Abstract

This paper uses M-ary chip symbols to increase the spectral efficiency of optical code division multiple access (OCDMA) even though, when compared to its binary counterpart, M-ary OCDMA can have significantly increased multi-user interference (MUI). The presented numerical channel capacity results show that appropriate non-uniform M-ary signaling allows the reduction of MUI and the achievement of improved OCDMA spectral efficiency. Using insights from the OCDMA capacity results, we design a coded M-ary OCDMA transmission scheme that can achieve very high spectral efficiencies. The proposed time/wavelength OCDMA scheme relies on intensity modulation, appropriate non-uniform signaling, M-ary trellis-coded modulation and error-correcting codes. The corresponding receiver is based on direct detection of optical signals, soft-decision single-user demodulation and error-control decoding. The presented simulation results illustrate that the designed OCDMA scheme can support hundreds of active users at the target bit error rate of 10−9 and is robust to impairments encountered in optical transmission (shot noise, thermal noise). The achieved spectral efficiency of up to 1.34 bits per OCDMA chip is significantly better than the best binary OCDMA result reported in the literature.

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References

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  1. P. R. Prucnal, Optical Code Division Multiple Access: Fundamentals and Applications, CRC Press, 2006.
  2. G. C. Yang and W. C. Kwong, Prime Codes With Applications to CDMA Optical and Wireless Networks, Artech House, 2002.
  3. L. Tančevski and I. Andonovic, "Wavelength hopping/time spreading code division multiple access systems," Electron. Lett. 30, 1388‒1390 (1994).
    [CrossRef]
  4. M. R. Dale and R. M. Gagliardi, "Channel coding for asynchronous fiber optic CDMA communications," IEEE Trans. Commun. 43, 2485‒2492 (1995).
    [CrossRef]
  5. K.-I. Kitayama, "Code division multiplexing lightwave networks based upon optical code conversion," IEEE J. Sel. Areas Commun. 16, 1309‒1319 (1998).
    [CrossRef]
  6. H. M. H. Shalaby, "Chip-level detection in optical code division multiple access," J. Lightwave Technol. 16, 1077‒1087 (1998).
    [CrossRef]
  7. H. M. H. Shalaby, "Complexities, error probabilities, and capacities of optical OOK-CDMA communication systems," IEEE Trans. Commun. 50, 2009‒2017 (2002).
    [CrossRef]
  8. E. Inaty, H. Shalaby, P. Fortier, and L. A. Rusch, "Multirate optical fast frequency hopping CDMA system using power control," J. Lightwave Technol. 20, 166‒176 (2002).
    [CrossRef]
  9. P. Azni, M. Nasiri-Kenari, and J. A. Salehi, "Soft-input decoder for decoding of internally channel coded fiber-optic CDMA communication systems," IEEE Trans. Commun. 50, 1994‒2002 (2002).
    [CrossRef]
  10. R. M. H. Yim, J. Bajcsy, and L. R. Chen, "A new family of 2-D wavelength-time codes for optical CDMA with differential detection," IEEE Photonics Technol. Lett. 15, 165‒167 (2003).
    [CrossRef]
  11. A. J. Mendez, R. M. Gagliardi, V. J. Hernandez, C. V. Bennett, and W. J. Lennon, "Design and performance analysis of wavelength/time (W/T) matrix codes for optical CDMA," J. Lightwave Technol. 21, 2524‒2533 (2003).
    [CrossRef]
  12. C.-S. Bres, I. Glesk, R. J. Runser, T. Banwell, P. R. Prucnal, and W. C. Kwong, "Novel M-ary architecture for optical CDMA using pulse position modulation," Proc. of the 18th Annu. Meeting of the IEEE Lasers and Electro-Optics Society, 2005, pp. 982‒983.
  13. T. Miyazawa and I. Sasase, "Enhancement of tolerance to MAIs by the synergistic effect between M-ary PAM and the chip-level receiver for optical CDMA systems," J. Lightwave Technol. 24, 658‒666 (2006).
    [CrossRef]
  14. A. A. Garba, R. M. H. Yim, J. Bajcsy, and L. R. Chen, "Analysis of optical CDMA signal transmission: capacity limits and simulation results," EURASIP J. Appl. Signal Process. 2005, 1603‒1616.
  15. A. A. Garba and J. Bajcsy, "A new approach to achieve high spectral efficiency in wavelength-time OCDMA network transmission," IEEE Photonics Technol. Lett. 19, 131‒133 (2007).
    [CrossRef]
  16. C. Berrou, A. Glavieux, and P. Thitimajshima, "Near Shannon limit error-correcting coding and decoding: turbo-codes," Proc. of the IEEE Int. Conf. Communications, 1993, pp. 1064‒1070.
  17. S. B. Wicker and V. K. Bhargava, ed., Reed-Solomon Codes and Their Applications, IEEE Press, 1994.
  18. G. Ungerboeck, "Channel coding with multilevel/phase signals," IEEE Trans. Inf. Theory 28, 55‒67 (1982).
    [CrossRef]
  19. L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, "Optimal decoding of linear codes for minimizing symbol error rate," IEEE Trans. Inf. Theory 20, 284‒287 (1974).
    [CrossRef]

