Abstract

A symbolic-substitution-based optical numeric processor that uses recoded and nonrecoded trinary signed-digit (TSD) number representations is proposed. Also, we propose new joint spatial encodings for the TSD numbers that reduce the symbolic-substitution computation rules involved in the processor. Optoelectronic implementation of the proposed recoded adder is feasible. Also, the nonrecoded TSD addition can be performed optically in two steps. Both the proposed recoded and nonrecoded adders are more compact than a recently reported modified signed-digit counterpart and use fewer correlators and spatial light modulators.

© 1998 Optical Society of America

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  1. E. L. Johnson, M. A. Karim, Digital Design: A Pragmatic Approach (PWS-Kent, Boston, Mass., 1987).
  2. K. Hwang, Computer Arithmetic Principles: Architecture and Design (Wiley, New York, 1979).
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    [CrossRef]
  4. M. M. Mirsalehi, T. K. Gaylord, “Logical minimization of multilevel coded functions,” Appl. Opt. 25, 3078–3088 (1986).
    [CrossRef] [PubMed]
  5. A. P. Gautzouilis, E. C. Malarkey, D. K. Davies, J. C. Bradley, P. R. Beaudet, “Optical processing with residue LED/LD lookup tables,” Appl. Opt. 27, 1674–1681 (1988).
    [CrossRef]
  6. G. A. De Biase, A. Massini, “High efficiency redundant binary number representations for parallel arithmetic on optical computers,” Opt. Laser Technol. 26, 219–224 (1994).
    [CrossRef]
  7. N. Takagi, S. Yajima, “Modular multiplication hardware algorithms with redundant representation and their application to RSA cryptosystem,” IEEE Trans. Comput. 41, 887–891 (1992).
    [CrossRef]
  8. T. Stouraitis, C. Chen, “Hybrid signed-digit logarithmic number system processor,” Proc. Inst. Electr. Eng. Part E 140, 205–210 (1993).
  9. W. Balakrishnan, N. Burgess, “Very-high-speed VLSI 2s-complement multiplier using signed binary digits,” Proc. Inst. Electr. Eng. Part E 139, 29–34 (1992).
  10. F. Li, M. Morisue, “A novel Josephson adder without carry propagation delay,” IEEE Trans. Appl. Supercond. 3, 2683–2686 (1993).
    [CrossRef]
  11. P. Srinivasan, F. E. Petry, “Constant-division algorithms,” Proc. Inst. Electr. Eng. Part E 141, 334–340 (1994).
  12. N. Burgess, “Radix-2 SRT division algorithm with simple quotient digit selection,” Electron. Lett. 27, 1910–1911 (1991).
    [CrossRef]
  13. 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]
  14. A. K. Cherri, M. A. Karim, “Modified signed digit arithmetic using an efficient symbolic substitution,” Appl. Opt. 27, 3824–3827 (1988).
    [CrossRef] [PubMed]
  15. H. Huang, M. Itoh, T. Yatagai, “Modified signed-digit arithmetic based on redundant bit representation,” Appl. Opt. 33, 6146–6156 (1994).
    [CrossRef] [PubMed]
  16. A. A. S. Awwal, M. N. Islam, M. A. Karim, “Modified signed-digit trinary arithmetic by using symbolic substitution,” Appl. Opt. 31, 1687–1694 (1994).
    [CrossRef]
  17. A. K. Cherri, M. A. Habib, M. S. Alam, “Efficient implementation of arithmetic units based on polarization-encoded optical shadow-casting,” Opt. Eng. 36, 94–101 (1997).
    [CrossRef]
  18. 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, 5614–5622 (1992).
    [CrossRef] [PubMed]
  19. A. K. Cherri, “Symmetrically recoded modified signed-digit optical addition and subtraction,” Appl. Opt. 33, 4378–4382 (1994).
    [CrossRef] [PubMed]
  20. K. Hwang, D. K. Panda, “High-radix symbolic substitution and superposition techniques for optical matrix algebraic computations,” Opt. Eng. 31, 2422–2433 (1992).
    [CrossRef]
  21. A. K. Cherri, N. I. Khachab, “Canonical quaternary signed-digit arithmetic using optoelectronics symbolic substitution,” Opt. Laser Technol. 28, 397–403 (1996).
    [CrossRef]
  22. 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]
  23. 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]
  24. D. P. Casasent, E. C. Botha, “Multifunctional optical processor based on symbolic substitution,” Opt. Eng. 28, 425–433 (1989).
    [CrossRef]
  25. E. Botha, J. Richards, D. P. Casasent, “Optical laboratory morphological inspection processor,” Appl. Opt. 28, 5342–5350 (1989).
    [CrossRef] [PubMed]
  26. D. P. Casasent, R. Sturgill, “Optical hit-or-miss morphological transforms for ATR,” in Applications of Digital Image Processing XII, J. Neffs, ed., Proc. SPIE1153, 500–510 (1989).
    [CrossRef]
  27. D. P. Casasent, E. C. Botha, “Optical correlator production system neural net,” Appl. Opt. 31, 1030–1040 (1992).
    [CrossRef] [PubMed]
  28. K. Al-Ghoneim, D. P. Casasent, “High-accuracy pipelined iterative-tree optical multiplication,” Appl. Opt. 33, 1517–1527 (1994).
    [CrossRef] [PubMed]
  29. M. S. Alam, “Parallel optical computing using recoded trinary signed-digit numbers,” Appl. Opt. 33, 4392–4397 (1994).
    [CrossRef] [PubMed]
  30. D. P. Casasent, P. Woodford, “Symbolic substitution modified signed-digit optical adder,” Appl. Opt. 33, 1498–1506 (1994).
    [CrossRef] [PubMed]
  31. A. Huang, “Parallel algorithms for optical digital computers,” in Proceedings of the Tenth International Optical Computing Conference, S. Horvitz, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1983), pp. 13–17.
    [CrossRef]
  32. A. Louri, “Throughput enhancement for optical symbolic substitution systems,” Appl. Opt. 29, 2979–2980 (1990).
    [CrossRef] [PubMed]
  33. A. Avizienis, “Signed-digit number representations for fast parallel arithmetic,” IRE Trans. Electron. Comput. EC-10, 389–400 (1961).
    [CrossRef]
  34. H. E. Michel, A. A. S. Awwal, “Noise and cross talk study in an optical neural network,” in Proceedings of the National and Aerospace Electronics Conference (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1996), pp. 662–669.
  35. R. P. Webb, “Performance of an optoelectronic neural network in the presence of noise,” Appl. Opt. 34, 5230–5240 (1995).
    [CrossRef] [PubMed]
  36. S. Yamamoto, R. Sekura, J. Yamanaka, T. Ebihara, N. Kato, H. Hosi, “Optical recognition with LAPS-SLM,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 273–283 (1990).
    [CrossRef]
  37. M. Roe, K. Schnehrer, “High-speed and high-contrast operation of ferroelectric liquid crystal optically addressed spatial light modulators,” Opt. Eng. 32, 1662–1667 (1993).
    [CrossRef]
  38. J. Gallant, “Futurebus+ standards spur commercial products,” Electron. Design News 37(18), 51–64 (1992).
  39. M. Jerome, “The fastest PCs in the world,” PC Computing 15 (8), 183–192 (1997).

