Abstract

In this paper, an iterative parallel interference cancellation (Iter-PIC) technique is developed for optical code-division multiple-access (OCDMA) systems relying on shot-noise limited Poisson photon-counting reception. The novel semi-analytical tool of extrinsic information transfer (EXIT) charts is used for analysing both the bit error rate (BER) performance as well as the channel capacity of these systems and the results are verified by Monte Carlo simulations. The proposed Iter-PIC OCDMA system is capable of achieving two orders of magnitude BER improvements and a 0.1 nats of capacity improvement over the conventional chip-level OCDMA systems at a coding rate of 1/10.

© 2013 OSA

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References

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  1. L. Hanzo, H. Haas, S. Imre, D. O’Brien, M. Rupp, and L. Gyongyosi, “Wireless myths, realities, and futures: from 3G/4G to optical and quantum wireless,” Proc. IEEE 100, 1853–1888 (2012).
    [Crossref]
  2. Z. Wang, W. Zhong, C. Yu, and S. Fu, “Performance improvement of on-off-keying free-space optical transmission systems by a co-propagating reference continuous wave light,” Opt. Express 20(8), 9284–9295 (2012).
    [Crossref] [PubMed]
  3. F. Yang and J. Cheng, “Coherent free-space optical communications in Lognormal-Rician turbulence,” IEEE Commun. Lett. 16(11), 1872–1875 (2012).
    [Crossref]
  4. V. W. S. Chan, “Free-space optical communications,” J. Lightwave Technol. 24(12), 4750–4762 (2006).
    [Crossref]
  5. X. Wang, Z. Gao, N. Kataoka, and N. Wada, “Time domain spectral phase encoding/DPSK data modulation using single phase modulator for OCDMA application,” Opt. Express 18(10), 9879–9890 (2010).
    [Crossref] [PubMed]
  6. H. M. H. Shalaby, “Complexities, error probabilities, and capacities of optical OOK-CDMA communication systems,” IEEE Trans. Commun. 50(12), 2009–2017 (2002).
    [Crossref]
  7. M. Jazayerifar and J. A. Salehi, “Atmospheric optical CDMA communication systems via optical orthogonal codes,” IEEE Trans. Commun. 54(9), 1614–1623 (2006).
    [Crossref]
  8. R. Zhang and L. Hanzo, “Three design aspects of multicarrier interleave division multiple access,” IEEE T. Veh. Technol. 57(6), 3607–3617 (2008).
    [Crossref]
  9. X. Zhou, D. Zhang, R. Zhang, and L. Hanzo, “A photon-counting spatial-diversity-and-multiplexing MIMO scheme for Poisson atmospheric channels relying on Q-ary PPM,” Opt. Express 20(24), 26379–26393 (2012).
    [Crossref] [PubMed]
  10. X. Zhou, Y. Yang, Y. Shao, and J. Liu, “Photon-counting chip-interleaved iterative PIC detector over atmospheric turbulence channels,” Chin. Opt. Lett. 10(11), 110603.1–110603.4 (2012).
  11. Li Ping, L. Liu, and W. K. Leung, “Interleave-division multiple-access,” IEEE Trans. Wireless Commun. 5(4), 938–947 (2006).
    [Crossref]
  12. C. Berrou and A. Glavieux, “Near optimum limit error correcting coding and decoding: Turbo-codes,” IEEE Trans. Commun. 44(10), 1261–1271 (1996).
    [Crossref]
  13. W. Gappmair and S. S. Muhammad, “Error performance of PPM/Poisson channels in turbulent atmosphere with Gamma-Gamma distribution,” Electron. Lett. 43(16), 880–882 (2007).
    [Crossref]
  14. K. Chakraborty, S. Dey, and M. Franceschetti, “Outage capacity of the MIMO Poisson fading channels,” IEEE Trans. Infor. Theory 54(11), 4887–4907 (2008).
    [Crossref]
  15. M. L. B. Riediger, R. Schober, and L. Lampe, “Multiple-symbol detection for photon-counting MIMO free-space optical communications,” IEEE Trans. Wireless Commun. 7(12), 5369–5379 (2008).
    [Crossref]
  16. S. ten Brink, “Convergence behavior of iteratively decoded parallel concatenated codes,” IEEE Trans. Commun. 49(10), 1727–1737 (2001).
    [Crossref]
  17. T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 1991).
    [Crossref]
  18. R. W. Hamming, Coding and Information Theory (Prentice-Hall, 1986).
  19. G. Casella and R. L. Berger, Statistical Inference (Duxbury Press, 2001).
  20. D. Karlis and I. Ntzoufras, “Analysis of sports data by using bivariate Poisson models,” J. R. Statist. Soc. D 52(3), 381–393 (2003).
    [Crossref]

