Symmetry Study for Delta- Operator-Based 2-D Digital Filters

The complexity in the design and implementation of two-dimensional (2-D) filters can be considerably reduced if we utilize the symmetries that might be present in the frequency response of these filters. As the delta-operator formulation of digital filters offers better numerical accuracy and lower coefficient sensitivity in narrowband filter designs when compared to the traditional shift-operator formulation, it is desirable to have efficient design and implementation techniques in gamma-domain which utilize the various symmetries in filter specifications. With this motivation, we comprehensively establish the theory of constraints for delta-operator formulated discrete-time real-coefficient polynomials and functions, arising out of the many types of symmetries in their magnitude responses. We also show that as sampling time tends to zero, the gamma-domain symmetry constraints merge with those of s-domain symmetry constraints. We then present a least square error criterion based procedure to design 2-D digital filters in gamma-domain that utilizes the symmetry properties of the magnitude specification. A design example is provided to illustrate the savings in computational complexity resulting from the use of the gamma-domain symmetry constraints