Abstract

A novel, to our knowledge, two-step digit-set-restricted modified signed-digit (MSD) addition–subtraction algorithm is proposed. With the introduction of the reference digits, the operand words are mapped into an intermediate carry word with all digits restricted to the set {1̅, 0} and an intermediate sum word with all digits restricted to the set {0, 1}, which can be summed to form the final result without carry generation. The operation can be performed in parallel by use of binary logic. An optical system that utilizes an electron-trapping device is suggested for accomplishing the required binary logic operations. By programming of the illumination of data arrays, any complex logic operations of multiple variables can be realized without additional temporal latency of the intermediate results. This technique has a high space–bandwidth product and signal-to-noise ratio. The main structure can be stacked to construct a compact optoelectronic MSD adder–subtracter.

© 1999 Optical Society of America

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  1. F. T. S. Yu, S. Jutamulia, Optical Signal Processing, Computing, and Neural Networks (Wiley, New York, 1992).
  2. A. P. Goutzoulis, D. K. Davies, E. C. Malarkey, “Prototype position-encoded residue look-up table using laser diodes,” Opt. Commun. 61, 302–308 (1987).
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    [CrossRef]
  4. P. S. Guilfoyle, “Systolic acousto-optic binary convolver,” Opt. Eng. 23, 020–025 (1984).
    [CrossRef]
  5. G. Li, L. Liu, L. Shao, Y. Yin, “Modified direct two’s complement parallel array multiplication algorithm for optical complex matrix operation,” Appl. Opt. 34, 1321–1328 (1995).
    [CrossRef] [PubMed]
  6. G. Li, L. Liu, L. Shao, Z. Wang, “Negabinary arithmetic algorithms for digital parallel optical computation,” Opt. Lett. 19, 1337–1339 (1994).
    [CrossRef] [PubMed]
  7. A. Avizienis, “Signed-digit number representations for fast parallel arithmetic,” IRE Trans. Electron. Comput. EC-10, 389–400 (1961).
    [CrossRef]
  8. 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, 38–43 (1986).
    [CrossRef]
  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. A. K. Cherri, M. A. Karim, “Modified-signed digit arithmetic using an efficient symbolic substitution,” Appl. Opt. 27, 3824–3827 (1988).
    [CrossRef] [PubMed]
  11. A. A. S. Awwal, “Recoded signed-digit binary addition–subtraction using optoelectronic symbolic substitution,” Appl. Opt. 31, 3205–3208 (1992).
    [CrossRef] [PubMed]
  12. M. S. Alam, M. A. Karim, A. A. S. Awwal, J. J. Westerkamp, “Optical processing based on conditional higher-order trinary modified signed-digit symbolic substitution,” Appl. Opt. 31, 5614–5621 (1992).
    [CrossRef] [PubMed]
  13. G. Li, F. Qian, H. Ruan, L. Liu, “Parallel optical negabinary signed-digit computing: algorithm and optical implementation,” Opt. Eng. 38, 403–414 (1999).
    [CrossRef]
  14. M. S. Alam, “Efficient binary signed-digit symbolic arithmetic,” Opt. Lett. 19, 353–355 (1994).
    [CrossRef] [PubMed]
  15. A. K. Cherri, M. K. Habib, M. S. Alam, “Optoelectronic recoded and nonrecoded binary signed-digit adder that uses optical correlation,” Appl. Opt. 37, 2153–2163 (1998).
    [CrossRef]
