Abstract

Symmetrically recoded modified signed-digit number algorithms are proposed for carry-free arithmetic. The superiority of the new algorithms is established over all other two-step-based symbolic-substitution optical implementations.

© 1994 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), pp. 143–148.
  2. J. R. Jump, S. R. Ahuja, “Effective pipelining of digital systems,” IEEE Trans. Comput. C-27, 855–862 (1978).
    [CrossRef]
  3. A. Avizienis, “Signed-digit number representations for fast parallel arithmetic,” IRE Trans. Electron. Comput. EC-10, 389–400 (1961).
    [CrossRef]
  4. 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).
  5. 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]
  6. 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]
  7. A. K. Cherri, M. A. Karim, “Modified signed digit arithmetic using an efficient symbolic substitution,” Appl. Opt. 27, 3824–3827 (1988).
    [CrossRef] [PubMed]
  8. A. K. Cherri, M. A. Karim, “Simplified arithmetic for optical symbolic substitution,” Microwave Opt. Tech. Lett. 1, 310–312 (1988).
    [CrossRef]
  9. A. K. Cherri, M. A. Karim, “Symbolic substitution based flagged arithmetic using polarization-encoded optical shadow-casting system,” Opt. Commun. 70, 455–461 (1989).
    [CrossRef]
  10. Y. Li, D. 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]
  11. A. A. S. Awwal, M. A. Karim, “Polarization-encoded optical shadow-casting: design of a carry-free adder,” Appl. Opt. 28, 785–790 (1989).
    [CrossRef] [PubMed]
  12. S. Barua, “Single-stage optical adder/subtracter,” Opt. Eng. 30, 265–270 (1991).
    [CrossRef]
  13. M. S. Alam, M. A. Karim, A. A. S. Awwal, J. J. Westerkamp, “Optical processing based on conditional higher-ordertrinary modified signed-digit symbolic substitution,” Appl. Opt. 31, 5614–5621 (1992).
    [CrossRef] [PubMed]
  14. F. T. S. Yu, S. Jutamulia, “Implementation of symbolic substitution logic using optical associative memories,” Appl. Opt. 26, 2293–2294 (1987).
    [CrossRef] [PubMed]
  15. T. K. Gaylord, M. M. Mirsalehi, “Truth-table look-up processing: number representation, multilevel coding, and logical minimization,” Opt. Eng. 25, 22–33 (1986).
  16. M. M. Mirsalehi, T. K. Gaylord, “Logical minimization of multilevel coded functions,” Appl. Opt. 25, 3078–3088 (1986).
    [CrossRef] [PubMed]
  17. A. Huang, “Parallel algorithm for optical digital computer,” Proc. IEEE 71, 13–17 (1983).
  18. B. Parhami, “Carry-free addition of recoded binary signed-digit numbers,” IEEE Trans. Comput. C-7, 1470–1476 (1988).
    [CrossRef]
  19. A. A. S. Awwal, “Recoded signed-digit binary addition-subtraction using optoelectronic symbolic substitution,” Appl. Opt. 31, 3205–3208 (1992).
    [CrossRef] [PubMed]

1992 (2)

1991 (1)

S. Barua, “Single-stage optical adder/subtracter,” Opt. Eng. 30, 265–270 (1991).
[CrossRef]

1989 (3)

1988 (3)

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

A. K. Cherri, M. A. Karim, “Simplified arithmetic for optical symbolic substitution,” Microwave Opt. Tech. Lett. 1, 310–312 (1988).
[CrossRef]

B. Parhami, “Carry-free addition of recoded binary signed-digit numbers,” IEEE Trans. Comput. C-7, 1470–1476 (1988).
[CrossRef]

1987 (2)

1986 (4)

T. K. Gaylord, M. M. Mirsalehi, “Truth-table look-up processing: number representation, multilevel coding, and logical minimization,” Opt. Eng. 25, 22–33 (1986).

M. M. Mirsalehi, T. K. Gaylord, “Logical minimization of multilevel coded functions,” Appl. Opt. 25, 3078–3088 (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, 38–43 (1986).

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

1983 (1)

A. Huang, “Parallel algorithm for optical digital computer,” Proc. IEEE 71, 13–17 (1983).

1978 (1)

J. R. Jump, S. R. Ahuja, “Effective pipelining of digital systems,” IEEE Trans. Comput. C-27, 855–862 (1978).
[CrossRef]

1961 (1)

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

Ahuja, S. R.

J. R. Jump, S. R. Ahuja, “Effective pipelining of digital systems,” IEEE Trans. Comput. C-27, 855–862 (1978).
[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.

Barua, S.

S. Barua, “Single-stage optical adder/subtracter,” Opt. Eng. 30, 265–270 (1991).
[CrossRef]

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).

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]

Cherri, A. K.

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

A. K. Cherri, M. A. Karim, “Simplified arithmetic for optical symbolic substitution,” Microwave Opt. Tech. Lett. 1, 310–312 (1988).
[CrossRef]

Drake, B. L.

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

B. L. Drake, R. P. Bocker, M. E. Lasher, R. H. Patterson, W. J. Miceli, “Photonic computing using the modified signed-digit number representation,” Opt. Eng. 25, 38–43 (1986).

Eichmann, G.

Gaylord, T. K.

