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 xT, 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 to is an arbitrary parameter, and hT is a periodic summation of h, defined by:

This operation is a periodic convolution of functions xT and hT. When xT 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

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