Abstract

Optical linear algebra processors that involve solutions of linear algebraic equations have significant potential in adaptive and inference machines. We present an algorithm that includes constraints on the accuracy of the processor and improves the accuracy of the results obtained from such analog processors. The constraint algorithm matches the problem to the accuracy of the processor. Calculation of the adaptive weights in a phased array radar is used as a case study. Simulation results prove the benefits advertised. The desensitization of the calculated weights to computational errors in the processor is quantified. Ridge regression is used to determine the parameter needed in the algorithm.

© 1988 Optical Society of America

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References

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  1. Special Issue on Optical Interconnections, Opt. Eng. 25, No. 10 (Oct.1986).
  2. D. Psaltis, N. Farhat, “Optical Information Processing Based on an Associative-Memory Model of Neural Nets with Thresholding and Feedback,” Opt. Lett. 10, 98 (1985).
    [CrossRef] [PubMed]
  3. R. Krishnapuram, D. Casasent, “Optical Associative Processor for General Linear Transformations,” App. Opt. 26, 3641 (1987).
    [CrossRef]
  4. Special Issue on Optical Computing, Proc. IEEE (July1984).
  5. D. Psaltis, R. A. Athale, “High Accuracy Computation with Linear Analog Optical Systems: a Critical Study,” Appl. Opt. 25, 3071 (1986).
    [CrossRef] [PubMed]
  6. E. Pochapsky, D. Casasent, “Complex Data Handling in Analog and High-Accuracy Optical Linear Algebra Processors,” Proc. Soc. Photo-Opt. Instrum. Eng. 752, 155 (1987).
  7. C. Neuman, D. Casasent, R. Baumbick, “An Electro-Optical Processor for the Optimal Control of F100 Aircraft Engines,” in Proceedings, EOSD (Nov.1981), pp. 311–320.
  8. E. Gilbert, S. Morgan, “Optimum Design of Directive Antenna Arrays Subject to Random Variations,” Bell Syst. Tech. J. 34, 637 (1955).
  9. B. Repasky, B. Breed, “Application of Ridge Regression Analysis in Optimum Array Processing,” in Proceedings, ICASSP 80 (CH1559-4/80) (Apr.1980), Vol. 1, pp. 299–302.
  10. D. Casasent, J. Jackson, “Space and Frequency-Multiplexed Optical Linear Algebra Processor: Fabrication and Initial Tests,” Appl. Opt. 25, 2258 (1986).
    [CrossRef] [PubMed]
  11. D. Casasent, A. Ghosh, “Optical Linear Algebra Processors: Noise and Error-Source Modeling,” Opt. Lett. 10, 252 (1985).
    [CrossRef] [PubMed]
  12. D. Casasent, A. Ghosh, C. Neuman, “A Quadratic Matrix Algorithm for Linear Algebra Procesors,” J. Large-Scale Syst. 9, 35 (1985).
  13. D. Casasent, A. Ghosh, “LU and Cholesky Decomposition on an Optical Systolic Array Processor,” Opt. Commun. 46, 270 (1983).
    [CrossRef]
  14. A. Ghosh, D. Casasent, C. Neuman, “Performance of Direct and Iterative Algorithms on an Optical Systolic Processor,” Appl. Opt. 24, 3883 (1985).
    [CrossRef] [PubMed]

1987 (2)

R. Krishnapuram, D. Casasent, “Optical Associative Processor for General Linear Transformations,” App. Opt. 26, 3641 (1987).
[CrossRef]

E. Pochapsky, D. Casasent, “Complex Data Handling in Analog and High-Accuracy Optical Linear Algebra Processors,” Proc. Soc. Photo-Opt. Instrum. Eng. 752, 155 (1987).

1986 (3)

1985 (4)

1984 (1)

Special Issue on Optical Computing, Proc. IEEE (July1984).

1983 (1)

D. Casasent, A. Ghosh, “LU and Cholesky Decomposition on an Optical Systolic Array Processor,” Opt. Commun. 46, 270 (1983).
[CrossRef]

1955 (1)

E. Gilbert, S. Morgan, “Optimum Design of Directive Antenna Arrays Subject to Random Variations,” Bell Syst. Tech. J. 34, 637 (1955).

