Circular convolution ($c\circledast x$)

The wraparound version of convolution for two finite sequences, computed by a circulant matrix.

For two length- $N$ sequences $c$ and $x$ , their circular convolution $c\circledast x$ is what a circulant matrix built from $c$ produces when it multiplies $x$ -- convolution where indices wrap around modulo $N$ instead of running off to infinity. The discrete convolution theorem states that circular convolution in the sample domain corresponds to ordinary multiplication of DFTs in the frequency domain.

Examples

Circular convolution of two length-4 sequences never needs indices outside $0,1,2,3$ -- everything that would fall off the end wraps back around.