Abstract

An efficient symbolic substitution scheme for modified-signed digit arithmetic operations is introduced. In this technique, an additional bit is used along with each pair of the input bits so that the nth additional bit is a characteristic of the (n − 1)th pair of input bits. The truth table minimization thereby shows that relatively fewer minterms are to be included in the corresponding optical content-addressable memory.

© 1988 Optical Society of America

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  1. E. L. Johnson, M. A. Karim, Digital Design: A Pragmatic Approach (PWS Publishers, Boston, 1987).
  2. M. M. Mirsalehi, T. K. Gaylord, “Logical Minimization of Multilevel Coded Function,” Appl. Opt. 25, 3078 (1986).
    [CrossRef] [PubMed]
  3. C. C. Guest, T. K. Gaylord, “Truth Table Look-Up Optical Processing Utilizing Binary and Residue Arithmetic,” Appl. Opt. 19, 1201 (1980).
    [CrossRef] [PubMed]
  4. M. M. Mirsalehi, C. C. Guest, T. K. Gaylord, “Residue Number System Holographic Truth-Table Look-Up Processing: Detector Threshold Setting and Probability of Error Due to Amplitude and Phase Variations,” Appl. Opt. 22, 3583 (1983).
    [CrossRef] [PubMed]
  5. M. M. Mirsalehi, T. K. Gaylord, “Truth-Table Look-Up Parallel Processing Using an Optical Content-Addressable Memory,” Appl. Opt. 25, 2277 (1986).
    [CrossRef] [PubMed]
  6. G. Eichmann, Y. Li, R. R. Alfano, “Optical Binary Coded Ternary Arithmetic and Logic,” Appl. Opt. 25, 3113 (1986).
    [CrossRef] [PubMed]
  7. T. T. Dao, D. M. Campbell, “Multiple-Valued Logic: An Implementation,” Opt. Eng. 25, 14 (1986).
    [CrossRef]
  8. A. Huang, Y. Tsunida, J. W. Goodman, S. Ishihara, “Optical Computation Using Residue Arithmetic,” Appl. Opt. 18, 149 (1979).
    [CrossRef] [PubMed]
  9. D. Psaltis, D. Casasent, “Optical Residue Arithmetic: A Correlation Approach,” Appl. Opt. 18, 163 (1979).
    [CrossRef] [PubMed]
  10. A. Avizienis, “Signed-Digit Number Representation for Fast Parallel Arithmetic,” IRE Trans. Electron. Comput. EC-10, 389 (1961).
    [CrossRef]
  11. B. L. Drake, R. P. Bocker, M. E. Lasher, R. H. Patterson, W. J. Miceli, “Photonic Computing Using the Modified-Signed-Digit Number Representation,” Opt. Eng. 25, 38 (1986).
    [CrossRef]
  12. R. P. Bocker, B. L. Drake, M. E. Lasher, T. B. Henderson, “Modified-Signed Digit Addtion and Subtraction Using Optical Symbolic Substitution,” Appl. Opt. 25, 2456 (1986).
    [CrossRef] [PubMed]
  13. N. Takagi, H. Yasuura, S. Yajima, “High-Speed VLSI Multiplication Algorithm with a Redundant Binary Addition Tree,” IEEE Trans. Comput. C-34, 789 (1985).
    [CrossRef]
  14. Y. Li, G. Eichmann, “Conditional Symbolic Modified-Signed Digit Arithmetic Using Optical Content-Addressable Memory Logic Elements,” Appl. Opt. 26, 2328 (1987).
    [CrossRef] [PubMed]

1987

1986

1985

N. Takagi, H. Yasuura, S. Yajima, “High-Speed VLSI Multiplication Algorithm with a Redundant Binary Addition Tree,” IEEE Trans. Comput. C-34, 789 (1985).
[CrossRef]

1983

1980

1979

1961

A. Avizienis, “Signed-Digit Number Representation for Fast Parallel Arithmetic,” IRE Trans. Electron. Comput. EC-10, 389 (1961).
[CrossRef]

Alfano, R. R.

Avizienis, A.

A. Avizienis, “Signed-Digit Number Representation for Fast Parallel Arithmetic,” IRE Trans. Electron. Comput. EC-10, 389 (1961).
[CrossRef]

Bocker, R. P.

R. P. Bocker, B. L. Drake, M. E. Lasher, T. B. Henderson, “Modified-Signed Digit Addtion and Subtraction Using Optical Symbolic Substitution,” Appl. Opt. 25, 2456 (1986).
[CrossRef] [PubMed]

B. L. Drake, R. P. Bocker, M. E. Lasher, R. H. Patterson, W. J. Miceli, “Photonic Computing Using the Modified-Signed-Digit Number Representation,” Opt. Eng. 25, 38 (1986).
[CrossRef]

Campbell, D. M.

T. T. Dao, D. M. Campbell, “Multiple-Valued Logic: An Implementation,” Opt. Eng. 25, 14 (1986).
[CrossRef]

Casasent, D.

Dao, T. T.

T. T. Dao, D. M. Campbell, “Multiple-Valued Logic: An Implementation,” Opt. Eng. 25, 14 (1986).
[CrossRef]

Drake, B. L.

B. L. Drake, R. P. Bocker, M. E. Lasher, R. H. Patterson, W. J. Miceli, “Photonic Computing Using the Modified-Signed-Digit Number Representation,” Opt. Eng. 25, 38 (1986).
[CrossRef]

R. P. Bocker, B. L. Drake, M. E. Lasher, T. B. Henderson, “Modified-Signed Digit Addtion and Subtraction Using Optical Symbolic Substitution,” Appl. Opt. 25, 2456 (1986).
[CrossRef] [PubMed]

Eichmann, G.