2007 (1)

A. A. Garba and J. Bajcsy, "A new approach to achieve high spectral efficiency in wavelength-time OCDMA network transmission," IEEE Photonics Technol. Lett. 19, 131‒133 (2007).
[CrossRef]

2006 (1)

2003 (2)

R. M. H. Yim, J. Bajcsy, and L. R. Chen, "A new family of 2-D wavelength-time codes for optical CDMA with differential detection," IEEE Photonics Technol. Lett. 15, 165‒167 (2003).
[CrossRef]

A. J. Mendez, R. M. Gagliardi, V. J. Hernandez, C. V. Bennett, and W. J. Lennon, "Design and performance analysis of wavelength/time (W/T) matrix codes for optical CDMA," J. Lightwave Technol. 21, 2524‒2533 (2003).
[CrossRef]

2002 (3)

H. M. H. Shalaby, "Complexities, error probabilities, and capacities of optical OOK-CDMA communication systems," IEEE Trans. Commun. 50, 2009‒2017 (2002).
[CrossRef]

E. Inaty, H. Shalaby, P. Fortier, and L. A. Rusch, "Multirate optical fast frequency hopping CDMA system using power control," J. Lightwave Technol. 20, 166‒176 (2002).
[CrossRef]

P. Azni, M. Nasiri-Kenari, and J. A. Salehi, "Soft-input decoder for decoding of internally channel coded fiber-optic CDMA communication systems," IEEE Trans. Commun. 50, 1994‒2002 (2002).
[CrossRef]

1998 (2)

K.-I. Kitayama, "Code division multiplexing lightwave networks based upon optical code conversion," IEEE J. Sel. Areas Commun. 16, 1309‒1319 (1998).
[CrossRef]

H. M. H. Shalaby, "Chip-level detection in optical code division multiple access," J. Lightwave Technol. 16, 1077‒1087 (1998).
[CrossRef]

1995 (1)

M. R. Dale and R. M. Gagliardi, "Channel coding for asynchronous fiber optic CDMA communications," IEEE Trans. Commun. 43, 2485‒2492 (1995).
[CrossRef]

1994 (1)

L. Tančevski and I. Andonovic, "Wavelength hopping/time spreading code division multiple access systems," Electron. Lett. 30, 1388‒1390 (1994).
[CrossRef]

1982 (1)

G. Ungerboeck, "Channel coding with multilevel/phase signals," IEEE Trans. Inf. Theory 28, 55‒67 (1982).
[CrossRef]

1974 (1)

L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, "Optimal decoding of linear codes for minimizing symbol error rate," IEEE Trans. Inf. Theory 20, 284‒287 (1974).
[CrossRef]

Andonovic, I.

