Abstract

High accuracy optical processors based on the algorithm of digital multiplication by analog convolution (DMAC) are studied for ultimate performance limitations. Variations of optical processors that perform high accuracy vector–vector inner products are studied in abstract and with specific examples. It is concluded that the use of linear analog optical processors in performing digital computations with DMAC leads to impractical requirements for the accuracy of analog optical systems and the complexity of postprocessing electronics.

© 1986 Optical Society of America

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References

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  1. L. J. Cutrona, E. N. Leith, L. J. Porcello, W. E. Vivian, “On the Application of Coherent Optical Processing Techniques to Synthetic Aperture Radar,” Proc. IEEE 54, 1026 (1966).
    [CrossRef]
  2. T. M. Turpin, “Spectrum Analysis Using Optical Processing,” Proc. IEEE, 79 (1981).
    [CrossRef]
  3. Special Issue on Acoustooptics, Proc. IEEE (Jan.1981).
  4. N. J. Berg, J. N. Lee, M. W. Casseday, E. Katzen, in Ultrasonics Symposium Proceedings, IEEE Catalog No. 78 CH 134-1SU (1978), p. 91.
  5. A. Huang, Y. Tsunoda, J. W. Goodman, S. Ishihara, “Optical Computation Using Residue Arithmetic,” Appl. Opt. 18, 149 (1979).
    [CrossRef] [PubMed]
  6. A. Tai, I. Cindrich, J. R. Fienup, C. C. Aleksoff, “Optical Residue Arithmetic Computer with Programmable Computation Modules,” Appl. Opt. 18, 2812 (1979).
    [CrossRef] [PubMed]
  7. D. Psaltis, D. Casasent, “Optical Residue Arithmetic: A Correlation Approach,” Appl. Opt. 18, 163 (1979).
    [CrossRef] [PubMed]
  8. S. A. Collins, “Numerical Optical Data Processing,” Proc. Soc. Photo-Opt. Instrum. Eng. 128, 313 (1977).
  9. C. C. Guest, T. K. Gaylord, “Truth-Table Look-up Optical Processing Utilizing Binary and Residue Arithmetic,” Appl. Opt. 19, 12011980.
    [CrossRef] [PubMed]
  10. R. Arrathoon, M. N. Hassoun, “Optical Threshold Logic Elements for Digital Computation,” Opt. Lett. 9, 143 (1984).
    [CrossRef] [PubMed]
  11. “Vedic Mathematics,” Shankaracharya of Govardhana Pitha, Motilal Banarsidass Pub., New Delhi, ISBN: 0-89581-416-1 (1965).
  12. E. E. Schwartzlander, “The Quasi-serial Multiplier,” IEEE Trans. Comput. C-22, 317 (1973).
    [CrossRef]
  13. H. J. Whitehouse, J. Speiser, “Linear Signal Processing Architectures,” in Aspects of Signal Processing with Emphasis on Underwater Acoustics, Vol. 2, G. Tacconi, Ed. (Reidel, Hingham, MA, 1977).
    [CrossRef]
  14. D. Psaltis et al., “Accurate Numerical Computation by Optical Convolution,” Proc. Soc. Photo-Opt. Instrum. Eng. 232, 151 (1980).
  15. P. S. Guilfoyle, “Systolic Acousto-optic Binary Convolver,” Opt. Eng. 23, 20 (1984).
    [CrossRef]
  16. W. C. Collins, R. A. Athale, P. D. Stilwell, “Improved Accuracy for Optical Iterative Processor,” Proc. Soc. Photo-Opt. Instrum. Eng. 352, 59 (1983).
  17. R. P. Bocker, “Optical Digital RUBIC (Rapid Unbiased Bipolar Incoherent Calculator) Cube Processor,” Opt. Eng. 23, 26 (1984).
    [CrossRef]
  18. K. Wagner, D. Psaltis, “A Space-integrating Acousto-optic Matrix Multiplier,” Opt. Commun. 52, 173 (1984).
    [CrossRef]
  19. A. P. Goutzoulis, “Systolic Time-Integrating Acoustooptic Binary Processor,” Appl. Opt. 23, 4095 (1984).
    [CrossRef] [PubMed]
  20. S. Cartwright, S. C. Gustafson, “Convolver-based Optical Systolic Architectures,” Opt. Eng. 26, 59 (1985).
  21. C. M. Verber, “Integrated Optical Architectures for Matrix Multiplications,” Opt. Eng. 24, 19 (1985).
    [CrossRef]
  22. J. Jackson, D. Casasent, “Optical Systolic Array Processor Using Residue Arithmetic,” Appl. Opt., 22, p. 2817 (1983).
    [CrossRef] [PubMed]
  23. M. S. Mort, “Modified Quasi-Serial Multiplier,” Appl. Opt. 24, 1396 (1985).
    [CrossRef] [PubMed]
  24. R. A. Athale, W. C. Collins, P. D. Stilwell, “High Accuracy Matrix Multiplication with Outer Product Optical Processor,” Appl. Opt. 22, 368 (1983).
    [CrossRef] [PubMed]
  25. S. L. Hurst, “Multiple Valued Logic—Its Status and Its Future,” IEEE Trans. Comput. C-33, 1160 (1984).
    [CrossRef]

