Abstract

Addition is the most primitive arithmetic operation in digital computation. Other arithmetic operations such as subtraction, multiplication, and division can all be performed by addition together with some logic operations. With the binary number system, addition speed is inevitably limited by the carry-propagation schemes. On the other hand, carry-free addition is possible when the modified signed-digit (MSD) number representation is used. We propose a novel optoelectronic scheme to handle the parallel MSD addition and subtraction operations. An optoelectronic shared content-addressable memroy is introduced. The shared content-addressable memory uses free-space optical processing to handle the large amount of parallel memory access operations and uses electronics to postprocess and derive logic decisions. We analyze the accuracy that the required optical hardware can deliver by using a statistical cross-talk-rate model that we propose. We also evaluate other important device and system performance parameters, such as the memory capacity or the maximum number of parallel bits the adder can handle in terms of a given cross-talk rate at a certain repetition rate, the corresponding diffraction-limited memory density, and the system’s power efficiency. To confirm the underlining operational principles of the proposed optoelectronic shared content-addressable-memory MSD adder, we design and perform initial experiments for handling 8-bit MSD number addition and subtraction and present the results.

© 1994 Optical Society of America

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References

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  1. A. Hwang, Y. Tsunida, J. W. Goodman, S. Ishihara, “Optical computation using residue arithmetic,” Appl. Opt. 18, 149–162 (1979).
    [CrossRef]
  2. A. Avizienis, “Signed-digit number representation for fast parallel arithmetic,” IRE Trans. Electron. Comput. EC-10, 389–400 (1961).
    [CrossRef]
  3. N. Takagi, H. Tasuura, S. Yajima, “High-speed VLSI multiplication algorithm with a redundant binary addition tree,” IEEE Trans. Comput. C-34, 789–796 (1985).
    [CrossRef]
  4. R. P. Borker, B. L. Drake, M. E. Lasher, T. B. Henderson, “Modified signed-digit addition and subtraction using optical symbolic substitution,” Appl. Opt. 25, 2456–2457 (1986).
    [CrossRef]
  5. M. M. Mirsalehi, T. K. Gaylord, “Truth-table look-up parallel processing using an optical content-addressable memory,” Appl. Opt. 25, 2277–2283 (1986).
    [CrossRef] [PubMed]
  6. Y. Li, G. Eichmann, “Conditional symbolic modified signed-digit arithmetic using optical content-addressable memory logic elements,” Appl. Opt. 26, 2328–2333 (1987).
    [CrossRef] [PubMed]
  7. A. K. Cherri, M. A. Karim, “Modified signed-digital arithmetic using an efficient symbolic substitution,” Appl. Opt. 27, 3824–3827 (1988).
    [CrossRef] [PubMed]
  8. B. Parhami“Carry-free addition of recorded binary signed-digital numbers,” IEEE Trans. Comput. C37, 1470–1476 (1988).
