Abstract

A numerical optical processor is described that performs operations in residue arithmetic. The position coding used to represent decimal and residue numbers allows one to describe the various conversions and operations in a correlation formulation. This description of residue arithmetic leads directly to novel residue adder and decimal/residue/decimal converter designs, which are described and experimentally demonstrated. The accuracy, dynamic range and space bandwidth of an optical residue arithmetic processor are also discussed.

© 1979 Optical Society of America

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References

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  1. H. L. Garner, IRE Trans. Electron. Comput. EC-8, 140 (1959).
    [CrossRef]
  2. N. S. Szabo, R. I. Tanaka, Residue Arithmetic and Its Applications to Computer Technology (McGraw-Hill, New York, 1967).
  3. D. E. Knuth, Seminumerical Algorithmsx, Vol. 2 (Addison-Wesley, Reading, Mass., 1969), pp. 248–256.
  4. A. Huang, in Proceedings of the International Optical Computing Conference Washington, D.C., April 1975, X. X. Blank, Ed. (IEEENew York, 1975), pp. 14–18.
  5. A. Huang, Proc. Electro-Optical Systems Design Conference, Anaheim, October 1977, pp. 208–212.
  6. S. Collins, Proc. Soc. Photo-Opt. Instrum. Eng. 128, (1977).
  7. A. Huang, Y. Tsunoda, J. W. Goodman, Appl. Opt. 18, 15Jan1979.
    [CrossRef] [PubMed]
  8. A. Huang, Y. Tsunoda, J. W. Goodman, Stanford Electronics Lab Rept. 6422-1 (March1978).
  9. D. Casasent, T. Luu, Proc. Electro-Optical Systems Design Conference, Anaheim, October 1977, pp. 208–212.
  10. D. Casasent, T. Luu, Appl. Opt. 17, 1701 (1978).
    [CrossRef] [PubMed]
  11. D. Casasent, Proc. IEEE 65, 143 (1977).
    [CrossRef]
  12. C. Thomas, Appl. Opt. 5, 1782 (1966).
    [CrossRef] [PubMed]
  13. This alternate raster recording scheme was suggested to us by L. Weiner of Ampex Corp.

1979 (1)

A. Huang, Y. Tsunoda, J. W. Goodman, Appl. Opt. 18, 15Jan1979.
[CrossRef] [PubMed]

1978 (1)

1977 (2)

D. Casasent, Proc. IEEE 65, 143 (1977).
[CrossRef]

S. Collins, Proc. Soc. Photo-Opt. Instrum. Eng. 128, (1977).

1966 (1)

1959 (1)

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

Casasent, D.

D. Casasent, T. Luu, Appl. Opt. 17, 1701 (1978).
[CrossRef] [PubMed]

D. Casasent, Proc. IEEE 65, 143 (1977).
[CrossRef]

D. Casasent, T. Luu, Proc. Electro-Optical Systems Design Conference, Anaheim, October 1977, pp. 208–212.

Collins, S.

S. Collins, Proc. Soc. Photo-Opt. Instrum. Eng. 128, (1977).

Garner, H. L.

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

Goodman, J. W.

A. Huang, Y. Tsunoda, J. W. Goodman, Appl. Opt. 18, 15Jan1979.
[CrossRef] [PubMed]

A. Huang, Y. Tsunoda, J. W. Goodman, Stanford Electronics Lab Rept. 6422-1 (March1978).

Huang, A.

A. Huang, Y. Tsunoda, J. W. Goodman, Appl. Opt. 18, 15Jan1979.
[CrossRef] [PubMed]

A. Huang, Y. Tsunoda, J. W. Goodman, Stanford Electronics Lab Rept. 6422-1 (March1978).

A. Huang, in Proceedings of the International Optical Computing Conference Washington, D.C., April 1975, X. X. Blank, Ed. (IEEENew York, 1975), pp. 14–18.

A. Huang, Proc. Electro-Optical Systems Design Conference, Anaheim, October 1977, pp. 208–212.

Knuth, D. E.

D. E. Knuth, Seminumerical Algorithmsx, Vol. 2 (Addison-Wesley, Reading, Mass., 1969), pp. 248–256.

Luu, T.

D. Casasent, T. Luu, Appl. Opt. 17, 1701 (1978).
[CrossRef] [PubMed]

D. Casasent, T. Luu, Proc. Electro-Optical Systems Design Conference, Anaheim, October 1977, pp. 208–212.

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).

Thomas, C.

Tsunoda, Y.

A. Huang, Y. Tsunoda, J. W. Goodman, Appl. Opt. 18, 15Jan1979.
[CrossRef] [PubMed]

A. Huang, Y. Tsunoda, J. W. Goodman, Stanford Electronics Lab Rept. 6422-1 (March1978).

Appl. Opt. (3)

IRE Trans. Electron. Comput. (1)

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

Proc. IEEE (1)

D. Casasent, Proc. IEEE 65, 143 (1977).
[CrossRef]

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

S. Collins, Proc. Soc. Photo-Opt. Instrum. Eng. 128, (1977).

Other (7)

This alternate raster recording scheme was suggested to us by L. Weiner of Ampex Corp.

