Design of two-dimensional digital filters on the basis of quadrantal and octagonal symmetry

Two-dimensional recursive digital filters have the advantages of flexibility and accuracy, typical of digital processing systems, and the ability to perform the desired filtering with significantly fewer operations than nonrecursive filters. In this paper some useful symmetries are exploited in the design of filters with circularly symmetric magnitude responses. With the imposition of quadrantal symmetry, followed by further symmetry about the 45° line, a particular filter structure is derived using a cascade of causal second-order sections. Each section uses only four independent coefficients and possesses a separable denominator, with resulting simplification of stability testing and stabilization, which can now be done by using one-dimensional techniques. The Fletcher-Powell nonlinear optimization routine is used with a mean-squared error criterion, which enables the optimum value of the gain to be incorporated into the objective function to be minimized. Linear phase is approximated by designing all-pass equalizers which can be cascaded with the filters designed.

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