2012 (5)

L. Hanzo, H. Haas, S. Imre, D. O’Brien, M. Rupp, and L. Gyongyosi, “Wireless myths, realities, and futures: from 3G/4G to optical and quantum wireless,” Proc. IEEE 100, 1853–1888 (2012).
[Crossref]

Z. Wang, W. Zhong, C. Yu, and S. Fu, “Performance improvement of on-off-keying free-space optical transmission systems by a co-propagating reference continuous wave light,” Opt. Express 20(8), 9284–9295 (2012).
[Crossref] [PubMed]

F. Yang and J. Cheng, “Coherent free-space optical communications in Lognormal-Rician turbulence,” IEEE Commun. Lett. 16(11), 1872–1875 (2012).
[Crossref]

X. Zhou, D. Zhang, R. Zhang, and L. Hanzo, “A photon-counting spatial-diversity-and-multiplexing MIMO scheme for Poisson atmospheric channels relying on Q-ary PPM,” Opt. Express 20(24), 26379–26393 (2012).
[Crossref] [PubMed]

X. Zhou, Y. Yang, Y. Shao, and J. Liu, “Photon-counting chip-interleaved iterative PIC detector over atmospheric turbulence channels,” Chin. Opt. Lett. 10(11), 110603.1–110603.4 (2012).

2010 (1)

2008 (3)

R. Zhang and L. Hanzo, “Three design aspects of multicarrier interleave division multiple access,” IEEE T. Veh. Technol. 57(6), 3607–3617 (2008).
[Crossref]

K. Chakraborty, S. Dey, and M. Franceschetti, “Outage capacity of the MIMO Poisson fading channels,” IEEE Trans. Infor. Theory 54(11), 4887–4907 (2008).
[Crossref]

M. L. B. Riediger, R. Schober, and L. Lampe, “Multiple-symbol detection for photon-counting MIMO free-space optical communications,” IEEE Trans. Wireless Commun. 7(12), 5369–5379 (2008).
[Crossref]

2007 (1)

W. Gappmair and S. S. Muhammad, “Error performance of PPM/Poisson channels in turbulent atmosphere with Gamma-Gamma distribution,” Electron. Lett. 43(16), 880–882 (2007).
[Crossref]

2006 (3)

Li Ping, L. Liu, and W. K. Leung, “Interleave-division multiple-access,” IEEE Trans. Wireless Commun. 5(4), 938–947 (2006).
[Crossref]

M. Jazayerifar and J. A. Salehi, “Atmospheric optical CDMA communication systems via optical orthogonal codes,” IEEE Trans. Commun. 54(9), 1614–1623 (2006).
[Crossref]

V. W. S. Chan, “Free-space optical communications,” J. Lightwave Technol. 24(12), 4750–4762 (2006).
[Crossref]

2003 (1)

D. Karlis and I. Ntzoufras, “Analysis of sports data by using bivariate Poisson models,” J. R. Statist. Soc. D 52(3), 381–393 (2003).
[Crossref]

2002 (1)

H. M. H. Shalaby, “Complexities, error probabilities, and capacities of optical OOK-CDMA communication systems,” IEEE Trans. Commun. 50(12), 2009–2017 (2002).
[Crossref]

2001 (1)

S. ten Brink, “Convergence behavior of iteratively decoded parallel concatenated codes,” IEEE Trans. Commun. 49(10), 1727–1737 (2001).
[Crossref]

1996 (1)

C. Berrou and A. Glavieux, “Near optimum limit error correcting coding and decoding: Turbo-codes,” IEEE Trans. Commun. 44(10), 1261–1271 (1996).
[Crossref]

Berger, R. L.

