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The effect of roundoff noise in a digital controller is analysed 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 realisations can be solved using an existing procedure. Noting that the optimal realisations are fully parametrised, based on a polynomial operator approach a new sparse controller realisation is derived. This realisation is a generalisation of the direct forms in the classical shift operator and the prevailing delta-operator. It provides more degrees of freedom with which to reduce the roundoff noise. The problem of finding optimal polynomial operators can be solved by an exhaustive search, and a design example is given. It is shown that with the proposed sparse realisation the optimal polynomial operators can outperform the shift- and delta-operators.