Abstract

The use of a holographic content-addressable memory system for parallel truth-table look-up digital data processing is analyzed. For binary-coded residue numbers, the operations of 4-, 8-, 12-, and 16-bit addition and multiplication are treated. The minimum probability of error that can be achieved and the corresponding detector threshold settings are determined in each case allowing for the effects of Gaussian distributions in the amplitude and the phase in the recording beams. Resultant probabilities of error for practical conditions are found to be very competitive with those from state-of-the-art nonparallel technologies.

© 1983 Optical Society of America

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References

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  1. T. L. Booth, Digital Networks and Computer Systems (Wiley, New York, 1971), p. 136.
  2. S. Muroga, Logical Design and Switching Theory (Wiley, New York, 1979), p. 92.
  3. C. C. Guest, T. K. Gaylord, Appl. Opt. 19, 1201 (1980).
    [CrossRef] [PubMed]
  4. A. Svoboda, M. Valach, in Stroje na Zpraccovani Informaci, Sbornik V (Nakl. CSZV, Prague, 1957), p. 9 (in English).
  5. H. L. Garner, IRE Trans. Electron. Comput. 8, 140 (1959).
    [CrossRef]
  6. N. S. Szabo, R. I. Tanaka, Residue Arithmetic and Its Applications to Computer Technology (McGraw-Hill, New York, 1967).
  7. C. C. Guest, M. M. Mirsalehi, T. K. Gaylord, IEEE Trans. Comput. submitted for publication.
  8. T. Kohonen, Associative Memory (Springer, New York, 1977).
    [CrossRef]
  9. P. E. Tverdokhleb, in Optical Information Processing, Vol. 2, E. S. Barrekette, G. W. Stroke, Y. E. Nesterikhin, W. E. Kock, Eds. (Plenum, New York, 1978), p. 283.
  10. G. R. Knight, Appl. Opt. 14, 1088 (1975).
    [CrossRef] [PubMed]
  11. K. Preston, Coherent Optical Computers (McGraw-Hill, New York, 1972), Chap. 8.
  12. H. J. Gallagher, T. K. Gaylord, M. G. Moharam, C. C. Guest, Appl. Opt. 20, 300 (1981).
    [CrossRef] [PubMed]
  13. J. V. Uspensky, Introduction to Mathematical Probability (McGraw-Hill, New York, 1937), p. 283.
  14. For example, A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), p. 211.
  15. Ref. 14, p. 189.
  16. For example, M. Schwartz, Information Transmission, Modulation, and Noise (McGraw-Hill, New York, 1980), p. 317.
  17. D. L. Staebler, W. J. Burke, W. Phillips, J. J. Amodei, Appl. Phys. Lett. 26, 182 (1975).
    [CrossRef]

1981 (1)

1980 (1)

1975 (2)

D. L. Staebler, W. J. Burke, W. Phillips, J. J. Amodei, Appl. Phys. Lett. 26, 182 (1975).
[CrossRef]

G. R. Knight, Appl. Opt. 14, 1088 (1975).
[CrossRef] [PubMed]

1959 (1)

H. L. Garner, IRE Trans. Electron. Comput. 8, 140 (1959).
[CrossRef]

Amodei, J. J.

D. L. Staebler, W. J. Burke, W. Phillips, J. J. Amodei, Appl. Phys. Lett. 26, 182 (1975).
[CrossRef]

Booth, T. L.

T. L. Booth, Digital Networks and Computer Systems (Wiley, New York, 1971), p. 136.

Burke, W. J.

D. L. Staebler, W. J. Burke, W. Phillips, J. J. Amodei, Appl. Phys. Lett. 26, 182 (1975).
[CrossRef]

Gallagher, H. J.

Garner, H. L.

H. L. Garner, IRE Trans. Electron. Comput. 8, 140 (1959).
[CrossRef]

Gaylord, T. K.

Guest, C. C.

Knight, G. R.

Kohonen, T.

T. Kohonen, Associative Memory (Springer, New York, 1977).
[CrossRef]

Mirsalehi, M. M.

C. C. Guest, M. M. Mirsalehi, T. K. Gaylord, IEEE Trans. Comput. submitted for publication.

Moharam, M. G.

Muroga, S.

S. Muroga, Logical Design and Switching Theory (Wiley, New York, 1979), p. 92.

Papoulis, A.

For example, A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), p. 211.

Phillips, W.

D. L. Staebler, W. J. Burke, W. Phillips, J. J. Amodei, Appl. Phys. Lett. 26, 182 (1975).
[CrossRef]

Preston, K.

