Family of generalised multi-polarity complex Hadamard transforms

A family of generalised complex Hadamard transforms using the concept of polarity is introduced. Forward and inverse transformation kernels and methods of recursive generation of transform matrices using Kronecker products of elementary matrices are shown. Mutual relationships among transform matrices and spectra for arbitrary polarities are presented. Efficient ways of calculating spectra for logic functions through decision diagrams are also shown. The half-spectrum property is used to reduce further the computational requirements for both fast transforms and decision diagram based calculations.

[1]  A. Street,et al.  Combinatorics: room squares, sum-free sets, Hadamard matrices , 1972 .

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

[3]  Suku Nair,et al.  Efficient calculation of spectral coefficients and their applications , 1995, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[4]  B. J. Falkowski,et al.  Properties and ways of calculation of multi-polarity generalized Walsh transforms , 1994 .

[5]  D. Myers.  Digital Signal ProcessingEfficient Convolution and Fourier Transform Techniques , 1990 .

[6]  E. A. Trachtenberg,et al.  Design automation tools for efficient implementation of logic functions by decomposition , 1989, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[7]  E. Macii,et al.  Predicting the complexity of large combinational circuits through symbolic spectral analysis of their functional specifications , 1997 .

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

[9]  Chip-Hong Chang,et al.  Forward and Inverse Transformations Between Haar Spectra and Ordered Binary Decision Diagrams of Boolean Functions , 1997, IEEE Trans. Computers.

[10]  Leonid P. Yaroslavsky,et al.  Digital Picture Processing , 1985 .

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

[12]  Leonid P. Yaroslavsky,et al.  Digital Picture Processing: An Introduction , 1985 .

[13]  B. J. Falkowski.  Recursive relationships, fast transforms, generalisations and VLSI iterative architecture for Gray code ordered Walsh functions , 1995 .

[14]  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..

[15]  Mitchell A. Thornton,et al.  BDD-based spectral approach for Reed-Muller circuit realisation , 1996 .

[16]  E. Macii,et al.  Using symbolic Rademacher-Walsh spectral transforms to evaluate the agreement between Boolean functions , 1996 .

[17]  A. Richard Newton,et al.  Selected Papers on Logic Synthesis for Integrated Circuit Design , 1987 .

[18]  Bogdan J. Falkowski,et al.  Walsh-like functions and their relations , 1996 .

[19]  Mark G. Karpovsky,et al.  Spectral Techniques and Fault Detection , 1985 .

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

[21]  J. Astola,et al.  Binary polynomial transforms and nonlinear digital filters , 1995 .

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