Abstract

Carry-free addition and borrow-free subtraction of modified signed-digit trinary numbers with optical symbolic substitution are presented. The proposed two-step and three-step algorithms can be easily implemented by using phase-only holograms, optical content-addressable memories, a multichannel correlator, or a polarization-encoded optical shadow-casting system.

© 1992 Optical Society of America

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References

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  1. T. T. Dao, D. M. Campbell, “Multiple valued logic: an implementation,” Opt. Eng. 25, 14–21 (1986).
  2. S. L. Hurst, “Multiple valued logic: its status and its realization,” Opt. Eng. 25, 44–55 (1986).
  3. A. Huang, “Parallel algorithms for optical digital computers,” in Proceedings of the Tenth International Computing Conference (Institute of Electrical and Electronics Engineers, New York, 1983), pp. 13–17.
    [CrossRef]
  4. E. L. Johnson, M. A. Karim, Digital Design: a Pragmatic Approach (PWS-Kent, Boston, Mass., 1987).
  5. A. Avizienis, “Signed-digit number representation for fast parallel arithmetic,” IRE Trans. Electron, Comput. EC-10, 389–400 (1961).
    [CrossRef]
  6. R. P. Bocker, B. L. Drake, M. E. Lasher, T. B. Henderson, “Modified signed-digit addition and subtraction using optical symbolic substitution,” Appl. Opt. 25, 2456–2457 (1986).
    [CrossRef] [PubMed]
  7. B. L. Drake, R. P. Bocker, M. E. Lasher, T. B. Henderson, “Photonic computing using modified signed digit number representation,” Opt. Eng. 25, 38–43 (1986).
  8. A. K. Cherri, M. A. Karim, “Modified signed-digit arithmetic using an efficient symbolic substitution,” Appl. Opt. 27, 3824–3827 (1988).
    [CrossRef] [PubMed]
  9. Y. Li, G. Eichmann, “Conditional symbolic modified-signed digit arithmetic using optical content-addressable memory logic elements,” Appl. Opt. 26, 2328–2333 (1987).
    [CrossRef] [PubMed]
  10. N. Takagi, H. Yasuura, S. Yajima, “High speed VLSI multiplication algorithm with a redundant binary addition tree,” IEEE Trans. Comput. C-34, 789–795 (1985).
    [CrossRef]
  11. T. K. Gaylord, M. M. Mirsalehi, “Truth-table look-up processing: number representation, multilevel coding, and logical minimization,” Opt. Eng. 25, 22–28 (1986).
  12. T. K. Gaylord, M. M. Mirsalehi, C. C. Guest, “Optical digital truth-table look-up processing,” Opt. Eng. 24, 48–58 (1985).
  13. M. M. Mirsalehi, T. K. Gaylord, “Truth-table look-up parallel data processing using an optical content-addressable memory,” Appl. Opt. 25, 2277–2283 (1986).
    [CrossRef] [PubMed]
  14. M. M. Mirsalehi, T. K. Gaylord, “Logical minimization of multilevel coded functions,” Appl. Opt. 25, 3078–3088 (1986).
    [CrossRef] [PubMed]
  15. F. T. S. Yu, S. Jutamulia, “Implementation of symbolic substitution logic using optical associative memories,” Appl. Opt. 26, 2293–2294 (1987).
    [CrossRef] [PubMed]
  16. D. Gabor, “Associative holographic memories,” IBM J. Res. Dev. 13, 156–159 (1969).
    [CrossRef]
  17. J. Hong, D. Psaltis, “Storage capacity of holographic associative memories,” Opt. Lett. 11, 812–814 (1986).
    [CrossRef] [PubMed]
  18. E. Botha, D. Casasent, E. Barnard, “Optical symbolic substitution using multichannel correlators,” Appl. Opt. 27, 817–818 (1988).
    [CrossRef] [PubMed]
  19. M. S. Kim, M. R. Feldman, C. C. Guest, “Optimization of binary phase-only filters with simulated annealing algorithms,” in Optical Computing, Vol. 9 of OSA 1989 Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 342–345.
  20. J. N. Mait, K. H. Brenner, “Optical symbolic substitution: system design using phase-only holograms,” Appl. Opt. 27, 1692–1700 (1988).
    [CrossRef] [PubMed]
  21. A. K. Cherri, M. A. Karim, “Symbolic-substitution-based operations using holograms: multiplication and histogram equalization,” Opt. Eng. 28, 638–642 (1989).
  22. R. Thalman, G. Pedrini, B. Acklin, R. Dandliker, “Optical symbolic substitution using diffraction grating,” in Optical Computing ’88, P. Chavel, J. W. Goodman, G. Roblin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.963, 1–7 (1988).
  23. Y. Lang, D. Zhao, H. K. Liu, “Multifocus dichromated gelatin hololens,” Appl. Opt. 22, 3451–3456 (1983).
    [CrossRef]
  24. A. A. S. Awwal, M. A. Karim, “Polarization-encoded optical shadow-casting: design of a carry-free adder,” Appl. Opt. 28, 785–790 (1989).
    [CrossRef] [PubMed]
  25. A. K. Cherri, M. A. Karim, “Symbolic substitution based flagged arithmetic using polarization-encoded optical shadow-casting,” Opt. Commun. 70, 455–461 (1989).
    [CrossRef]
  26. A. A. S. Awwal, “Computations and image processing techniques using polarization-encoded optical shadow-casting,” Ph.D. dissertation (University of Dayton, Dayton, Ohio, April1989).
  27. A. K. Cherri, “Computations and Image Processing Techniques Using Optical Symbolic Substitution,” Ph.D. dissertation (University of Dayton, Dayton, Ohio, 1989).
  28. J. Tanida, Y. Ichioka, “Optical logic array processor using shadowgrams,” J. Opt. Soc. Am. 73, 800–809 (1983).
    [CrossRef]
  29. A. A. S. Awwal, M. A. Karim, S. R. Jahan, “Improved correlation discrimination using an amplitude-modulated phase-only filter,” Appl. Opt. 29, 233–236 (1990).
    [CrossRef] [PubMed]

