
This Article  
 
Share  
Bibliographic References  
Add to:  
Digg Furl Spurl Blink Simpy Del.icio.us Y!MyWeb  
Search  
 
ASCII Text  x  
Bogdan J. Falkowski, ChipHong Chang, "Forward and Inverse Transformations Between Haar Spectra and Ordered Binary Decision Diagrams of Boolean Functions," IEEE Transactions on Computers, vol. 46, no. 11, pp. 12721279, November, 1997.  
BibTex  x  
@article{ 10.1109/12.644301, author = {Bogdan J. Falkowski and ChipHong Chang}, title = {Forward and Inverse Transformations Between Haar Spectra and Ordered Binary Decision Diagrams of Boolean Functions}, journal ={IEEE Transactions on Computers}, volume = {46}, number = {11}, issn = {00189340}, year = {1997}, pages = {12721279}, doi = {http://doi.ieeecomputersociety.org/10.1109/12.644301}, publisher = {IEEE Computer Society}, address = {Los Alamitos, CA, USA}, }  
RefWorks Procite/RefMan/Endnote  x  
TY  JOUR JO  IEEE Transactions on Computers TI  Forward and Inverse Transformations Between Haar Spectra and Ordered Binary Decision Diagrams of Boolean Functions IS  11 SN  00189340 SP1272 EP1279 EPD  12721279 A1  Bogdan J. Falkowski, A1  ChipHong Chang, PY  1997 KW  Boolean functions KW  Haar spectrum KW  Haar transform KW  ordered binary decision diagram KW  Shannon decomposition KW  spectral techniques. VL  46 JA  IEEE Transactions on Computers ER   
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.
[1] N. Ahmed and K.R. Rao, Orthogonal Transforms for Digital Signal Processing.Berlin: SpringerVerlag, 1975.
[2] S.B. Akers, "Binary Decision Diagram," IEEE Trans. Computers, vol. 27, no. 6, pp. 509516, June 1978.
[3] R.E. Bryant, "GraphBased Algorithms for Boolean Function Manipulation," IEEE Trans. Computers, Vol. C35, No. 8, Aug. 1986, pp. 667690.
[4] A.M. Buron, J.A. Michell, and J.M. Solana, "Single Chip Fast Haar Transform at Megahertz Rates," Theory and Applications of Spectral Techniques, C. Moraga, ed., pp. 817. Univ. Dortmund Press, Oct. 1988.
[5] S. Chakravarty, "A Characterization of Binary Decision Diagrams," IEEE Trans. Computers, vol. 42, no. 2, pp. 129137, Feb. 1993.
[6] E.M. Clarke, K.L. McMillan, X. Zhao, M. Fujita, and J. Yang, "Spectral Transforms for Large Boolean Functions with Applications to Technology Mapping," Proc. 30th Design Automation Conf., pp. 5460, 1993.
[7] B.J. Falkowski, I. Schäfer, and M.A. Perkowski, “Effective Computer Methods for the Calculation of RademacherWalsh Spectrum for Completely and Incompletely Specified Boolean Functions,” IEEE Trans. ComputerAided Design, vol. 11, no. 10, pp. 12071226, 1992.
[8] B.J. Falkowski and CH Chang, "A Novel Paired Haar Based Transform: Algorithms and Interpretations in Boolean Domain," Proc. 36th Midwest Symp. Circuits and Systems,Detroit, pp. 1,1011,104, Aug. 1993.
[9] , "Efficient Algorithms for the Forward and Inverse Transformations Between Haar Spectrum and Binary Decision Diagram," Proc. 13th IEEE Int'l Phoenix Conf. Computers and Communications, pp. 497503, Phoenix, Apr. 1994.
[10] S.J. Friedman and K.J. Supowit, "Finding the Optimal Variable Ordering for Binary Decision Diagrams," IEEE Trans. Computers, vol. 39, no. 5, pp. 710713, May 1990.
[11] J. Gergov and C. Meinel, Efficient Analysis and Manipulation of OBDDs Can Be Extended to FBDDs IEEE Trans. Computers, vol. 43, pp. 11971209, 1994.
[12] A. Haar, "Zur Theorie der othorgonalen Funktionsysteme," Math. Ann., vol. 69, pp. 331371, 1910.