L. Tančevski and I. Andonovic, "Wavelength hopping/time spreading code division multiple access systems," Electron. Lett. 30, 1388‒1390 (1994).
[CrossRef]

Azni, P.

P. Azni, M. Nasiri-Kenari, and J. A. Salehi, "Soft-input decoder for decoding of internally channel coded fiber-optic CDMA communication systems," IEEE Trans. Commun. 50, 1994‒2002 (2002).
[CrossRef]

Bahl, L.

L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, "Optimal decoding of linear codes for minimizing symbol error rate," IEEE Trans. Inf. Theory 20, 284‒287 (1974).
[CrossRef]

Bajcsy, J.

A. A. Garba and J. Bajcsy, "A new approach to achieve high spectral efficiency in wavelength-time OCDMA network transmission," IEEE Photonics Technol. Lett. 19, 131‒133 (2007).
[CrossRef]

R. M. H. Yim, J. Bajcsy, and L. R. Chen, "A new family of 2-D wavelength-time codes for optical CDMA with differential detection," IEEE Photonics Technol. Lett. 15, 165‒167 (2003).
[CrossRef]

A. A. Garba, R. M. H. Yim, J. Bajcsy, and L. R. Chen, "Analysis of optical CDMA signal transmission: capacity limits and simulation results," EURASIP J. Appl. Signal Process. 2005, 1603‒1616.

Banwell, T.

C.-S. Bres, I. Glesk, R. J. Runser, T. Banwell, P. R. Prucnal, and W. C. Kwong, "Novel M-ary architecture for optical CDMA using pulse position modulation," Proc. of the 18th Annu. Meeting of the IEEE Lasers and Electro-Optics Society, 2005, pp. 982‒983.

Bennett, C. V.

Berrou, C.

C. Berrou, A. Glavieux, and P. Thitimajshima, "Near Shannon limit error-correcting coding and decoding: turbo-codes," Proc. of the IEEE Int. Conf. Communications, 1993, pp. 1064‒1070.

Bres, C.-S.

C.-S. Bres, I. Glesk, R. J. Runser, T. Banwell, P. R. Prucnal, and W. C. Kwong, "Novel M-ary architecture for optical CDMA using pulse position modulation," Proc. of the 18th Annu. Meeting of the IEEE Lasers and Electro-Optics Society, 2005, pp. 982‒983.

Chen, L. R.

R. M. H. Yim, J. Bajcsy, and L. R. Chen, "A new family of 2-D wavelength-time codes for optical CDMA with differential detection," IEEE Photonics Technol. Lett. 15, 165‒167 (2003).
[CrossRef]

A. A. Garba, R. M. H. Yim, J. Bajcsy, and L. R. Chen, "Analysis of optical CDMA signal transmission: capacity limits and simulation results," EURASIP J. Appl. Signal Process. 2005, 1603‒1616.

Cocke, J.

L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, "Optimal decoding of linear codes for minimizing symbol error rate," IEEE Trans. Inf. Theory 20, 284‒287 (1974).
[CrossRef]

Dale, M. R.

M. R. Dale and R. M. Gagliardi, "Channel coding for asynchronous fiber optic CDMA communications," IEEE Trans. Commun. 43, 2485‒2492 (1995).
[CrossRef]

Fortier, P.

Gagliardi, R. M.

Garba, A. A.

A. A. Garba and J. Bajcsy, "A new approach to achieve high spectral efficiency in wavelength-time OCDMA network transmission," IEEE Photonics Technol. Lett. 19, 131‒133 (2007).
[CrossRef]

A. A. Garba, R. M. H. Yim, J. Bajcsy, and L. R. Chen, "Analysis of optical CDMA signal transmission: capacity limits and simulation results," EURASIP J. Appl. Signal Process. 2005, 1603‒1616.

Glavieux, A.

C. Berrou, A. Glavieux, and P. Thitimajshima, "Near Shannon limit error-correcting coding and decoding: turbo-codes," Proc. of the IEEE Int. Conf. Communications, 1993, pp. 1064‒1070.

