Unified theory of symmetry for two-dimensional complex polynomials using delta discrete-time operator

• Published in: Multidimensional systems and signal processing 2011-03, Vol.22 (1-3), p.147-172
• Main Authors: Khoo, I.-Hung, Reddy, Hari C., Rajan, P. K.
• Format: Article
• Language: English
• Subjects: Artificial Intelligence; Circuits and Systems; Electrical Engineering; Engineering; Signal,Image and Speech Processing
• ISSN: 0923-6082; 1573-0824
• DOI: 10.1007/s11045-010-0133-0

The complexity in the design and implementation of 2-D filters can be reduced considerably if the symmetries that might be present in the frequency responses of these filters are utilized. As the delta operator (γ-domain) formulation of digital filters offers better numerical accuracy and lower coefficient sensitivity in narrow-band filter designs when compared to the traditional shift-operator formulation, it is desirable to have efficient design and implementation techniques in γ-domain which utilize the various symmetries in the filter specifications. Furthermore, with the delta operator formulation, the discrete-time systems and results converge to their continuous-time counterparts as the sampling periods tend to zero. So a unifying theory can be established for both discrete- and continuous-time systems using the delta operator approach. With these motivations, we comprehensively establish the unifying symmetry theory for delta-operator formulated discrete-time complex-coefficient 2-D polynomials and functions, arising out of the many types of symmetries in their magnitude responses. The derived symmetry results merge with the s-domain results when the sampling periods tend to zero, and are more general than the real-coefficient results presented earlier. An example is provided to illustrate the use of the symmetry constraints in the design of a 2-D IIR filter with complex coefficients. For the narrow-band filter in the example, it can be seen that the γ-domain transfer function possesses better sensitivity to coefficient rounding than the z-domain counterpart.