1997 (2)

A. K. Cherri, M. A. Habib, M. S. Alam, “Efficient implementation of arithmetic units based on polarization-encoded optical shadow-casting,” Opt. Eng. 36, 94–101 (1997).
[CrossRef]

M. Jerome, “The fastest PCs in the world,” PC Computing 15 (8), 183–192 (1997).

1996 (2)

A. K. Cherri, N. I. Khachab, “Canonical quaternary signed-digit arithmetic using optoelectronics symbolic substitution,” Opt. Laser Technol. 28, 397–403 (1996).
[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]

1995 (1)

1994 (9)

1993 (3)

T. Stouraitis, C. Chen, “Hybrid signed-digit logarithmic number system processor,” Proc. Inst. Electr. Eng. Part E 140, 205–210 (1993).

F. Li, M. Morisue, “A novel Josephson adder without carry propagation delay,” IEEE Trans. Appl. Supercond. 3, 2683–2686 (1993).
[CrossRef]

M. Roe, K. Schnehrer, “High-speed and high-contrast operation of ferroelectric liquid crystal optically addressed spatial light modulators,” Opt. Eng. 32, 1662–1667 (1993).
[CrossRef]

1992 (6)

J. Gallant, “Futurebus+ standards spur commercial products,” Electron. Design News 37(18), 51–64 (1992).

D. P. Casasent, E. C. Botha, “Optical correlator production system neural net,” Appl. Opt. 31, 1030–1040 (1992).
[CrossRef] [PubMed]

W. Balakrishnan, N. Burgess, “Very-high-speed VLSI 2s-complement multiplier using signed binary digits,” Proc. Inst. Electr. Eng. Part E 139, 29–34 (1992).

