Abstract
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.
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Late Professor N. K. Bose through his fundamental and pioneering contributions as well as through his many writings has influenced the works of a large number of researchers in multidimensional Circuits and Systems theory including the authors of this paper. This paper is thus dedicated to his memory.
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Khoo, IH., Reddy, H.C. & Rajan, P.K. Unified theory of symmetry for two-dimensional complex polynomials using delta discrete-time operator. Multidim Syst Sign Process 22, 147–172 (2011). https://doi.org/10.1007/s11045-010-0133-0
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DOI: https://doi.org/10.1007/s11045-010-0133-0