Glesk, I.

C.-S. Bres, I. Glesk, R. J. Runser, T. Banwell, P. R. Prucnal, and W. C. Kwong, "Novel M-ary architecture for optical CDMA using pulse position modulation," Proc. of the 18th Annu. Meeting of the IEEE Lasers and Electro-Optics Society, 2005, pp. 982‒983.

Hernandez, V. J.

Inaty, E.

Jelinek, F.

L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, "Optimal decoding of linear codes for minimizing symbol error rate," IEEE Trans. Inf. Theory 20, 284‒287 (1974).
[CrossRef]

Kitayama, K.-I.

K.-I. Kitayama, "Code division multiplexing lightwave networks based upon optical code conversion," IEEE J. Sel. Areas Commun. 16, 1309‒1319 (1998).
[CrossRef]

Kwong, W. C.

G. C. Yang and W. C. Kwong, Prime Codes With Applications to CDMA Optical and Wireless Networks, Artech House, 2002.

C.-S. Bres, I. Glesk, R. J. Runser, T. Banwell, P. R. Prucnal, and W. C. Kwong, "Novel M-ary architecture for optical CDMA using pulse position modulation," Proc. of the 18th Annu. Meeting of the IEEE Lasers and Electro-Optics Society, 2005, pp. 982‒983.

Lennon, W. J.

Mendez, A. J.

Miyazawa, T.

Nasiri-Kenari, M.

P. Azni, M. Nasiri-Kenari, and J. A. Salehi, "Soft-input decoder for decoding of internally channel coded fiber-optic CDMA communication systems," IEEE Trans. Commun. 50, 1994‒2002 (2002).
[CrossRef]

Prucnal, P. R.

P. R. Prucnal, Optical Code Division Multiple Access: Fundamentals and Applications, CRC Press, 2006.

C.-S. Bres, I. Glesk, R. J. Runser, T. Banwell, P. R. Prucnal, and W. C. Kwong, "Novel M-ary architecture for optical CDMA using pulse position modulation," Proc. of the 18th Annu. Meeting of the IEEE Lasers and Electro-Optics Society, 2005, pp. 982‒983.

Raviv, J.

L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, "Optimal decoding of linear codes for minimizing symbol error rate," IEEE Trans. Inf. Theory 20, 284‒287 (1974).
[CrossRef]

Runser, R. J.

C.-S. Bres, I. Glesk, R. J. Runser, T. Banwell, P. R. Prucnal, and W. C. Kwong, "Novel M-ary architecture for optical CDMA using pulse position modulation," Proc. of the 18th Annu. Meeting of the IEEE Lasers and Electro-Optics Society, 2005, pp. 982‒983.

Rusch, L. A.

Salehi, J. A.

P. Azni, M. Nasiri-Kenari, and J. A. Salehi, "Soft-input decoder for decoding of internally channel coded fiber-optic CDMA communication systems," IEEE Trans. Commun. 50, 1994‒2002 (2002).
[CrossRef]

Sasase, I.

Shalaby, H.

Shalaby, H. M. H.

H. M. H. Shalaby, "Complexities, error probabilities, and capacities of optical OOK-CDMA communication systems," IEEE Trans. Commun. 50, 2009‒2017 (2002).
[CrossRef]

H. M. H. Shalaby, "Chip-level detection in optical code division multiple access," J. Lightwave Technol. 16, 1077‒1087 (1998).
[CrossRef]

Tancevski, L.

L. Tančevski and I. Andonovic, "Wavelength hopping/time spreading code division multiple access systems," Electron. Lett. 30, 1388‒1390 (1994).
[CrossRef]

Thitimajshima, P.

C. Berrou, A. Glavieux, and P. Thitimajshima, "Near Shannon limit error-correcting coding and decoding: turbo-codes," Proc. of the IEEE Int. Conf. Communications, 1993, pp. 1064‒1070.

Ungerboeck, G.