N. Takagi, S. Yajima, “Modular multiplication hardware algorithms with redundant representation and their application to RSA cryptosystem,” IEEE Trans. Comput. 41, 887–891 (1992).
[CrossRef]

K. Hwang, D. K. Panda, “High-radix symbolic substitution and superposition techniques for optical matrix algebraic computations,” Opt. Eng. 31, 2422–2433 (1992).
[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, 5614–5622 (1992).
[CrossRef] [PubMed]

1991 (1)

N. Burgess, “Radix-2 SRT division algorithm with simple quotient digit selection,” Electron. Lett. 27, 1910–1911 (1991).
[CrossRef]

1990 (1)

1989 (2)

D. P. Casasent, E. C. Botha, “Multifunctional optical processor based on symbolic substitution,” Opt. Eng. 28, 425–433 (1989).
[CrossRef]

E. Botha, J. Richards, D. P. Casasent, “Optical laboratory morphological inspection processor,” Appl. Opt. 28, 5342–5350 (1989).
[CrossRef] [PubMed]

1988 (2)

1986 (3)

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.

A. K. Cherri, M. A. Habib, M. S. Alam, “Efficient implementation of arithmetic units based on polarization-encoded optical shadow-casting,” Opt. Eng. 36, 94–101 (1997).
[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]

M. S. Alam, “Parallel optical computing using recoded trinary signed-digit numbers,” Appl. Opt. 33, 4392–4397 (1994).
[CrossRef] [PubMed]

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, 5614–5622 (1992).
[CrossRef] [PubMed]

Al-Ghoneim, K.

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.

Balakrishnan, W.

W. Balakrishnan, N. Burgess, “Very-high-speed VLSI 2s-complement multiplier using signed binary digits,” Proc. Inst. Electr. Eng. Part E 139, 29–34 (1992).

Beaudet, P. R.

Bocker, R. P.

Botha, E.

Botha, E. C.

D. P. Casasent, E. C. Botha, “Optical correlator production system neural net,” Appl. Opt. 31, 1030–1040 (1992).
[CrossRef] [PubMed]

D. P. Casasent, E. C. Botha, “Multifunctional optical processor based on symbolic substitution,” Opt. Eng. 28, 425–433 (1989).
[CrossRef]

Bradley, J. C.

Burgess, N.

W. Balakrishnan, N. Burgess, “Very-high-speed VLSI 2s-complement multiplier using signed binary digits,” Proc. Inst. Electr. Eng. Part E 139, 29–34 (1992).

N. Burgess, “Radix-2 SRT division algorithm with simple quotient digit selection,” Electron. Lett. 27, 1910–1911 (1991).
[CrossRef]

Casasent, D. P.

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.

T. Stouraitis, C. Chen, “Hybrid signed-digit logarithmic number system processor,” Proc. Inst. Electr. Eng. Part E 140, 205–210 (1993).

Cherri, A. K.

A. K. Cherri, M. A. Habib, M. S. Alam, “Efficient implementation of arithmetic units based on polarization-encoded optical shadow-casting,” Opt. Eng. 36, 94–101 (1997).
[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, N. I. Khachab, “Canonical quaternary signed-digit arithmetic using optoelectronics symbolic substitution,” Opt. Laser Technol. 28, 397–403 (1996).
[CrossRef]

A. K. Cherri, “Symmetrically recoded modified signed-digit optical addition and subtraction,” Appl. Opt. 33, 4378–4382 (1994).
[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]

Davies, D. K.

De Biase, G. A.

G. A. De Biase, A. Massini, “High efficiency redundant binary number representations for parallel arithmetic on optical computers,” Opt. Laser Technol. 26, 219–224 (1994).
[CrossRef]

Drake, B. L.

Ebihara, T.

S. Yamamoto, R. Sekura, J. Yamanaka, T. Ebihara, N. Kato, H. Hosi, “Optical recognition with LAPS-SLM,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 273–283 (1990).
[CrossRef]

Gallant, J.

J. Gallant, “Futurebus+ standards spur commercial products,” Electron. Design News 37(18), 51–64 (1992).

Gautzouilis, A. P.

