Abstract

A sampling theorem is developed to reduce integration error in matrix–vector and linear multiplexing processors that perform discrete versions of continuous linear operations. By simply filtering the operation kernel before sampling, one can perform integration-error-free processing on inputs sampled at their Nyquist rate. Example applications to Laplace and Hilbert transformation are presented.

© 1981 Optical Society of America

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References

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  1. M. A. Monahan, K. Bromley, R. P. Bocker, “Incoherent optical correlators,” Proc. IEEE 65, 121 (1977).
    [CrossRef]
  2. J. F. Walkup, “Space-variant coherent optical processing,” Opt. Eng. 19, 339 (1980).
  3. R. P. Bocker, “Matrix multipication using incoherent optical techniques,” Appl. Opt. 13, 1670 (1974).
    [CrossRef] [PubMed]
  4. W. Schneider, W. Fink, “Incoherent optical matrix multiplication,” Opt. Acta 22, 879 (1975).
    [CrossRef]
  5. P. N. Tamura, J. C. Wyant, “Two-dimensional matrix multiplication using coherent optical techniques,” Opt. Eng. 18, 198 (1979).
  6. M. A. Monahan et al., “Incoherent electro-optical processing with CCD’s,” in Proceedings of Digital International Computing Conference (Institute of Electrical and Electronics Engineers, New York, 1975).
  7. A. R. Dias, “Incoherent matrix-vector multiplication for high-speed data processing,” Ph.D. thesis (Stanford U. Press, Stanford, Calif., 1980).
  8. J. W. Goodman, “The matrix-vector multiplication incoherent processor,” presented at Workshop on Future Directions for Optical Information Processing, May 20–22, 1980, Lubbock, Texas.
  9. T. F. Krile et al., “Holographic representation of space-variant systems using phase-coded reference beams,” Appl. Opt. 16, 3131 (1977).
    [CrossRef] [PubMed]
  10. T. F. Krile et al., “Multiplex holography with chirp-modulated binary phase-coded reference-beam masks,” Appl. Opt. 18, 52 (1979).
    [CrossRef] [PubMed]
  11. M. I. Jones, J. F. Walkup, M. O. Hagler, “Multiplex holography for space-variant optical computing,” Proc. Soc. Photo-Opt. Instrum. Eng. 177, 16 (1979).
  12. R. J. Marks, “Two-dimensional space-variant processing using temporal holography: processor theory,” Appl. Opt. 18, 3670 (1979).
    [CrossRef]
  13. R. Kasturi, T. F. Krile, J. F. Walkup, “Space-variant 2-D processing using a sampled input/sampled transfer function approach,” presented at 1980 International Optical Computing Conference, April 7–11, 1980, Washington, D.C.
  14. R. J. Marks, J. F. Walkup, M. O. Hagler, “A sampling theorem for space-variant systems,” J. Opt. Soc. Am. 66, 918 (1976).
    [CrossRef]
  15. R. J. Marks, J. F. Walkup, M. O. Hagler, “Sampling theorems for linear shift-variant systems,” IEEE Trans. Circuits Syst. CAS-25, 228 (1978).
    [CrossRef]
  16. R. J. Marks, J. F. Walkup, M. O. Hagler, “Methods of linear system characterization through response cataloging,” Appl. Opt. 18, 655 (1979).
    [CrossRef]
  17. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

1980 (1)

J. F. Walkup, “Space-variant coherent optical processing,” Opt. Eng. 19, 339 (1980).

1979 (5)

1978 (1)

R. J. Marks, J. F. Walkup, M. O. Hagler, “Sampling theorems for linear shift-variant systems,” IEEE Trans. Circuits Syst. CAS-25, 228 (1978).
[CrossRef]

1977 (2)

1976 (1)

1975 (1)

W. Schneider, W. Fink, “Incoherent optical matrix multiplication,” Opt. Acta 22, 879 (1975).
[CrossRef]

1974 (1)

Bocker, R. P.

M. A. Monahan, K. Bromley, R. P. Bocker, “Incoherent optical correlators,” Proc. IEEE 65, 121 (1977).
[CrossRef]

R. P. Bocker, “Matrix multipication using incoherent optical techniques,” Appl. Opt. 13, 1670 (1974).
[CrossRef] [PubMed]

Bromley, K.

M. A. Monahan, K. Bromley, R. P. Bocker, “Incoherent optical correlators,” Proc. IEEE 65, 121 (1977).
[CrossRef]

Dias, A. R.