G. Ungerboeck, "Channel coding with multilevel/phase signals," IEEE Trans. Inf. Theory 28, 55‒67 (1982).
[CrossRef]

Yang, G. C.

G. C. Yang and W. C. Kwong, Prime Codes With Applications to CDMA Optical and Wireless Networks, Artech House, 2002.

Yim, R. M. H.

R. M. H. Yim, J. Bajcsy, and L. R. Chen, "A new family of 2-D wavelength-time codes for optical CDMA with differential detection," IEEE Photonics Technol. Lett. 15, 165‒167 (2003).
[CrossRef]

A. A. Garba, R. M. H. Yim, J. Bajcsy, and L. R. Chen, "Analysis of optical CDMA signal transmission: capacity limits and simulation results," EURASIP J. Appl. Signal Process. 2005, 1603‒1616.

Electron. Lett. (1)

L. Tančevski and I. Andonovic, "Wavelength hopping/time spreading code division multiple access systems," Electron. Lett. 30, 1388‒1390 (1994).
[CrossRef]

EURASIP J. Appl. Signal Process. (1)

A. A. Garba, R. M. H. Yim, J. Bajcsy, and L. R. Chen, "Analysis of optical CDMA signal transmission: capacity limits and simulation results," EURASIP J. Appl. Signal Process. 2005, 1603‒1616.

IEEE J. Sel. Areas Commun. (1)

K.-I. Kitayama, "Code division multiplexing lightwave networks based upon optical code conversion," IEEE J. Sel. Areas Commun. 16, 1309‒1319 (1998).
[CrossRef]

IEEE Photonics Technol. Lett. (2)

A. A. Garba and J. Bajcsy, "A new approach to achieve high spectral efficiency in wavelength-time OCDMA network transmission," IEEE Photonics Technol. Lett. 19, 131‒133 (2007).
[CrossRef]

R. M. H. Yim, J. Bajcsy, and L. R. Chen, "A new family of 2-D wavelength-time codes for optical CDMA with differential detection," IEEE Photonics Technol. Lett. 15, 165‒167 (2003).
[CrossRef]

IEEE Trans. Commun. (3)

P. Azni, M. Nasiri-Kenari, and J. A. Salehi, "Soft-input decoder for decoding of internally channel coded fiber-optic CDMA communication systems," IEEE Trans. Commun. 50, 1994‒2002 (2002).
[CrossRef]

H. M. H. Shalaby, "Complexities, error probabilities, and capacities of optical OOK-CDMA communication systems," IEEE Trans. Commun. 50, 2009‒2017 (2002).
[CrossRef]

M. R. Dale and R. M. Gagliardi, "Channel coding for asynchronous fiber optic CDMA communications," IEEE Trans. Commun. 43, 2485‒2492 (1995).
[CrossRef]

IEEE Trans. Inf. Theory (2)

G. Ungerboeck, "Channel coding with multilevel/phase signals," IEEE Trans. Inf. Theory 28, 55‒67 (1982).
[CrossRef]

L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, "Optimal decoding of linear codes for minimizing symbol error rate," IEEE Trans. Inf. Theory 20, 284‒287 (1974).
[CrossRef]

J. Lightwave Technol. (4)

Other (5)

C.-S. Bres, I. Glesk, R. J. Runser, T. Banwell, P. R. Prucnal, and W. C. Kwong, "Novel M-ary architecture for optical CDMA using pulse position modulation," Proc. of the 18th Annu. Meeting of the IEEE Lasers and Electro-Optics Society, 2005, pp. 982‒983.

C. Berrou, A. Glavieux, and P. Thitimajshima, "Near Shannon limit error-correcting coding and decoding: turbo-codes," Proc. of the IEEE Int. Conf. Communications, 1993, pp. 1064‒1070.