K. Preston, Coherent Optical Computers (McGraw-Hill, New York, 1972), Chap. 8.

Schwartz, M.

For example, M. Schwartz, Information Transmission, Modulation, and Noise (McGraw-Hill, New York, 1980), p. 317.

Staebler, D. L.

D. L. Staebler, W. J. Burke, W. Phillips, J. J. Amodei, Appl. Phys. Lett. 26, 182 (1975).
[CrossRef]

Svoboda, A.

A. Svoboda, M. Valach, in Stroje na Zpraccovani Informaci, Sbornik V (Nakl. CSZV, Prague, 1957), p. 9 (in English).

Szabo, N. S.

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

Tanaka, R. I.

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

Tverdokhleb, P. E.

P. E. Tverdokhleb, in Optical Information Processing, Vol. 2, E. S. Barrekette, G. W. Stroke, Y. E. Nesterikhin, W. E. Kock, Eds. (Plenum, New York, 1978), p. 283.

Uspensky, J. V.

J. V. Uspensky, Introduction to Mathematical Probability (McGraw-Hill, New York, 1937), p. 283.

Valach, M.

A. Svoboda, M. Valach, in Stroje na Zpraccovani Informaci, Sbornik V (Nakl. CSZV, Prague, 1957), p. 9 (in English).

Appl. Opt. (3)

Appl. Phys. Lett. (1)

D. L. Staebler, W. J. Burke, W. Phillips, J. J. Amodei, Appl. Phys. Lett. 26, 182 (1975).
[CrossRef]

IRE Trans. Electron. Comput. (1)

H. L. Garner, IRE Trans. Electron. Comput. 8, 140 (1959).
[CrossRef]

Other (12)

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

C. C. Guest, M. M. Mirsalehi, T. K. Gaylord, IEEE Trans. Comput. submitted for publication.

T. Kohonen, Associative Memory (Springer, New York, 1977).
[CrossRef]

P. E. Tverdokhleb, in Optical Information Processing, Vol. 2, E. S. Barrekette, G. W. Stroke, Y. E. Nesterikhin, W. E. Kock, Eds. (Plenum, New York, 1978), p. 283.

A. Svoboda, M. Valach, in Stroje na Zpraccovani Informaci, Sbornik V (Nakl. CSZV, Prague, 1957), p. 9 (in English).

T. L. Booth, Digital Networks and Computer Systems (Wiley, New York, 1971), p. 136.

S. Muroga, Logical Design and Switching Theory (Wiley, New York, 1979), p. 92.

K. Preston, Coherent Optical Computers (McGraw-Hill, New York, 1972), Chap. 8.

J. V. Uspensky, Introduction to Mathematical Probability (McGraw-Hill, New York, 1937), p. 283.

For example, A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), p. 211.

Ref. 14, p. 189.

For example, M. Schwartz, Information Transmission, Modulation, and Noise (McGraw-Hill, New York, 1980), p. 317.

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Figures (6)

Fig. 1
Fig. 1

Holographic truth-table look-up processor: (a) recording the truth-table holograms. (b) processing of binary input data.

Fig. 2
Fig. 2

Probability densities in the 1-D case. A general-threshold setting (xth) and the optimum-threshold setting (x0) are shown.

Fig. 3
Fig. 3

(a) Example phasor diagram showing the complex amplitude of the wave front for each bit. (b) Phasor diagram showing all possible resultant amplitudes at a particular detector. Numbers indicate degeneracy of phasor.

Fig. 4
Fig. 4

Probability density function of the resultant complex amplitude corresponding to the binary pattern 1010. The optimum-threshold amplitude setting is shown. This figure is plotted for the σa = σθ = 0.10 case.

Fig. 5
Fig. 5

Probability densities along the real axis of Fig. 4. The position of the optimum amplitude threshold setting (a0), and the two amplitude settings that produce twice the optimum error ( a 2 p + and a 2 p ) are indicated.

Fig. 6
Fig. 6

Characteristic probability of error curves corresponding to the logically reduced truth table for the addition operation with modulus 4. The parameters of curves 1–4 are (1) R = 1, Nnm = 6, Nd = 1; (2) R = 1, Nnm = 8,Nd = 2; (3) R = 2, Nnm = 4,Nd = 0; and (4) R = 4, Nnm = 4, Nd = 0. The corresponding degeneracies of curves 1–4 are 4, 2,1,1, respectively. The dotted curve corresponds to the probability of error for the common-threshold setting technique. This figure is plotted for the σa = σθ = 0.05 case.