A. R. Dias, “Incoherent matrix-vector multiplication for high-speed data processing,” Ph.D. thesis (Stanford U. Press, Stanford, Calif., 1980).

Fink, W.

W. Schneider, W. Fink, “Incoherent optical matrix multiplication,” Opt. Acta 22, 879 (1975).
[CrossRef]

Goodman, J. W.

J. W. Goodman, “The matrix-vector multiplication incoherent processor,” presented at Workshop on Future Directions for Optical Information Processing, May 20–22, 1980, Lubbock, Texas.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Hagler, M. O.

R. J. Marks, J. F. Walkup, M. O. Hagler, “Methods of linear system characterization through response cataloging,” Appl. Opt. 18, 655 (1979).
[CrossRef]

M. I. Jones, J. F. Walkup, M. O. Hagler, “Multiplex holography for space-variant optical computing,” Proc. Soc. Photo-Opt. Instrum. Eng. 177, 16 (1979).

R. J. Marks, J. F. Walkup, M. O. Hagler, “Sampling theorems for linear shift-variant systems,” IEEE Trans. Circuits Syst. CAS-25, 228 (1978).
[CrossRef]

R. J. Marks, J. F. Walkup, M. O. Hagler, “A sampling theorem for space-variant systems,” J. Opt. Soc. Am. 66, 918 (1976).
[CrossRef]

Jones, M. I.

M. I. Jones, J. F. Walkup, M. O. Hagler, “Multiplex holography for space-variant optical computing,” Proc. Soc. Photo-Opt. Instrum. Eng. 177, 16 (1979).

Kasturi, R.

R. Kasturi, T. F. Krile, J. F. Walkup, “Space-variant 2-D processing using a sampled input/sampled transfer function approach,” presented at 1980 International Optical Computing Conference, April 7–11, 1980, Washington, D.C.

Krile, T. F.

T. F. Krile et al., “Multiplex holography with chirp-modulated binary phase-coded reference-beam masks,” Appl. Opt. 18, 52 (1979).
[CrossRef] [PubMed]

T. F. Krile et al., “Holographic representation of space-variant systems using phase-coded reference beams,” Appl. Opt. 16, 3131 (1977).
[CrossRef] [PubMed]

R. Kasturi, T. F. Krile, J. F. Walkup, “Space-variant 2-D processing using a sampled input/sampled transfer function approach,” presented at 1980 International Optical Computing Conference, April 7–11, 1980, Washington, D.C.

Marks, R. J.

Monahan, M. A.

M. A. Monahan, K. Bromley, R. P. Bocker, “Incoherent optical correlators,” Proc. IEEE 65, 121 (1977).
[CrossRef]

M. A. Monahan et al., “Incoherent electro-optical processing with CCD’s,” in Proceedings of Digital International Computing Conference (Institute of Electrical and Electronics Engineers, New York, 1975).

Schneider, W.

W. Schneider, W. Fink, “Incoherent optical matrix multiplication,” Opt. Acta 22, 879 (1975).
[CrossRef]

Tamura, P. N.

P. N. Tamura, J. C. Wyant, “Two-dimensional matrix multiplication using coherent optical techniques,” Opt. Eng. 18, 198 (1979).

Walkup, J. F.

J. F. Walkup, “Space-variant coherent optical processing,” Opt. Eng. 19, 339 (1980).

M. I. Jones, J. F. Walkup, M. O. Hagler, “Multiplex holography for space-variant optical computing,” Proc. Soc. Photo-Opt. Instrum. Eng. 177, 16 (1979).

R. J. Marks, J. F. Walkup, M. O. Hagler, “Methods of linear system characterization through response cataloging,” Appl. Opt. 18, 655 (1979).
[CrossRef]

R. J. Marks, J. F. Walkup, M. O. Hagler, “Sampling theorems for linear shift-variant systems,” IEEE Trans. Circuits Syst. CAS-25, 228 (1978).
[CrossRef]

R. J. Marks, J. F. Walkup, M. O. Hagler, “A sampling theorem for space-variant systems,” J. Opt. Soc. Am. 66, 918 (1976).
[CrossRef]

R. Kasturi, T. F. Krile, J. F. Walkup, “Space-variant 2-D processing using a sampled input/sampled transfer function approach,” presented at 1980 International Optical Computing Conference, April 7–11, 1980, Washington, D.C.

Wyant, J. C.

