Complex composite spectra of Unified Complex Hadamard transform for logic functions

A method to evaluate the Unified Complex Hadamard spectra of AND, OR, and XOR for Boolean functions, directly from the spectra of the functions, is presented. The results are given using a general coding scheme, and different possible codings of Boolean functions are also discussed. A new definition of the convolution operation called complex convolution is derived. Different properties of such a convolution are presented. A theorem giving final formulas for the composite Unified Complex Hadamard spectra of Boolean functions is stated in terms of the complex convolution. Efficient representations of the spectra in the form of decision diagrams are presented. An application of Unified Complex Hadamard Transform in image watermarking is also discussed.

[1]  H. E. Chrestenson.  A class of generalized Walsh functions , 1955 .

[2]  Robert J. Lechner.  HARMONIC ANALYSIS OF SWITCHING FUNCTIONS , 1971 .

[3]  C. Moraga.  Introducing disjoint spectral translation in spectral multiple-valued logic design , 1978 .

[4]  C. Moraga.  Characterisation of ternary threshold functions using a partial spectrum , 1979 .

[5]  Jon C. Muzio.  Composite Spectra and the Analysis of Switching Circuits , 1980, IEEE Transactions on Computers.

[6]  Parameter spectrum in spectral multiple-valued logic design , 1983 .

[7]  Stanley L. Hurst,et al.  Spectral techniques in digital logic , 1985 .

[8]  Bogdan J. Falkowski,et al.  Effective computer methods for the calculation of Rademacher-Walsh spectrum for completely and incompletely specified Boolean functions , 1992, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[9]  Masahiro Fujita,et al.  Spectral Transforms for Large Boolean Functions with Applications to Technology Mapping , 1993, 30th ACM/IEEE Design Automation Conference.

[10]  Giovanni De Micheli,et al.  Synthesis and Optimization of Digital Circuits , 1994 .

[11]  Sarma Vrudhula,et al.  EVBDD-based algorithms for integer linear programming, spectral transformation, and function decomposition , 1994, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[12]  Francis M. Boland,et al.  Phase watermarking of digital images , 1996, Proceedings of 3rd IEEE International Conference on Image Processing.

[13]  Composite complex Hadamard spectra of Boolean functions , 1996, 1996 IEEE International Symposium on Circuits and Systems. Circuits and Systems Connecting the World. ISCAS 96.

[14]  Bogdan J. Falkowski,et al.  Complex decision diagrams , 1996 .

[15]  Susanto Rahardja,et al.  Classifications and graph-based representations of switching functions using a novel complex spectral technique , 1997 .

[16]  S. Rahardja,et al.  Detection of Boolean symmetries using complex Hadamard transformation , 1997, Proceedings of 1997 IEEE International Symposium on Circuits and Systems. Circuits and Systems in the Information Age ISCAS '97.

[17]  B. J. Falkowski,et al.  Image watermarking using the complex Hadamard transform , 2000, 2000 IEEE International Symposium on Circuits and Systems. Emerging Technologies for the 21st Century. Proceedings (IEEE Cat No.00CH36353).