1990 (1)

1989 (3)

A. A. S. Awwal, M. A. Karim, “Polarization-encoded optical shadow-casting: design of a carry-free adder,” Appl. Opt. 28, 785–790 (1989).
[CrossRef] [PubMed]

A. K. Cherri, M. A. Karim, “Symbolic substitution based flagged arithmetic using polarization-encoded optical shadow-casting,” Opt. Commun. 70, 455–461 (1989).
[CrossRef]

A. K. Cherri, M. A. Karim, “Symbolic-substitution-based operations using holograms: multiplication and histogram equalization,” Opt. Eng. 28, 638–642 (1989).

1988 (3)

1987 (2)

1986 (8)

M. M. Mirsalehi, T. K. Gaylord, “Truth-table look-up parallel data processing using an optical content-addressable memory,” Appl. Opt. 25, 2277–2283 (1986).
[CrossRef] [PubMed]

M. M. Mirsalehi, T. K. Gaylord, “Logical minimization of multilevel coded functions,” Appl. Opt. 25, 3078–3088 (1986).
[CrossRef] [PubMed]

T. K. Gaylord, M. M. Mirsalehi, “Truth-table look-up processing: number representation, multilevel coding, and logical minimization,” Opt. Eng. 25, 22–28 (1986).

T. T. Dao, D. M. Campbell, “Multiple valued logic: an implementation,” Opt. Eng. 25, 14–21 (1986).

S. L. Hurst, “Multiple valued logic: its status and its realization,” Opt. Eng. 25, 44–55 (1986).

R. P. Bocker, B. L. Drake, M. E. Lasher, T. B. Henderson, “Modified signed-digit addition and subtraction using optical symbolic substitution,” Appl. Opt. 25, 2456–2457 (1986).
[CrossRef] [PubMed]

B. L. Drake, R. P. Bocker, M. E. Lasher, T. B. Henderson, “Photonic computing using modified signed digit number representation,” Opt. Eng. 25, 38–43 (1986).

J. Hong, D. Psaltis, “Storage capacity of holographic associative memories,” Opt. Lett. 11, 812–814 (1986).
[CrossRef] [PubMed]

1985 (2)

T. K. Gaylord, M. M. Mirsalehi, C. C. Guest, “Optical digital truth-table look-up processing,” Opt. Eng. 24, 48–58 (1985).