  16. M. S. Alam, Y. Ahuja, A. K. Cherri, A. Chatterjea, “Symmetrically recoded quaternary signed-digit arithmetic using a shared content-addressable memory,” Opt. Eng. 35, 1141–1149 (1996).
    [CrossRef]
  17. G. Li, L. Liu, H. Cheng, H. Jing, “Simplified quaternary signed-digit arithmetic and its optical implementation,” Opt. Commun. 137, 389–396 (1997).
    [CrossRef]
  18. D. Casasent, P. Woodford, “Symbolic substitution modified signed-digit optical adder,” Appl. Opt. 33, 1498–1506 (1994).
    [CrossRef] [PubMed]
  19. M. M. Mirsalehi, T. K. Gaylord, “Logical minimization of multilevel coded functions,” Appl. Opt. 25, 3078–3088 (1986).
    [CrossRef] [PubMed]
  20. B. Ha, Y. Li, “Parallel modified signed-digit arithmetic using an optoelectronic shared content-addressable-memory processor,” Appl. Opt. 33, 3647–3662 (1994).
    [CrossRef] [PubMed]
  21. M. Karim, A. A. S. Awwal, A. K. Cherri, “Polarization-encoded optical shadow-casting logic units: design,” Appl. Opt. 28, 2720–2725 (1987).
    [CrossRef]
  22. K. W. Wong, L. M. Cheng, “Optical modiofied signed-digit addition based on binary logical operations,” Opt. Laser Technol. 26, 213–217 (1994).
    [CrossRef]
  23. Y. Wu, Z. Zhang, L. Liu, Z. Wang, “Arithmetic operations using binary encoding modified-signed-digit system,” Opt. Commun. 100, 53–58 (1993).
    [CrossRef]
  24. K. Hwang, A. Louri, “Optical multiplication and division using modified-signed-digit symbolic substitution,” Opt. Eng. 28, 364–372 (1989).
    [CrossRef]
  25. M. D. Ercegovac, T. Lang, “On-the-fly conversion of redundant into conventional representation,” IEEE Trans. Comput. C-36, 895–897 (1987).
    [CrossRef]
  26. Y. Li, J. Zhu, G. Eichmann, “Optical on-the-fly conversion of a modified signed digit into two’s complement binary number representation,” Opt. Lett. 13, 294–296 (1988).
    [CrossRef] [PubMed]
  27. M. M. Mirsalehi, “Modified signed-digit to binary conversion using symbolic substitution,” in Optical Information and Processing Architectures IV, B. Javidi, ed., Proc. SPIE1772, 374–376 (1992).
    [CrossRef]
  28. S. Yen, C. Laih, C. Chen, J. Lee, “An efficient redundant-binary number to binary number converter,” IEEE J. Solid-State Circuits 27, 109–112 (1992).
    [CrossRef]
  29. M. A. Thornton, “Signed binary addition circuitry with inherent even parity outputs,” IEEE Trans. Comput. 46, 811–816 (1997).
    [CrossRef]
  30. A. D. McAulay, “Logic and arithmetic with luminescent rebroadcasting devices,” in Advances in Optical Information Processing III, D. R. Pape, ed., Proc. SPIE936, 321–326 (1988).
    [CrossRef]
  31. S. Jutamula, G. M. Storti, J. Lindmayer, “Use of electron trapping materials in optical signal processing. 1. Parallel Boolean logic,” Appl. Opt. 29, 4806–4811 (1990).
    [CrossRef]
  32. X. Yang, C. Y. Wrigley, J. Lindmayer, “Three-dimensional optical memory based on transparent electron thin film,” in Photonics for Computers, Neural Networks, and Memories, W. J. Miceli, A. Neff, T. Kowel, eds., Proc. SPIE1773, 413–422 (1992).
    [CrossRef]
  33. A. D. McAulay, J. Wang, X. Xu, “Optical perception learning for binary classification with spatial light rebroadcasters,” Appl. Opt. 32, 1346–1353 (1993).
    [CrossRef] [PubMed]

1999 (1)

G. Li, F. Qian, H. Ruan, L. Liu, “Parallel optical negabinary signed-digit computing: algorithm and optical implementation,” Opt. Eng. 38, 403–414 (1999).
[CrossRef]

1998 (1)

1997 (2)