Gaylord, T. K.

Goodman, J. W.

Guest, C. C.

Henderson, T. B.

Huang, A.

Ishihara, S.

Johnson, E. L.

E. L. Johnson, M. A. Karim, Digital Design: A Pragmatic Approach (PWS Publishers, Boston, 1987).

Karim, M. A.

E. L. Johnson, M. A. Karim, Digital Design: A Pragmatic Approach (PWS Publishers, Boston, 1987).

Lasher, M. E.

R. P. Bocker, B. L. Drake, M. E. Lasher, T. B. Henderson, “Modified-Signed Digit Addtion and Subtraction Using Optical Symbolic Substitution,” Appl. Opt. 25, 2456 (1986).
[CrossRef] [PubMed]

B. L. Drake, R. P. Bocker, M. E. Lasher, R. H. Patterson, W. J. Miceli, “Photonic Computing Using the Modified-Signed-Digit Number Representation,” Opt. Eng. 25, 38 (1986).
[CrossRef]

Li, Y.

Miceli, W. J.

B. L. Drake, R. P. Bocker, M. E. Lasher, R. H. Patterson, W. J. Miceli, “Photonic Computing Using the Modified-Signed-Digit Number Representation,” Opt. Eng. 25, 38 (1986).
[CrossRef]

Mirsalehi, M. M.

Patterson, R. H.

B. L. Drake, R. P. Bocker, M. E. Lasher, R. H. Patterson, W. J. Miceli, “Photonic Computing Using the Modified-Signed-Digit Number Representation,” Opt. Eng. 25, 38 (1986).
[CrossRef]

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 (1985).
[CrossRef]

Tsunida, Y.

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 (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 (1985).
[CrossRef]

Appl. Opt.

A. Huang, Y. Tsunida, J. W. Goodman, S. Ishihara, “Optical Computation Using Residue Arithmetic,” Appl. Opt. 18, 149 (1979).
[CrossRef] [PubMed]

D. Psaltis, D. Casasent, “Optical Residue Arithmetic: A Correlation Approach,” Appl. Opt. 18, 163 (1979).
[CrossRef] [PubMed]

C. C. Guest, T. K. Gaylord, “Truth Table Look-Up Optical Processing Utilizing Binary and Residue Arithmetic,” Appl. Opt. 19, 1201 (1980).
[CrossRef] [PubMed]

M. M. Mirsalehi, C. C. Guest, T. K. Gaylord, “Residue Number System Holographic Truth-Table Look-Up Processing: Detector Threshold Setting and Probability of Error Due to Amplitude and Phase Variations,” Appl. Opt. 22, 3583 (1983).
[CrossRef] [PubMed]

M. M. Mirsalehi, T. K. Gaylord, “Truth-Table Look-Up Parallel Processing Using an Optical Content-Addressable Memory,” Appl. Opt. 25, 2277 (1986).
[CrossRef] [PubMed]

R. P. Bocker, B. L. Drake, M. E. Lasher, T. B. Henderson, “Modified-Signed Digit Addtion and Subtraction Using Optical Symbolic Substitution,” Appl. Opt. 25, 2456 (1986).
[CrossRef] [PubMed]

M. M. Mirsalehi, T. K. Gaylord, “Logical Minimization of Multilevel Coded Function,” Appl. Opt. 25, 3078 (1986).
[CrossRef] [PubMed]

G. Eichmann, Y. Li, R. R. Alfano, “Optical Binary Coded Ternary Arithmetic and Logic,” Appl. Opt. 25, 3113 (1986).
[CrossRef] [PubMed]

Y. Li, G. Eichmann, “Conditional Symbolic Modified-Signed Digit Arithmetic Using Optical Content-Addressable Memory Logic Elements,” Appl. Opt. 26, 2328 (1987).
[CrossRef] [PubMed]

IEEE Trans. Comput.

N. Takagi, H. Yasuura, S. Yajima, “High-Speed VLSI Multiplication Algorithm with a Redundant Binary Addition Tree,” IEEE Trans. Comput. C-34, 789 (1985).
[CrossRef]

IRE Trans. Electron. Comput.

A. Avizienis, “Signed-Digit Number Representation for Fast Parallel Arithmetic,” IRE Trans. Electron. Comput. EC-10, 389 (1961).
[CrossRef]

Opt. Eng.

B. L. Drake, R. P. Bocker, M. E. Lasher, R. H. Patterson, W. J. Miceli, “Photonic Computing Using the Modified-Signed-Digit Number Representation,” Opt. Eng. 25, 38 (1986).
[CrossRef]

T. T. Dao, D. M. Campbell, “Multiple-Valued Logic: An Implementation,” Opt. Eng. 25, 14 (1986).
[CrossRef]

Other

E. L. Johnson, M. A. Karim, Digital Design: A Pragmatic Approach (PWS Publishers, Boston, 1987).

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Figures (1)

Fig. 1
Fig. 1

MSD addition (subtraction) scheme using modified symbolic substitution.

Tables (5)

Tables Icon

Table I Conditional Symbolic Substitution Rules for Addition and Subtraction (see Ref. 14 for Additional Details)

Tables Icon

Table II Modification of Table I(a): (a) with Changed Entries; (b) Reduced Rule Table.

Tables Icon

Table III Modified Truth Table for MSD Subtraction: (a) Intermediate Rule Table; (b) Reduced Rule Table

Tables Icon

Table IV Minterms for MSD Addition and Subtraction

Tables Icon

Table V Comparison of Various MSD Symbolic Substitution Schemas

Equations (1)

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D = j = 0 n - 1 X i 2 j ,

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