S. B. Wicker and V. K. Bhargava, ed., Reed-Solomon Codes and Their Applications, IEEE Press, 1994.

P. R. Prucnal, Optical Code Division Multiple Access: Fundamentals and Applications, CRC Press, 2006.

G. C. Yang and W. C. Kwong, Prime Codes With Applications to CDMA Optical and Wireless Networks, Artech House, 2002.

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Figures (19)

Fig. 1
Fig. 1

Schematic block diagram of OCDMA network transmission with SUD at the receiver.

Fig. 2
Fig. 2

(Color online) Theoretically achievable rates for an interference-only OCDMA network transmission with SUD.

Fig. 3
Fig. 3

(Color online) Comparison of the amount of OCDMA MUI in the cases of uniform and non-uniform channel input PMF from Eq. (7). The MUI is shown for different time instants t for K = 50 active OCDMA users and M = 16 modulation levels.

Fig. 4
Fig. 4

(Color online) Histogram of MUI for 16-ary ( M = 16 ) OCDMA transmission with K = 300 users. (a) Non-Gaussian MUI for non-uniform signaling p 0 = 0 . 9967 ; p 1 = = p 15 = 0 . 00022 . (b) Gaussian interference with uniform PMF: p 0 = p 1 = = p 15 = 1 / 16 .

Fig. 5
Fig. 5

(Color online) Achievable throughput as a function of the number of users for selected M-ary OCDMA transmission with SUD at the receiver when the average number of transmitted photons is μ = 20 per M-ary level.

Fig. 6
Fig. 6

(Color online) Achievable throughput as a function of μ, the number of transmitted photons per M-ary level, when the number of users is K = 500 for selected M-ary OCDMA transmission with SUD at the receiver. (The horizontal dashed lines represent the limiting cases without shot noise for the considered M-ary OCDMA systems.)

Fig. 7
Fig. 7

(Color online) Achievable throughput limits on 16 -ary OCDMA transmission with SUD at the receiver for different numbers of users K and numbers of photons per OCDMA level μ.

Fig. 8
Fig. 8

(Color online) Histogram of MUI and shot noise for 16-ary ( M = 16 ) OCDMA transmission with K = 50 users and μ = 1 . (a) Non-Gaussian MUI for near-optimal non-uniform channel input PMF: p 0 = 0 . 98 ; p 1 = = p 15 = 0 . 013 . (b) Gaussian interference with uniform PMF: p 0 = p 1 = = p 15 = 1 / 16 .

Fig. 9
Fig. 9

(Color online) Histogram of MUI and shot noise for 16-ary ( M = 16 ) CDMA transmission with K = 300 users and μ = 20 . (a) Non-Gaussian MUI with non-uniform, near-optimal channel input PMF: p 0 = 0 . 9967 ; p 1 = = p 15 = 0 . 00022 . (b) Gaussian interference with uniform PMF: p 0 = p 1 = = p 15 = 1 / 16 .

Fig. 10
Fig. 10

(Color online) Achievable throughput against number of users for the selected M-ary OCDMA transmission with SUD at the receiver at 20 dB chip SNR.

Fig. 11
Fig. 11

(Color online) Achievable throughput as a function of chip SNR for the selected 50-user M-ary CDMA transmission with SUD at the receiver.

Fig. 12
Fig. 12

(Color online) Schematic block diagram of the proposed OCDMA transmission scheme with SUD at the receiver. (Electro-optic and opto-electric signal conversion is performed by the E/O and O/E modules and data interleaving/de-interleaving is performed by modules denoted by p i i , π i , Π i , π i 1 , Π i 1 , p i i 1 ).

Fig. 13
Fig. 13

Detailed operation of the proposed M-ary wavelength–time OCDMA modulator: twelve symbols that were error-control encoded and 16-ary TCM encoded are rearranged into a time/wavelength array with parameters T = 3 and Λ = 4 , padded with Z = 5 all-zero columns and time-interleaved prior to being sent through the optical channel.

Fig. 14
Fig. 14

(Color online) One stage of the designed trellis diagram for the rate two, 16-ary TCM encoder. (Note: all trellis edges entering a given state have the same input labels.)

