This paper treats error spectrum shaping for two-dimensional recursive digital filters, described by a rational transfer function as well as the Roesser local state-space model. For each description of the filter, a technique is developed for obtaining the optimal error-feedback (EF) coefficients that minimize the noise variance at the filter output. The optimal solution can be characterized in a closed form which minimizes a quadratic form subject to constraints such as symmetry, antisymmetry, quadrantal symmetry, or quadrantal antisymmetry of the EF coefficients. In an unconstrained case, the optimal EF corresponds to a special case of the above optimal solution. Finally, two numerical examples are given to illustrate the utility of the proposed technique. In the numerical examples, the discrete sub-optimal coefficients of the EF are obtained by applying an existing procedure for the least-squares discrete-space optimization.
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