G. Casella and R. L. Berger, Statistical Inference (Duxbury Press, 2001).

Berrou, C.

C. Berrou and A. Glavieux, “Near optimum limit error correcting coding and decoding: Turbo-codes,” IEEE Trans. Commun. 44(10), 1261–1271 (1996).
[Crossref]

Casella, G.

G. Casella and R. L. Berger, Statistical Inference (Duxbury Press, 2001).

Chakraborty, K.

K. Chakraborty, S. Dey, and M. Franceschetti, “Outage capacity of the MIMO Poisson fading channels,” IEEE Trans. Infor. Theory 54(11), 4887–4907 (2008).
[Crossref]

Chan, V. W. S.

Cheng, J.

F. Yang and J. Cheng, “Coherent free-space optical communications in Lognormal-Rician turbulence,” IEEE Commun. Lett. 16(11), 1872–1875 (2012).
[Crossref]

Cover, T. M.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 1991).
[Crossref]

Dey, S.

K. Chakraborty, S. Dey, and M. Franceschetti, “Outage capacity of the MIMO Poisson fading channels,” IEEE Trans. Infor. Theory 54(11), 4887–4907 (2008).
[Crossref]

Franceschetti, M.

K. Chakraborty, S. Dey, and M. Franceschetti, “Outage capacity of the MIMO Poisson fading channels,” IEEE Trans. Infor. Theory 54(11), 4887–4907 (2008).
[Crossref]

Fu, S.

Gao, Z.

Gappmair, W.

W. Gappmair and S. S. Muhammad, “Error performance of PPM/Poisson channels in turbulent atmosphere with Gamma-Gamma distribution,” Electron. Lett. 43(16), 880–882 (2007).
[Crossref]

Glavieux, A.

C. Berrou and A. Glavieux, “Near optimum limit error correcting coding and decoding: Turbo-codes,” IEEE Trans. Commun. 44(10), 1261–1271 (1996).
[Crossref]

Gyongyosi, L.

L. Hanzo, H. Haas, S. Imre, D. O’Brien, M. Rupp, and L. Gyongyosi, “Wireless myths, realities, and futures: from 3G/4G to optical and quantum wireless,” Proc. IEEE 100, 1853–1888 (2012).
[Crossref]

Haas, H.

L. Hanzo, H. Haas, S. Imre, D. O’Brien, M. Rupp, and L. Gyongyosi, “Wireless myths, realities, and futures: from 3G/4G to optical and quantum wireless,” Proc. IEEE 100, 1853–1888 (2012).
[Crossref]

Hamming, R. W.

R. W. Hamming, Coding and Information Theory (Prentice-Hall, 1986).

Hanzo, L.

X. Zhou, D. Zhang, R. Zhang, and L. Hanzo, “A photon-counting spatial-diversity-and-multiplexing MIMO scheme for Poisson atmospheric channels relying on Q-ary PPM,” Opt. Express 20(24), 26379–26393 (2012).
[Crossref] [PubMed]

L. Hanzo, H. Haas, S. Imre, D. O’Brien, M. Rupp, and L. Gyongyosi, “Wireless myths, realities, and futures: from 3G/4G to optical and quantum wireless,” Proc. IEEE 100, 1853–1888 (2012).
[Crossref]

R. Zhang and L. Hanzo, “Three design aspects of multicarrier interleave division multiple access,” IEEE T. Veh. Technol. 57(6), 3607–3617 (2008).
[Crossref]

Imre, S.

L. Hanzo, H. Haas, S. Imre, D. O’Brien, M. Rupp, and L. Gyongyosi, “Wireless myths, realities, and futures: from 3G/4G to optical and quantum wireless,” Proc. IEEE 100, 1853–1888 (2012).
[Crossref]

Jazayerifar, M.

