# Circular convolution

The circular convolution $$\mathbf{z} = \mathbf{x} \circledast \mathbf{y}$$ between two vectors $$\mathbf{x}$$ and $$\mathbf{y}$$ in $$\mathbf{C}^N$$ is defined by: $z_n = \sum_{k=0}^{N-1} x_{(k - n\ \mathrm{mod. } N)} y_k$ for $$0 \leq n < N$$.

It can be interpreted as the product of a circular matrix and a vector. With $$N = 4$$ : $\left( \begin{array}{c} z_0 \\ z_1 \\ z_2 \\ z_3 \end{array} \right) = \left( \begin{array}{cccc} x_0 & x_{3} & x_2 & x_1 \\ x_1 & x_0 & x_3 & x_2 \\ x_2 & x_1 & x_0 & x_3 \\ x_3 & x_2 & x_1 & x_2 \end{array} \right) \left( \begin{array}{c} y_0 \\ y_1 \\ y_2 \\ y_3 \end{array} \right)$ or equivalently, using the commutativity of the convolution: $\left( \begin{array}{c} z_0 \\ z_1 \\ z_2 \\ z_3 \end{array} \right) = \left( \begin{array}{cccc} y_0 & y_{3} & y_2 & y_1 \\ y_1 & y_0 & y_3 & y_2 \\ y_2 & y_1 & y_0 & y_3 \\ y_3 & y_2 & y_1 & y_2 \end{array} \right) \left( \begin{array}{c} x_0 \\ x_1 \\ x_2 \\ x_3 \end{array} \right)$

Consequence : coefficients in the tails of the input signals will influence the head of the result.