    [CrossRef]
  9. K. Hwang, A. Louri, “Optical multiplication and division using modified signed-digit symbolic substitution,” Opt. Eng 28, 364–372 (1989).
  10. Y. Li, D. H. Kim, A. Kostrzewski, G. Eichmann, “Content-addressable-memory-based optical modified-signed-digit arithmetic,” Opt. Lett. 14, 1254–1256 (1989).
    [CrossRef] [PubMed]
  11. T. Kohonen, Content-Addressable Memories (Springer-Verlag, New York, 1980), pp. 1–55.
    [CrossRef]
  12. K. Hwang, F. A. Briggs, Computer Architecture and Parallel Processing (McGraw-Hill, New York, 1984), Chap. 5, pp.325–388.
  13. T. K. Gaylord, M. M. Mirsalehi, C. C. Guest, “Optical digit truth-table look-up processing,” Opt. Eng. 24, 48–58 (1985).
  14. M. M. Mirsalihi, T. K. Gaylord, “Logical minimization of multilevel coded functions,” Appl. Opt. 25, 3078–3088 (1986).
    [CrossRef]
  15. P. B. Berra, K. H. Brenner, W. T. Cathey, H. J. Caulfield, S. H. Lee, H. Szu, “Optical database/knowledgebase machines,” Appl. Opt. 28, 195–205 (1990).
    [CrossRef]
  16. A. Louri, “Optical content-addressable parallel processor: architecture, algorithms, and design concepts,” Appl. Opt. 31, 3241–3258 (1992).
    [CrossRef] [PubMed]
  17. A. A. S. Awwal, “Recoded signal-digit binary addition–subtraction using optoelectronic symbolic substitution,” Appl. Opt. 31, 3205–3208 (1992).
    [CrossRef] [PubMed]
  18. G. Gheen, “Optical matrix–matrix multiplier,” Appl. Opt. 29, 886–887 (1990).
  19. J. Hong, P. Yeh, “Photorefractive parallel matrix–matrix multiplier,” Opt. Lett. 16, 1343–1345 (1991).
    [CrossRef] [PubMed]
  20. R. A. Athale, “Optical matrix processors,” in Optical and Hybrid Computing, H. H. Szu, ed., Proc. Soc. Photo-Opt. Instrum. Eng634, 96–111 (1986).
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    [CrossRef] [PubMed]
  22. H. Hwang, L. Liu, Z. Wang, “Parallel multiple matrix multiplication using an orthogonal shadow-casting and imaging system,” Opt. Lett. 15, 1085–1087 (1990).
    [CrossRef]
  23. R. A. Athale, K. Raj, V. A. Savkar, “Fully parallel analog optical calculation of multiple outer products,” Opt. Lett. 18, 989–991 (1993).
    [CrossRef] [PubMed]
  24. A. Papoulis, Probabilities, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984), Chap. 6, pp. 225–249.
  25. J. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 3, pp. 89–167.
  26. R. K. Kostuk, “Simulation of board-level free-space optical interconnects for electronic processing,” Appl. Opt. 31, 2438–2445 (1992).
    [CrossRef] [PubMed]
  27. C. W. Stirk, “Bit error rate of optical logic: fan-in, threshold, and contrast,” Appl. Opt. 31, 5632–5641 (1992).
    [CrossRef] [PubMed]
  28. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 5, pp. 620–685.
  29. R. Linke, “Power distribution in a planar-waveguide-based broadcast STAR network,” IEEE Photon. Tech. Lett. 3, 850–852 (1991).
    [CrossRef]
  30. Programmable Logic Devices Databook and Design Guide (National Semiconductor, Santa Clara, Calif., 1992).