1985 (3)

S. Cartwright, S. C. Gustafson, “Convolver-based Optical Systolic Architectures,” Opt. Eng. 26, 59 (1985).

C. M. Verber, “Integrated Optical Architectures for Matrix Multiplications,” Opt. Eng. 24, 19 (1985).
[CrossRef]

M. S. Mort, “Modified Quasi-Serial Multiplier,” Appl. Opt. 24, 1396 (1985).
[CrossRef] [PubMed]

1984 (6)

A. P. Goutzoulis, “Systolic Time-Integrating Acoustooptic Binary Processor,” Appl. Opt. 23, 4095 (1984).
[CrossRef] [PubMed]

S. L. Hurst, “Multiple Valued Logic—Its Status and Its Future,” IEEE Trans. Comput. C-33, 1160 (1984).
[CrossRef]

R. Arrathoon, M. N. Hassoun, “Optical Threshold Logic Elements for Digital Computation,” Opt. Lett. 9, 143 (1984).
[CrossRef] [PubMed]

R. P. Bocker, “Optical Digital RUBIC (Rapid Unbiased Bipolar Incoherent Calculator) Cube Processor,” Opt. Eng. 23, 26 (1984).
[CrossRef]

K. Wagner, D. Psaltis, “A Space-integrating Acousto-optic Matrix Multiplier,” Opt. Commun. 52, 173 (1984).
[CrossRef]

P. S. Guilfoyle, “Systolic Acousto-optic Binary Convolver,” Opt. Eng. 23, 20 (1984).
[CrossRef]

1983 (3)

1981 (2)

T. M. Turpin, “Spectrum Analysis Using Optical Processing,” Proc. IEEE, 79 (1981).
[CrossRef]

Special Issue on Acoustooptics, Proc. IEEE (Jan.1981).

1980 (2)

D. Psaltis et al., “Accurate Numerical Computation by Optical Convolution,” Proc. Soc. Photo-Opt. Instrum. Eng. 232, 151 (1980).

C. C. Guest, T. K. Gaylord, “Truth-Table Look-up Optical Processing Utilizing Binary and Residue Arithmetic,” Appl. Opt. 19, 12011980.
[CrossRef] [PubMed]

1979 (3)

1977 (1)

S. A. Collins, “Numerical Optical Data Processing,” Proc. Soc. Photo-Opt. Instrum. Eng. 128, 313 (1977).

1973 (1)

E. E. Schwartzlander, “The Quasi-serial Multiplier,” IEEE Trans. Comput. C-22, 317 (1973).
[CrossRef]

1966 (1)

L. J. Cutrona, E. N. Leith, L. J. Porcello, W. E. Vivian, “On the Application of Coherent Optical Processing Techniques to Synthetic Aperture Radar,” Proc. IEEE 54, 1026 (1966).
[CrossRef]

Aleksoff, C. C.

Arrathoon, R.

Athale, R. A.

W. C. Collins, R. A. Athale, P. D. Stilwell, “Improved Accuracy for Optical Iterative Processor,” Proc. Soc. Photo-Opt. Instrum. Eng. 352, 59 (1983).

R. A. Athale, W. C. Collins, P. D. Stilwell, “High Accuracy Matrix Multiplication with Outer Product Optical Processor,” Appl. Opt. 22, 368 (1983).
[CrossRef] [PubMed]

Berg, N. J.

N. J. Berg, J. N. Lee, M. W. Casseday, E. Katzen, in Ultrasonics Symposium Proceedings, IEEE Catalog No. 78 CH 134-1SU (1978), p. 91.

Bocker, R. P.

R. P. Bocker, “Optical Digital RUBIC (Rapid Unbiased Bipolar Incoherent Calculator) Cube Processor,” Opt. Eng. 23, 26 (1984).
[CrossRef]

Cartwright, S.

S. Cartwright, S. C. Gustafson, “Convolver-based Optical Systolic Architectures,” Opt. Eng. 26, 59 (1985).

Casasent, D.

Casseday, M. W.

N. J. Berg, J. N. Lee, M. W. Casseday, E. Katzen, in Ultrasonics Symposium Proceedings, IEEE Catalog No. 78 CH 134-1SU (1978), p. 91.

Cindrich, I.

Collins, S. A.

