Abstract

Information theory shows that one can communicate with arbitrarily low error probability over a noisy channel. By recognizing that some types of computations can be cast as communications problems, it may be possible to compute accurately with an inexact processor. Traditional analog optical processors, for example, offer advantages of parallelism and speed but suffer from significant inaccuracies. We propose an algorithm whereby the accuracy of matched filter processors can be improved significantly at the cost of a modest increase in computational resources. Errors that are due to noisy data and/or inexact computing can be detected and in some cases corrected.

© 1987 Optical Society of America

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References

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  1. A. A. Sawchuk, T. C. Strand, Proc. IEEE 72, 758 (1984).
    [CrossRef]
  2. Special issue on digital optical computing, Opt. Eng. 25, (1986).
  3. C. E. Shannon, Bell. Syst. Tech. J. 27, 379 (1948).
  4. R. E. Blahut, Theory and Practice of Error Control Codes (Addision-Wesley, Reading, Mass., 1983).
  5. Also called synthetic discriminant function and linear combination filter.
  6. Q. B. Braunecker, R. Hauck, A. W. Lohmann, Appl. Opt. 18, 2746 (1980).
    [CrossRef]
  7. B. V. K. V. Kumar, Appl. Opt. 22, 1445 (1983).
    [CrossRef] [PubMed]
  8. D. Casasent, Appl. Opt. 23, 1620 (1984).
    [CrossRef] [PubMed]
  9. H. J. Caulfield, R. S. Putnam, Opt. Eng. 24, 65 (1985).
  10. N. H. Farhat, D. Psaltis, A. Prata, E. Paek, Appl. Opt. 24, 1469 (1985).
    [CrossRef] [PubMed]
  11. H. J. Larson, Introduction to Probability Theory and Statistical Inference (Wiley, New York, 1969).
  12. R. H. Kuhn, in VLSI and Modern Signal Processing, S. Kung, H. Whitehouse, T. Kailath, eds. (Prentice-Hall, Englewood Cliffs, N.J., 1985), pp. 178–184.
  13. R. C. Aubusson, I. Catt, IEEE J. Solid-State Circuits CS-15, 677 (1980).
  14. H. L. VanTrees, Detection, Estimation and Modulation Theory, Part I (Wiley, New York, 1968).

1986 (1)

Special issue on digital optical computing, Opt. Eng. 25, (1986).

1985 (2)

1984 (2)

D. Casasent, Appl. Opt. 23, 1620 (1984).
[CrossRef] [PubMed]

A. A. Sawchuk, T. C. Strand, Proc. IEEE 72, 758 (1984).
[CrossRef]

1983 (1)

1980 (2)

R. C. Aubusson, I. Catt, IEEE J. Solid-State Circuits CS-15, 677 (1980).

Q. B. Braunecker, R. Hauck, A. W. Lohmann, Appl. Opt. 18, 2746 (1980).
[CrossRef]

1948 (1)

C. E. Shannon, Bell. Syst. Tech. J. 27, 379 (1948).

Aubusson, R. C.

R. C. Aubusson, I. Catt, IEEE J. Solid-State Circuits CS-15, 677 (1980).

Blahut, R. E.

R. E. Blahut, Theory and Practice of Error Control Codes (Addision-Wesley, Reading, Mass., 1983).

Braunecker, Q. B.

Casasent, D.

Catt, I.

R. C. Aubusson, I. Catt, IEEE J. Solid-State Circuits CS-15, 677 (1980).

Caulfield, H. J.

H. J. Caulfield, R. S. Putnam, Opt. Eng. 24, 65 (1985).

Farhat, N. H.

Hauck, R.

Kuhn, R. H.

R. H. Kuhn, in VLSI and Modern Signal Processing, S. Kung, H. Whitehouse, T. Kailath, eds. (Prentice-Hall, Englewood Cliffs, N.J., 1985), pp. 178–184.

Kumar, B. V. K. V.

Larson, H. J.

H. J. Larson, Introduction to Probability Theory and Statistical Inference (Wiley, New York, 1969).

Lohmann, A. W.

Paek, E.

Prata, A.

Psaltis, D.

Putnam, R. S.

H. J. Caulfield, R. S. Putnam, Opt. Eng. 24, 65 (1985).

Sawchuk, A. A.

A. A. Sawchuk, T. C. Strand, Proc. IEEE 72, 758 (1984).
[CrossRef]

Shannon, C. E.

C. E. Shannon, Bell. Syst. Tech. J. 27, 379 (1948).

Strand, T. C.