1993 (2)

1992 (4)

1991 (2)

R. Linke, “Power distribution in a planar-waveguide-based broadcast STAR network,” IEEE Photon. Tech. Lett. 3, 850–852 (1991).
[CrossRef]

J. Hong, P. Yeh, “Photorefractive parallel matrix–matrix multiplier,” Opt. Lett. 16, 1343–1345 (1991).
[CrossRef] [PubMed]

1990 (3)

G. Gheen, “Optical matrix–matrix multiplier,” Appl. Opt. 29, 886–887 (1990).

H. Hwang, L. Liu, Z. Wang, “Parallel multiple matrix multiplication using an orthogonal shadow-casting and imaging system,” Opt. Lett. 15, 1085–1087 (1990).
[CrossRef]

P. B. Berra, K. H. Brenner, W. T. Cathey, H. J. Caulfield, S. H. Lee, H. Szu, “Optical database/knowledgebase machines,” Appl. Opt. 28, 195–205 (1990).
[CrossRef]

1989 (2)

K. Hwang, A. Louri, “Optical multiplication and division using modified signed-digit symbolic substitution,” Opt. Eng 28, 364–372 (1989).

Y. Li, D. H. Kim, A. Kostrzewski, G. Eichmann, “Content-addressable-memory-based optical modified-signed-digit arithmetic,” Opt. Lett. 14, 1254–1256 (1989).
[CrossRef] [PubMed]

1988 (2)

A. K. Cherri, M. A. Karim, “Modified signed-digital arithmetic using an efficient symbolic substitution,” Appl. Opt. 27, 3824–3827 (1988).
[CrossRef] [PubMed]

B. Parhami“Carry-free addition of recorded binary signed-digital numbers,” IEEE Trans. Comput. C37, 1470–1476 (1988).
[CrossRef]

1987 (1)

1986 (3)

1985 (2)

T. K. Gaylord, M. M. Mirsalehi, C. C. Guest, “Optical digit truth-table look-up processing,” Opt. Eng. 24, 48–58 (1985).

N. Takagi, H. Tasuura, S. Yajima, “High-speed VLSI multiplication algorithm with a redundant binary addition tree,” IEEE Trans. Comput. C-34, 789–796 (1985).
[CrossRef]

1979 (1)

1961 (1)

A. Avizienis, “Signed-digit number representation for fast parallel arithmetic,” IRE Trans. Electron. Comput. EC-10, 389–400 (1961).
[CrossRef]

Athale, R. A.

R. A. Athale, K. Raj, V. A. Savkar, “Fully parallel analog optical calculation of multiple outer products,” Opt. Lett. 18, 989–991 (1993).
[CrossRef] [PubMed]

R. A. Athale, “Optical matrix processors,” in Optical and Hybrid Computing, H. H. Szu, ed., Proc. Soc. Photo-Opt. Instrum. Eng634, 96–111 (1986).

Avizienis, A.

A. Avizienis, “Signed-digit number representation for fast parallel arithmetic,” IRE Trans. Electron. Comput. EC-10, 389–400 (1961).
[CrossRef]

Awwal, A. A. S.

Berra, P. B.

P. B. Berra, K. H. Brenner, W. T. Cathey, H. J. Caulfield, S. H. Lee, H. Szu, “Optical database/knowledgebase machines,” Appl. Opt. 28, 195–205 (1990).
[CrossRef]

Borker, R. P.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 5, pp. 620–685.

Brenner, K. H.

P. B. Berra, K. H. Brenner, W. T. Cathey, H. J. Caulfield, S. H. Lee, H. Szu, “Optical database/knowledgebase machines,” Appl. Opt. 28, 195–205 (1990).
[CrossRef]

Briggs, F. A.

K. Hwang, F. A. Briggs, Computer Architecture and Parallel Processing (McGraw-Hill, New York, 1984), Chap. 5, pp.325–388.

Campbell, S.

Cathey, W. T.

P. B. Berra, K. H. Brenner, W. T. Cathey, H. J. Caulfield, S. H. Lee, H. Szu, “Optical database/knowledgebase machines,” Appl. Opt. 28, 195–205 (1990).
[CrossRef]

Caulfield, H. J.

P. B. Berra, K. H. Brenner, W. T. Cathey, H. J. Caulfield, S. H. Lee, H. Szu, “Optical database/knowledgebase machines,” Appl. Opt. 28, 195–205 (1990).
[CrossRef]

Cherri, A. K.

Drake, B. L.

Eichmann, G.

Gaylord, T. K.

Gheen, G.

G. Gheen, “Optical matrix–matrix multiplier,” Appl. Opt. 29, 886–887 (1990).

Goodman, J.

J. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 3, pp. 89–167.

Goodman, J. W.

Gu, C.

Guest, C. C.

T. K. Gaylord, M. M. Mirsalehi, C. C. Guest, “Optical digit truth-table look-up processing,” Opt. Eng. 24, 48–58 (1985).

Henderson, T. B.

Hong, J.

Hwang, A.

Hwang, H.

Hwang, K.

K. Hwang, A. Louri, “Optical multiplication and division using modified signed-digit symbolic substitution,” Opt. Eng 28, 364–372 (1989).

