Local convolution of l-adic sheaves on the torus

For K and L two l-adic perverse sheaves on the one-dimensional torus over the algebraic closure of a finite field, we show that the local monodromies of their convolution K*L at its points of non-smoothness is completely determined by the local monodromies of K and L. This generalizes a previous result of N. Katz for the case where K and L are smooth, tame at 0 and totally wild at infinity.