P. N. Tamura, J. C. Wyant, “Two-dimensional matrix multiplication using coherent optical techniques,” Opt. Eng. 18, 198 (1979).

Appl. Opt. (5)

IEEE Trans. Circuits Syst. (1)

R. J. Marks, J. F. Walkup, M. O. Hagler, “Sampling theorems for linear shift-variant systems,” IEEE Trans. Circuits Syst. CAS-25, 228 (1978).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Acta (1)

W. Schneider, W. Fink, “Incoherent optical matrix multiplication,” Opt. Acta 22, 879 (1975).
[CrossRef]

Opt. Eng. (2)

P. N. Tamura, J. C. Wyant, “Two-dimensional matrix multiplication using coherent optical techniques,” Opt. Eng. 18, 198 (1979).

J. F. Walkup, “Space-variant coherent optical processing,” Opt. Eng. 19, 339 (1980).

Proc. IEEE (1)

M. A. Monahan, K. Bromley, R. P. Bocker, “Incoherent optical correlators,” Proc. IEEE 65, 121 (1977).
[CrossRef]

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

M. I. Jones, J. F. Walkup, M. O. Hagler, “Multiplex holography for space-variant optical computing,” Proc. Soc. Photo-Opt. Instrum. Eng. 177, 16 (1979).

Other (5)

M. A. Monahan et al., “Incoherent electro-optical processing with CCD’s,” in Proceedings of Digital International Computing Conference (Institute of Electrical and Electronics Engineers, New York, 1975).

A. R. Dias, “Incoherent matrix-vector multiplication for high-speed data processing,” Ph.D. thesis (Stanford U. Press, Stanford, Calif., 1980).

J. W. Goodman, “The matrix-vector multiplication incoherent processor,” presented at Workshop on Future Directions for Optical Information Processing, May 20–22, 1980, Lubbock, Texas.

R. Kasturi, T. F. Krile, J. F. Walkup, “Space-variant 2-D processing using a sampled input/sampled transfer function approach,” presented at 1980 International Optical Computing Conference, April 7–11, 1980, Washington, D.C.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

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Tables (2)

Tables Icon

Table 1 Laplace Transform of u(ξ) = (d/dξ) sin πξ/ξ

Tables Icon

Table 2 Laplace Transform of u(ξ) = [sin(πξ/2)/ξ]2

Equations (21)

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g ( x ) = u ( ξ ) h ( x ; ξ ) d ξ
g ( x ) Δ n u ( n Δ ) h ( x ; n Δ ) ,
u ( ξ ) = 2 B u ( η ) sinc 2 B ( ξ η ) d η ,
B W ,
g ( x ) = u ( η ) h ˆ ( x ; η ) d η ,
h ˆ ( x ; η ) = 2 B h ( x ; ξ ) sinc 2 B ( ξ η ) d ξ .
u ( η ) = n = u ( n Δ ) sinc ( n η / Δ ) ,
h ˆ ( x ; η ) = m = h ˆ ( x ; m Δ ) sinc ( m η / Δ ) ,
Δ 1 / 2 B 1 / 2 W .
g ( x ) = n m u ( n Δ ) h ˆ ( x ; m Δ ) ×   sinc ( n η / Δ ) sinc ( m η / Δ ) d η = Δ n u ( n Δ ) h ˆ ( x ; n Δ ) .
E u = | u ( η ) | 2 d η = Δ n = | u ( n Δ ) | 2 <
E h ( x ) = | h ˆ ( x ; η ) | 2 d η = Δ n = | h ˆ ( x ; n Δ ) | 2 < .
| g ( x ) | 2 E u E h ( x ) .
g ( x ) = 0 u ( ξ ) e x ξ d ξ .
h ( x ; ξ ) = e x ξ μ ( ξ )
h ˆ ( x ; η ) = 0 e x ξ sinc ( ξ η ) d ξ ,
g ( x ) = 1 π u ( ξ ) d ξ x ξ
h ( x ; ξ ) = 1 π ( x ξ ) .
h ˆ ( x ; η ) = W W [ 1 π exp ( j 2 π f ξ ) x ξ d ξ ] e j 2 π f η d f = j W W sgn f e j 2 π f ( x η ) d f = 2 W sin π W ( x η ) sinc W ( x η ) .
g ( m Δ ) = n u ( n Δ ) sin π 2 ( m n ) sinc 1 2 ( m n ) .
g ( m Δ ) = 2 π m n odd u ( n Δ ) m n .

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