Fig. 15
Fig. 15

(Color online) BER performance of the designed 16-ary OCDMA transmission corrupted by Poisson shot noise, when the number of photons per 16-ary level is μ = 20 : (a) after convolutional decoding and (b) after both convolutional and Reed–Solomon decoding.

Fig. 16
Fig. 16

(Color online) BER performance for the designed 16-ary OCDMA transmission scheme with non-uniform signaling in the presence of Poisson shot noise. The number of photons per 16-ary level is μ = 40 and the BER is shown after (a) convolutional decoding and (b) both convolutional and Reed–Solomon decoding.

Fig. 17
Fig. 17

(Color online) BER results of the designed 16-ary OCDMA transmission scheme with non-uniform signaling when shot noise with μ = 80 photons per 16-ary level corrupts the transmission after (a) convolutional decoding and (b) both convolutional and Reed–Solomon decoding.

Fig. 18
Fig. 18

(Color online) BER of the proposed 16-ary OCDMA transmission system with AWGN thermal noise and non-uniform input signaling after (a) iterative decoding and (b) iterative decoding followed by RS decoding.

Fig. 19
Fig. 19

(Color online) The number of actively interfering users K t when the total number of users is K = 300 for the proposed OCDMA scheme, when M = 16 and the channel access probability is P c h = 0 . 0032 : (a) K t is shown at different time instants t; (b) histogram of K t .

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

Y = X 1 + X 2 + + X K ,
P Y | X 1 Y = y | X 1 = x 1 = ( x 2 , x 3 , , x K ) { 0 , 1 , 2 , , M 1 } K 1 s.t. x 2 + x 3 + + x K = y x 1 p x 2 p x 3 p x K ,
P Y | X 1 , , X K Y = y | X 1 = x 1 , , X K = x K = e μ s μ s y y ! ,
P Y | X 1 Y = y | X 1 = x 1 = s = 0 K M 1 e μ s μ s y y ! x 2 , x 3 , , x K 0 , 1 , 2 , , M 1 K 1 s.t. x 2 + x 3 + + x K = s x 1 p x 2 p x 3 p x K ,
Y = X 1 + X 2 + + X K + υ .
f Y | X 1 y | x 1 = 1 2 π σ x 2 = 0 M 1 x 3 = 0 M 1 x K = 0 M 1 e 1 2 σ 2 y i = 1 K x i 2 i = 2 K p x i ,
C = K sup P X 1 I X 1 ; Y .
p 0 = 1 1 K and p 1 = p 2 = = p M 1 = 1 K M 1 ,
Z = T × K M 1 M M .
P c h = M K M 1 .
P X 1 | Y x 1 t , λ = m | y t , λ = α j = 0 K t M 1 e μ j μ j y t , λ y t , λ ! P j | m
P X 1 | Y x 1 t , λ = m | y ( t , λ ) = α j = 0 K t 1 M 1 e 1 2 σ 2 y ( t , λ ) m + j 2 P j | m .
P j | m = x 2 , x 3 , , x K 0 , 1 , 2 , , M 1 K 1 s.t. x 2 + x 3 + + x K = y x 1 p x 2 p x 3 p x K .
η = K Number of users × R c ECC Code rate × R mod Rate of the OCDMA modulator  bits/ OCDMA chip,
η = K × R C C × R T C M × R R S × P c h × 62 64  bits/OCDMA chip .
η = 298 × 1 2 × 2 × 239 255 × 62 64 × P c h = 0 . 86  bits/OCDMA chip .
η = 368 × 1 2 × 2 × 239 255 × 62 64 × P c h = 1 . 07  bits/ OCDMA chip .
η = 418 × 1 2 × 2 × 239 255 × P c h × 62 64 = 1 . 21  bits/OCDMA chip .
η = 462 × 1 2 × 2 × 239 255 × 62 64 × P c h = 1 . 34 bits/OCDMA chip .
P K t = j = K 1 j 1 P c h K 1 j P c h j .
P K t = j 1 j ! M M 1 j e M M 1 ,