Abstract

Optical matrix processors using acoustooptic transducers are described with emphasis on new systolic array architectures using frequency multiplexing in addition to space and time multiplexing. A Kalman filtering application is considered as our case study from which the operations required on such a system can be defined. This also serves as a new and powerful application for iterative optical processors. The importance of pipelining the data flow and the ordering of the operations performed in a specific application of such a system are also noted. Several examples of how to effectively achieve this are included. A new technique for handling bipolar data on such architectures is also described.

© 1983 Optical Society of America

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References

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  1. J. Goodman, A. Dias, L. Woody, Opt. Lett. 2, 1 (1978).
    [CrossRef] [PubMed]
  2. A. Dias, in “Optical Information Processing for Aerospace Applications,” NASA Conf. Publ. 2207 (NTIS, Springfield, Va., 1981).
  3. M. Carlotto, D. Casasent, Appl. Opt. 21, 147 (1982).
    [CrossRef] [PubMed]
  4. D. Casasent, Proc. IEEE 65, 143 (1977).
    [CrossRef]
  5. H. Caulfield et al., Opt. Commun. 40, 86 (1981).
    [CrossRef]
  6. D. Casasent, Appl. Opt. 21, 1859 (1982).
    [CrossRef] [PubMed]
  7. A. Warner et al., J. Appl. Phys. 43, 4489 (1972).
    [CrossRef]
  8. D. Casasent et al., Proc. Soc. Photo-Opt. Instrum. Eng. 295, 176 (1981).
  9. C. Neuman et al., Proc. Electro-Opt. Syst. Des. Conf. 311, (1981).
  10. A. Bryson, Y. C. Ho, Applied Optimal Control (Blaisdell, Waltham, Mass., 1969), Chap. 12.
  11. D. Young, Iterative Solution of Large Linear Systems (Academic, New York, 1971), pp. 94 and 361–365.
  12. D. Casasent, M. Carlotto, Opt. Eng. 21, 814 (Sept.1982).
    [CrossRef]
  13. D. Casasent, C. Neuman, in “Proceedings, Optical Data Processing for Aerospace Applications,” NASA Conf. Publ. 2207 (NTIS, Springfield, Va., 1981).

1982 (3)

1981 (3)

D. Casasent et al., Proc. Soc. Photo-Opt. Instrum. Eng. 295, 176 (1981).

C. Neuman et al., Proc. Electro-Opt. Syst. Des. Conf. 311, (1981).

H. Caulfield et al., Opt. Commun. 40, 86 (1981).
[CrossRef]

1978 (1)

1977 (1)

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

1972 (1)

A. Warner et al., J. Appl. Phys. 43, 4489 (1972).
[CrossRef]

Bryson, A.

A. Bryson, Y. C. Ho, Applied Optimal Control (Blaisdell, Waltham, Mass., 1969), Chap. 12.

Carlotto, M.

D. Casasent, M. Carlotto, Opt. Eng. 21, 814 (Sept.1982).
[CrossRef]

M. Carlotto, D. Casasent, Appl. Opt. 21, 147 (1982).
[CrossRef] [PubMed]

Casasent, D.

M. Carlotto, D. Casasent, Appl. Opt. 21, 147 (1982).
[CrossRef] [PubMed]

D. Casasent, Appl. Opt. 21, 1859 (1982).
[CrossRef] [PubMed]

D. Casasent, M. Carlotto, Opt. Eng. 21, 814 (Sept.1982).
[CrossRef]

D. Casasent et al., Proc. Soc. Photo-Opt. Instrum. Eng. 295, 176 (1981).

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

D. Casasent, C. Neuman, in “Proceedings, Optical Data Processing for Aerospace Applications,” NASA Conf. Publ. 2207 (NTIS, Springfield, Va., 1981).

Caulfield, H.

H. Caulfield et al., Opt. Commun. 40, 86 (1981).
[CrossRef]

Dias, A.

J. Goodman, A. Dias, L. Woody, Opt. Lett. 2, 1 (1978).
[CrossRef] [PubMed]

A. Dias, in “Optical Information Processing for Aerospace Applications,” NASA Conf. Publ. 2207 (NTIS, Springfield, Va., 1981).

Goodman, J.

Ho, Y. C.

A. Bryson, Y. C. Ho, Applied Optimal Control (Blaisdell, Waltham, Mass., 1969), Chap. 12.

Neuman, C.

C. Neuman et al., Proc. Electro-Opt. Syst. Des. Conf. 311, (1981).

D. Casasent, C. Neuman, in “Proceedings, Optical Data Processing for Aerospace Applications,” NASA Conf. Publ. 2207 (NTIS, Springfield, Va., 1981).

Warner, A.

A. Warner et al., J. Appl. Phys. 43, 4489 (1972).
[CrossRef]

Woody, L.

Young, D.