Tables (6)

Tables Icon

Table I Operational Characteristics for Addition with Residue Number System Holographic Truth-Table Look-Up Processora

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Table II Operational Characteristics for Multiplication with Residue Number System Holographic Truth-Table Look-Up Processora

Tables Icon

Table III Operational Characteristics for Logically Reduced Addition with Residue Number System Holographic Truth-Table Look-Up Processora

Tables Icon

Table IV Operational Characteristics for Logically Reduced Multiplication with Residue Number System Holographic Truth-Table Look-Up Processora

Tables Icon

Table V Operational Characteristics for 4-, 8-, 12-, and 16-Bit Addition and Multiplication with Residue Number System Holographic Truth-Table Look-Up Processora

Tables Icon

Table VI Operational Characteristics for Logically Reduced 4-, 8-, 12-, and 16-Bit Addition and Multiplication with Residue Number System Holographic Truth-Table Look-Up Processora

Equations (21)

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p i ( a , θ ) = 1 2 π σ θ i σ a i exp [ ( a a ¯ i ) 2 / 2 σ a i 2 ] exp [ ( θ θ ¯ i ) 2 / 2 σ θ i 2 ] ,
σ θ ( 1 ) = σ θ ( 0 ) = σ θ ( R ) σ θ .
σ a ( 1 ) = σ a ( 0 ) σ a .
σ a ( R ) = ( R ) 1 / 2 σ a .
p ( x , y ) = + p 1 ( x 1 , y 1 ) p 2 ( x x 1 , y y 1 ) d x 1 d y 1 .
p i ( x , y ) = 1 2 π ( x 2 + y 2 ) 1 / 2 σ a i σ θ i × exp { [ ( x 2 + y 2 ) 1 / 2 ā i ] 2 / 2 σ a i 2 } × exp { [ tan 1 ( y / x ) θ ¯ i ] 2 / 2 σ θ i 2 } .
p i ( x , y ) = 1 2 π σ x i σ y i exp [ ( x x ¯ i ) 2 / 2 σ x i 2 ] exp ( y 2 / 2 σ y i 2 ) ,
p 1 ( x , y ) = 1 2 π σ x 1 σ y 1 exp [ ( x x ¯ 1 ) 2 / 2 σ x 1 2 ] exp ( y 2 / 2 σ y 1 2 ) ,
p 2 ( x , y ) = 1 2 π σ x 2 σ y 2 exp [ ( x x ¯ 2 ) 2 / 2 σ x 2 2 ] exp ( y 2 / 2 σ y 2 2 ) ,
p ( x , y ) = p 1 ( x , y ) * p 2 ( x , y ) = 1 2 π ( σ x 1 2 + σ x 2 2 ) 1 / 2 ( σ y 1 2 + σ y 2 2 ) 1 / 2 × exp { [ x ( x ¯ 1 + x ¯ 2 ) ] 2 / 2 ( σ x 1 2 + σ x 2 2 ) } × exp [ y 2 / 2 ( σ y 1 2 + σ y 2 2 ) ] .
P f a = P 1 x t h p 1 ( x ) d x , P m = P 0 x t h + p 0 ( x ) d x ,
P e = P 1 x t h p 1 ( x ) d x + P 0 x t h + p 0 ( x ) d x .
P m = P 0 a = a t h θ = 0 2 π p 0 ( a , θ ) d θ d a ,
P fa = i = 1 N m P i a = 0 a t h θ = 0 2 π p i ( a , θ ) d θ da ,
P e = P 0 a = a t h θ = 0 2 π p 0 ( a , θ ) d θ da + i = 1 N m P i a = 0 a t h θ = 0 2 π p i ( a , θ ) d θ da .
P 0 θ = 0 2 π p 0 ( a 0 , θ ) d θ = i = 1 N m P i θ = 0 2 π p i ( a 0 , θ ) d θ .
θ = 0 2 π p 0 ( a 0 , θ ) d θ = i = 1 N m θ = 0 2 π p i ( a 0 , θ ) d θ .
P c = j = 1 J k = 1 K j [ 1 P e ( j k ) ] ,
P e = 1 P c = 1 j = 1 J k = 1 K j [ 1 P e ( j k ) ] .
P t e = 1 i = 1 I j = 1 J i k = 1 K i j [ 1 P e ( ijk ) ] ,
P t e i = 1 I j = 1 J i k = 1 K i j P e ( ijk ) .

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