Abstract

Trinary signed-digit (TSD) symbolic-substitution-based (SS-based) optical adders, which were recently proposed, are used as the basic modules for designing highly parallel optical multiplications by use of cascaded optical correlators. The proposed multiplications perform carry-free generation of the multiplication partial products of two words in constant time. Also, three different multiplication designs are presented, and new joint spatial encodings for the TSD numbers are introduced. The proposed joint spatial encodings allow one to reduce the SS computation rules involved in optical multiplication. In addition, the proposed joint spatial encodings increase the space–bandwidth product of the spatial light modulators of the optical system. This increase is achieved by reduction of the numbers of pixels in the joint spatial encodings for the input TSD operands as well as reduction of the number of pixels used in the proposed matched spatial filters for the optical multipliers.

© 1999 Optical Society of America

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References

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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).
  3. 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]
  4. 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]
  5. K. Hwang, D. K. Panda, “High-radix symbolic substitution and superposition techniques for optical matrix algebraic computations,” Opt. Eng. 31, 2422–2433 (1992).
    [CrossRef]
  6. A. K. Cherri, N. I. Khachab, “Canonical quaternary signed-digit arithmetic using optoelectronics symbolic substitution,” Opt. Laser Technol. 28, 397–403 (1996).
    [CrossRef]
  7. 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]
  8. A. Avizienis, “Signed-digit number representations for fast parallel arithmetic,” IRE Trans. Electronic Computers, EC-10, 389–400 (1961).
  9. A. Huang, “Parallel algorithms for optical digital computers,” in Proceedings of the Tenth International Optical Computing Conference, S. Horvitz, ed. (IEEE Computer Society, Los Alamitos, Calif., 1983), pp. 13–17.
    [CrossRef]
  10. 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]
  11. A. K. Cherri, M. A. Karim, “Modified signed digit arithmetic using an efficient symbolic substitution,” Appl. Opt. 27, 3824–3827 (1988).
    [CrossRef] [PubMed]
  12. H. Huang, M. Itoh, T. Yatagai, “Modified signed-digit arithmetic based on redundant bit representation,” Appl. Opt. 33, 6146–6156 (1994).
    [CrossRef] [PubMed]
  13. 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]
  14. Y. Li, H. Kim, A. Kostrzewski, G. Eichmann, “Content-addressable-memory-based single-stage optical modified signed-digit arithmetic,” Opt. Lett. 14, 1254–1256 (1989).
    [CrossRef] [PubMed]
  15. L. Liu, “Optoelectronics implementation of mathmatical morphology,” Opt. Lett. 14, 482–485 (1989).
    [CrossRef] [PubMed]
  16. S. Zhou, S. Campbell, W. Wu, P. Yeh, H.-K. Liu, “Modified signed-digit arithmetic for multi-input digital computing,” Appl. Opt. 33, 1507–1516 (1994).
    [CrossRef] [PubMed]
  17. G. Eichmann, A. Kostrzewski, D. H. Kim, Y. Li, “Optical higher-order symbolic recognition,” Appl. Opt. 29, 2135–2147 (1996).
    [CrossRef]
  18. G. Li, L. Liu, L. Shao, Y. Yin, “Optical negabinary addition with higher-order substitution rules,” Opt. Commun. 129, 323–330 (1996).
    [CrossRef]
  19. G. Li, L. Liu, H. Cheng, H. Jing, “Simplified quaternary signed-digit arithmetic and its optical implementation,” Opt. Commun. 173, 389–396 (1997).
    [CrossRef]
  20. G. Li, L. Liu, H. Cheng, X. Yan, “Parallel optical quaternary signed-digit multiplication and its use for matrix-vector operation,” Optik (Stuttgart/Weimar) 107, 165–172 (1998).
  21. A. K. Cherri, M. K. Habib, M. S. Alam, “Optoelectronic recoded and nonrecoded trinary signed-digit adder using optical correlation,” Appl. Opt. 37, 2153–2163 (1998).
    [CrossRef]
  22. A. K. Cherri, N. I. Khachab, E. H. Ismail, “One-step optical trinary signed-digit arithmetic using redundant bit representations,” Opt. Laser Technol. 29, 281–290 (1997).
    [CrossRef]
  23. 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]
  24. D. P. Casasent, P. Woodford, “Symbolic substitution modified signed-digit optical adder,” Appl. Opt. 33, 1498–1506 (1994).
    [CrossRef] [PubMed]
  25. M. S. Alam, “Parallel optical computing using recoded trinary signed-digit numbers,” Appl. Opt. 33, 4392–4397 (1994).
    [CrossRef] [PubMed]
  26. D. P. Casasent, E. C. Botha, “Multifunctional optical processor based on symbolic substitution,” Opt. Eng. 28, 425–433 (1989).
    [CrossRef]
  27. D. P. Casasent, E. C. Botha, “Optical correlator production system neural net,” Appl. Opt. 31, 1030–1040 (1992).
    [CrossRef] [PubMed]
  28. E. Botha, J. Richards, D. P. Casassent, “Optical laboratory morphological inspection processor,” Appl. Opt. 28, 5342–5350 (1989).
    [CrossRef] [PubMed]
  29. 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]
  30. K. Al-Ghoneim, D. P. Casasent, “High-accuracy pipelined iterative-tree optical multiplication,” Appl. Opt. 33, 4392–4397 (1994).
    [CrossRef]
  31. A. K. Cherri, M. A. Karim, “Symbolic substitution based operations using holograms: multiplication and histogram equalization,” Opt. Eng. 28, 638–644 (1989).
    [CrossRef]
  32. A. K. Cherri, M. A. Karim, “Edge detection using symbolic substitution,” Opt. Commun. 74, 10–14 (1989).
    [CrossRef]
  33. A. K. Cherri, M. A. Karim, “Optical symbolic substitution: edge detection using Prewitt, Sobel, and Roberts operators,” Appl. Opt. 28, 4644–4648 (1989).
    [CrossRef] [PubMed]
  34. A. Louri, “Throughput enhancement for optical symbolic substitution systems,” Appl. Opt. 29, 2979–2980 (1990).
    [CrossRef] [PubMed]