N. Takagi, H. Yasuura, S. Yajima, “High speed VLSI multiplication algorithm with a redundant binary addition tree,” IEEE Trans. Comput. C-34, 789–795 (1985).
[CrossRef]

1983 (2)

1969 (1)

D. Gabor, “Associative holographic memories,” IBM J. Res. Dev. 13, 156–159 (1969).
[CrossRef]

1961 (1)

A. Avizienis, “Signed-digit number representation for fast parallel arithmetic,” IRE Trans. Electron, Comput. EC-10, 389–400 (1961).
[CrossRef]

Acklin, B.

R. Thalman, G. Pedrini, B. Acklin, R. Dandliker, “Optical symbolic substitution using diffraction grating,” in Optical Computing ’88, P. Chavel, J. W. Goodman, G. Roblin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.963, 1–7 (1988).

Avizienis, A.

A. Avizienis, “Signed-digit number representation for fast parallel arithmetic,” IRE Trans. Electron, Comput. EC-10, 389–400 (1961).
[CrossRef]

Awwal, A. A. S.

Barnard, E.

Bocker, R. P.

B. L. Drake, R. P. Bocker, M. E. Lasher, T. B. Henderson, “Photonic computing using modified signed digit number representation,” Opt. Eng. 25, 38–43 (1986).

R. P. Bocker, B. L. Drake, M. E. Lasher, T. B. Henderson, “Modified signed-digit addition and subtraction using optical symbolic substitution,” Appl. Opt. 25, 2456–2457 (1986).
[CrossRef] [PubMed]

Botha, E.

Brenner, K. H.

Campbell, D. M.

T. T. Dao, D. M. Campbell, “Multiple valued logic: an implementation,” Opt. Eng. 25, 14–21 (1986).

Casasent, D.

Cherri, A. K.

A. K. Cherri, M. A. Karim, “Symbolic substitution based flagged arithmetic using polarization-encoded optical shadow-casting,” Opt. Commun. 70, 455–461 (1989).
[CrossRef]

A. K. Cherri, M. A. Karim, “Symbolic-substitution-based operations using holograms: multiplication and histogram equalization,” Opt. Eng. 28, 638–642 (1989).

A. K. Cherri, M. A. Karim, “Modified signed-digit arithmetic using an efficient symbolic substitution,” Appl. Opt. 27, 3824–3827 (1988).
[CrossRef] [PubMed]

A. K. Cherri, “Computations and Image Processing Techniques Using Optical Symbolic Substitution,” Ph.D. dissertation (University of Dayton, Dayton, Ohio, 1989).

Dandliker, R.

R. Thalman, G. Pedrini, B. Acklin, R. Dandliker, “Optical symbolic substitution using diffraction grating,” in Optical Computing ’88, P. Chavel, J. W. Goodman, G. Roblin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.963, 1–7 (1988).

Dao, T. T.

T. T. Dao, D. M. Campbell, “Multiple valued logic: an implementation,” Opt. Eng. 25, 14–21 (1986).

Drake, B. L.

R. P. Bocker, B. L. Drake, M. E. Lasher, T. B. Henderson, “Modified signed-digit addition and subtraction using optical symbolic substitution,” Appl. Opt. 25, 2456–2457 (1986).
[CrossRef] [PubMed]

B. L. Drake, R. P. Bocker, M. E. Lasher, T. B. Henderson, “Photonic computing using modified signed digit number representation,” Opt. Eng. 25, 38–43 (1986).

Eichmann, G.

Feldman, M. R.

M. S. Kim, M. R. Feldman, C. C. Guest, “Optimization of binary phase-only filters with simulated annealing algorithms,” in Optical Computing, Vol. 9 of OSA 1989 Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 342–345.

Gabor, D.

D. Gabor, “Associative holographic memories,” IBM J. Res. Dev. 13, 156–159 (1969).
[CrossRef]

Gaylord, T. K.

M. M. Mirsalehi, T. K. Gaylord, “Logical minimization of multilevel coded functions,” Appl. Opt. 25, 3078–3088 (1986).
[CrossRef] [PubMed]

M. M. Mirsalehi, T. K. Gaylord, “Truth-table look-up parallel data processing using an optical content-addressable memory,” Appl. Opt. 25, 2277–2283 (1986).
[CrossRef] [PubMed]

T. K. Gaylord, M. M. Mirsalehi, “Truth-table look-up processing: number representation, multilevel coding, and logical minimization,” Opt. Eng. 25, 22–28 (1986).