M. Jazayerifar and J. A. Salehi, “Atmospheric optical CDMA communication systems via optical orthogonal codes,” IEEE Trans. Commun. 54(9), 1614–1623 (2006).
[Crossref]

Karlis, D.

D. Karlis and I. Ntzoufras, “Analysis of sports data by using bivariate Poisson models,” J. R. Statist. Soc. D 52(3), 381–393 (2003).
[Crossref]

Kataoka, N.

Lampe, L.

M. L. B. Riediger, R. Schober, and L. Lampe, “Multiple-symbol detection for photon-counting MIMO free-space optical communications,” IEEE Trans. Wireless Commun. 7(12), 5369–5379 (2008).
[Crossref]

Leung, W. K.

Li Ping, L. Liu, and W. K. Leung, “Interleave-division multiple-access,” IEEE Trans. Wireless Commun. 5(4), 938–947 (2006).
[Crossref]

Liu, J.

X. Zhou, Y. Yang, Y. Shao, and J. Liu, “Photon-counting chip-interleaved iterative PIC detector over atmospheric turbulence channels,” Chin. Opt. Lett. 10(11), 110603.1–110603.4 (2012).

Liu, L.

Li Ping, L. Liu, and W. K. Leung, “Interleave-division multiple-access,” IEEE Trans. Wireless Commun. 5(4), 938–947 (2006).
[Crossref]

Muhammad, S. S.

W. Gappmair and S. S. Muhammad, “Error performance of PPM/Poisson channels in turbulent atmosphere with Gamma-Gamma distribution,” Electron. Lett. 43(16), 880–882 (2007).
[Crossref]

Ntzoufras, I.

D. Karlis and I. Ntzoufras, “Analysis of sports data by using bivariate Poisson models,” J. R. Statist. Soc. D 52(3), 381–393 (2003).
[Crossref]

O’Brien, D.

L. Hanzo, H. Haas, S. Imre, D. O’Brien, M. Rupp, and L. Gyongyosi, “Wireless myths, realities, and futures: from 3G/4G to optical and quantum wireless,” Proc. IEEE 100, 1853–1888 (2012).
[Crossref]

Ping, Li

Li Ping, L. Liu, and W. K. Leung, “Interleave-division multiple-access,” IEEE Trans. Wireless Commun. 5(4), 938–947 (2006).
[Crossref]

Riediger, M. L. B.

M. L. B. Riediger, R. Schober, and L. Lampe, “Multiple-symbol detection for photon-counting MIMO free-space optical communications,” IEEE Trans. Wireless Commun. 7(12), 5369–5379 (2008).
[Crossref]

Rupp, M.

L. Hanzo, H. Haas, S. Imre, D. O’Brien, M. Rupp, and L. Gyongyosi, “Wireless myths, realities, and futures: from 3G/4G to optical and quantum wireless,” Proc. IEEE 100, 1853–1888 (2012).
[Crossref]

Salehi, J. A.

M. Jazayerifar and J. A. Salehi, “Atmospheric optical CDMA communication systems via optical orthogonal codes,” IEEE Trans. Commun. 54(9), 1614–1623 (2006).
[Crossref]

Schober, R.

M. L. B. Riediger, R. Schober, and L. Lampe, “Multiple-symbol detection for photon-counting MIMO free-space optical communications,” IEEE Trans. Wireless Commun. 7(12), 5369–5379 (2008).
[Crossref]

Shalaby, H. M. H.

H. M. H. Shalaby, “Complexities, error probabilities, and capacities of optical OOK-CDMA communication systems,” IEEE Trans. Commun. 50(12), 2009–2017 (2002).
[Crossref]

Shao, Y.

X. Zhou, Y. Yang, Y. Shao, and J. Liu, “Photon-counting chip-interleaved iterative PIC detector over atmospheric turbulence channels,” Chin. Opt. Lett. 10(11), 110603.1–110603.4 (2012).

ten Brink, S.