1998 (2)

G. Li, L. Liu, H. Cheng, X. Yan, “Parallel optical quaternary signed-digit multiplication and its use for matrix-vector operation,” Optik (Stuttgart/Weimar) 107, 165–172 (1998).

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

1997 (2)

A. K. Cherri, N. I. Khachab, E. H. Ismail, “One-step optical trinary signed-digit arithmetic using redundant bit representations,” Opt. Laser Technol. 29, 281–290 (1997).
[CrossRef]

G. Li, L. Liu, H. Cheng, H. Jing, “Simplified quaternary signed-digit arithmetic and its optical implementation,” Opt. Commun. 173, 389–396 (1997).
[CrossRef]

1996 (4)

G. Eichmann, A. Kostrzewski, D. H. Kim, Y. Li, “Optical higher-order symbolic recognition,” Appl. Opt. 29, 2135–2147 (1996).
[CrossRef]

G. Li, L. Liu, L. Shao, Y. Yin, “Optical negabinary addition with higher-order substitution rules,” Opt. Commun. 129, 323–330 (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]

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]

1994 (8)

1992 (2)

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

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

1990 (1)

1989 (7)

1988 (2)

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

1986 (1)

1961 (1)

A. Avizienis, “Signed-digit number representations for fast parallel arithmetic,” IRE Trans. Electronic Computers, EC-10, 389–400 (1961).

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.

Al-Ghoneim, K.

Avizienis, A.

A. Avizienis, “Signed-digit number representations for fast parallel arithmetic,” IRE Trans. Electronic Computers, EC-10, 389–400 (1961).

Awwal, A. A. S.

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.

Campbell, S.

S. Zhou, S. Campbell, W. Wu, P. Yeh, H.-K. Liu, “Modified signed-digit arithmetic for multi-input digital computing,” Appl. Opt. 33, 1507–1516 (1994).
[CrossRef] [PubMed]

Casasent, D. P.

K. Al-Ghoneim, D. P. Casasent, “High-accuracy pipelined iterative-tree optical multiplication,” Appl. Opt. 33, 4392–4397 (1994).
[CrossRef]

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

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]

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]

Casassent, 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]

Cheng, H.

G. Li, L. Liu, H. Cheng, X. Yan, “Parallel optical quaternary signed-digit multiplication and its use for matrix-vector operation,” Optik (Stuttgart/Weimar) 107, 165–172 (1998).

G. Li, L. Liu, H. Cheng, H. Jing, “Simplified quaternary signed-digit arithmetic and its optical implementation,” Opt. Commun. 173, 389–396 (1997).
[CrossRef]

Cherri, A. K.