K. Hwang, F. A. Briggs, Computer Architecture and Parallel Processing (McGraw-Hill, New York, 1984), Chap. 5, pp.325–388.

Ishihara, S.

Karim, M. A.

Kim, D. H.

Kohonen, T.

T. Kohonen, Content-Addressable Memories (Springer-Verlag, New York, 1980), pp. 1–55.
[CrossRef]

Kostrzewski, A.

Kostuk, R. K.

Lasher, M. E.

Lee, S. H.

P. B. Berra, K. H. Brenner, W. T. Cathey, H. J. Caulfield, S. H. Lee, H. Szu, “Optical database/knowledgebase machines,” Appl. Opt. 28, 195–205 (1990).
[CrossRef]

Li, Y.

Linke, R.

R. Linke, “Power distribution in a planar-waveguide-based broadcast STAR network,” IEEE Photon. Tech. Lett. 3, 850–852 (1991).
[CrossRef]

Liu, L.

Louri, A.

A. Louri, “Optical content-addressable parallel processor: architecture, algorithms, and design concepts,” Appl. Opt. 31, 3241–3258 (1992).
[CrossRef] [PubMed]

K. Hwang, A. Louri, “Optical multiplication and division using modified signed-digit symbolic substitution,” Opt. Eng 28, 364–372 (1989).

Mirsalehi, M. M.

M. M. Mirsalehi, T. K. Gaylord, “Truth-table look-up parallel processing using an optical content-addressable memory,” Appl. Opt. 25, 2277–2283 (1986).
[CrossRef] [PubMed]

T. K. Gaylord, M. M. Mirsalehi, C. C. Guest, “Optical digit truth-table look-up processing,” Opt. Eng. 24, 48–58 (1985).

Mirsalihi, M. M.

Papoulis, A.

A. Papoulis, Probabilities, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984), Chap. 6, pp. 225–249.

Parhami, B.

B. Parhami“Carry-free addition of recorded binary signed-digital numbers,” IEEE Trans. Comput. C37, 1470–1476 (1988).
[CrossRef]

Raj, K.

Savkar, V. A.

Stirk, C. W.

Szu, H.

P. B. Berra, K. H. Brenner, W. T. Cathey, H. J. Caulfield, S. H. Lee, H. Szu, “Optical database/knowledgebase machines,” Appl. Opt. 28, 195–205 (1990).
[CrossRef]

Takagi, N.

N. Takagi, H. Tasuura, S. Yajima, “High-speed VLSI multiplication algorithm with a redundant binary addition tree,” IEEE Trans. Comput. C-34, 789–796 (1985).
[CrossRef]

Tasuura, H.

N. Takagi, H. Tasuura, S. Yajima, “High-speed VLSI multiplication algorithm with a redundant binary addition tree,” IEEE Trans. Comput. C-34, 789–796 (1985).
[CrossRef]

Tsunida, Y.

Wang, Z.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 5, pp. 620–685.

Yajima, S.

N. Takagi, H. Tasuura, S. Yajima, “High-speed VLSI multiplication algorithm with a redundant binary addition tree,” IEEE Trans. Comput. C-34, 789–796 (1985).
[CrossRef]

Yeh, P.

Appl. Opt. (12)

R. P. Borker, B. L. Drake, M. E. Lasher, T. B. Henderson, “Modified signed-digit addition and subtraction using optical symbolic substitution,” Appl. Opt. 25, 2456–2457 (1986).
[CrossRef]

M. M. Mirsalehi, T. K. Gaylord, “Truth-table look-up parallel processing using an optical content-addressable memory,” Appl. Opt. 25, 2277–2283 (1986).
[CrossRef] [PubMed]

Y. Li, G. Eichmann, “Conditional symbolic modified signed-digit arithmetic using optical content-addressable memory logic elements,” Appl. Opt. 26, 2328–2333 (1987).
[CrossRef] [PubMed]