T. K. Gaylord, M. M. Mirsalehi, C. C. Guest, “Optical digital truth-table look-up processing,” Opt. Eng. 24, 48–58 (1985).

Guest, C. C.

T. K. Gaylord, M. M. Mirsalehi, C. C. Guest, “Optical digital truth-table look-up processing,” Opt. Eng. 24, 48–58 (1985).

M. S. Kim, M. R. Feldman, C. C. Guest, “Optimization of binary phase-only filters with simulated annealing algorithms,” in Optical Computing, Vol. 9 of OSA 1989 Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 342–345.

Henderson, T. B.

B. L. Drake, R. P. Bocker, M. E. Lasher, T. B. Henderson, “Photonic computing using modified signed digit number representation,” Opt. Eng. 25, 38–43 (1986).

R. P. Bocker, B. L. Drake, M. E. Lasher, T. B. Henderson, “Modified signed-digit addition and subtraction using optical symbolic substitution,” Appl. Opt. 25, 2456–2457 (1986).
[CrossRef] [PubMed]

Hong, J.

Huang, A.

A. Huang, “Parallel algorithms for optical digital computers,” in Proceedings of the Tenth International Computing Conference (Institute of Electrical and Electronics Engineers, New York, 1983), pp. 13–17.
[CrossRef]

Hurst, S. L.

S. L. Hurst, “Multiple valued logic: its status and its realization,” Opt. Eng. 25, 44–55 (1986).

Ichioka, Y.

Jahan, S. R.

Johnson, E. L.

E. L. Johnson, M. A. Karim, Digital Design: a Pragmatic Approach (PWS-Kent, Boston, Mass., 1987).

Jutamulia, S.

Karim, M. A.

A. A. S. Awwal, M. A. Karim, S. R. Jahan, “Improved correlation discrimination using an amplitude-modulated phase-only filter,” Appl. Opt. 29, 233–236 (1990).
[CrossRef] [PubMed]

A. K. Cherri, M. A. Karim, “Symbolic substitution based flagged arithmetic using polarization-encoded optical shadow-casting,” Opt. Commun. 70, 455–461 (1989).
[CrossRef]

A. K. Cherri, M. A. Karim, “Symbolic-substitution-based operations using holograms: multiplication and histogram equalization,” Opt. Eng. 28, 638–642 (1989).

A. A. S. Awwal, M. A. Karim, “Polarization-encoded optical shadow-casting: design of a carry-free adder,” Appl. Opt. 28, 785–790 (1989).
[CrossRef] [PubMed]

A. K. Cherri, M. A. Karim, “Modified signed-digit arithmetic using an efficient symbolic substitution,” Appl. Opt. 27, 3824–3827 (1988).
[CrossRef] [PubMed]

E. L. Johnson, M. A. Karim, Digital Design: a Pragmatic Approach (PWS-Kent, Boston, Mass., 1987).

Kim, M. S.

M. S. Kim, M. R. Feldman, C. C. Guest, “Optimization of binary phase-only filters with simulated annealing algorithms,” in Optical Computing, Vol. 9 of OSA 1989 Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 342–345.

Lang, Y.

Lasher, M. E.

R. P. Bocker, B. L. Drake, M. E. Lasher, T. B. Henderson, “Modified signed-digit addition and subtraction using optical symbolic substitution,” Appl. Opt. 25, 2456–2457 (1986).
[CrossRef] [PubMed]

B. L. Drake, R. P. Bocker, M. E. Lasher, T. B. Henderson, “Photonic computing using modified signed digit number representation,” Opt. Eng. 25, 38–43 (1986).

Li, Y.

Liu, H. K.

Mait, J. N.

Mirsalehi, M. M.

M. M. Mirsalehi, T. K. Gaylord, “Logical minimization of multilevel coded functions,” Appl. Opt. 25, 3078–3088 (1986).
[CrossRef] [PubMed]

T. K. Gaylord, M. M. Mirsalehi, “Truth-table look-up processing: number representation, multilevel coding, and logical minimization,” Opt. Eng. 25, 22–28 (1986).