A. A. Sawchuk, T. C. Strand, Proc. IEEE 72, 758 (1984).
[CrossRef]

VanTrees, H. L.

H. L. VanTrees, Detection, Estimation and Modulation Theory, Part I (Wiley, New York, 1968).

Appl. Opt. (4)

Bell. Syst. Tech. J. (1)

C. E. Shannon, Bell. Syst. Tech. J. 27, 379 (1948).

IEEE J. Solid-State Circuits (1)

R. C. Aubusson, I. Catt, IEEE J. Solid-State Circuits CS-15, 677 (1980).

Opt. Eng. (2)

Special issue on digital optical computing, Opt. Eng. 25, (1986).

H. J. Caulfield, R. S. Putnam, Opt. Eng. 24, 65 (1985).

Proc. IEEE (1)

A. A. Sawchuk, T. C. Strand, Proc. IEEE 72, 758 (1984).
[CrossRef]

Other (5)

R. E. Blahut, Theory and Practice of Error Control Codes (Addision-Wesley, Reading, Mass., 1983).

Also called synthetic discriminant function and linear combination filter.

H. L. VanTrees, Detection, Estimation and Modulation Theory, Part I (Wiley, New York, 1968).

H. J. Larson, Introduction to Probability Theory and Statistical Inference (Wiley, New York, 1969).

R. H. Kuhn, in VLSI and Modern Signal Processing, S. Kung, H. Whitehouse, T. Kailath, eds. (Prentice-Hall, Englewood Cliffs, N.J., 1985), pp. 178–184.

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Figures (1)

Fig. 1
Fig. 1

Monte Carlo simulation of CMF performance without (solid line) and with (dashed line) error correction. The plots are percent of correct results versus input noise standard deviation σF. Error bars represent 90% confidence intervals.

Tables (2)

Tables Icon

Table 1 Coding Map for Four CMF’s

Tables Icon

Table 2 Error-Correction Coding Map for Four CMF’s

Equations (12)

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h 0 = f 1 + f 3 + f 5 + f 7 + f 9 + f 11 + f 13 + f 15 , h 1 = f 2 + f 3 + f 6 + f 7 + f 10 + f 11 + f 14 + f 15 , h 2 = f 4 + f 5 + f 6 + f 7 + f 12 + f 13 + f 14 + f 15 , h 3 = f 8 + f 9 + f 10 + f 11 + f 12 + f 13 + f 14 + f 15 .
f n T h m = Δ ^ n p { 1 n s p 0 n s p .
d 4 = f 1 + f 2 + f 4 + f 7 + f 8 + f 11 + f 13 + f 14 .
τ 0 = t 1 , τ 1 = t 2 , h 0 = t 3 = f 1 + f 3 + f 5 + f 7 + f 9 + f 11 + f 13 + f 15 , τ 2 = t 4 , h 1 = t 5 = f 2 + f 3 + f 6 + f 7 + f 10 + f 11 + f 14 + f 15 , h 2 = t 6 = f 4 + f 5 + f 6 + f 7 + f 12 + f 13 + f 14 + f 15 , h 3 = t 7 = f 8 + f 9 + f 10 + f 11 + f 12 + f 13 + f 14 + f 15 .
τ 0 = t 1 = f 1 + f 2 + f 5 + f 6 + f 8 + f 11 + f 12 + f 15 , τ 1 = t 2 = f 1 + f 3 + f 4 + f 6 + f 8 + f 10 + f 13 + f 15 , t 3 = f 1 + f 3 + f 5 + f 7 + f 9 + f 11 + f 13 + f 15 , τ 2 = t 4 = f 2 + f 3 + f 4 + f 5 + f 8 + f 9 + f 14 + f 15 , t 5 = f 2 + f 3 + f 6 + f 7 + f 10 + f 11 + f 14 + f 15 , t 6 = f 4 + f 5 + f 6 + f 7 + f 12 + f 13 + f 14 + f 15 , t 7 = f 8 + f 9 + f 10 + f 11 + f 12 + f 13 + f 14 + f 15 .
f 9 T [ t 1 t 2 t 3 t 4 t 5 t 6 t 7 ] = [ 0011001 ] .
[ 0011101 ] .
s p = n = 1 N a p m f n             1 p P ,
s p T f m = { 1 f m t p 0 f m t p = Δ ^ p m .
Δ ^ = [ 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 0 1 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 ] .
n = 1 N a p n r n m = Δ ^ p m ,
A = Δ ^ R - 1 .

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