A. K. Cherri, M. A. Karim, “Modified signed-digital arithmetic using an efficient symbolic substitution,” Appl. Opt. 27, 3824–3827 (1988).
[CrossRef] [PubMed]

A. Hwang, Y. Tsunida, J. W. Goodman, S. Ishihara, “Optical computation using residue arithmetic,” Appl. Opt. 18, 149–162 (1979).
[CrossRef]

M. M. Mirsalihi, T. K. Gaylord, “Logical minimization of multilevel coded functions,” Appl. Opt. 25, 3078–3088 (1986).
[CrossRef]

P. B. Berra, K. H. Brenner, W. T. Cathey, H. J. Caulfield, S. H. Lee, H. Szu, “Optical database/knowledgebase machines,” Appl. Opt. 28, 195–205 (1990).
[CrossRef]

A. Louri, “Optical content-addressable parallel processor: architecture, algorithms, and design concepts,” Appl. Opt. 31, 3241–3258 (1992).
[CrossRef] [PubMed]

A. A. S. Awwal, “Recoded signal-digit binary addition–subtraction using optoelectronic symbolic substitution,” Appl. Opt. 31, 3205–3208 (1992).
[CrossRef] [PubMed]

G. Gheen, “Optical matrix–matrix multiplier,” Appl. Opt. 29, 886–887 (1990).

R. K. Kostuk, “Simulation of board-level free-space optical interconnects for electronic processing,” Appl. Opt. 31, 2438–2445 (1992).
[CrossRef] [PubMed]

C. W. Stirk, “Bit error rate of optical logic: fan-in, threshold, and contrast,” Appl. Opt. 31, 5632–5641 (1992).
[CrossRef] [PubMed]

IEEE Photon. Tech. Lett. (1)

R. Linke, “Power distribution in a planar-waveguide-based broadcast STAR network,” IEEE Photon. Tech. Lett. 3, 850–852 (1991).
[CrossRef]

IEEE Trans. Comput. (2)

N. Takagi, H. Tasuura, S. Yajima, “High-speed VLSI multiplication algorithm with a redundant binary addition tree,” IEEE Trans. Comput. C-34, 789–796 (1985).
[CrossRef]

B. Parhami“Carry-free addition of recorded binary signed-digital numbers,” IEEE Trans. Comput. C37, 1470–1476 (1988).
[CrossRef]

IRE Trans. Electron. Comput. (1)

A. Avizienis, “Signed-digit number representation for fast parallel arithmetic,” IRE Trans. Electron. Comput. EC-10, 389–400 (1961).
[CrossRef]

Opt. Eng (1)

K. Hwang, A. Louri, “Optical multiplication and division using modified signed-digit symbolic substitution,” Opt. Eng 28, 364–372 (1989).

Opt. Eng. (1)

T. K. Gaylord, M. M. Mirsalehi, C. C. Guest, “Optical digit truth-table look-up processing,” Opt. Eng. 24, 48–58 (1985).

Opt. Lett. (5)

Other (7)

A. Papoulis, Probabilities, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984), Chap. 6, pp. 225–249.

J. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 3, pp. 89–167.

Programmable Logic Devices Databook and Design Guide (National Semiconductor, Santa Clara, Calif., 1992).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 5, pp. 620–685.

T. Kohonen, Content-Addressable Memories (Springer-Verlag, New York, 1980), pp. 1–55.
[CrossRef]

K. Hwang, F. A. Briggs, Computer Architecture and Parallel Processing (McGraw-Hill, New York, 1984), Chap. 5, pp.325–388.

R. A. Athale, “Optical matrix processors,” in Optical and Hybrid Computing, H. H. Szu, ed., Proc. Soc. Photo-Opt. Instrum. Eng634, 96–111 (1986).