S. ten Brink, “Convergence behavior of iteratively decoded parallel concatenated codes,” IEEE Trans. Commun. 49(10), 1727–1737 (2001).
[Crossref]

Thomas, J. A.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 1991).
[Crossref]

Wada, N.

Wang, X.

Wang, Z.

Yang, F.

F. Yang and J. Cheng, “Coherent free-space optical communications in Lognormal-Rician turbulence,” IEEE Commun. Lett. 16(11), 1872–1875 (2012).
[Crossref]

Yang, Y.

X. Zhou, Y. Yang, Y. Shao, and J. Liu, “Photon-counting chip-interleaved iterative PIC detector over atmospheric turbulence channels,” Chin. Opt. Lett. 10(11), 110603.1–110603.4 (2012).

Yu, C.

Zhang, D.

Zhang, R.

Zhong, W.

Zhou, X.

X. Zhou, D. Zhang, R. Zhang, and L. Hanzo, “A photon-counting spatial-diversity-and-multiplexing MIMO scheme for Poisson atmospheric channels relying on Q-ary PPM,” Opt. Express 20(24), 26379–26393 (2012).
[Crossref] [PubMed]

X. Zhou, Y. Yang, Y. Shao, and J. Liu, “Photon-counting chip-interleaved iterative PIC detector over atmospheric turbulence channels,” Chin. Opt. Lett. 10(11), 110603.1–110603.4 (2012).

Chin. Opt. Lett. (1)

X. Zhou, Y. Yang, Y. Shao, and J. Liu, “Photon-counting chip-interleaved iterative PIC detector over atmospheric turbulence channels,” Chin. Opt. Lett. 10(11), 110603.1–110603.4 (2012).

Electron. Lett. (1)

W. Gappmair and S. S. Muhammad, “Error performance of PPM/Poisson channels in turbulent atmosphere with Gamma-Gamma distribution,” Electron. Lett. 43(16), 880–882 (2007).
[Crossref]

IEEE Commun. Lett. (1)

F. Yang and J. Cheng, “Coherent free-space optical communications in Lognormal-Rician turbulence,” IEEE Commun. Lett. 16(11), 1872–1875 (2012).
[Crossref]

IEEE T. Veh. Technol. (1)

R. Zhang and L. Hanzo, “Three design aspects of multicarrier interleave division multiple access,” IEEE T. Veh. Technol. 57(6), 3607–3617 (2008).
[Crossref]

IEEE Trans. Commun. (4)

H. M. H. Shalaby, “Complexities, error probabilities, and capacities of optical OOK-CDMA communication systems,” IEEE Trans. Commun. 50(12), 2009–2017 (2002).
[Crossref]

M. Jazayerifar and J. A. Salehi, “Atmospheric optical CDMA communication systems via optical orthogonal codes,” IEEE Trans. Commun. 54(9), 1614–1623 (2006).
[Crossref]

S. ten Brink, “Convergence behavior of iteratively decoded parallel concatenated codes,” IEEE Trans. Commun. 49(10), 1727–1737 (2001).
[Crossref]

C. Berrou and A. Glavieux, “Near optimum limit error correcting coding and decoding: Turbo-codes,” IEEE Trans. Commun. 44(10), 1261–1271 (1996).
[Crossref]

IEEE Trans. Infor. Theory (1)

K. Chakraborty, S. Dey, and M. Franceschetti, “Outage capacity of the MIMO Poisson fading channels,” IEEE Trans. Infor. Theory 54(11), 4887–4907 (2008).
[Crossref]

IEEE Trans. Wireless Commun. (2)

M. L. B. Riediger, R. Schober, and L. Lampe, “Multiple-symbol detection for photon-counting MIMO free-space optical communications,” IEEE Trans. Wireless Commun. 7(12), 5369–5379 (2008).
[Crossref]

Li Ping, L. Liu, and W. K. Leung, “Interleave-division multiple-access,” IEEE Trans. Wireless Commun. 5(4), 938–947 (2006).
[Crossref]