A. Huang, Y. Tsunoda, J. W. Goodman, Stanford Electronics Lab Rept. 6422-1 (March1978).

D. Casasent, T. Luu, Proc. Electro-Optical Systems Design Conference, Anaheim, October 1977, pp. 208–212.

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

D. E. Knuth, Seminumerical Algorithmsx, Vol. 2 (Addison-Wesley, Reading, Mass., 1969), pp. 248–256.

A. Huang, in Proceedings of the International Optical Computing Conference Washington, D.C., April 1975, X. X. Blank, Ed. (IEEENew York, 1975), pp. 14–18.

A. Huang, Proc. Electro-Optical Systems Design Conference, Anaheim, October 1977, pp. 208–212.

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram of a decimal-to-residue optical converter.

Fig. 2
Fig. 2

Schematic diagram of a residue-to-decimal optical converter.

Fig. 3
Fig. 3

General correlator schematic for a decimal/residue converter.

Fig. 4
Fig. 4

Schematic diagram of a residue arithmetic optical adder.

Fig. 5
Fig. 5

Experimental demonstration of the conversion of the 21 decimal input numbers 0 to 20 into residue numbers modulo 7 using the optical system in Fig. 1: (a) input P0 pattern; (b) output P2 pattern.

Fig. 6
Fig. 6

Experimental demonstration of the conversion of the residue number (1,1,3) with moduli (3,4,5) into the decimal number 13 using the optical system of Fig. 2: (a) input P0 pattern; (b) output P2 pattern.

Fig. 7
Fig. 7

Schematic diagram of a high dynamic range decimal to residue converter using raster recorded input data.

Equations (39)

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M = i = 1 N m i .
m i = ( 5 , 7 , 9 , 2 ) 13 = ( 3 , 6 , 4 , 1 ) + 59 = ( 4 , 3 , 5 , 1 ) 72 = ( 2 , 2 , 0 , 0 ) ,
m i = ( 5 , 7 , 9 , 2 ) 13 = ( 3 , 6 , 4 , 1 ) - 13 = ( 2 , 1 , 5 , 1 ) .
72 = ( 2 , 2 , 0 , 0 ) - 13 = ( 2 , 1 , 5 , 1 ) 59 = ( 4 , 3 , 5 , 1 ) .
m i = ( 5 , 7 , 9 , 2 ) 19 = ( 4 , 5 , 1 , 1 ) 12 = ( 2 , 5 , 3 , 0 ) 19 × 12 = ( 3 , 4 , 3 , 0 ) = 228 ,
g ( x 0 ) = δ ( x 0 - X Δ x ) .
G ( u ) = exp ( j 2 π X Δ x ) .
x 1 = λ f L u ,
H i ( u ) = n = - exp ( j 2 π u n m i Δ x ) ,
m i Δ x = u g i λ f L .
G H i = n = - exp [ j 2 π u ( X - n m i ) Δ x ] .
f 2 ( x 2 ) = n = - δ [ x 2 - ( X - n m i ) Δ x ] ,
x 2 = ( X - n m i ) Δ x .
X = n m i + R m i ,
0 R m i m i - 1.
R m i = X - n m i .
0 R m i Δ x ( m i - 1 ) Δ x ,
R m i Δ x = ( X - n m i ) Δ x .
0 x 2 ( m i - 1 ) Δ x .
h ( x 0 ) = n δ ( x 0 - n m i Δ x ) ,
g h = n δ ( x 0 - X Δ x ) δ ( x 0 - n m i Δ x + x 2 ) d x 0 = n δ ( x 0 + x 2 ) δ [ x 0 - ( X - n m i ) Δ x ] d x 0 = n δ [ x 2 - ( X - n m i ) Δ x ] .
a = δ ( x 0 - R a Δ x )
b = δ ( x 0 - R b Δ x ) .
R a + R b m i ,
a b = - δ ( x - R a Δ x ) δ ( x + R b Δ x + x ) d x = - δ [ x - ( R a + R b ) Δ x ] δ ( x + x ) d x = δ [ x - ( R a + R b ) Δ x ] .
t 0 ( x 0 ) = δ ( x 0 - x a ) + δ ( x 0 + x b ) ,
f 1 ( u ) = exp ( j 2 π u x a ) + exp ( - j 2 π u x b ) .
t 1 = exp ( j 2 π u x a ) + exp ( - j 2 π u x b ) 2 = 2 + 2 cos [ 2 π u ( x a + x b ) ] .
t 2 = n exp ( - j 2 π u n m i Δ x ) .
t 1 t 2 = n exp [ j 2 π u ( x a + x b - n m i Δ x ) ] .
f 3 = n δ [ x 3 - ( x a + x b - n m i Δ x ) ] = n δ [ x 3 - ( R a + R b - n m i ) Δ x ] .
Δ x = 0.7 mm = u g λ f L / m i
M = A / Δ x .
N = n = 0 B a b 2 b ,
N m i = ( a 0 2 0 m i + a 1 2 1 m i + a 2 2 2 m i + ) m i .
N x 0 - K N L x 0 = ( N - K N L ) x 0 .
N - K N L = ( n - r ) m i + R m i - R m i ,
R m i = ( N - K N L ) - ( n - r ) m i
18 5 = ( 168 - 150 ) 5 = 168 5 - 150 5 = 168 5 .

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