Gaylord, T. K.

Ha, B.

Habib, M. A.

A. K. Cherri, M. A. Habib, M. S. Alam, “Efficient implementation of arithmetic units based on polarization-encoded optical shadow-casting,” Opt. Eng. 36, 94–101 (1997).
[CrossRef]

Henderson, T. B.

Hosi, H.

S. Yamamoto, R. Sekura, J. Yamanaka, T. Ebihara, N. Kato, H. Hosi, “Optical recognition with LAPS-SLM,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 273–283 (1990).
[CrossRef]

Huang, A.

A. Huang, “Parallel algorithms for optical digital computers,” in Proceedings of the Tenth International Optical Computing Conference, S. Horvitz, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1983), pp. 13–17.
[CrossRef]

Huang, H.

Hurst, S. L.

S. L. Hurst, “Multiple-valued threshold logic: its status and its realization,” Opt. Eng. 25, 44–53 (1986).
[CrossRef]

Hwang, K.

K. Hwang, D. K. Panda, “High-radix symbolic substitution and superposition techniques for optical matrix algebraic computations,” Opt. Eng. 31, 2422–2433 (1992).
[CrossRef]

K. Hwang, Computer Arithmetic Principles: Architecture and Design (Wiley, New York, 1979).

Islam, M. N.

Itoh, M.

Jerome, M.

M. Jerome, “The fastest PCs in the world,” PC Computing 15 (8), 183–192 (1997).

Johnson, E. L.

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

Karim, M. A.

Kato, N.

S. Yamamoto, R. Sekura, J. Yamanaka, T. Ebihara, N. Kato, H. Hosi, “Optical recognition with LAPS-SLM,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 273–283 (1990).
[CrossRef]

Khachab, N. I.

A. K. Cherri, N. I. Khachab, “Canonical quaternary signed-digit arithmetic using optoelectronics symbolic substitution,” Opt. Laser Technol. 28, 397–403 (1996).
[CrossRef]

Lasher, M. E.

Li, F.

F. Li, M. Morisue, “A novel Josephson adder without carry propagation delay,” IEEE Trans. Appl. Supercond. 3, 2683–2686 (1993).
[CrossRef]

Li, Y.

Louri, A.

Malarkey, E. C.

Massini, A.

G. A. De Biase, A. Massini, “High efficiency redundant binary number representations for parallel arithmetic on optical computers,” Opt. Laser Technol. 26, 219–224 (1994).
[CrossRef]

Michel, H. E.

H. E. Michel, A. A. S. Awwal, “Noise and cross talk study in an optical neural network,” in Proceedings of the National and Aerospace Electronics Conference (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1996), pp. 662–669.

Mirsalehi, M. M.

Morisue, M.

F. Li, M. Morisue, “A novel Josephson adder without carry propagation delay,” IEEE Trans. Appl. Supercond. 3, 2683–2686 (1993).
[CrossRef]

Panda, D. K.

K. Hwang, D. K. Panda, “High-radix symbolic substitution and superposition techniques for optical matrix algebraic computations,” Opt. Eng. 31, 2422–2433 (1992).
[CrossRef]

Petry, F. E.

P. Srinivasan, F. E. Petry, “Constant-division algorithms,” Proc. Inst. Electr. Eng. Part E 141, 334–340 (1994).

Richards, J.

Roe, M.

M. Roe, K. Schnehrer, “High-speed and high-contrast operation of ferroelectric liquid crystal optically addressed spatial light modulators,” Opt. Eng. 32, 1662–1667 (1993).
[CrossRef]

Schnehrer, K.

M. Roe, K. Schnehrer, “High-speed and high-contrast operation of ferroelectric liquid crystal optically addressed spatial light modulators,” Opt. Eng. 32, 1662–1667 (1993).
[CrossRef]

Sekura, R.

S. Yamamoto, R. Sekura, J. Yamanaka, T. Ebihara, N. Kato, H. Hosi, “Optical recognition with LAPS-SLM,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 273–283 (1990).
[CrossRef]

Srinivasan, P.

P. Srinivasan, F. E. Petry, “Constant-division algorithms,” Proc. Inst. Electr. Eng. Part E 141, 334–340 (1994).

Stouraitis, T.

T. Stouraitis, C. Chen, “Hybrid signed-digit logarithmic number system processor,” Proc. Inst. Electr. Eng. Part E 140, 205–210 (1993).

