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14
Forward and inverse transformations between Haar spectra and ordered binary decision diagrams of Boolean functions
 IEEE Trans. on Comp
, 1997
"... 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 ..."
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Cited by 13 (7 self)
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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. Index Terms—Boolean functions, Haar spectrum, Haar transform, ordered binary decision diagram, Shannon decomposition, spectral
Modified Haar Transform Calculation Using Digital Circuit Output Probabilities
 IN INTERNATIONAL CONFERENCE ON INFORMATION, COMMUNICATION & SIGNAL PROCESSING
, 1997
"... A method for the computation of Haar spectral coefficients is described using the OBDD representation of a function. An algebraic relationship between the circuit output probabilities and the Haar spectral coefficients is derived and used. The circuit output probabilities are computed by applying gr ..."
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Cited by 11 (4 self)
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A method for the computation of Haar spectral coefficients is described using the OBDD representation of a function. An algebraic relationship between the circuit output probabilities and the Haar spectral coefficients is derived and used. The circuit output probabilities are computed by applying graph algorithms to the OBDDs. A fundamental dependence between the spectral coefficients and n + 1 simple Boolean equivalence relationships is noted.
TOWARDS EFFICIENT CALCULATION OF INFORMATION MEASURES FOR REORDERING OF BINARY DECISION DIAGRAMS
, 2002
"... This paper introduces new technique for efficient calculation of different Shannon information measures which operates Binary Decision Diagrams (BDDs). We offer an algorithm of BDD reordering which demonstrates the improvement of the obtaining outcomes over the existing reordering approaches. The te ..."
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Cited by 3 (3 self)
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This paper introduces new technique for efficient calculation of different Shannon information measures which operates Binary Decision Diagrams (BDDs). We offer an algorithm of BDD reordering which demonstrates the improvement of the obtaining outcomes over the existing reordering approaches. The technique and the reordering algorithm have been implemented, and the results on circuits ’ benchmarks are analyzed. We point out that the results are quite promising, the algorithm is very fast, and it is easy to implement. Finally, we show that our approach to BDD reordering can yield to reduction in the power dissipation for the circuits derived from BDDs. 1.
Antisymmetries in the realization of Boolean functions
 in Proc. Int. Symp. on Circuits and Systems, ISCAS 2002, Scottsdale Princess Resort
"... New symmetries of degree two are introduced, along with spectral techniques for identifying these symmetries. Some applications of these symmetries are discussed, in particular their application to the construction of binary decision diagrams and the implementation of Boolean functions. 1. ..."
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Cited by 1 (0 self)
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New symmetries of degree two are introduced, along with spectral techniques for identifying these symmetries. Some applications of these symmetries are discussed, in particular their application to the construction of binary decision diagrams and the implementation of Boolean functions. 1.
Boolean Function Representation and Spectral Characterization Using AND/OR Graphs
 INTEGRATION, The VLSI journal
, 2000
"... Methods based on AND/OR graph representations of Boolean relations provide a promising new way of approaching VLSI CAD design automation problems. AND/OR graphs can represent any Boolean network and they allow for systematic reasoning through the application of the technique of recursive learning ..."
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Cited by 1 (0 self)
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Methods based on AND/OR graph representations of Boolean relations provide a promising new way of approaching VLSI CAD design automation problems. AND/OR graphs can represent any Boolean network and they allow for systematic reasoning through the application of the technique of recursive learning. An approach to build and analyze AND/OR graphs that makes use of hashing techniques in a way similar to that for modern Decision Diagram (DD) packages is described. Additionally, the problem of extracting spectral information from AND/OR graphs is also examined. Spectral information can be used for many CAD system tasks including synthesis, verification and test vector generation. It is shown that spectral information may be calculated directly from output probabilities and a method for estimating output probabilities from AND/OR graphs is presented. Experimental results regarding the AND/OR graph package efficiency and the extraction of spectral information are provided. 1 Introdu...
Logic Synthesis Based on the Structure of an Ordered DD
 In Int'l Workshop on Logic Synthesis
, 1999
"... Introduction: Digital logic synthesis is an integral step in modern integrated circuit design. Coupled with the popular usage of synthesis techniques is the use of BDDs, which are data structures that represent Boolean functions [1]. These directed acyclic graphs (DAGs) have generated great interest ..."