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

A. K. Cherri, N. I. Khachab, E. H. Ismail, “One-step optical trinary signed-digit arithmetic using redundant bit representations,” Opt. Laser Technol. 29, 281–290 (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, M. A. Karim, “Symbolic substitution based operations using holograms: multiplication and histogram equalization,” Opt. Eng. 28, 638–644 (1989).
[CrossRef]

A. K. Cherri, M. A. Karim, “Edge detection using symbolic substitution,” Opt. Commun. 74, 10–14 (1989).
[CrossRef]

A. K. Cherri, M. A. Karim, “Optical symbolic substitution: edge detection using Prewitt, Sobel, and Roberts operators,” Appl. Opt. 28, 4644–4648 (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]

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.

Eichmann, G.

Gautzouilis, A. P.

Ha, B.

Habib, M. K.

Henderson, T. B.

Huang, A.

A. Huang, “Parallel algorithms for optical digital computers,” in Proceedings of the Tenth International Optical Computing Conference, S. Horvitz, ed. (IEEE Computer Society, Los Alamitos, Calif., 1983), pp. 13–17.
[CrossRef]

Huang, H.

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.

Ismail, E. H.

A. K. Cherri, N. I. Khachab, E. H. Ismail, “One-step optical trinary signed-digit arithmetic using redundant bit representations,” Opt. Laser Technol. 29, 281–290 (1997).
[CrossRef]

Itoh, M.

Jing, H.

G. Li, L. Liu, H. Cheng, H. Jing, “Simplified quaternary signed-digit arithmetic and its optical implementation,” Opt. Commun. 173, 389–396 (1997).
[CrossRef]

Johnson, E. L.

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

Karim, M. A.

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]

A. K. Cherri, M. A. Karim, “Optical symbolic substitution: edge detection using Prewitt, Sobel, and Roberts operators,” Appl. Opt. 28, 4644–4648 (1989).
[CrossRef] [PubMed]

A. K. Cherri, M. A. Karim, “Symbolic substitution based operations using holograms: multiplication and histogram equalization,” Opt. Eng. 28, 638–644 (1989).
[CrossRef]

A. K. Cherri, M. A. Karim, “Edge detection using symbolic substitution,” Opt. Commun. 74, 10–14 (1989).
[CrossRef]

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).

Khachab, N. I.

A. K. Cherri, N. I. Khachab, E. H. Ismail, “One-step optical trinary signed-digit arithmetic using redundant bit representations,” Opt. Laser Technol. 29, 281–290 (1997).
[CrossRef]

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

Kim, D. H.

Kim, H.

Kostrzewski, A.

Lasher, M. E.

Li, G.

G. Li, L. Liu, H. Cheng, X. Yan, “Parallel optical quaternary signed-digit multiplication and its use for matrix-vector operation,” Optik (Stuttgart/Weimar) 107, 165–172 (1998).

G. Li, L. Liu, H. Cheng, H. Jing, “Simplified quaternary signed-digit arithmetic and its optical implementation,” Opt. Commun. 173, 389–396 (1997).
[CrossRef]

G. Li, L. Liu, L. Shao, Y. Yin, “Optical negabinary addition with higher-order substitution rules,” Opt. Commun. 129, 323–330 (1996).
[CrossRef]

Li, Y.

Liu, H.-K.

S. Zhou, S. Campbell, W. Wu, P. Yeh, H.-K. Liu, “Modified signed-digit arithmetic for multi-input digital computing,” Appl. Opt. 33, 1507–1516 (1994).
[CrossRef] [PubMed]

Liu, L.

G. Li, L. Liu, H. Cheng, X. Yan, “Parallel optical quaternary signed-digit multiplication and its use for matrix-vector operation,” Optik (Stuttgart/Weimar) 107, 165–172 (1998).

G. Li, L. Liu, H. Cheng, H. Jing, “Simplified quaternary signed-digit arithmetic and its optical implementation,” Opt. Commun. 173, 389–396 (1997).
[CrossRef]

G. Li, L. Liu, L. Shao, Y. Yin, “Optical negabinary addition with higher-order substitution rules,” Opt. Commun. 129, 323–330 (1996).
[CrossRef]

L. Liu, “Optoelectronics implementation of mathmatical morphology,” Opt. Lett. 14, 482–485 (1989).
[CrossRef] [PubMed]

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]

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]

Richards, J.

Shao, L.

G. Li, L. Liu, L. Shao, Y. Yin, “Optical negabinary addition with higher-order substitution rules,” Opt. Commun. 129, 323–330 (1996).
[CrossRef]

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]

Woodford, P.

Wu, W.

