of two aperiodic functions occurs when one of them is convolved in the normal way with a periodic summation of the other function. That situation arises in the context of the Circular convolution theorem. The identical operation can also be expressed in terms of the periodic summations of both functions, if the infinite integration interval is reduced to just one period. That situation arises in the context of the discrete-time Fourier transform (DTFT) and is also called periodic convolution. In particular, the transform (DTFT) of the product of two discrete sequences is the periodic convolution of the transforms of the individual sequences.

For a periodic function x_T, with period T, the convolution with another function, h, is also periodic, and can be expressed in terms of integration over a finite interval as follows:

$\begin{align} (x_T * h)(t)\quad &\stackrel{\mathrm{def}}{=} \ \int_{-\infty}^\infty h(\tau)\cdot x_T(t - \tau)\,d\tau \\ &= \int_{t_o}^{t_o+T} h_T(\tau)\cdot x_T(t - \tau)\,d\tau, \end{align}$

where t_o is an arbitrary parameter, and h_T is a periodic summation of h, defined by:

This operation is a periodic convolution of functions x_T and h_T. When x_T is expressed as the periodic summation of another function, x, the same operation may also be referred to as a circular convolution of functions h and x.

Read more about Circular Convolution: Discrete Sequences, Example

Famous quotes containing the word circular:

“‘A thing is called by a certain name because it instantiates a certain universal’ is obviously circular when particularized, but it looks imposing when left in this general form. And it looks imposing in this general form largely because of the inveterate philosophical habit of treating the shadows cast by words and sentences as if they were separately identifiable. Universals, like facts and propositions, are such shadows.”
—David Pears (b. 1921)