T. K. Gaylord, M. M. Mirsalehi, “Truth-table look-up processing: number representation, multilevel coding, and logical minimization,” Opt. Eng. 25, 22–33 (1986).

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

Henderson, T. B.

Huang, A.

A. Huang, “Parallel algorithm for optical digital computer,” Proc. IEEE 71, 13–17 (1983).

Johnson, E. L.

E. L. Johnson, M. A. Karim, Digital Design: A Pragmatic Approach (PWS-Kent, Boston, Mass., 1987), pp. 143–148.

Jump, J. R.

J. R. Jump, S. R. Ahuja, “Effective pipelining of digital systems,” IEEE Trans. Comput. C-27, 855–862 (1978).
[CrossRef]

Jutamulia, S.

Karim, M. A.

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

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

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

A. K. Cherri, M. A. Karim, “Simplified arithmetic for optical symbolic substitution,” Microwave Opt. Tech. Lett. 1, 310–312 (1988).
[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), pp. 143–148.

Kim, D. H.

Kostrzewski, A.

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).

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]

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, 38–43 (1986).

Mirsalehi, M. M.

T. K. Gaylord, M. M. Mirsalehi, “Truth-table look-up processing: number representation, multilevel coding, and logical minimization,” Opt. Eng. 25, 22–33 (1986).

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

Parhami, B.

B. Parhami, “Carry-free addition of recoded binary signed-digit numbers,” IEEE Trans. Comput. C-7, 1470–1476 (1988).
[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).

Westerkamp, J. J.

Yu, F. T. S.

Appl. Opt. (8)

IEEE Trans. Comput. (2)

B. Parhami, “Carry-free addition of recoded binary signed-digit numbers,” IEEE Trans. Comput. C-7, 1470–1476 (1988).
[CrossRef]

J. R. Jump, S. R. Ahuja, “Effective pipelining of digital systems,” IEEE Trans. Comput. C-27, 855–862 (1978).
[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]

Microwave Opt. Tech. Lett. (1)

A. K. Cherri, M. A. Karim, “Simplified arithmetic for optical symbolic substitution,” Microwave Opt. Tech. Lett. 1, 310–312 (1988).
[CrossRef]

Opt. Commun. (1)

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

Opt. Eng. (3)

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).

S. Barua, “Single-stage optical adder/subtracter,” Opt. Eng. 30, 265–270 (1991).
[CrossRef]

T. K. Gaylord, M. M. Mirsalehi, “Truth-table look-up processing: number representation, multilevel coding, and logical minimization,” Opt. Eng. 25, 22–33 (1986).

Opt. Lett. (1)

Proc. IEEE (1)

A. Huang, “Parallel algorithm for optical digital computer,” Proc. IEEE 71, 13–17 (1983).

Other (1)

E. L. Johnson, M. A. Karim, Digital Design: A Pragmatic Approach (PWS-Kent, Boston, Mass., 1987), pp. 143–148.

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Figures (1)

Fig. 1
Fig. 1

(a) OCAM implementation of the SRMSD numbers; (b) triple-rail encodings for 0, 1, 1 ¯, and don’t cares. Nonholographic CAM for generating the output digit 1 for (c) the symmetrical recoding and (d) the addition truth tables.

Tables (4)

Tables Icon

Table 1 Recoding Truth Table for Modified Signed-Digit Numbers

Tables Icon

Table 2 Addition Truth Table for Recoded Modified Signed-Digit Numbers

Tables Icon

Table 3 Truth Table for Recoded Modified Signed-Digit Algorithm that Shows the Nonsymmetrical Nature of the Scheme

Tables Icon

Table 4 Symmetrical Recoding Truth Table for Modified Signed-Digit Numbersa

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

0 1 ¯ 1 ¯ 00 1 ¯ 00111 1 ¯ 1 ¯ 0 0 0 MSD = 1 ¯ 1 1 ¯ 00 1 ¯ 010 1 ¯ 01 1 ¯ 0 RMSD = 3175 10 , 0 011 1 ¯ 1 ¯ 001 1 ¯ 1 ¯ 11 0 0 0 MSD = 01 1 ¯ 01 1 ¯ 000100 1 ¯ 0 RMSD = 1159 10 , 0 1 ¯ 00000100000 MSD = 2016 10 ,
0 1 ¯ 1 ¯ 00 1 ¯ 00111 1 ¯ 1 ¯ 0 0 0 MSD = 1 ¯ 010 1 ¯ 1010 1 ¯ 001 0 SRMSD = 3175 10 , 0 011 1 ¯ 1 ¯ 001 1 ¯ 1 ¯ 11 0 0 0 MSD = 01 1 ¯ 001000100 1 ¯ 0 SRMSD = 1159 10 , 1 ¯ 100000100000 MSD = 2016 10 ,
0 1 ¯ 1 ¯ 00 1 ¯ 00111 1 ¯ 1 ¯ 0 0 0 MSD = 1 ¯ 1 1 ¯ 00 1 ¯ 010 1 ¯ 01 1 ¯ 0 SRMSD = 3175 10 , 0 011 1 ¯ 1 ¯ 001 1 ¯ 1 ¯ 11 0 0 0 MSD = 01 1 ¯ 01 1 ¯ 000101 1 ¯ 0 SRMSD = 1159 10 , 0 1 ¯ 00000100000 MSD = 2016 10 ,

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