Design of Circularly Symmetric Two-dimensional Digital Filters Using Schur Decomposition and Singular Perturbation Approximation
Rabah W. Aldhaheri · 2003
This paper presents a new and numerically efficient technique for the design of a low order 2-D IIR digital filter with approximately linear-phase in the passband. This technique is based on two steps: First, the 2-D impulse response specification is decomposed by Schur method into a parallel realization of K sections; each consists of two cascaded SISO 1-D FIR filters. Second, the singular perturbation approximation is used to convert the constituent N-dimensional FIR filters to n-dimensional IIR filters, where n < N. The reduced IIR filters are obtained without computing the balancing transformation, but by finding the orthonormal eigen spaces associated with the large eigenvalues of the cross-Gramian matrix WCO. It is shown that using the diagonal symmetry of the class of filters that we are considering, octagonally symmetric, and the fact that the left and right eigen spaces obtained by the Schur decomposition are transpose of each other, the design problem of two 1-D digital filters in each section is reduced to the design problem of only one 1-D digital filter. Thus, the computational effort is reduced to approximately one half. Moreover, the Schur decomposition is applied only to a submatrix of the impulse response matrix, the leading principal minor sub matrix.