Spectral Methods for Boolean and Multiple-Valued Input Logic Functions

Abstract

Spectral techniques in digital logic design have been known for more than thirty years. They have been used for Boolean function classification, disjoint decomposition, parallel and serial linear decomposition, spectral translation synthesis (extraction of 2 linear pre-and post-filters), multiplexer synthesis, prime implicant extraction by spectral summation, threshold logic synthesis, estimation of logic complexity, testing, and state assignment.This dissertation resolves many important issues concerning the efficient application of spectral methods used in the computer-aided design of digital circuits. The main obstacles in these applications were, up to now, memory requirements for computer systems and lack of the possibility of calculating spectra direcdy from Boolean equations.By using the algorithms presented here these obstacles have been overcome. Moreover, the methods presented in this dissertation can be regarded as representatives of a whole family of methods and the approach presented can 1Y: easily adapted to other orthogonal transforms used in digital logic design. Algorithms are shown for Adding, Arithmetic, and Reed-Muller transforms. However, the main focus of this dissertation is on the efficient computer calculation of Rademacher-Walsh spectra of Boolean functions, since this particular ordering of Walsh transforms is most frequendy used in digital logic design.A theory has been developed to calculate the Rademacher-Walsh transform from a cube array specification of incompletely specified Boolean functions. The importance of representing Boolean functions as arrays of disjoint ON-and DC-cubes has been pointed out, and an efficient new algorithm to generate disjoint cubes from non-disjoint ones has been designed. The transform algorithm makes use of the properties of an array of disjoint cubes and allows the determination of the spectral coefficients in an independent way. By such an approach each spectral coefficient can be calculated separately or all the coefficients can be calculated in parallel. These advantages are absent in the existing methods. The possibility of calculating only some coefficients is very important since there are many spectral methods in digital logic design for which the values of only a few selected coefficients are needed.Most of the current methods used in the spectral domain deal only with completely specified Boolean functions. On the other hand, all of the algorithms introduced here are valid, not only for completely specified Boolean functions, but for functions with don't cares. Don't care minterms are simply represented in the form of disjoint cubes.The links between spectral and classical methods used for designing digital cir- A new algorithm is shown that converts the disjoint cube representation of Boolean functions into fixed-polarity Generalized Reed-Muller Expansions (GRME).Since the known fast algorithm that generates the GR..\1E, based on the factorization of 4 the Reed-Muller transform matrix, always starts from the truth table (mintenns) of a Boole...

Orthogonal and Nonorthogonal Expansions for Multi-Level Logic Synthesis for Nearly Linear Functions and their Application to Field Programmable Gate Array Mapping

“…The inverse transform is given by [12,27,71,73,74,90,92,[94][95][96][97]. In a disjoint representation of an mv function every minterm is covered only once.…”

“…Other formulas derived from Equations (IV.2) and (IV.3) can be found in [73,74]. The R or S spectrum for a local transform can be obtained from the T-spectrum by…”

“…Moreover, more emphasis has been recently concentrated on minimizers that find more optimal solutions, with the trade-off of increased computation time. Thus, spectral methods in logic synthesis are becoming feasible.In the last couple of years it has been attempted to decrease the computational complexity and memory requirement due to the necessary minterm representation of the Based on the previous research on the computation of spectra from a disjoint cover [72,74] this dissertation introduces the computation of spectra from Ordered Decision Diagrams (ODDs) [10,[76][77][78][79][80]. Their use as well as the modification is introduced to make them suitable for the application to spectral methods.…”

“…Another advantage is the possibility of the fast computation of spectra for functions which can be represented by relatively few disjoint terms. An algorithm and its implementation has been presented in [74,98,99] for the generation of a set of disjoint cubes for single output Boolean functions without respect to the minimality of the representation. As mentioned in the introduction, the calculation complexity of a spectrum increases linearly with the number of cubes in the disjoint representation of an mv function.…”

“…However, such a spectrum does not have to be unique. The vector of spectral coefficients C for an incompletely 35 specified n-variable mv function can be described according to Equation ( Chapter VII will present such a transform for the Actel ACT™ macrocells.The Walsh-type transforms have been investigated most in logic design [62,63,65,70,74]. Therefore, the calculation of the Rademacher-Walsh spectrum, which is one of the several Walsh-type transforms, is used to illustrate the computation of a spectrum by matrix calculation.…”