Sturgill, R.

D. P. Casasent, R. Sturgill, “Optical hit-or-miss morphological transforms for ATR,” in Applications of Digital Image Processing XII, J. Neffs, ed., Proc. SPIE1153, 500–510 (1989).
[CrossRef]

Takagi, N.

N. Takagi, S. Yajima, “Modular multiplication hardware algorithms with redundant representation and their application to RSA cryptosystem,” IEEE Trans. Comput. 41, 887–891 (1992).
[CrossRef]

Webb, R. P.

Westerkamp, J. J.

Woodford, P.

Yajima, S.

N. Takagi, S. Yajima, “Modular multiplication hardware algorithms with redundant representation and their application to RSA cryptosystem,” IEEE Trans. Comput. 41, 887–891 (1992).
[CrossRef]

Yamamoto, S.

S. Yamamoto, R. Sekura, J. Yamanaka, T. Ebihara, N. Kato, H. Hosi, “Optical recognition with LAPS-SLM,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 273–283 (1990).
[CrossRef]

Yamanaka, J.

S. Yamamoto, R. Sekura, J. Yamanaka, T. Ebihara, N. Kato, H. Hosi, “Optical recognition with LAPS-SLM,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 273–283 (1990).
[CrossRef]

Yatagai, T.

Appl. Opt. (16)

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

A. P. Gautzouilis, E. C. Malarkey, D. K. Davies, J. C. Bradley, P. R. Beaudet, “Optical processing with residue LED/LD lookup tables,” Appl. Opt. 27, 1674–1681 (1988).
[CrossRef]

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]

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

H. Huang, M. Itoh, T. Yatagai, “Modified signed-digit arithmetic based on redundant bit representation,” Appl. Opt. 33, 6146–6156 (1994).
[CrossRef] [PubMed]

A. A. S. Awwal, M. N. Islam, M. A. Karim, “Modified signed-digit trinary arithmetic by using symbolic substitution,” Appl. Opt. 31, 1687–1694 (1994).
[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, 5614–5622 (1992).
[CrossRef] [PubMed]

A. K. Cherri, “Symmetrically recoded modified signed-digit optical addition and subtraction,” Appl. Opt. 33, 4378–4382 (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]

D. P. Casasent, E. C. Botha, “Optical correlator production system neural net,” Appl. Opt. 31, 1030–1040 (1992).
[CrossRef] [PubMed]

K. Al-Ghoneim, D. P. Casasent, “High-accuracy pipelined iterative-tree optical multiplication,” Appl. Opt. 33, 1517–1527 (1994).
[CrossRef] [PubMed]

M. S. Alam, “Parallel optical computing using recoded trinary signed-digit numbers,” Appl. Opt. 33, 4392–4397 (1994).
[CrossRef] [PubMed]

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

E. Botha, J. Richards, D. P. Casasent, “Optical laboratory morphological inspection processor,” Appl. Opt. 28, 5342–5350 (1989).
[CrossRef] [PubMed]

A. Louri, “Throughput enhancement for optical symbolic substitution systems,” Appl. Opt. 29, 2979–2980 (1990).
[CrossRef] [PubMed]

R. P. Webb, “Performance of an optoelectronic neural network in the presence of noise,” Appl. Opt. 34, 5230–5240 (1995).
[CrossRef] [PubMed]

Electron. Design News (1)

J. Gallant, “Futurebus+ standards spur commercial products,” Electron. Design News 37(18), 51–64 (1992).

Electron. Lett. (1)

N. Burgess, “Radix-2 SRT division algorithm with simple quotient digit selection,” Electron. Lett. 27, 1910–1911 (1991).
[CrossRef]

IEEE Trans. Appl. Supercond. (1)

F. Li, M. Morisue, “A novel Josephson adder without carry propagation delay,” IEEE Trans. Appl. Supercond. 3, 2683–2686 (1993).
[CrossRef]

IEEE Trans. Comput. (1)

N. Takagi, S. Yajima, “Modular multiplication hardware algorithms with redundant representation and their application to RSA cryptosystem,” IEEE Trans. Comput. 41, 887–891 (1992).
[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. Eng. (6)

S. L. Hurst, “Multiple-valued threshold logic: its status and its realization,” Opt. Eng. 25, 44–53 (1986).
[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]

D. P. Casasent, E. C. Botha, “Multifunctional optical processor based on symbolic substitution,” Opt. Eng. 28, 425–433 (1989).
[CrossRef]

