In this paper, the effect of roundoff noise in a digital controller is analyzed for a digital feedback control system. An analytical expression for the roundoff noise gain, defined as the ratio between the variances of the output error and the rounding error, is obtained. The problem of identifying the minimum roundoff noise realizations can be solved using an existing procedure. Noting that the optimal realizations are fully parametrized, based on a polynomial operator approach a new sparse controller realization is derived. This realization is a generalization of the direct forms in the classical shift operator and the prevailing delta operator. It provides us more degrees of freedom to reduce the roundoff noise. The problem of finding optimal polynomial operators can be solved with exhaustive search, and a design example is given. It is shown that with the proposed sparse realization the optimal polynomial operators can outperform the shift- and delta-operators.