Abstract

There has been difficulty in achieving a fully parallel, digital optical adder or multiplier. The primary obstacle is the carry operation inherent in any fixed-radix number system. The concepts of residue number representation and symbolic substitution can be combined to produce a parallel optical arithmetic/logic unit.

© 1988 Optical Society of America

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References

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  1. See, for example, Technical Digest, Topical Meeting on Optical Computing (Optical Society of America, Washington, DC, 1987).
  2. D. Casasent, “Hybrid Optical/Digital Image Pattern Recognition: A Review,” Proc. Soc. Photo-Opt. Instrum. Eng. 528, 64 (1985).
  3. H. J. Whitehouse, J. Speiser, “Linear Signal Processing Architectures,” in Aspects of Signal Processing with Emphasis of Underwater Acoustics, Vol. 2, G. Tacconi, Ed. (Reidel, Hinghan, MA, 1977).
    [CrossRef]
  4. D. Psaltis, R. A. Athale, “High Accuracy Computation wit Linear Analog Optical Systems: A Critical Study,” Appl. Opt. 25, 3071 (1986).
    [CrossRef] [PubMed]
  5. A. Huang, “Parallel Algorithms for Optical Digital Computers,” in Technical Digest, IEEE Tenth International Optical Computing Conference (1983), p. 13.
  6. K.-H. Brenner, A. Huang, N. Streibl, “Digital Optical Computing With Symbolic Substitution,” Appl. Opt. 25, 3054 (1986).
    [CrossRef] [PubMed]
  7. 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 (1986).
    [CrossRef] [PubMed]
  8. N. S. Szabo, R. I. Tanaka, Residue Arithmetic and Its Applications to Computer Technology (McGraw-Hill, New York, 1967).
  9. I. Niven, H. S. Zuckerman, An Introduction to the Theory of Numbers (Wiley, New York, 1980).

1986 (3)

1985 (1)

D. Casasent, “Hybrid Optical/Digital Image Pattern Recognition: A Review,” Proc. Soc. Photo-Opt. Instrum. Eng. 528, 64 (1985).

Athale, R. A.

Bocker, R. P.

Brenner, K.-H.

Casasent, D.

D. Casasent, “Hybrid Optical/Digital Image Pattern Recognition: A Review,” Proc. Soc. Photo-Opt. Instrum. Eng. 528, 64 (1985).

Drake, B. L.

Henderson, T. B.

Huang, A.

K.-H. Brenner, A. Huang, N. Streibl, “Digital Optical Computing With Symbolic Substitution,” Appl. Opt. 25, 3054 (1986).
[CrossRef] [PubMed]

A. Huang, “Parallel Algorithms for Optical Digital Computers,” in Technical Digest, IEEE Tenth International Optical Computing Conference (1983), p. 13.

Lasher, M. E.

Niven, I.

I. Niven, H. S. Zuckerman, An Introduction to the Theory of Numbers (Wiley, New York, 1980).

Psaltis, D.

Speiser, J.

H. J. Whitehouse, J. Speiser, “Linear Signal Processing Architectures,” in Aspects of Signal Processing with Emphasis of Underwater Acoustics, Vol. 2, G. Tacconi, Ed. (Reidel, Hinghan, MA, 1977).
[CrossRef]

Streibl, N.

Szabo, N. S.

N. S. Szabo, R. I. Tanaka, Residue Arithmetic and Its Applications to Computer Technology (McGraw-Hill, New York, 1967).

Tanaka, R. I.

N. S. Szabo, R. I. Tanaka, Residue Arithmetic and Its Applications to Computer Technology (McGraw-Hill, New York, 1967).

Whitehouse, H. J.

H. J. Whitehouse, J. Speiser, “Linear Signal Processing Architectures,” in Aspects of Signal Processing with Emphasis of Underwater Acoustics, Vol. 2, G. Tacconi, Ed. (Reidel, Hinghan, MA, 1977).
[CrossRef]

Zuckerman, H. S.

I. Niven, H. S. Zuckerman, An Introduction to the Theory of Numbers (Wiley, New York, 1980).

Appl. Opt. (3)

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

D. Casasent, “Hybrid Optical/Digital Image Pattern Recognition: A Review,” Proc. Soc. Photo-Opt. Instrum. Eng. 528, 64 (1985).

Other (5)

H. J. Whitehouse, J. Speiser, “Linear Signal Processing Architectures,” in Aspects of Signal Processing with Emphasis of Underwater Acoustics, Vol. 2, G. Tacconi, Ed. (Reidel, Hinghan, MA, 1977).
[CrossRef]

A. Huang, “Parallel Algorithms for Optical Digital Computers,” in Technical Digest, IEEE Tenth International Optical Computing Conference (1983), p. 13.

N. S. Szabo, R. I. Tanaka, Residue Arithmetic and Its Applications to Computer Technology (McGraw-Hill, New York, 1967).

I. Niven, H. S. Zuckerman, An Introduction to the Theory of Numbers (Wiley, New York, 1980).

See, for example, Technical Digest, Topical Meeting on Optical Computing (Optical Society of America, Washington, DC, 1987).

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Figures (7)

Fig. 1
Fig. 1

Operation table for a cyclic group with m elements.

Fig. 2
Fig. 2

Examples of cyclic groups: (a) addition modulo 5; (b) multiplication modulo 5.

Fig. 3
Fig. 3

Symbolic substitution to perform binary addition.

Fig. 4
Fig. 4

Example of one-of-many representation for five elements.

Fig. 5
Fig. 5

Symbolic substitution for operation on a cyclic group of m elements.

Fig. 6
Fig. 6

Conceptual diagram for a parallel modulo 5 adder.

Fig. 7
Fig. 7

Conceptual diagram for a parallel modulo 5 multiplier.

Equations (8)

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

X = i = 0 n 1 a i b i , 0 a i < b .
x i = X modulo m i = | X | m i .
M = i = 1 n m i .
s i = | S | m i = | X + Y | m = | | X | m i + | Y | m i | m i = | x i + y i | m i ,
p i = | P | m i = | X * Y | m i = | | X | m i * | Y | m i | m i = | x i * y i | m i .
a b = c ,
D = ( i + j + 1 ) * x ,
k = | ( D / x 1 ) | m .

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