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Cited by 1 (1 self)
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Introduction: Digital logic synthesis is an integral step in modern integrated circuit design. Coupled with the popular usage of synthesis techniques is the use of BDDs, which are data structures that represent Boolean functions [1]. These directed acyclic graphs (DAGs) have generated great interest due to their ability to represent many Boolean functions in a very compact manner and the existence of associated algorithms for their manipulation. Although BDDs have traditionally been viewed as purely behavioral entities, it is well known that they can represent "trees of multiplexers" due to the correspondence between their structure and the Shannon decomposition. With this point of view, a minimized BDD can represent a correspondingly small structural representation of a circuit in the form of a network of multiplexers. The motivation for developing a BDD based synthesis tool based on the structure of the DAG, is given by the fact that many good variable ordering heuristics have
Parity Function Detection and Realization Using a Small Set of Spectral Coefficients
 of Spectral Coefficients”, IEEE/ACM International Workshop on Logic Synthesis (IWLS
, 1995
"... A technique to detect and realize a parity logic function using a linear number of spectral coefficients is presented. Recent advances in computation methods for the spectra of a Boolean function have resulted in the determination of a single coefficient very efficiently. The use of a small set of s ..."
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Cited by 1 (0 self)
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A technique to detect and realize a parity logic function using a linear number of spectral coefficients is presented. Recent advances in computation methods for the spectra of a Boolean function have resulted in the determination of a single coefficient very efficiently. The use of a small set of spectral coefficients offers low cost and attractive alternatives to more traditional digital logic design and analysis techniques when coupled with the use of the new spectral computation approach. The detection and realization of the class of parity functions is an important problem since it generally requires the computation of 2 n Walsh coefficients when spectral methods are used, or, extensive symbolic algebraic manipulation when other methods are used. The method presented in this paper only requires the computation of n + 2 Walsh coefficients and n + 1 nonWalsh coefficients to detect and realize a parity function. 1 Introduction With the recent advent of an efficient method for th...
Efficient Approximation of Spectral and Autocorrelation Coefficients
 Department of Computer Science at James Cook University, Townsville Australia
, 1996
"... In this paper we provide polynomial time approximation techniques which allow us to calculate, to arbitrary levels of accuracy and with high probability of success, the spectral coefficients and autocorrelation coefficients of Boolean functions, given that those functions are expressed in either Sum ..."
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Cited by 1 (1 self)
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In this paper we provide polynomial time approximation techniques which allow us to calculate, to arbitrary levels of accuracy and with high probability of success, the spectral coefficients and autocorrelation coefficients of Boolean functions, given that those functions are expressed in either SumofProducts or ProductofSums form. 1 Introduction The focus of this paper is on the generation of spectral coefficients and autocorrelation coefficients for Boolean functions. Efficient calculation of those coefficients would allow digital logic analysts to draw on the large body of research already effectively employed in the area of signal processing. Utilisation of coefficientbased techniques in areas such as logic testing and synthesis has traditionally been hampered by the computational requirements for coefficient calculation. To reduce the computational demands, we use an approximation technique to estimate the coefficient values. As arbitrary levels of accuracy can still be obta...
Unified algorithm to generate Walsh functions in four different orderings and its programmable hardware implementations
, 2005
"... This paper presents an algorithm to generate Walsh functions in four different orderings: Hadamard, Harmuth, Paley and strict sequency. By the analysis of the properties and mutual relations among these four orderings, the authors found a unified approach to generate any of the orderings from the ..."
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Cited by 1 (0 self)
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This paper presents an algorithm to generate Walsh functions in four different orderings: Hadamard, Harmuth, Paley and strict sequency. By the analysis of the properties and mutual relations among these four orderings, the authors found a unified approach to generate any of the orderings from the primary set of Rademacher functions. By using these properties, the authors developed a programmable Walsh function generator for 64 outputs by both field programmable gate arrays and lookup table cascades to estimate the amount of hardware and performance. Such a programmable Walsh function generator can be used in VLSI testing, CDMA, pattern recognition, as well as image and signal processing.
MUTUAL RELATIONS AND PROPERTIES OF SYMMETRIC FUNCTIONS IN WALSH SPECTRAL DOMAIN
"... This article explains the properties of the symmetric functions and their mutual relations in terms of the Walsh spectral coefficients. Equivalent, nonequivalent, complement equivalent and complement nonequivalent symmetric functions have been considered. 1. ..."
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Cited by 1 (0 self)
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This article explains the properties of the symmetric functions and their mutual relations in terms of the Walsh spectral coefficients. Equivalent, nonequivalent, complement equivalent and complement nonequivalent symmetric functions have been considered. 1.