G. Li, L. Liu, H. Cheng, H. Jing, “Simplified quaternary signed-digit arithmetic and its optical implementation,” Opt. Commun. 137, 389–396 (1997).
[CrossRef]

M. A. Thornton, “Signed binary addition circuitry with inherent even parity outputs,” IEEE Trans. Comput. 46, 811–816 (1997).
[CrossRef]

1996 (1)

M. S. Alam, Y. Ahuja, A. K. Cherri, A. Chatterjea, “Symmetrically recoded quaternary signed-digit arithmetic using a shared content-addressable memory,” Opt. Eng. 35, 1141–1149 (1996).
[CrossRef]

1995 (1)

1994 (5)

1993 (2)

Y. Wu, Z. Zhang, L. Liu, Z. Wang, “Arithmetic operations using binary encoding modified-signed-digit system,” Opt. Commun. 100, 53–58 (1993).
[CrossRef]

A. D. McAulay, J. Wang, X. Xu, “Optical perception learning for binary classification with spatial light rebroadcasters,” Appl. Opt. 32, 1346–1353 (1993).
[CrossRef] [PubMed]

1992 (3)

1990 (1)

1989 (1)

K. Hwang, A. Louri, “Optical multiplication and division using modified-signed-digit symbolic substitution,” Opt. Eng. 28, 364–372 (1989).
[CrossRef]

1988 (2)

1987 (4)

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]

M. D. Ercegovac, T. Lang, “On-the-fly conversion of redundant into conventional representation,” IEEE Trans. Comput. C-36, 895–897 (1987).
[CrossRef]

M. Karim, A. A. S. Awwal, A. K. Cherri, “Polarization-encoded optical shadow-casting logic units: design,” Appl. Opt. 28, 2720–2725 (1987).
[CrossRef]

A. P. Goutzoulis, D. K. Davies, E. C. Malarkey, “Prototype position-encoded residue look-up table using laser diodes,” Opt. Commun. 61, 302–308 (1987).
[CrossRef]

1986 (2)

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, 38–43 (1986).
[CrossRef]

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

1984 (1)

P. S. Guilfoyle, “Systolic acousto-optic binary convolver,” Opt. Eng. 23, 020–025 (1984).
[CrossRef]

1961 (1)

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

Ahuja, Y.

M. S. Alam, Y. Ahuja, A. K. Cherri, A. Chatterjea, “Symmetrically recoded quaternary signed-digit arithmetic using a shared content-addressable memory,” Opt. Eng. 35, 1141–1149 (1996).
[CrossRef]

Alam, M. S.

Avizienis, A.

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

Awwal, A. A. S.

Bocker, R. P.

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, 38–43 (1986).
[CrossRef]

Carlotto, M.

D. Psaltis, D. Casasent, D. Neft, M. Carlotto, “Accurate numerical computation by optical convolution,” in 1980 International Optical Computing Conference II, W. T. Rhodes, ed., Proc. SPIE232, 151–156 (1980).
[CrossRef]

Casasent, D.

D. Casasent, P. Woodford, “Symbolic substitution modified signed-digit optical adder,” Appl. Opt. 33, 1498–1506 (1994).
[CrossRef] [PubMed]

D. Psaltis, D. Casasent, D. Neft, M. Carlotto, “Accurate numerical computation by optical convolution,” in 1980 International Optical Computing Conference II, W. T. Rhodes, ed., Proc. SPIE232, 151–156 (1980).
[CrossRef]

Chatterjea, A.

M. S. Alam, Y. Ahuja, A. K. Cherri, A. Chatterjea, “Symmetrically recoded quaternary signed-digit arithmetic using a shared content-addressable memory,” Opt. Eng. 35, 1141–1149 (1996).
[CrossRef]

Chen, C.

S. Yen, C. Laih, C. Chen, J. Lee, “An efficient redundant-binary number to binary number converter,” IEEE J. Solid-State Circuits 27, 109–112 (1992).
[CrossRef]

Cheng, H.