Athale, R. A.

Baumbick, R.

C. Neuman, D. Casasent, R. Baumbick, “An Electro-Optical Processor for the Optimal Control of F100 Aircraft Engines,” in Proceedings, EOSD (Nov.1981), pp. 311–320.

Breed, B.

B. Repasky, B. Breed, “Application of Ridge Regression Analysis in Optimum Array Processing,” in Proceedings, ICASSP 80 (CH1559-4/80) (Apr.1980), Vol. 1, pp. 299–302.

Casasent, D.

E. Pochapsky, D. Casasent, “Complex Data Handling in Analog and High-Accuracy Optical Linear Algebra Processors,” Proc. Soc. Photo-Opt. Instrum. Eng. 752, 155 (1987).

R. Krishnapuram, D. Casasent, “Optical Associative Processor for General Linear Transformations,” App. Opt. 26, 3641 (1987).
[CrossRef]

D. Casasent, J. Jackson, “Space and Frequency-Multiplexed Optical Linear Algebra Processor: Fabrication and Initial Tests,” Appl. Opt. 25, 2258 (1986).
[CrossRef] [PubMed]

D. Casasent, A. Ghosh, “Optical Linear Algebra Processors: Noise and Error-Source Modeling,” Opt. Lett. 10, 252 (1985).
[CrossRef] [PubMed]

D. Casasent, A. Ghosh, C. Neuman, “A Quadratic Matrix Algorithm for Linear Algebra Procesors,” J. Large-Scale Syst. 9, 35 (1985).

A. Ghosh, D. Casasent, C. Neuman, “Performance of Direct and Iterative Algorithms on an Optical Systolic Processor,” Appl. Opt. 24, 3883 (1985).
[CrossRef] [PubMed]

D. Casasent, A. Ghosh, “LU and Cholesky Decomposition on an Optical Systolic Array Processor,” Opt. Commun. 46, 270 (1983).
[CrossRef]

C. Neuman, D. Casasent, R. Baumbick, “An Electro-Optical Processor for the Optimal Control of F100 Aircraft Engines,” in Proceedings, EOSD (Nov.1981), pp. 311–320.

Farhat, N.

Ghosh, A.

D. Casasent, A. Ghosh, C. Neuman, “A Quadratic Matrix Algorithm for Linear Algebra Procesors,” J. Large-Scale Syst. 9, 35 (1985).

D. Casasent, A. Ghosh, “Optical Linear Algebra Processors: Noise and Error-Source Modeling,” Opt. Lett. 10, 252 (1985).
[CrossRef] [PubMed]

A. Ghosh, D. Casasent, C. Neuman, “Performance of Direct and Iterative Algorithms on an Optical Systolic Processor,” Appl. Opt. 24, 3883 (1985).
[CrossRef] [PubMed]

D. Casasent, A. Ghosh, “LU and Cholesky Decomposition on an Optical Systolic Array Processor,” Opt. Commun. 46, 270 (1983).
[CrossRef]

Gilbert, E.

E. Gilbert, S. Morgan, “Optimum Design of Directive Antenna Arrays Subject to Random Variations,” Bell Syst. Tech. J. 34, 637 (1955).

Jackson, J.

Krishnapuram, R.

R. Krishnapuram, D. Casasent, “Optical Associative Processor for General Linear Transformations,” App. Opt. 26, 3641 (1987).
[CrossRef]

Morgan, S.

E. Gilbert, S. Morgan, “Optimum Design of Directive Antenna Arrays Subject to Random Variations,” Bell Syst. Tech. J. 34, 637 (1955).

Neuman, C.

A. Ghosh, D. Casasent, C. Neuman, “Performance of Direct and Iterative Algorithms on an Optical Systolic Processor,” Appl. Opt. 24, 3883 (1985).
[CrossRef] [PubMed]

D. Casasent, A. Ghosh, C. Neuman, “A Quadratic Matrix Algorithm for Linear Algebra Procesors,” J. Large-Scale Syst. 9, 35 (1985).

