This paper proposes a new expression of frequency transformation for state-space digital filters. The proposed frequency transformation has the property of preserving the controllability Gramian and the observability Gramian of prototype state-space digital filters. This property shows that the proposed frequency transformation preserves the useful realizations of prototype digital filters such as minimum roundoff noise realizations, balanced realizations and limit cycle free realizations. Moreover, it is proved that the proposed frequency transformation retains an important inequality property with respect to the norm of the state transition matrix of limit cycle free digital filters. These results are derived from a modified formulation of Mullis and Roberts' frequency transformation and balanced realizations of allpass filters.