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

IEEE Transactions on Computers

Volumn 46, Issue 11, 1997, Pages 1272-1279

  Falkowski, Bogdan J ^a,b   Chang, Chip Hong ^a,c  

^a IEEE   (Singapore)

^b NANYANG TECHNOLOGICAL UNIVERSITY   (Singapore)

^c SCHOOL OF ENGINEERING   (Singapore)

Author keywords

Boolean functions; Haar spectrum; Haar transform; Ordered binary decision diagram; Shannon decomposition; Spectral techniques

Indexed keywords

BINARY SEQUENCES; DECISION THEORY; MATHEMATICAL TRANSFORMATIONS;

HAAR SPECTRA; ORDERED BINARY DECISION DIAGRAMS (OBDD); SHANNON DECOMPOSITION;

BOOLEAN FUNCTIONS;

EID: 0031349454     PISSN: 00189340     EISSN: None     Source Type: Journal    
DOI: 10.1109/12.644301     Document Type: Article

References (34)

1. N. Ahmed and K.R. Rao, Orthogonal Transforms for Digital Signal Processing. Berlin: Springer-Verlag, 1975.
2. S.B. Akers, "Binary Decision Diagram," IEEE Trans. Computers, vol. 27, no. 6, pp. 509-516, June 1978.
3. R.E. Bryant, "Graph-Based Algorithm for Boolean Function Manipulation," IEEE Trans. Computers, vol. 35, no. 8, pp. 677-691, Aug. 1986.
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. 8-17. Univ. Dortmund Press, Oct. 1988.
5. S. Chakravarty, "A Characterization of Binary Decision Diagrams," IEEE Trans. Computers, vol. 42, no. 2, pp. 129-137, Feb. 1993.
6. E.M. Clarke, K.L. McMillian, X. Zhao, M. Fujita, and J. Yang, "Spectral Transforms for Large Boolean Functions with Applications to Technology Mapping," Proc. 30th ACM/IEEE Design Automation Conf., pp. 54-60, June 1993.
7. B.J. Falkowski, I. Schaefer, and M.A. Perkowski, "Effective Computer Methods for the Calculation of Rademacher-Walsh Spectrum for Completely and Incompletely Specified Boolean Functions," IEEE Trans. Computer-Aided Design, vol. 11, no. 10, pp. 1,207-1,226, Oct. 1992.
8. B.J. Falkowski and C-H Chang, "A Novel Paired Haar Based Transform: Algorithms and Interpretations in Boolean Domain," Proc. 36th Midwest Symp. Circuits and Systems, Detroit, pp. 1,101-1,104, 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. 497-503, 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. 710-713, May 1990.
11. J. Gergov and C. Meinel, "Efficient Boolean Manipulation with OBDDs Can Be Extended to FBDDs," IEEE Trans. Computers, vol. 43, no. 10, pp. 1,197-1,209, Oct. 1994.
12. A. Haar, "Zur Theorie der othorgonalen Funktionsysteme," Math. Ann., vol. 69, pp. 331-371, 1910.
13. H.F. Harmuth, Transmission of Information by Orthogonal Functions. Berlin: Springer-Verlag, 1972.
14. S.L. Hurst, D.M. Miller, and J.C. Muzio, Spectral Techniques in Digital Logic. London: 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 Binary-Decision Diagrams," Bell System Technical J., vol. 38, p p. 985-999, July 1959.
19. M.R. Mercer, R. Kapur and D.E. Ross, "Functional Approaches to Generating Orderings for Efficient Symbolic Representations," Proc. 29th ACM/IEEE Design Automation Conf., pp. 624-627, June 1992.
20. D.M. Miller, "Spectral Transformation of Multiple-Valued Decision Diagrams," Proc. IEEE 24th Int'l Symp. Multiple-Valued Logic, pp. 89-96, Boston, May 1994.
21. B.M.E. Moret, "Decision Trees and Diagrams," ACM Computing Surveys, vol. 14, no. 4, pp. 593-622, Dec. 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. 231-249, 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. 59-65, 1989.
24. S. Purwar, "An Efficient Method of Computing Generalized Reed-Muller Expansion from Binary Decision Diagram," IEEE Trans. Computers, vol. 40, no. 11, pp. 1,298-1,301, Nov. 1991.
25. P.R. Roeser and M.E. Jernigan, "Fast Haar Transform Algorithms," IEEE Trans. Computers, vol. 31, no. 2, pp. 175-177, 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. 97-106. Univ. Dortmund Press, Oct. 1988.
27. G. Ruiz, J.A. Michell, and A. Buron, "Switch-Level Fault Detection and Diagnosis Environment for MOS Digital Circuits Using Spectral Techniques," IEE Proc., Part E, vol. 139, no. 4, pp. 293-307, 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. 471-495. IEEE CS Press, 1992.
29. J.E. Shore, "On the Applications of Haar Functions," IEEE Trans. Comm., vol. 21, pp. 206-216, 1973.
30. M.R. Stojic, M.S. Stankovic, and R.S. Stankovic, Discrete Transforms and Their Applications (in Serbian). Beograd: Nauka, 1993.
31. M.A. Thornton and V.S.S. Nair, "Efficient Calculation of Spectral Coefficients and Their Applications," IEEE Trans. Computer-Aided Design Integrated Circuits and Systems, vol. 4, no. 11, pp. 1,328-1,341, Nov. 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: Springer-Verlag, 1985.
34. L.A. Zalmanzon, Fourier, Walsh and Haar Transforms and Their Application in Control, Communication and Other Fields (in Russian). Moscow: Nauka, 1989.