C. Neuman, D. Casasent, R. Baumbick, “An Electro-Optical Processor for the Optimal Control of F100 Aircraft Engines,” in Proceedings, EOSD (Nov.1981), pp. 311–320.

Pochapsky, E.

E. Pochapsky, D. Casasent, “Complex Data Handling in Analog and High-Accuracy Optical Linear Algebra Processors,” Proc. Soc. Photo-Opt. Instrum. Eng. 752, 155 (1987).

Psaltis, D.

Repasky, B.

B. Repasky, B. Breed, “Application of Ridge Regression Analysis in Optimum Array Processing,” in Proceedings, ICASSP 80 (CH1559-4/80) (Apr.1980), Vol. 1, pp. 299–302.

App. Opt. (1)

R. Krishnapuram, D. Casasent, “Optical Associative Processor for General Linear Transformations,” App. Opt. 26, 3641 (1987).
[CrossRef]

Appl. Opt. (3)

Bell Syst. Tech. J. (1)

E. Gilbert, S. Morgan, “Optimum Design of Directive Antenna Arrays Subject to Random Variations,” Bell Syst. Tech. J. 34, 637 (1955).

J. Large-Scale Syst. (1)

D. Casasent, A. Ghosh, C. Neuman, “A Quadratic Matrix Algorithm for Linear Algebra Procesors,” J. Large-Scale Syst. 9, 35 (1985).

Opt. Commun. (1)

D. Casasent, A. Ghosh, “LU and Cholesky Decomposition on an Optical Systolic Array Processor,” Opt. Commun. 46, 270 (1983).
[CrossRef]

Opt. Eng. (1)

Special Issue on Optical Interconnections, Opt. Eng. 25, No. 10 (Oct.1986).

Opt. Lett. (2)

Proc. IEEE (1)

Special Issue on Optical Computing, Proc. IEEE (July1984).

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

E. Pochapsky, D. Casasent, “Complex Data Handling in Analog and High-Accuracy Optical Linear Algebra Processors,” Proc. Soc. Photo-Opt. Instrum. Eng. 752, 155 (1987).

Other (2)

C. Neuman, D. Casasent, R. Baumbick, “An Electro-Optical Processor for the Optimal Control of F100 Aircraft Engines,” in Proceedings, EOSD (Nov.1981), pp. 311–320.

B. Repasky, B. Breed, “Application of Ridge Regression Analysis in Optimum Array Processing,” in Proceedings, ICASSP 80 (CH1559-4/80) (Apr.1980), Vol. 1, pp. 299–302.

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Figures (3)

Fig. 1
Fig. 1

Frequency-multiplexed optical matrix–vector processor.

Fig. 2
Fig. 2

Antenna patterns for isotropic noise with (solid lines) and without (dashed lines) optical system errors in the calculation of the weights for the LDL T algorithm: (a) with k = 0 (no desensitivity) and (b) with the Lagrange parameter k = 0.1.

Fig. 3
Fig. 3

Antenna patterns for directive noise (a jammer at 30°) plus additive antenna noise with (solid line) and without (dashed line) optical system errors in the calculation of the weights for the LDL T solution using our desensitivity algorithm with k = 0.01.

Tables (3)

Tables Icon

Table I Effect of the Lagrange Parameter on the Adaptive Weight Problem

Tables Icon

Table II Examples of the Algorithm’s Desensitivity to Optical System Error Sources with Isotropic Antenna Noise

Tables Icon

Table III Example of the Algorithm’s Desensitivity to Optical System Error Sources for the LDL T Solution with Directional and Isotropic Noise

Equations (7)

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

p ( w ) = | w H s | 2 / ( w H R w ) ,
R w = s ,
w = R 1 s .
E { P ( θ ) } / P * ( θ ) = P 0 ( θ ) / P * ( θ ) + c σ n 2 α w ,
1 / p ( w ) + k ( α w α 0 ) ,
( R + k I ) w ˆ = s ,
α = s H [ R + k I ] H [ R + k I ] 1 s | s H [ R + k I ] H s | 2 ,

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