An analytical approach for the synthesis of two-dimensional state-space filter structures with minimum weighted sensitivity

This paper considers the problem of synthesizing the finite-word-length (FWL) two-dimensional (2-D) state-space filter structures with minimum weighted sensitivity. Two kinds of frequency-weighted sensitivity measures, one based on a mixture of L₁/L₂ norms and the other a pure L₂ norm, are defined in place of the usual sensitivity measure and an upper bound expressed in terms of 2D weighted Gramians is used to evaluate the weighted L₁/L₂ mixed sensitivity. A simple technique is then developed for obtaining a set of filter structures with very low weighted L₁/L₂-sensitivity. In this connection, the optimal coordinate transformation is characterized in a closed form. Next, an iterative procedure is proposed to obtain the optimal coordinate transformation that minimizes the weighted L sensitivity measure. Once the initial value is given, the estimate at each iteration can be calculated analytically. Finally, two numerical examples are given to illustrate the utility of the proposed technique