G. Li, L. Liu, H. Cheng, H. Jing, “Simplified quaternary signed-digit arithmetic and its optical implementation,” Opt. Commun. 137, 389–396 (1997).
[CrossRef]

Cheng, L. M.

K. W. Wong, L. M. Cheng, “Optical modiofied signed-digit addition based on binary logical operations,” Opt. Laser Technol. 26, 213–217 (1994).
[CrossRef]

Cherri, A. K.

A. K. Cherri, M. K. Habib, M. S. Alam, “Optoelectronic recoded and nonrecoded binary signed-digit adder that uses optical correlation,” Appl. Opt. 37, 2153–2163 (1998).
[CrossRef]

M. S. Alam, Y. Ahuja, A. K. Cherri, A. Chatterjea, “Symmetrically recoded quaternary signed-digit arithmetic using a shared content-addressable memory,” Opt. Eng. 35, 1141–1149 (1996).
[CrossRef]

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

M. Karim, A. A. S. Awwal, A. K. Cherri, “Polarization-encoded optical shadow-casting logic units: design,” Appl. Opt. 28, 2720–2725 (1987).
[CrossRef]

Davies, D. K.

A. P. Goutzoulis, D. K. Davies, E. C. Malarkey, “Prototype position-encoded residue look-up table using laser diodes,” Opt. Commun. 61, 302–308 (1987).
[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, 38–43 (1986).
[CrossRef]

Eichmann, G.

Ercegovac, M. D.

M. D. Ercegovac, T. Lang, “On-the-fly conversion of redundant into conventional representation,” IEEE Trans. Comput. C-36, 895–897 (1987).
[CrossRef]

Gaylord, T. K.

Goutzoulis, A. P.

A. P. Goutzoulis, D. K. Davies, E. C. Malarkey, “Prototype position-encoded residue look-up table using laser diodes,” Opt. Commun. 61, 302–308 (1987).
[CrossRef]

Guilfoyle, P. S.

P. S. Guilfoyle, “Systolic acousto-optic binary convolver,” Opt. Eng. 23, 020–025 (1984).
[CrossRef]

Ha, B.

Habib, M. K.

Hwang, K.

K. Hwang, A. Louri, “Optical multiplication and division using modified-signed-digit symbolic substitution,” Opt. Eng. 28, 364–372 (1989).
[CrossRef]

Jing, H.

G. Li, L. Liu, H. Cheng, H. Jing, “Simplified quaternary signed-digit arithmetic and its optical implementation,” Opt. Commun. 137, 389–396 (1997).
[CrossRef]

Jutamula, S.

Jutamulia, S.

F. T. S. Yu, S. Jutamulia, Optical Signal Processing, Computing, and Neural Networks (Wiley, New York, 1992).

Karim, M.

M. Karim, A. A. S. Awwal, A. K. Cherri, “Polarization-encoded optical shadow-casting logic units: design,” Appl. Opt. 28, 2720–2725 (1987).
[CrossRef]

Karim, M. A.

Laih, C.

S. Yen, C. Laih, C. Chen, J. Lee, “An efficient redundant-binary number to binary number converter,” IEEE J. Solid-State Circuits 27, 109–112 (1992).
[CrossRef]

Lang, T.

M. D. Ercegovac, T. Lang, “On-the-fly conversion of redundant into conventional representation,” IEEE Trans. Comput. C-36, 895–897 (1987).
[CrossRef]

Lasher, M. E.

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, 38–43 (1986).
[CrossRef]

Lee, J.

S. Yen, C. Laih, C. Chen, J. Lee, “An efficient redundant-binary number to binary number converter,” IEEE J. Solid-State Circuits 27, 109–112 (1992).
[CrossRef]

Li, G.

