Abstract-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 L 1 =L 2 norms and the other a pure L 2 norm, are defined in place of the usual sensitivity measure and an upper bound expressed in terms of 2-D weighted Gramians is used to evaluate the weighted L 1 =L 2 mixed sensitivity. A simple technique is then developed for obtaining a set of filter structures with very low weighted L 1 =L 2 -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 2 -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.Index Terms-Finite word length, optimal realization, Roesser model, two-dimensional IIR digital filter, weighted coefficient sensitivity.

Based on the Fornasini-Marchesini second local state-space model, the problem of synthesizing the optimal finite word length 2-D state-space fllter structures is considered. First, a frequency-weighted sensitivity measure is defined in place of the usual sensitivity measure. Two techniques are then developed for finding the set of optimal filter structures that minimize this frequency-weighted sensitivity measure over all the similarity transformations. One is analytically found by applying Lagurange's method and the other is iteratively done by using a gradient method. Finally, a numerical example is given to illustrate the utility of the proposed two techniques.

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ABSTRUCT An efficient technique is developed for synthesizing two dimensional (2-D) separable-denominator filter structures with low sensitivity. A 2-D separable-denominator fIlter is decoupled into two 1-D fIlters. The sensitivity of a 2-D separable denominator filter is analyzed in term of each 1-D filter and is evaluated independently by using two performance measures. By minimizing two performance measures individually, 2-D separahle-denominator filter structures with low sensitivity are synthesized in unconstraint case and in the case of scaling constraints on the state variables. It is also clarified that the balanced realization is not a minimum sensitivity realization for a 2-D separable-denominator filter. Finally, a numerical example is given to illustrate the utility of the proposed technique.

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