M. M. Mirsalehi, T. K. Gaylord, “Truth-table look-up parallel data processing using an optical content-addressable memory,” Appl. Opt. 25, 2277–2283 (1986).
[CrossRef] [PubMed]

T. K. Gaylord, M. M. Mirsalehi, C. C. Guest, “Optical digital truth-table look-up processing,” Opt. Eng. 24, 48–58 (1985).

Pedrini, G.

R. Thalman, G. Pedrini, B. Acklin, R. Dandliker, “Optical symbolic substitution using diffraction grating,” in Optical Computing ’88, P. Chavel, J. W. Goodman, G. Roblin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.963, 1–7 (1988).

Psaltis, D.

Takagi, N.

N. Takagi, H. Yasuura, S. Yajima, “High speed VLSI multiplication algorithm with a redundant binary addition tree,” IEEE Trans. Comput. C-34, 789–795 (1985).
[CrossRef]

Tanida, J.

Thalman, R.

R. Thalman, G. Pedrini, B. Acklin, R. Dandliker, “Optical symbolic substitution using diffraction grating,” in Optical Computing ’88, P. Chavel, J. W. Goodman, G. Roblin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.963, 1–7 (1988).

Yajima, S.

N. Takagi, H. Yasuura, S. Yajima, “High speed VLSI multiplication algorithm with a redundant binary addition tree,” IEEE Trans. Comput. C-34, 789–795 (1985).
[CrossRef]

Yasuura, H.

N. Takagi, H. Yasuura, S. Yajima, “High speed VLSI multiplication algorithm with a redundant binary addition tree,” IEEE Trans. Comput. C-34, 789–795 (1985).
[CrossRef]

Yu, F. T. S.

Zhao, D.

Appl. Opt. (11)

Y. Lang, D. Zhao, H. K. Liu, “Multifocus dichromated gelatin hololens,” Appl. Opt. 22, 3451–3456 (1983).
[CrossRef]

M. M. Mirsalehi, T. K. Gaylord, “Truth-table look-up parallel data processing using an optical content-addressable memory,” Appl. Opt. 25, 2277–2283 (1986).
[CrossRef] [PubMed]

R. P. Bocker, B. L. Drake, M. E. Lasher, T. B. Henderson, “Modified signed-digit addition and subtraction using optical symbolic substitution,” Appl. Opt. 25, 2456–2457 (1986).
[CrossRef] [PubMed]

M. M. Mirsalehi, T. K. Gaylord, “Logical minimization of multilevel coded functions,” Appl. Opt. 25, 3078–3088 (1986).
[CrossRef] [PubMed]

F. T. S. Yu, S. Jutamulia, “Implementation of symbolic substitution logic using optical associative memories,” Appl. Opt. 26, 2293–2294 (1987).
[CrossRef] [PubMed]

Y. Li, G. Eichmann, “Conditional symbolic modified-signed digit arithmetic using optical content-addressable memory logic elements,” Appl. Opt. 26, 2328–2333 (1987).
[CrossRef] [PubMed]

J. N. Mait, K. H. Brenner, “Optical symbolic substitution: system design using phase-only holograms,” Appl. Opt. 27, 1692–1700 (1988).
[CrossRef] [PubMed]

A. K. Cherri, M. A. Karim, “Modified signed-digit arithmetic using an efficient symbolic substitution,” Appl. Opt. 27, 3824–3827 (1988).
[CrossRef] [PubMed]

A. A. S. Awwal, M. A. Karim, “Polarization-encoded optical shadow-casting: design of a carry-free adder,” Appl. Opt. 28, 785–790 (1989).
[CrossRef] [PubMed]

A. A. S. Awwal, M. A. Karim, S. R. Jahan, “Improved correlation discrimination using an amplitude-modulated phase-only filter,” Appl. Opt. 29, 233–236 (1990).
[CrossRef] [PubMed]

E. Botha, D. Casasent, E. Barnard, “Optical symbolic substitution using multichannel correlators,” Appl. Opt. 27, 817–818 (1988).
[CrossRef] [PubMed]

IBM J. Res. Dev. (1)