S. A. Collins, “Numerical Optical Data Processing,” Proc. Soc. Photo-Opt. Instrum. Eng. 128, 313 (1977).

Collins, W. C.

W. C. Collins, R. A. Athale, P. D. Stilwell, “Improved Accuracy for Optical Iterative Processor,” Proc. Soc. Photo-Opt. Instrum. Eng. 352, 59 (1983).

R. A. Athale, W. C. Collins, P. D. Stilwell, “High Accuracy Matrix Multiplication with Outer Product Optical Processor,” Appl. Opt. 22, 368 (1983).
[CrossRef] [PubMed]

Cutrona, L. J.

L. J. Cutrona, E. N. Leith, L. J. Porcello, W. E. Vivian, “On the Application of Coherent Optical Processing Techniques to Synthetic Aperture Radar,” Proc. IEEE 54, 1026 (1966).
[CrossRef]

Fienup, J. R.

Gaylord, T. K.

Goodman, J. W.

Goutzoulis, A. P.

Guest, C. C.

Guilfoyle, P. S.

P. S. Guilfoyle, “Systolic Acousto-optic Binary Convolver,” Opt. Eng. 23, 20 (1984).
[CrossRef]

Gustafson, S. C.

S. Cartwright, S. C. Gustafson, “Convolver-based Optical Systolic Architectures,” Opt. Eng. 26, 59 (1985).

Hassoun, M. N.

Huang, A.

Hurst, S. L.

S. L. Hurst, “Multiple Valued Logic—Its Status and Its Future,” IEEE Trans. Comput. C-33, 1160 (1984).
[CrossRef]

Ishihara, S.

Jackson, J.

Katzen, E.

N. J. Berg, J. N. Lee, M. W. Casseday, E. Katzen, in Ultrasonics Symposium Proceedings, IEEE Catalog No. 78 CH 134-1SU (1978), p. 91.

Lee, J. N.

N. J. Berg, J. N. Lee, M. W. Casseday, E. Katzen, in Ultrasonics Symposium Proceedings, IEEE Catalog No. 78 CH 134-1SU (1978), p. 91.

Leith, E. N.

L. J. Cutrona, E. N. Leith, L. J. Porcello, W. E. Vivian, “On the Application of Coherent Optical Processing Techniques to Synthetic Aperture Radar,” Proc. IEEE 54, 1026 (1966).
[CrossRef]

Mort, M. S.

Porcello, L. J.

L. J. Cutrona, E. N. Leith, L. J. Porcello, W. E. Vivian, “On the Application of Coherent Optical Processing Techniques to Synthetic Aperture Radar,” Proc. IEEE 54, 1026 (1966).
[CrossRef]

Psaltis, D.

K. Wagner, D. Psaltis, “A Space-integrating Acousto-optic Matrix Multiplier,” Opt. Commun. 52, 173 (1984).
[CrossRef]

D. Psaltis et al., “Accurate Numerical Computation by Optical Convolution,” Proc. Soc. Photo-Opt. Instrum. Eng. 232, 151 (1980).

D. Psaltis, D. Casasent, “Optical Residue Arithmetic: A Correlation Approach,” Appl. Opt. 18, 163 (1979).
[CrossRef] [PubMed]

Schwartzlander, E. E.

E. E. Schwartzlander, “The Quasi-serial Multiplier,” IEEE Trans. Comput. C-22, 317 (1973).
[CrossRef]

Speiser, J.

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

Stilwell, P. D.

W. C. Collins, R. A. Athale, P. D. Stilwell, “Improved Accuracy for Optical Iterative Processor,” Proc. Soc. Photo-Opt. Instrum. Eng. 352, 59 (1983).

R. A. Athale, W. C. Collins, P. D. Stilwell, “High Accuracy Matrix Multiplication with Outer Product Optical Processor,” Appl. Opt. 22, 368 (1983).
[CrossRef] [PubMed]

Tai, A.

Tsunoda, Y.

Turpin, T. M.

T. M. Turpin, “Spectrum Analysis Using Optical Processing,” Proc. IEEE, 79 (1981).
[CrossRef]

Verber, C. M.

C. M. Verber, “Integrated Optical Architectures for Matrix Multiplications,” Opt. Eng. 24, 19 (1985).
[CrossRef]

Vivian, W. E.

L. J. Cutrona, E. N. Leith, L. J. Porcello, W. E. Vivian, “On the Application of Coherent Optical Processing Techniques to Synthetic Aperture Radar,” Proc. IEEE 54, 1026 (1966).
[CrossRef]

Wagner, K.

K. Wagner, D. Psaltis, “A Space-integrating Acousto-optic Matrix Multiplier,” Opt. Commun. 52, 173 (1984).
[CrossRef]

Whitehouse, H. J.