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Figures (17)

Fig. 1
Fig. 1

Schematic diagram of a three-step 5-bit MSD adder in which output zi is affected by six input variables: xi, yi xi−1, yi−1, xi−2, yi−2.

Fig. 2
Fig. 2

Schematic diagram of a single-step n-bit MSD adder in which a single six-variable gate replaces the 11 gates shown in the Fig. 1 setup.

Fig. 3
Fig. 3

Schematic diagram of a S-CAM n-bit MSD adder in which a single four-variable CAm adder is shared by n + 1 groups of four-variable input addends with the help of a space multiplexer and demultiplexer.

Fig. 4
Fig. 4

(a) Encoding rule for the input data; (b) the encoding rule for MSD CAM operation; (c) an example of an encoded minterm, d 0 1 ¯ 10 d 1 1 ¯ (d) the encoded input data matrix representing two input addends, 101010 1 ¯ 0 and 100100 1 ¯ 0; (e) the encoded CAM MSD addition masks for generating 1 and −1.

Fig. 5
Fig. 5

Schematic O-E S-CAM MSD adder. Input matrix A is formed from input addends X and Y. The product of input matrix A and S-CAM matrix B is further processed with parallel electronics.

Fig. 6
Fig. 6

(a) 5-f triple-matrix multiplier, with the three matrices located on planes I, A, and B, attaching to the lenses; (b) 6-f triple-matrix multiplier, with the three matrices located on planes I, A, and B, detaching from the lenses.

Fig. 7
Fig. 7

CAM MSD adder architecture based on electrically addressed reflective SLM’s: VCSE, vertical cavity surface-emitting laser array; LCSLM’s liquid-crystal spatial light modulators.

Fig. 8
Fig. 8

Typical Gaussian probability-density functions of the low- and the high-level input signals.

Fig. 9
Fig. 9

Probability-density function of the multiplication result of the two inputs shown: the zero intensity results from the combination of the products of low × low and low × high; the one intensity results from the product of two high-level inputs.

Fig. 10
Fig. 10

(a) Probability-density function of the summed variable of the two variables defined in Fig. 9, resulting in three terms; (b) the probability-density function of the summed variable of the four variables defined in Fig. 9, resulting in five terms. For example, the result two (intensity) is the summation of two low and two high levels.

Fig. 11
Fig. 11

Probability-density function of the summed variable of the 12 variables defined in Fig. 9. The overlapping area of zero intensity (the sum of 12 low levels) and one intensity (the sum of 11 low levels and 1 high level) is used to evaluate the cross-talk rate.

Fig. 12
Fig. 12

Cross-talk rate as a result of using matrix dimension M, where a is rmask-cell aperture, w is the half-width of the diffraction main lobe, and R is the related intensity ratio of the low and the high levels.

Fig. 13
Fig. 13

Selections of the diffraction-limited mask-cell apertures, with λ as the reference wavelength.

Fig. 14
Fig. 14

Element bit rate of the MSD adder, where N is the number of bits processed.

Fig. 15
Fig. 15

Photograph of our bench-top experimental setup. The setup can be used to support many S-CAM operations other than the MSD arithmetic.

Fig. 16
Fig. 16

Experimental results of the MSD addition 101010 1 ¯ 0 + 100100 1 ¯ 0 = 100111 1 ¯ 00; (a) experimental results of the matrix multiplication (a) before and (b) after the threshold operation.

Fig. 17
Fig. 17

Experimental results of the MSD subtraction 101010 1 ¯ 0 − 100 1 ¯ 0010 = 101010 1 ¯ 0 + 1 ¯ 001000 1 ¯ 0 = 00111 1 ¯ 00: (a) experimental results of the matrix multiplication (a) before and (b) after the threshold operation.

