Abstract

In this paper, we consider the use of residue arithmetic to increase the accuracy and reduce the dynamic range requirements of optical matrix–vector processors. We show that matrix–vector operations and iterative algorithms can be performed totally in residue notation. We suggest an architecture using a frequency-multiplexed optical systolic array feedback processor. As our case study, we consider the solution of three simultaneous equations.

© 1983 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. Caulfield et al., Opt. Commun. 40, 86 (1981).
    [CrossRef]
  2. D. Casasent, Appl. Opt. 21, 1859 (1982).
    [CrossRef] [PubMed]
  3. D. Casasent, J. Jackson, C. Neuman, Appl. Opt. 22, 115 (1983).
    [CrossRef] [PubMed]
  4. N. Szabo, R. Tanaka, Residue Arithmetic and Its Applications to Computer Technology (McGraw-Hill, New York, 1967).
  5. A. Huang, T. Tsunoda, J. W. Goodman, Appl. Opt. 18, 149 (1979).
    [CrossRef] [PubMed]
  6. D. Psaltis, D. Casasent, Appl. Opt. 18, 163 (1979).
    [CrossRef] [PubMed]
  7. A. M. Tai et al., Appl. Opt. 18, 2812 (1979).
    [CrossRef] [PubMed]
  8. Proc. Soc. Photo-Opt. Instrum. Eng. 232, 109–173 (1980).
  9. W. K. Jenkins, IEEE Trans. Acoust. Speech Signal Process. ASSP-27, 19 (1979).
    [CrossRef]
  10. L. F. Richardson, Philos. Trans. R. Soc. London Ser. A 210, 307 (1910).
  11. L. A. Hageman, D. M. Young, Applied Iterative Methods (Academic, New York, 1981).
  12. A. Oppenheim, R. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J. (Prentice-Hall, 1975).

1983 (1)

1982 (1)

1981 (1)

H. Caulfield et al., Opt. Commun. 40, 86 (1981).
[CrossRef]

1980 (1)

Proc. Soc. Photo-Opt. Instrum. Eng. 232, 109–173 (1980).

1979 (4)

1910 (1)

L. F. Richardson, Philos. Trans. R. Soc. London Ser. A 210, 307 (1910).

Casasent, D.

Caulfield, H.

H. Caulfield et al., Opt. Commun. 40, 86 (1981).
[CrossRef]

Goodman, J. W.

Hageman, L. A.

L. A. Hageman, D. M. Young, Applied Iterative Methods (Academic, New York, 1981).

Huang, A.

Jackson, J.

Jenkins, W. K.

W. K. Jenkins, IEEE Trans. Acoust. Speech Signal Process. ASSP-27, 19 (1979).
[CrossRef]

Neuman, C.

Oppenheim, A.

A. Oppenheim, R. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J. (Prentice-Hall, 1975).

Psaltis, D.

Richardson, L. F.

L. F. Richardson, Philos. Trans. R. Soc. London Ser. A 210, 307 (1910).

Schafer, R.

A. Oppenheim, R. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J. (Prentice-Hall, 1975).

Szabo, N.

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

Tai, A. M.

Tanaka, R.

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

Tsunoda, T.

Young, D. M.

L. A. Hageman, D. M. Young, Applied Iterative Methods (Academic, New York, 1981).

Appl. Opt. (5)

IEEE Trans. Acoust. Speech Signal Process. (1)

W. K. Jenkins, IEEE Trans. Acoust. Speech Signal Process. ASSP-27, 19 (1979).
[CrossRef]

Opt. Commun. (1)

H. Caulfield et al., Opt. Commun. 40, 86 (1981).
[CrossRef]

Philos. Trans. R. Soc. London Ser. A (1)

L. F. Richardson, Philos. Trans. R. Soc. London Ser. A 210, 307 (1910).

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

Proc. Soc. Photo-Opt. Instrum. Eng. 232, 109–173 (1980).

Other (3)

L. A. Hageman, D. M. Young, Applied Iterative Methods (Academic, New York, 1981).

A. Oppenheim, R. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J. (Prentice-Hall, 1975).

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Schematic diagram of a frequency-multiplexed optical systolic matrix–matrix processor.

Fig. 2
Fig. 2

Example of data flow in the system of Fig. 1 for the multiplication of two 3 × 3 matrices.

Fig. 3
Fig. 3

One realization of an optical residue processor in which the residues for five moduli are space-multiplexed.

Fig. 4
Fig. 4

Functional block diagram of a residue extractor/quantizer.

Fig. 5
Fig. 5

Implementation of an iterative algorithm on the optical residue processor.

Fig. 6
Fig. 6

Convergence of the iterative algorithm using residue arithmetic.

Equations (15)

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

x = m i [ x / m i ] + | x | m i .
| x ± y | m = | | x | m ± | y | m | m ,
| x y | m = | | x | m | y | m | m .
| i = 1 N x i | m = | i = 1 N | x i | m | m ,
| i = 1 N x i | m = | i = 1 N | x i | m | m .
| a | m = | 1 b ̅ | m ,
| c / b | m = | 1 b ̅ c | m
d i j = k = 1 N a i k b k j .
| d i j | m = | k = 1 N a i k b k j | m = | k = 1 N | a i k | m | b k j | m | m .
B k + 1 = ( I ω H ) B k + ω C ,
s ( t B k + 1 ) = [ s ( I ω H ) ] ( t B k ) + [ ω s t C ]
H = [ 3.6 0.6 0.7 1.8 0.6 0.0 0.0 0.6 2.1 ] , c = [ 1.0 0.0 1.0 ] .
b T = ( 1.105 3.310 1.421 )
b T = ( 1.111 3.333 1.429 ) .
b T = ( 1.090 3.263 1.402 ) .

Metrics