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

Appl. Opt. (8)

IEEE Trans. Comput. (2)

E. E. Schwartzlander, “The Quasi-serial Multiplier,” IEEE Trans. Comput. C-22, 317 (1973).
[CrossRef]

S. L. Hurst, “Multiple Valued Logic—Its Status and Its Future,” IEEE Trans. Comput. C-33, 1160 (1984).
[CrossRef]

Opt. Commun. (1)

K. Wagner, D. Psaltis, “A Space-integrating Acousto-optic Matrix Multiplier,” Opt. Commun. 52, 173 (1984).
[CrossRef]

Opt. Eng. (4)

S. Cartwright, S. C. Gustafson, “Convolver-based Optical Systolic Architectures,” Opt. Eng. 26, 59 (1985).

C. M. Verber, “Integrated Optical Architectures for Matrix Multiplications,” Opt. Eng. 24, 19 (1985).
[CrossRef]

P. S. Guilfoyle, “Systolic Acousto-optic Binary Convolver,” Opt. Eng. 23, 20 (1984).
[CrossRef]

R. P. Bocker, “Optical Digital RUBIC (Rapid Unbiased Bipolar Incoherent Calculator) Cube Processor,” Opt. Eng. 23, 26 (1984).
[CrossRef]

Opt. Lett. (1)

Proc. IEEE (3)

L. J. Cutrona, E. N. Leith, L. J. Porcello, W. E. Vivian, “On the Application of Coherent Optical Processing Techniques to Synthetic Aperture Radar,” Proc. IEEE 54, 1026 (1966).
[CrossRef]

T. M. Turpin, “Spectrum Analysis Using Optical Processing,” Proc. IEEE, 79 (1981).
[CrossRef]

Special Issue on Acoustooptics, Proc. IEEE (Jan.1981).

Proc. Soc. Photo-Opt. Instrum. Eng. (3)

S. A. Collins, “Numerical Optical Data Processing,” Proc. Soc. Photo-Opt. Instrum. Eng. 128, 313 (1977).

D. Psaltis et al., “Accurate Numerical Computation by Optical Convolution,” Proc. Soc. Photo-Opt. Instrum. Eng. 232, 151 (1980).

W. C. Collins, R. A. Athale, P. D. Stilwell, “Improved Accuracy for Optical Iterative Processor,” Proc. Soc. Photo-Opt. Instrum. Eng. 352, 59 (1983).

Other (3)

“Vedic Mathematics,” Shankaracharya of Govardhana Pitha, Motilal Banarsidass Pub., New Delhi, ISBN: 0-89581-416-1 (1965).

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

N. J. Berg, J. N. Lee, M. W. Casseday, E. Katzen, in Ultrasonics Symposium Proceedings, IEEE Catalog No. 78 CH 134-1SU (1978), p. 91.

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Figures (9)

Fig. 1
Fig. 1

Block diagram of the DMAC multiples.

Fig. 2
Fig. 2

Schematic diagram of a generalized optical processor.

Fig. 3
Fig. 3

Schematic diagrams of optical systems for performing a vector–vector inner product (a) and scalar–vector product (b). A vector-matrix multiplication is performed by repeatedly performing either operation.

Fig. 4
Fig. 4

Two ways of performing digital multiplication via linear analog optical systems: (a) space-integrating convolver; (b) time-integrating convolver.

Fig. 5
Fig. 5

Schematic diagram of a system employing a vector–vector inner product with space-integrating convolution.

Fig. 6
Fig. 6

Schematic diagram of a system employing a scalar–vector product with time-integrating convolution.

Fig. 7
Fig. 7

Schematic diagram of a system employing a vector–vector inner product with time-integrating convolution.

Fig. 8
Fig. 8

Schematic diagram of a system employing a scalar–vector product with space-integrating convolution.

Fig. 9
Fig. 9

Space-integrating analog processor for performing vector–matrix multiplication.

Tables (2)

Tables Icon

Table I Design Parameter Common to all Architectures

Tables Icon

Table II Parameters for Processor Architectures

Equations (9)

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f = i = 0 L - 1 f i 2 i ;             g = i = 0 L - 1 g i 2 i .
h = f · g = i = 0 ( 2 L - 2 ) 2 i j = 0 ( L - 1 ) ( f j g i - j ) .
h i = j = 0 L - 1 f j g i - j .
h = f T g = k = 1 N f k g k = k = 1 N i = 0 ( 2 L - 2 ) 2 i j = 0 L - 1 f j , k g ( i - j ) , k .
h = i = 0 2 L - 2 2 i { k = 1 N j = 0 L - 1 f j , k g ( i - j ) , k } .
P = N 1 B 1 M / a L 2 multiplications / s .
C = N 2 B 2 A - D conversions / s .
R = P / C = ( N 1 B 1 M N 2 B 2 a L 2 ) multiplications / A - D conversion .
R < D R 2 / a L 2 .

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