G. Li, F. Qian, H. Ruan, L. Liu, “Parallel optical negabinary signed-digit computing: algorithm and optical implementation,” Opt. Eng. 38, 403–414 (1999).
[CrossRef]

G. Li, L. Liu, H. Cheng, H. Jing, “Simplified quaternary signed-digit arithmetic and its optical implementation,” Opt. Commun. 137, 389–396 (1997).
[CrossRef]

G. Li, L. Liu, L. Shao, Y. Yin, “Modified direct two’s complement parallel array multiplication algorithm for optical complex matrix operation,” Appl. Opt. 34, 1321–1328 (1995).
[CrossRef] [PubMed]

G. Li, L. Liu, L. Shao, Z. Wang, “Negabinary arithmetic algorithms for digital parallel optical computation,” Opt. Lett. 19, 1337–1339 (1994).
[CrossRef] [PubMed]

Li, Y.

Lindmayer, J.

S. Jutamula, G. M. Storti, J. Lindmayer, “Use of electron trapping materials in optical signal processing. 1. Parallel Boolean logic,” Appl. Opt. 29, 4806–4811 (1990).
[CrossRef]

X. Yang, C. Y. Wrigley, J. Lindmayer, “Three-dimensional optical memory based on transparent electron thin film,” in Photonics for Computers, Neural Networks, and Memories, W. J. Miceli, A. Neff, T. Kowel, eds., Proc. SPIE1773, 413–422 (1992).
[CrossRef]

Liu, L.

G. Li, F. Qian, H. Ruan, L. Liu, “Parallel optical negabinary signed-digit computing: algorithm and optical implementation,” Opt. Eng. 38, 403–414 (1999).
[CrossRef]

G. Li, L. Liu, H. Cheng, H. Jing, “Simplified quaternary signed-digit arithmetic and its optical implementation,” Opt. Commun. 137, 389–396 (1997).
[CrossRef]

G. Li, L. Liu, L. Shao, Y. Yin, “Modified direct two’s complement parallel array multiplication algorithm for optical complex matrix operation,” Appl. Opt. 34, 1321–1328 (1995).
[CrossRef] [PubMed]

G. Li, L. Liu, L. Shao, Z. Wang, “Negabinary arithmetic algorithms for digital parallel optical computation,” Opt. Lett. 19, 1337–1339 (1994).
[CrossRef] [PubMed]

Y. Wu, Z. Zhang, L. Liu, Z. Wang, “Arithmetic operations using binary encoding modified-signed-digit system,” Opt. Commun. 100, 53–58 (1993).
[CrossRef]

Louri, A.

K. Hwang, A. Louri, “Optical multiplication and division using modified-signed-digit symbolic substitution,” Opt. Eng. 28, 364–372 (1989).
[CrossRef]

Malarkey, E. C.

A. P. Goutzoulis, D. K. Davies, E. C. Malarkey, “Prototype position-encoded residue look-up table using laser diodes,” Opt. Commun. 61, 302–308 (1987).
[CrossRef]

McAulay, A. D.

A. D. McAulay, J. Wang, X. Xu, “Optical perception learning for binary classification with spatial light rebroadcasters,” Appl. Opt. 32, 1346–1353 (1993).
[CrossRef] [PubMed]

A. D. McAulay, “Logic and arithmetic with luminescent rebroadcasting devices,” in Advances in Optical Information Processing III, D. R. Pape, ed., Proc. SPIE936, 321–326 (1988).
[CrossRef]

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, 38–43 (1986).
[CrossRef]

Mirsalehi, M. M.

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

M. M. Mirsalehi, “Modified signed-digit to binary conversion using symbolic substitution,” in Optical Information and Processing Architectures IV, B. Javidi, ed., Proc. SPIE1772, 374–376 (1992).
[CrossRef]

Neft, D.

D. Psaltis, D. Casasent, D. Neft, M. Carlotto, “Accurate numerical computation by optical convolution,” in 1980 International Optical Computing Conference II, W. T. Rhodes, ed., Proc. SPIE232, 151–156 (1980).
[CrossRef]

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, 38–43 (1986).
[CrossRef]

Psaltis, D.