D. Gabor, “Associative holographic memories,” IBM J. Res. Dev. 13, 156–159 (1969).
[CrossRef]

IEEE Trans. Comput. (1)

N. Takagi, H. Yasuura, S. Yajima, “High speed VLSI multiplication algorithm with a redundant binary addition tree,” IEEE Trans. Comput. C-34, 789–795 (1985).
[CrossRef]

IRE Trans. Electron, Comput. (1)

A. Avizienis, “Signed-digit number representation for fast parallel arithmetic,” IRE Trans. Electron, Comput. EC-10, 389–400 (1961).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

A. K. Cherri, M. A. Karim, “Symbolic substitution based flagged arithmetic using polarization-encoded optical shadow-casting,” Opt. Commun. 70, 455–461 (1989).
[CrossRef]

Opt. Eng. (6)

A. K. Cherri, M. A. Karim, “Symbolic-substitution-based operations using holograms: multiplication and histogram equalization,” Opt. Eng. 28, 638–642 (1989).

T. K. Gaylord, M. M. Mirsalehi, “Truth-table look-up processing: number representation, multilevel coding, and logical minimization,” Opt. Eng. 25, 22–28 (1986).

T. K. Gaylord, M. M. Mirsalehi, C. C. Guest, “Optical digital truth-table look-up processing,” Opt. Eng. 24, 48–58 (1985).

B. L. Drake, R. P. Bocker, M. E. Lasher, T. B. Henderson, “Photonic computing using modified signed digit number representation,” Opt. Eng. 25, 38–43 (1986).

T. T. Dao, D. M. Campbell, “Multiple valued logic: an implementation,” Opt. Eng. 25, 14–21 (1986).

S. L. Hurst, “Multiple valued logic: its status and its realization,” Opt. Eng. 25, 44–55 (1986).

Opt. Lett. (1)

Other (6)

R. Thalman, G. Pedrini, B. Acklin, R. Dandliker, “Optical symbolic substitution using diffraction grating,” in Optical Computing ’88, P. Chavel, J. W. Goodman, G. Roblin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.963, 1–7 (1988).

A. A. S. Awwal, “Computations and image processing techniques using polarization-encoded optical shadow-casting,” Ph.D. dissertation (University of Dayton, Dayton, Ohio, April1989).

A. K. Cherri, “Computations and Image Processing Techniques Using Optical Symbolic Substitution,” Ph.D. dissertation (University of Dayton, Dayton, Ohio, 1989).

M. S. Kim, M. R. Feldman, C. C. Guest, “Optimization of binary phase-only filters with simulated annealing algorithms,” in Optical Computing, Vol. 9 of OSA 1989 Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 342–345.

A. Huang, “Parallel algorithms for optical digital computers,” in Proceedings of the Tenth International Computing Conference (Institute of Electrical and Electronics Engineers, New York, 1983), pp. 13–17.
[CrossRef]

E. L. Johnson, M. A. Karim, Digital Design: a Pragmatic Approach (PWS-Kent, Boston, Mass., 1987).

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Figures (6)

Fig. 1
Fig. 1

Set of An−1 and Bn−1 satisfying substitution rules for the type 2 input condition.

Fig. 2
Fig. 2

Input encoding for 2, 1, 0, 1 ¯, and 2 ¯.

Fig. 3
Fig. 3

(a) Symbolic substutition system based on a hololens and CGH, (b) possible optical system for trinary MSD addition.

Fig. 4
Fig. 4

Reflection mode shadow-casting setup.

Tables (7)

Tables Icon

Table I Truth Table for Trinary MSD Addition

Tables Icon

Table II Alternate Truth Table for Trinary MSD Addition

Tables Icon

Table III Truth Table for Trinary MSD Subtraction

Tables Icon

Table IV Alternate Truth Table for Trinary MSD Subtraction

Tables Icon

Table V Reduced Minterms for Three-step Trinary MSD Addition and Subtraction

Tables Icon

Table VI Reduced Minterms for Two-step Trinary MSD Addition and Subtraction

Tables Icon

Table VII Reduced Minterms for Two-step Trinary MSD Addition and Subtraction

Equations (3)

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X = i = 1 n x i 3 i ,
S n + C n - 1 < R ,
A n - B n = A n + B n ¯ .

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