K. Hwang, D. K. Panda, “High-radix symbolic substitution and superposition techniques for optical matrix algebraic computations,” Opt. Eng. 31, 2422–2433 (1992).
[CrossRef]

A. K. Cherri, M. A. Habib, M. S. Alam, “Efficient implementation of arithmetic units based on polarization-encoded optical shadow-casting,” Opt. Eng. 36, 94–101 (1997).
[CrossRef]

M. Roe, K. Schnehrer, “High-speed and high-contrast operation of ferroelectric liquid crystal optically addressed spatial light modulators,” Opt. Eng. 32, 1662–1667 (1993).
[CrossRef]

Opt. Laser Technol. (2)

A. K. Cherri, N. I. Khachab, “Canonical quaternary signed-digit arithmetic using optoelectronics symbolic substitution,” Opt. Laser Technol. 28, 397–403 (1996).
[CrossRef]

G. A. De Biase, A. Massini, “High efficiency redundant binary number representations for parallel arithmetic on optical computers,” Opt. Laser Technol. 26, 219–224 (1994).
[CrossRef]

PC Computing (1)

M. Jerome, “The fastest PCs in the world,” PC Computing 15 (8), 183–192 (1997).

Proc. Inst. Electr. Eng. Part E (3)

T. Stouraitis, C. Chen, “Hybrid signed-digit logarithmic number system processor,” Proc. Inst. Electr. Eng. Part E 140, 205–210 (1993).

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[CrossRef]

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[CrossRef]

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[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Diagram of a two-step signed-digit adder–subtracter where the functional blocks A and B employ the signed digital computation rules shown in Tables 3, 4, respectively.

Fig. 2
Fig. 2

Basic SS cascaded correlator architecture (after Ref. 30).

Fig. 3
Fig. 3

Four-pixel TSD encodings that can be used in both the X and Y matrices.

Fig. 4
Fig. 4

New superimposed 4-pixel TSD encodings that are used in the input X matrix.

Fig. 5
Fig. 5

Alternative 4-pixel encodings for the recoded TSD adder: encodings for the digits in (a) the Y matrix and (b) the X matrix.

Fig. 6
Fig. 6

Alternative 3-pixel encodings for the recoded TSD adder: encodings for the digits in (a) the Y matrix and (b) the X matrix.

Fig. 7
Fig. 7

Ten-pixel encodings for the first-step nonrecoded TSD adder: (a) the input TSD digits, (b) the superimposed inputs for the X matrix, (c) the encodings for the Y matrix.

Fig. 8
Fig. 8

Nine-pixel encodings for the first-step nonrecoded TSD adder: (a) the input TSD digits, (b) the superimposed inputs for the X matrix, (c) the encodings for the Y matrix.

Fig. 9
Fig. 9

Five-pixel encodings for the first-step nonrecoded TSD adder: (a) the input TSD digits, (b) the superimposed inputs for the X matrix, (c) the encodings for the Y matrix.

Tables (4)

Tables Icon

Table 1 Recoding Truth Table for TSD Numbersa

Tables Icon

Table 2 Truth Table for Recoded TSD Addition

Tables Icon

Table 3 Truth Table for TSD Addition: First-Step Rules

Tables Icon

Table 4 Truth Table for TSD Addition: Second-Step Rules

Equations (39)