D. Psaltis, D. Casasent, D. Neft, M. Carlotto, “Accurate numerical computation by optical convolution,” in 1980 International Optical Computing Conference II, W. T. Rhodes, ed., Proc. SPIE232, 151–156 (1980).
[CrossRef]

Qian, F.

G. Li, F. Qian, H. Ruan, L. Liu, “Parallel optical negabinary signed-digit computing: algorithm and optical implementation,” Opt. Eng. 38, 403–414 (1999).
[CrossRef]

Ruan, H.

G. Li, F. Qian, H. Ruan, L. Liu, “Parallel optical negabinary signed-digit computing: algorithm and optical implementation,” Opt. Eng. 38, 403–414 (1999).
[CrossRef]

Shao, L.

Storti, G. M.

Thornton, M. A.

M. A. Thornton, “Signed binary addition circuitry with inherent even parity outputs,” IEEE Trans. Comput. 46, 811–816 (1997).
[CrossRef]

Wang, J.

Wang, Z.

G. Li, L. Liu, L. Shao, Z. Wang, “Negabinary arithmetic algorithms for digital parallel optical computation,” Opt. Lett. 19, 1337–1339 (1994).
[CrossRef] [PubMed]

Y. Wu, Z. Zhang, L. Liu, Z. Wang, “Arithmetic operations using binary encoding modified-signed-digit system,” Opt. Commun. 100, 53–58 (1993).
[CrossRef]

Westerkamp, J. J.

Wong, K. W.

K. W. Wong, L. M. Cheng, “Optical modiofied signed-digit addition based on binary logical operations,” Opt. Laser Technol. 26, 213–217 (1994).
[CrossRef]

Woodford, P.

Wrigley, C. Y.

X. Yang, C. Y. Wrigley, J. Lindmayer, “Three-dimensional optical memory based on transparent electron thin film,” in Photonics for Computers, Neural Networks, and Memories, W. J. Miceli, A. Neff, T. Kowel, eds., Proc. SPIE1773, 413–422 (1992).
[CrossRef]

Wu, Y.

Y. Wu, Z. Zhang, L. Liu, Z. Wang, “Arithmetic operations using binary encoding modified-signed-digit system,” Opt. Commun. 100, 53–58 (1993).
[CrossRef]

Xu, X.

Yang, X.

X. Yang, C. Y. Wrigley, J. Lindmayer, “Three-dimensional optical memory based on transparent electron thin film,” in Photonics for Computers, Neural Networks, and Memories, W. J. Miceli, A. Neff, T. Kowel, eds., Proc. SPIE1773, 413–422 (1992).
[CrossRef]

Yen, S.

S. Yen, C. Laih, C. Chen, J. Lee, “An efficient redundant-binary number to binary number converter,” IEEE J. Solid-State Circuits 27, 109–112 (1992).
[CrossRef]

Yin, Y.

Yu, F. T. S.

F. T. S. Yu, S. Jutamulia, Optical Signal Processing, Computing, and Neural Networks (Wiley, New York, 1992).

Zhang, Z.

Y. Wu, Z. Zhang, L. Liu, Z. Wang, “Arithmetic operations using binary encoding modified-signed-digit system,” Opt. Commun. 100, 53–58 (1993).
[CrossRef]

Zhu, J.