[13] H.F. Harmuth, Transmission of Information by Orthogonal Functions.Berlin: SpringerVerlag, 1972.
[14] J.S.L. Hurst, D.M. Miller,, and J.C. Muzio,Spectral Techniques in Digital Logic. Orlando, Fla.: Academic Press, 1985.
[15] S. Kaczmarz and H. Steinhaus, Theory of Orthogonal Series (in German).New York: Chelsea, 1951.
[16] M.G. Karpovsky, Finite Orthogonal Series in the Design of Digital Devices.New York: John Wiley, 1976.
[17] W. Kulesza, Systems of Spectral Analysis of Digital Data (in Polish). Warsaw: WKL, 1984.
[18] C.Y. Lee, "Representation of Switching Circuits by BinaryDecision Diagrams," Bell System Technical J., vol. 38, pp. 985999, July 1959.
[19] M.R. Mercer, R. Kapur, and D.E. Ross, "Functional Approaches to Generating Orderings for Efficient Symbolic Representation," Proc. 29th ACM/IEEE Design Automation Conf, pp. 614619, 1992.
[20] D.M. Miller, “Spectral Transform of MultipleValued Decision Diagrams,” Proc. 24th Int'l Symp. MultipleValued Logic, pp. 8996, 1994.
[21] B.M.E. Moret,“Decision trees and diagrams,” Computing Surveys, vol. 14, no. 4, pp. 593623, 1982.
[22] J.C. Muzio and S.L. Hurst, "The Computation of Complete and Reduced Sets of Orthogonal Spectral Coefficients for Logic Design and Pattern Recognition Purpose," Int'l J. Computers and Electrical Eng., vol. 5, pp. 231249, 1978.
[23] S. Purwar and A.K. Susskind, "Computation of Walsh Spectrum from Binary Decision Diagram and Binary Decision Diagram from Walsh Spectrum," Int'l J. Computers and Electrical Eng., vol. 15, no. 2, pp. 5965, 1989.
[24] S. Purwar, "An Efficient Method of Computing Generalized ReedMuller Expansion from Binary Decision Diagram," IEEE Trans. Computers, vol. 40, no. 11, pp. 1,2981,301, Nov. 1991.
[25] P.R. Roeser and M.E. Jernigan, "Fast Haar Transform Algorithms," IEEE Trans. Computers, vol. 31, no. 2, pp. 175177, 1982.
[26] G. Ruiz, J.A. Michell, and A. Buron, "Fault Detection and Diagnosis for MOS Circuits from Haar and Walsh Spectrum Analysis: On the Fault Coverage of Haar Reduced Analysis," Theory and Applications of Spectral Techniques, C. Moraga, ed., pp. 97106. Univ. Dortmund Press, Oct. 1988.
[27] G. Ruiz, J.A. Michell, and A. Buron, "SwitchLevel Fault Detection and Diagnosis Environment for MOS Digital Circuits Using Spectral Techniques," IEE Proc., Part E, vol. 139, no. 4, pp. 293307, July 1992.
[28] C.E. Shannon, "A Symbolic Analysis of Relay and Switching Circuits," Collected Papers of Claude Elwood Shannon, N.J.A. Sloane and A.D. Wyner, eds., pp. 471495. IEEE CS Press, 1992.
[29] J.E. Shore, "On the Applications of Haar Functions," IEEE Trans. Comm., vol. 21, pp. 206216, 1973.
[30] M.R. Stojic, M.S. Stankovic, and R.S. Stankovic, Discrete Transforms and Their Applications (in Serbian). Beograd: Nauka, 1993.
[31] M. Thornton and V. Nair, “Efficient Calculation of Spectral Coefficients and Their Application,” IEEE Trans. ComputerAided Design, vol. 14, no. 11, pp. 13281341, 1995.
[32] G.G. Walter, Wavelets and Other Orthogonal Systems with Applications.Boca Raton, Fla.: CRC Press, 1994.
[33] L.P. Yaroslavsky, Digital Picture Processing.Berlin: SpringerVerlag, 1985.
[34] L.A. Zalmanzon, Fourier, Walsh and Haar Transforms and Their Application in Control, Communication and Other Fields (in Russian). Moscow: Nauka, 1989.