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D = j = 0 n - 1   b i 3 j ,
22 3 = 1000 2 = 8 10 ,
222222222222222 3 = 1101   1010   1111   0010   0110   1010 2 = 14348906 10 .
Step   one   x i + y i = 3 t i + 1 + w i ,
Step   two   S i = t i - 1 + w i ,
X = o o o z z z ob ob ob o z ob o z ob o z ob .
Y = t   o   z   o   z   ob   z   ob   tb .
X = o + o     o + z     o + ob     z + z     z + ob     ob + ob ,
Y = t   o   z   z   ob   tb .
X = 1 1 1 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 1 0 1 1 0 ,
Y = 0 1 0 0 1 0 1 1 0 0 0 0 0 0 1 1 1 1 0 0 1 1 0 0 .
M = 0.54 - 0.18 - 0.45 0.82 0.36 0.54 - 0.64 0.54 0.09 - 0.36 1.09 - 0.36 - 0.91 0.64 1.09 - 0.36 .
Y = MX = 0.36 0.73 - 0.09 0.36 0.91 0.09 0.91 0.82 0.27 - 0.09 0.27 - 0.27 - 0.27 0.45 0.82 0.73 0.82 1.18 - 0.27 0.45 0.82 0.73 - 0.18 0.18 .
Y = thresh MX = 0 1 0 0 1 0 1 1 0 0 0 0 0 0 1 1 1 1 0 0 1 1 0 0 .
X = 0 1 1 1 1 1 0 0 1 0 1 1 1 1 1 1 1 0 1 1 1 0 0 0 ,
Y = 1 0 1 1 1 0 0 0 0 0 1 1 0 1 1 1 0 1 1 1 0 0 0 0 ,
M = - 0.64 0.54 1.36 - 0.45 0.1   0.64 0.09 - 0.36 1.36 - 0.45 - 0.64 0.54 0.09 - 0.36 0.09 0.64 ,
Y = MX = 0.91 0.27 0.82 0.73 1.27 - 0.09 - 0.27 - 0.81 0.45 0.18 0.82 0.73 - 0.09 1.27 0.82 0.73 0.27 0.91 0.73 0.82 0.45 0.18 - 0.18 - 0.27 ,
Y = thresh MX = 1 0 1 1 1 0 0 0 0 0 1 1 0 1 1 1 0 1 1 1 0 0 0 0 .
Y = 1 1 0 0 0 1 0 1 0 1 0 0 0 1 1 ,
X = 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 ,
M = 0.75 0 - 0.25 1 - 1 1 - 0.25 0 0.75 ,
Y = MX = 0.75 0.75 0.5 - 0.25 - 0.25 1 0 1 0 1 - 0.25 - 0.25 0.5 0.75 0.75 .
Y = thresh MX = 1 1 0 0 0 1 0 1 0 1 0 0 0 1 1 .
Y = 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 0 0 0 1 0 0 0 1 0 0 1 ,
X = 1 1 1 1 0 1 0 0 0 0 0 1 1 1 0 1 1 0 0 0 0 0 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 ,
M = 0 0 0 - 1 0 0 0 1 0 0 - 2 1 - 2 1 - 2 2 1 - 1 2 - 1 1 - 1 1 0 1 - 1 0 0 - 1 1 1 - 1 1 1 0 0 1 - 1 - 1 0 - 1 0 - 2 0 - 1 1 - 1 1 1 1 - 1 1 0 - 1 0 0 1 0 1 - 1 .
Y = 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 ,
X = 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 ,
M = 0 0 0 - 1 0 0 1 0 0 0 1 - 1 1 0 1 - 1 1 - 1 0 - 1 0 0 0 0 0 0 1 0 - 1 - 1 1 - 1 2 - 1 1 0 1 0 1 0 1 - 2 1 - 2 1 - 1 1 - 1 - 1 0 1 0 2 - 1 .
Y = 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 ,
X = 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 ,
M = 0.75 0 0 - 0.5 0.25 - 0.17 0.5 0 0.5 - 0.17 0.25 - 0.5 0 0 0.75 0.75 - 1 0.5 0.5 - 0.25 0.33 0.5 - 1 0.5 0.33 - 0.25 0.5 0.5 - 1 0.75 ,
Y = MX = 0.75 0.75 0.75 0.25 0.5 - 0.25 - 0.25 - 0.25 0.25 - 0.17 0.33 0.33 0.83 0.67 0.83 0.33 0.33 - 0.17 0.25 - 0.25 - 0.25 - 0.25 0.5 0.25 0.75 0.75 0.75 0.75 - 0.25 0.25 0.75 0.5 - 0.25 0.75 0.25 - 0.25 0.33 0.83 - 0.17 0.33 0.67 0.33 - 0.17 0.83 0.33 - 0.25 0.25 0.75 - 0.25 0.5 0.75 0.25 - 0.25 0.75 .
Y = thresh MX = 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 .
200   pixels / mm 25   mm 2 100   μ s = 250 × 10 9 pixels / s .
250 × 10 9 pixels / s 3   pixels / bit = 83 × 10 9 bits / s .
250 × 10 9 pixels / s 5   pixels / bit = 50 × 10 9 bits / s .
83 × 10 9 bits / s 64   bits / word 2   words / addition = 0.65 × 10 9 additions / s , 50 × 10 9 bits / s 64   bits / word 2   words / addition = 0.39 × 10 9 additions / s

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