Appl. Opt. (12)

M. Karim, A. A. S. Awwal, A. K. Cherri, “Polarization-encoded optical shadow-casting logic units: design,” Appl. Opt. 28, 2720–2725 (1987).
[CrossRef]

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

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

S. Jutamula, G. M. Storti, J. Lindmayer, “Use of electron trapping materials in optical signal processing. 1. Parallel Boolean logic,” Appl. Opt. 29, 4806–4811 (1990).
[CrossRef]

M. S. Alam, M. A. Karim, A. A. S. Awwal, J. J. Westerkamp, “Optical processing based on conditional higher-order trinary modified signed-digit symbolic substitution,” Appl. Opt. 31, 5614–5621 (1992).
[CrossRef] [PubMed]

A. D. McAulay, J. Wang, X. Xu, “Optical perception learning for binary classification with spatial light rebroadcasters,” Appl. Opt. 32, 1346–1353 (1993).
[CrossRef] [PubMed]

D. Casasent, P. Woodford, “Symbolic substitution modified signed-digit optical adder,” Appl. Opt. 33, 1498–1506 (1994).
[CrossRef] [PubMed]

B. Ha, Y. Li, “Parallel modified signed-digit arithmetic using an optoelectronic shared content-addressable-memory processor,” Appl. Opt. 33, 3647–3662 (1994).
[CrossRef] [PubMed]

A. K. Cherri, M. K. Habib, M. S. Alam, “Optoelectronic recoded and nonrecoded binary signed-digit adder that uses optical correlation,” Appl. Opt. 37, 2153–2163 (1998).
[CrossRef]

G. Li, L. Liu, L. Shao, Y. Yin, “Modified direct two’s complement parallel array multiplication algorithm for optical complex matrix operation,” Appl. Opt. 34, 1321–1328 (1995).
[CrossRef] [PubMed]

A. A. S. Awwal, “Recoded signed-digit binary addition–subtraction using optoelectronic symbolic substitution,” Appl. Opt. 31, 3205–3208 (1992).
[CrossRef] [PubMed]

IEEE J. Solid-State Circuits (1)

S. Yen, C. Laih, C. Chen, J. Lee, “An efficient redundant-binary number to binary number converter,” IEEE J. Solid-State Circuits 27, 109–112 (1992).
[CrossRef]

IEEE Trans. Comput. (2)

M. A. Thornton, “Signed binary addition circuitry with inherent even parity outputs,” IEEE Trans. Comput. 46, 811–816 (1997).
[CrossRef]

M. D. Ercegovac, T. Lang, “On-the-fly conversion of redundant into conventional representation,” IEEE Trans. Comput. C-36, 895–897 (1987).
[CrossRef]

IRE Trans. Electron. Comput. (1)

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

Opt. Commun. (3)

A. P. Goutzoulis, D. K. Davies, E. C. Malarkey, “Prototype position-encoded residue look-up table using laser diodes,” Opt. Commun. 61, 302–308 (1987).
[CrossRef]

Y. Wu, Z. Zhang, L. Liu, Z. Wang, “Arithmetic operations using binary encoding modified-signed-digit system,” Opt. Commun. 100, 53–58 (1993).
[CrossRef]

G. Li, L. Liu, H. Cheng, H. Jing, “Simplified quaternary signed-digit arithmetic and its optical implementation,” Opt. Commun. 137, 389–396 (1997).
[CrossRef]

Opt. Eng. (5)

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Other (5)

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Figures (4)

Fig. 1
Fig. 1

Diagram of the three-step digit-set-restricted MSD adder–subtracter.

Fig. 2
Fig. 2

Tree structure for the two-step digit-set-restricted MSD addition–subtraction. The functional block A represents the computation rules generated from Eqs. (19) and (20), and the functional block B represents the computation rules generated from Eqs. (21) and (22).

Fig. 3
Fig. 3

Schematic diagram of the optical system for the digit-set-restricted MSD addition–subtraction with a single ET device. BS, beam splitter.

Fig. 4
Fig. 4

Experimental results corresponding to numerical examples 3 and 4. (a)–(c) Intermediate results. (d) Final results.

