Rounding of discrete variables

Let $X$ be a random variable that takes its values in $\frac{1}{q}\mathbb{Z}$, for some integer $q\ge2$, and consider $X$ rounded to an integer, either downwards or upwards or to the nearest integer. We give general formulas for the characteristic function and moments of the rounded variable. These formulas complement the related but different formulas in the case that $X$ has a continuous distribution, which was studied by Janson (2006).