We consider a commutative ring $R$ with identity and a positive integer $N$. We characterize all the 3-tuples $(L_1,L_2,L_3)$ of linear transforms over $R^N$, having the “circular convolution” property, i.eṡuch that $x*y=L_3(L_1\left(x\right)\otimes L_2\left(y\right))$ for all $x,y\in R^N$.