Tables (7)

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Table 1 Computation Rules

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Table 2 Computation Rules

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Table 3 Computation Rules

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Table 4 Computation rules

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Table 5 Encoding Scheme and Computation Rules for the Digit-Set-Restricted MSD Addition

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Table 6 Computation Rulesa

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Table 7 Encoding Scheme and Computation Rules for Digit-Set-Restricted MSD Subtraction

Equations (28)

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X=i=0N-1 xi2i,  xi1¯, 0, 1.
step 1: xi+yi=2ci+1+si,
step 2: zi=ci+si.
xi+yi=2ci+1+ri+1+si-ri.
X+Y=i=0N-1 2ixi+yi.
X+Y=i=0N-1 2i2ci+1+ri+1+si-ri=i=0N-1 2i+1ci+1+i=0N-1 2isi+i=0N-1 2i+1ri+1-i=0N-1 2iri.
i=0N-1 2isi=i=0N 2isi-2NsN.
i=0N-1 2i+1ri+1-i=0N-1 2iri=2NrN.
X+Y=i=0N 2ici+si+2NrN-sN.
2NrN-sN=2N1¯-1=2N+1cN+1.
2NrN-sN=2N+1cN+1.
2NrN-sN=2N+1cN+1.
X+Y=i=0N 2ici+si+2N+1cN+1 =i=0N+1 2ici+i=0N+1 2isi,
X+Y=C+S.
Xϕ 1¯ 0 1 1¯ 1¯ 0 1 1=-11710Yϕ 1 1¯ 0 0 1 1¯ 0 1=6910step 1:R0 0 1 0 0 0 1 1 ϕstep 2:C0 0 0 1¯ 1¯ 0 0 0 0 ϕSϕ 0 0 0 1 1 0 0 0 0step 3:Z0 0 0 1¯ 0 1 0 0 0 0=-4810.
X+Y=i=0N 2ici+i=0N 2isi.
Xϕ 1 0 1 1¯ 1¯ 1¯ 0 1=13310Yϕ 1 1¯ 1¯ 0 1 1¯ 1 1=3910step 1:R1 0 1 0 1 0 1 1 ϕstep 2:C0 0 1¯ 0 1¯ 1¯ 0 0 ϕS1 0 0 0 0 0 1 0 0step 3:Z1 0 1¯ 0 1¯ 1¯ 1 0 0=17210.
ri+1=wi.
ci+1=ui+vir¯i,
si=v¯iri+vir¯i.
ci+1=ui+viw¯i-1,
si=v¯iwi-1+viw¯i-1.
zi-=cis¯i,
zi+=sic¯i.
X1ϕ 1 1 1¯ 0 1¯ 1 0 0=15610Y1ϕ 0 1 0 0 1¯ 1¯ 1 1¯=5310U0 0 0 0 0 1 1 0 0V0 1 0 1 0 0 0 1 1W0 1 1 0 0 0 1 1 0step 1:C0 0 1 0 1 1 1 1 ϕS1 0 0 1 0 1 1 1 1step 2:Z-0 0 1 0 1 0 0 0 0Z+1 0 0 1 0 0 0 0 1Z1 0 1¯ 1 1¯ 0 0 0 1=20910.
X-Y=X+Y¯=i=0N-1xi+yi¯.
X2ϕ 1¯ 0 1 0 0 1¯ 1 1=-9710Y2ϕ 1¯ 1 1¯ 0 1¯ 1 0 1=-9910U0 1 0 0 0 0 1 0 1V0 0 1 0 0 1 0 1 0W0 1 0 1 0 1 0 1 1step 1:C1 0 0 0 1 1 0 1 ϕS1 0 0 0 1 1 1 0 0step 2:Z-0 0 0 0 0 0 0 1 0Z+0 0 0 0 0 0 1 0 0Z0 0 0 0 0 0 1 1¯ 0=210.
ui-1¯¯ vi-1w¯i-2¯¯ v¯iwi-1¯¯ viw¯i-1¯¯,v¯iwi-1¯¯ viw¯i-1¯¯ ui-1¯¯ vi-1w¯i-2¯¯,

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