Tables (2)

Tables Icon

Table 1 Diffractiona from Rectangular Cells

Tables Icon

Table 2 Parameters, Used for Cross-Talk-Rate Simulations

Equations (28)

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N MSD = i a i 2 i ,
7 10 = 0111 MSD = 100 1 ¯ MSD = 10 1 ¯ 1 MSD = 1 1 ¯ 11 MSD .
11 ¯ 11 , 0011 , 01 d 0 , 01 d 01 1 ¯ , 10 1 ¯ d 01 , 100 d             for generating 1 ,
11 11 ¯ , 00 11 ¯ , 0 1 ¯ d 0 , 0 1 ¯ d 0 1 ¯ 1 , 1 ¯ 01 d 0 1 ¯ , 1 ¯ 00 d             for generating 1 ¯ ,
[ 0 0 x 0 y 0 x 1 y 1 x 0 y 0 x 2 y 2 x 1 y 1 x 3 y 3 x 2 y 2 x 4 y 4 x 3 y 3 x 5 y 5 x 4 y 4 x 6 y 6 x 5 y 5 x 7 y 7 x 6 y 6 0 0 x 7 y 7 ] ,             [ 0 0 0 0 - 1 - 1 0 0 0 0 - 1 - 1 1 0 0 0 0 1 1 0 1 0 0 1 0 0 1 1 1 1 0 0 0 0 1 1 ] .
[ 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 ] .
[ Generating 1 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 1 1 0 0 1 1 0 0 1 1 1 1 0 1 0 1 1 1 0 0 1 0 0 0 0 0 1 1 1 1 1 0 1 0 1 1 0 1 0 0 0 0 1 1 0 0 Generating - 1 1 1 1 1 0 0 1 0 0 0 1 1 0 1 1 1 1 1 1 1 0 0 1 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 1 0 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 1 1 1 0 1 0 ]
[ Generating 1 4 4 2 1 2 1 2 4 1 3 3 2 4 2 2 2 2 2 3 3 2 3 1 0 3 2 0 1 3 3 3 2 3 3 1 0 3 1 1 2 2 2 4 4 1 2 2 1 2 0 2 2 2 2 Generating - 1 4 2 1 3 1 2 4 4 1 2 2 1 2 0 2 2 2 2 3 3 2 3 2 1 3 3 2 3 2 1 3 3 1 3 2 3 4 2 1 3 1 2 2 4 2 3 3 2 4 2 2 2 2 2 Final Result 0 0 - 1 1 1 1 0 0 1 ]
c i j = k = 0 M - 1 a i k b k j .
f c ( c ) = - 1 w f A ( w ) f B ( C w ) d w ,
f 0 ( O ) = f C 1 ( C 1 ) * f C 2 ( C 2 ) * f C 3 ( C 3 ) * * f C M ( C M ) ,
f ZERO ( I th ) = f ONE ( I th ) .
CTR = 0 th f one ( I ) d I + th f zero ( I ) d I 0 f zero ( I ) d I .
y i = ( sin x i x i ) 2 ,
s 2 w ,
Δ 1 = 2 ( y 1 + y 2 ) = 0.128.
Δ 4 = i = 3 6 y i = 0.019.
Δ 2 = i = 1 4 y i = 0.077 ,
Δ 3 = 2 ( y 3 + y 4 ) = 0.026.
μ = 1 4 ζ = 1 4 Δ ζ = 0.063.
σ d 2 = Σ ( Δ ζ ) 2 P ( I ζ ) = 0.076 2 .
w = f λ a ,
a > ( 1.8 λ f ) 1 / 2
a > ( 3.6 λ f ) 1 / 2 .
P bit = 10 , 000 h v = 10 , 000 h c λ = 10 , 000 ( 6.63 ) ( 10 - 34 ) 3 ( 10 8 ) 0.9 ( 10 - 6 ) = 2.2 ( 10 - 15 ) w .
P rec = P source ( 2 M - 1 ) 2 η ,
EBR = P rec P bit = P source η ( 2 M - 1 ) 2 P bit .
EBR = P rec P bit = P sour η ( N + M ) 2 P bit .

Metrics