S. Zhou, S. Campbell, W. Wu, P. Yeh, H.-K. Liu, “Modified signed-digit arithmetic for multi-input digital computing,” Appl. Opt. 33, 1507–1516 (1994).
[CrossRef] [PubMed]

Yan, X.

G. Li, L. Liu, H. Cheng, X. Yan, “Parallel optical quaternary signed-digit multiplication and its use for matrix-vector operation,” Optik (Stuttgart/Weimar) 107, 165–172 (1998).

Yatagai, T.

Yeh, P.

S. Zhou, S. Campbell, W. Wu, P. Yeh, H.-K. Liu, “Modified signed-digit arithmetic for multi-input digital computing,” Appl. Opt. 33, 1507–1516 (1994).
[CrossRef] [PubMed]

Yin, Y.

G. Li, L. Liu, L. Shao, Y. Yin, “Optical negabinary addition with higher-order substitution rules,” Opt. Commun. 129, 323–330 (1996).
[CrossRef]

Zhou, S.

S. Zhou, S. Campbell, W. Wu, P. Yeh, H.-K. Liu, “Modified signed-digit arithmetic for multi-input digital computing,” Appl. Opt. 33, 1507–1516 (1994).
[CrossRef] [PubMed]

Appl. Opt. (2)

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

S. Zhou, S. Campbell, W. Wu, P. Yeh, H.-K. Liu, “Modified signed-digit arithmetic for multi-input digital computing,” Appl. Opt. 33, 1507–1516 (1994).
[CrossRef] [PubMed]

Appl. Opt. (14)

G. Eichmann, A. Kostrzewski, D. H. Kim, Y. Li, “Optical higher-order symbolic recognition,” Appl. Opt. 29, 2135–2147 (1996).
[CrossRef]

H. Huang, M. Itoh, T. Yatagai, “Modified signed-digit arithmetic based on redundant bit representation,” Appl. Opt. 33, 6146–6156 (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. 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. K. Habib, M. S. Alam, “Optoelectronic recoded and nonrecoded trinary signed-digit adder using optical correlation,” Appl. Opt. 37, 2153–2163 (1998).
[CrossRef]

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]

D. P. Casasent, P. Woodford, “Symbolic substitution modified signed-digit optical adder,” Appl. Opt. 33, 1498–1506 (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, E. C. Botha, “Optical correlator production system neural net,” Appl. Opt. 31, 1030–1040 (1992).
[CrossRef] [PubMed]

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

K. Al-Ghoneim, D. P. Casasent, “High-accuracy pipelined iterative-tree optical multiplication,” Appl. Opt. 33, 4392–4397 (1994).
[CrossRef]

A. K. Cherri, M. A. Karim, “Optical symbolic substitution: edge detection using Prewitt, Sobel, and Roberts operators,” Appl. Opt. 28, 4644–4648 (1989).
[CrossRef] [PubMed]

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

IRE Trans. Electronic Computers (1)

A. Avizienis, “Signed-digit number representations for fast parallel arithmetic,” IRE Trans. Electronic Computers, EC-10, 389–400 (1961).

Opt. Commun. (2)

G. Li, L. Liu, L. Shao, Y. Yin, “Optical negabinary addition with higher-order substitution rules,” Opt. Commun. 129, 323–330 (1996).
[CrossRef]

G. Li, L. Liu, H. Cheng, H. Jing, “Simplified quaternary signed-digit arithmetic and its optical implementation,” Opt. Commun. 173, 389–396 (1997).
[CrossRef]

Opt. Laser Technol. (1)

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

Opt. Commun. (1)

A. K. Cherri, M. A. Karim, “Edge detection using symbolic substitution,” Opt. Commun. 74, 10–14 (1989).
[CrossRef]

Opt. Eng. (4)

A. K. Cherri, M. A. Karim, “Symbolic substitution based operations using holograms: multiplication and histogram equalization,” Opt. Eng. 28, 638–644 (1989).
[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]

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]

Opt. Laser Technol. (2)

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]

A. K. Cherri, N. I. Khachab, E. H. Ismail, “One-step optical trinary signed-digit arithmetic using redundant bit representations,” Opt. Laser Technol. 29, 281–290 (1997).
[CrossRef]

Opt. Lett. (2)

Optik (Stuttgart/Weimar) (1)

G. Li, L. Liu, H. Cheng, X. Yan, “Parallel optical quaternary signed-digit multiplication and its use for matrix-vector operation,” Optik (Stuttgart/Weimar) 107, 165–172 (1998).

