Multidim Syst Sign Process (2011) 22:147–172 DOI 10.1007/s11045-010-0133-0 Unified theory of symmetry for two-dimensional complex polynomials using delta discrete-time operator I.-Hung Khoo · Hari C. Reddy · P. K. Rajan Received: 3 April 2010 / Revised: 1 August 2010 / Accepted: 19 August 2010 / Published online: 9 September 2010 © Springer Science+Business Media, LLC 2010 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 bet- ter 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 mag- nitude 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. 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. I.-H. Khoo (B ) · H. C. Reddy Department of Electrical Engineering, California State University Long Beach, Long Beach, CA, USA e-mail: ikhoo@csulb.edu H. C. Reddy College of Computer Science, National Chiao-Tung University, HsinChu City, Taiwan P. K. Rajan Department of Electrical & Computer Engineering, Tennessee Technological University, Cookeville, TN, USA 123