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

  • Authors:

  • I. -Hung Khoo;Hari C. Reddy;P. K. Rajan

  • Affiliations:

  • Department of Electrical Engineering, California State University Long Beach, Long Beach, USA;Department of Electrical Engineering, California State University Long Beach, Long Beach, USA and College of Computer Science, National Chiao-Tung University, HsinChu City, Taiwan;Department of Electrical & Computer Engineering, Tennessee Technological University, Cookeville, USA

  • Venue:
  • Multidimensional Systems and Signal Processing
  • Year:
  • 2011

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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 (驴-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 驴-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 驴-domain transfer function possesses better sensitivity to coefficient rounding than the z-domain counterpart.