Other (4)

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

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

A. Huang, “Parallel algorithms for optical digital computers,” in Proceedings of the Tenth International Optical Computing Conference, S. Horvitz, ed. (IEEE Computer Society, Los Alamitos, Calif., 1983), pp. 13–17.
[CrossRef]

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]

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Figures (14)

Fig. 1
Fig. 1

Basic SS cascaded-correlator architecture (in the manner of Ref. 24).

Fig. 2
Fig. 2

Recoded TSD adder design 1: (a) Four-pixel TSD encodings that can be used in both the X and the Y matrices. (b) Joint spatial encodings that are used in the input X matrix. (c) Solution of the M–V equation.

Fig. 3
Fig. 3

Recoded TSD adder design 2: (a) Alternative 3-pixel TSD encodings that can be used in both the X and the Y matrices. (b) Joint spatial encodings that are used in the input X matrix. (c) Solution of the M–V equation.

Fig. 4
Fig. 4

Two 5-digit multiplications.

Fig. 5
Fig. 5

Recoded TSD multiplier design 1: (a) Six-pixel TSD encodings that can be used in both the X and the Y matrices. (b) Joint spatial encodings that are used in the input X matrix. (c) Solution of the M–V equation.

Fig. 6
Fig. 6

Recoded TSD multiplier design 2: (a) Alternative 4-pixel TSD encodings that can be used in both the X and the Y matrices. (b) Joint spatial encodings that are used in the input X matrix. (c) Solution of the M–V equation.

Fig. 7
Fig. 7

Example for recoded TSD multiplication showing the sequence of recoding and addition.

Fig. 8
Fig. 8

Nonrecoded TSD multiplication when only the multiplier is recoded in design 1: Twelve-pixel encodings that can be used in both the X and the Y matrices. (b) Joint spatial encodings that are used in the input X matrix. (c) Solution of the M–V equation.

Fig. 9
Fig. 9

Nonrecoded TSD multiplication when only the multiplier is recoded in design 2: (a) Alternative 8-pixel encodings that can be used in both the X and the Y matrices. (b) Joint spatial encodings that are used in the input X matrix. (c) Solution of the M–V equation.

Fig. 10
Fig. 10

Nonrecoded TSD multiplication when only the multiplier is recoded in design 3: Third alternative of 7-pixel encodings that can be used in both the X and the Y matrices. (b) Joint spatial encodings that are used in the input X matrix. (c) Solution of the M–V equation.

Fig. 11
Fig. 11

Nonrecoded TSD multiplication when only the multiplier is recoded in design 4: (a) Fourth alternative of 6-pixel encodings that can be used in both the X and the Y matrices. (b) Joint spatial encodings that are used in the input X matrix. (c) Solution of the M–V equation.

Fig. 12
Fig. 12

Four-digit nonrecoded TSD multiplication: p and c are the partial product and the partial carry, respectively.

Fig. 13
Fig. 13

Nonrecoded TSD multiplication design 1 with carries: (a) Fifteen-pixel and (b) 3-pixel pixel encodings that can be used in the X and the Y matrices, respectively. (c) Joint spatial encodings that are used in the input X matrix. (d) Solution of the M–V equation.

Fig. 14
Fig. 14

Nonrecoded TSD multiplication design 2 with carries: (a) Ten-pixel and (b) 3-pixel encodings that can be used in the X and the Y matrices, respectively. (c) Joint spatial encodings that are used in the input X matrix. (d) Solution of the M–V equation.

Tables (5)

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Table 1 Recoding Truth Table for TSD Numbers as Reported in Ref. 25

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Table 2 Truth Table for Recoded TSD Addition

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Table 3 Truth Table for Recoded TSD Multiplication

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Table 4 Truth Table for TSD Multiplication when Only the Multiplier is Recoded

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Table 5 Truth Table for Nonrecoded TSD Multiplication with Carries

Equations (5)

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X = 1 1 1 0 0 0 1 ¯ 1 ¯ 1 ¯ 1 0 1 ¯ 1 0 1 ¯ 1 0 1 ¯ ,   Y = 2 1 0 1 0 1 ¯ 0 1 ¯ 2 ¯ .
X = 1 + 1   1 + 0   1 + 1 ¯   0 + 0   0 + 1 ¯   1 ¯ + 1 ¯ ,   Y = 2     1     0     0     1 ¯     2 ¯ ,
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 .
Y = thresh MX = 1 1 0 0 0 1 0 1 0 1 0 0 0 1 1 .
P = AB = i = 0 2 n - 1   P i = i = 0 n - 1   pp i ,

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