Forward and inverse transformations between Haar spectra and ordered binary decision diagrams of Boolean functions

Abstract

Abstract-Unnormalized Haar spectra and Ordered Binary Decision Diagrams (OBDDs) are two standard representations of Boolean functions used in logic design. In this article, mutual relationships between those two representations have been derived. The method of calculating the Haar spectrum from OBDD has been presented. The decomposition of the Haar spectrum, in terms of the cofactors of Boolean functions, has been introduced. Based on the above decomposition, another method to synthesize OBDD directly from the Haar spectrum has been presented.

Matrix decomposition and butterfly diagrams for mutual relations between Hadamard-Haar and arithmetic spectra

“…The presented relations allow transferring known results of spectral logic design in Arithmetic domain to Hadamard-Haar domain and vice verse and compare efficiency of both approaches in different applications for large discrete functions. Finally it should be also noticed that presented derivations based on layered matrices and corresponding butterfly diagrams can be efficiently implemented in the form of operations on spectral decision diagrams [11], [13], [15], [17]- [19], using software or as hardware operations using look-up table cascades in a manner similar to Walsh transform [28]. Hence, all the presented results are not only very interesting theoretically but also are very important for the practical applications of Haar, Walsh, Hadamard-Haar and Arithmetic transforms in many research areas.…”

Minimization of Circuit Design Using Permutation of Haar Wavelet Series

“…Recently, such methods were introduced for calculation of the Haar spectrum from disjoint cubes, 10,11 and different types of decision diagrams. 10,[12][13][14][15][16] In this paper, we extend our results from Ref. 7 and discuss the efficiency of circuit synthesis through the minimized Haar series in the number of coefficients count.…”

Foundations for Applications of Gibbs Derivatives in Logic Design and VLSI

“…[8], DDs are efficiently used in calculation of different spectral transforms and related Kronecker product representable linear operators, see for example, Refs. [9,6,7,17,19,20,21,28,38,44,50,57,58,60,61,62,68].…”

“…The present state-of-art are VLSI CAD systems based on Boolean algebraic structures and related theories. However, alternative approaches are continuously presented from earlier times until now [17,18,20,30,31,33,35,37,48,75], and challenges of new technologies renewed interest in such research attempts. For more details, we refer to a discussion of the development of switching theory in Ref.…”