Optimal synthetic FWI design of state-space digital filters

The optimal finite word length (FWL) design problem of state-space filters is investigated. Instead of the usual L₁/L₂-mixed sensitivity measure, it is argued that a sensitivity measure based on the L₂ norm only is natural and reasonable. The minimization problem of this newly defined sensitivity measure is studied. The set of optimal realizations minimizing this measure is characterized. It is shown that the FWL effects can be synthesized by what is called FWL noise gain (FNG), which is a linear combination of the classical roundoff noise gain and the L₂ sensitivity measure. By minimizing the FNG with dynamical constraint, the optimal synthetic FWL state-space design problem is formulated. The existence of optimal realizations is shown. The necessary and sufficient condition equation that should be satisfied by the optimal realizations is given