Abstract
A nonlinear transformation, called the sign transform, is defined and studied. It maps all the ternary functions onto themselves, is uniquely invertible and is inherently related to Walsh and Walsh–Galois transforms. Decomposable Boolean functions and ternary product terms are mapped analytically into the sign domain. It is shown that sign coefficients describe the structure of a ternary output logic network implemented by a complete basis: EXOR of variables and sign function. Methods are discussed for direct transformation of switching functions without having to use the butterfly algorithm: Shannon's decomposition in the sign domain and, in general, merging sign spectra of subfunctions.As applications, spectra of arbitrary logic circuits is computed analytically by merging spectra of their gate components. Also, it is shown that testable realisations of incompletely specified Boolean functions can be designed by appropriate assignments of don't cares in the sign domain.
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