D. Young, Iterative Solution of Large Linear Systems (Academic, New York, 1971), pp. 94 and 361–365.

Appl. Opt. (2)

J. Appl. Phys. (1)

A. Warner et al., J. Appl. Phys. 43, 4489 (1972).
[CrossRef]

Opt. Commun. (1)

H. Caulfield et al., Opt. Commun. 40, 86 (1981).
[CrossRef]

Opt. Eng. (1)

D. Casasent, M. Carlotto, Opt. Eng. 21, 814 (Sept.1982).
[CrossRef]

Opt. Lett. (1)

Proc. Electro-Opt. Syst. Des. Conf. (1)

C. Neuman et al., Proc. Electro-Opt. Syst. Des. Conf. 311, (1981).

Proc. IEEE (1)

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

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

D. Casasent et al., Proc. Soc. Photo-Opt. Instrum. Eng. 295, 176 (1981).

Other (4)

A. Bryson, Y. C. Ho, Applied Optimal Control (Blaisdell, Waltham, Mass., 1969), Chap. 12.

D. Young, Iterative Solution of Large Linear Systems (Academic, New York, 1971), pp. 94 and 361–365.

A. Dias, in “Optical Information Processing for Aerospace Applications,” NASA Conf. Publ. 2207 (NTIS, Springfield, Va., 1981).

D. Casasent, C. Neuman, in “Proceedings, Optical Data Processing for Aerospace Applications,” NASA Conf. Publ. 2207 (NTIS, Springfield, Va., 1981).

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Figures (6)

Fig. 1
Fig. 1

Original optical systolic matrix–vector multiplier system (after Refs. 5 and 6).

Fig. 2
Fig. 2

Simplified frequency-multiplexed optical systolic matrix–vector multiplier using a point acoustooptic modulator.

Fig. 3
Fig. 3

Basic space-, time-, and frequency-multiplexed optical systolic matrix–vector processor system.

Fig. 4
Fig. 4

Basic space-, time-, and frequency-multiplexed optical systolic matrix–matrix multiplier.

Fig. 5
Fig. 5

Basic space-, time-, and frequency-multiplexed optical systolic matrix–matrix–matrix multiplier system.

Fig. 6
Fig. 6

General optical systolic iterative optical processor architecture (matrix inversion case study detailed).

Tables (5)

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Table I Kalman Filtering Algorithm

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Table II Control Parameter Notation

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Table III Time History (with Space and Frequency Multiplexing) of the Contents of the Components in the System in Fig. 4 for Matrix–matrix Multiplications

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Table IV Time History (with Space and Frequency Multiplexing) for the Contents of the Components in the System in Fig. 5 for Matrix–Matrix–Matrix Multiplications

Tables Icon

Table V Data and Operational Flow and Pipelining of the System in Fig. 6 for Matrix Inversion

Equations (29)

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[ b 11 b 21 b 22 b 31 b 32 b 33 b 42 b 43 b 44 b 53 b 54 b 55 · · · · · ] [ a 1 a 2 a 3 · · · ] = [ d 1 d 2 d 3 · · · ] .
f 1 ( t ) = 0 b 42 0 b 31 0 0 f 2 ( t ) = b 43 0 b 32 0 b 21 0 f 3 ( t ) = 0 b 33 0 b 22 0 b 11 g ( t ) = 0 a 3 0 a 2 0 a 1 ,
[ b 11 b 1 N · · · b m n · · · b M 1 b M N ] [ a 1 · · · a N ] = [ c 1 · · · c M ] .
[ b 1 b n b N ] a = c .
c i j = k = 1 N a i k b k j .
B k + 1 = [ I ω H ] B k + ω C ,
B = H 1 C .
B k + 1 / ω = [ I / ω H ] B k + C .
B k + 1 = ω [ A B k + C ] .
a b = ( a + a ) ( b + b ) = ( a + b + + a b ) ( a b + + a + b ) .
( a b ) + = ( a + b + + a b ) ,
( a b ) = ( a b + + a + b ) ,
a b = ( a b ) + ( a b ) ,
A ¯ B ¯ = [ a 11 + a 11 a 12 + a 12 a 11 a 11 + a 12 a 12 + a 21 + a 21 a 22 + a 22 a 21 a 21 + a 22 a 22 + ] [ b 11 + b 12 + b 11 b 12 b 21 + b 22 b 21 b 22 + ] = [ c 11 + c 12 + c 11 c 12 c 21 + c 22 + c 21 c 22 ] = C ¯ .
A B C = D .
D = D + D = ( A + A ) [ ( B C ) + ( B C ) ] = [ A + + ( B C ) ] [ A + ( B C ) + ) ] ,
D + = A + + ( B C ) ,
D = A + ( B C ) + .
x ¯ k + 1 = Φ ¯ k x ¯ k + Γ ¯ k w ¯ k
z ¯ k = H ¯ k x ¯ k + v ¯ k
w ¯ k N ( w ¯ ¯ k , Q ¯ k )
v ¯ k N ( O , R ¯ k )
E [ w ¯ k v ¯ j T ] = 0 for all j and k
x ¯ ¯ 0 and M ¯ 0
P ¯ k = ( M ¯ k 1 + H ¯ k T R ¯ k 1 H ¯ k ) 1 = M ¯ k M ¯ k H ¯ k T ( H ¯ k M ¯ k H ¯ k T + R ¯ k ) 1 H ¯ k M ¯ k
K ¯ k = P ¯ k H ¯ k T R ¯ k 1
x ¯ ˆ k = x ¯ ¯ k + K ¯ k ( z ¯ k H ¯ k x ¯ ¯ k )
x ¯ ¯ k + 1 = Φ ¯ k x ¯ ˆ k + Γ ¯ k w ¯ ¯ k
M ¯ k + 1 = Φ ¯ k P ¯ k Φ ¯ k T + Γ ¯ k Q ¯ k Γ ¯ k T

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