J. Lightwave Technol. (1)

J. R. Statist. Soc. D (1)

D. Karlis and I. Ntzoufras, “Analysis of sports data by using bivariate Poisson models,” J. R. Statist. Soc. D 52(3), 381–393 (2003).
[Crossref]

Opt. Express (3)

Proc. IEEE (1)

L. Hanzo, H. Haas, S. Imre, D. O’Brien, M. Rupp, and L. Gyongyosi, “Wireless myths, realities, and futures: from 3G/4G to optical and quantum wireless,” Proc. IEEE 100, 1853–1888 (2012).
[Crossref]

Other (3)

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 1991).
[Crossref]

R. W. Hamming, Coding and Information Theory (Prentice-Hall, 1986).

G. Casella and R. L. Berger, Statistical Inference (Duxbury Press, 2001).

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Figures (8)

Fig. 1
Fig. 1

Model of the Iter-PIC OCDMA system based on Poisson photon-counting reception and iterative detection.

Fig. 2
Fig. 2

Extrinsic information transfer trajectories of soft in/soft out decoder for repetition codes, K = 4 users, ns = 60, nb = 39, coding rates of Rc = 1/4, 1/8, 1/16.

Fig. 3
Fig. 3

The relation between the output mutual information of ESE and the average signal-photon count in a single-user system without iterations, nb = 39.

Fig. 4
Fig. 4

BER performance for the single-user, non-iterative system, with nb = 39.

Fig. 5
Fig. 5

BER performance obtained by simulation (color markers) and EXIT charts analysis (black solid lines) with Ninfo = 2048, Rc = 1/8 and nb = 39, different number of iterations and users. The single-user performance (black dashed lines) is also plotted for reference.

Fig. 6
Fig. 6

Capacity in nats per signal photon versus the average number of photons per bit for Poisson based Iter-PIC OCDMA systems, Rc = 1/10 and nb = 39, K = 2, 4, 6.

Fig. 7
Fig. 7

BER performance of Iter-PIC OCDMA systems with It = 50 iterations and conventional chip-level OCDMA systems, for K = 4 users, nb = 39.

Fig. 8
Fig. 8

Capacities of the Iter-PIC OCDMA systems associated with It = 50 iterations and of conventional chip-level OCDMA systems, for K = 4 users, nb = 39.

Tables (1)

Tables Icon

Table 1 OOK based Parallel Iterative Detection Algorithm

Equations (36)

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P r [ r ( l ) ] = { k = 1 K [ h k × x k ( l ) × ( 2 n s R c ) ] + ( n b R c ) } r ( l ) r ( l ) ! e { k = 1 K [ h k × x k ( l ) × ( 2 n s R c ) ] + ( n b R c ) } ,
λ ese , out [ x k ( l ) ] = r ( l ) ln [ 1 + 2 n s n i , k est ( l ) + n b ] 2 n s R c ,
n i , k est ( l ) = 2 n s k k K e λ ese , in [ x k ( l ) ] 1 + e λ ese , in [ x k ( l ) ] ,
d k ^ ( n d ) = sgn { i = 1 N c ( 1 ) i 1 λ dec , in [ c k ( n d N c N c + i ) ] } + 1 2 ,
λ dec , out [ c k ( n d N c N c + i ) ] = { j = 1 N c ( 1 ) j 1 λ dec , in [ c k ( n d N c N c + j ) ] } ( 1 ) i 1 λ dec , in [ c k ( n d N c N c + i ) ] .
I in = 1 2 x = 0 , 1 P r [ λ in | X = x ] log 2 [ 2 P r [ λ in | X = x ] P r [ λ in | X = 0 ] + P r [ λ in | X = 1 ] ] d λ in , 0 I in 1 ,
I out = 1 2 x = 0 , 1 P r [ λ out | X = x ] log 2 [ 2 P r [ λ out | X = x ] P r [ λ out | X = 0 ] + P r [ λ out | X = 1 ] ] d λ out , 0 I out 1.
{ I dec , in i = I ese , out i I ese , in i = I dec , out i 1 .
I ese , out = T ese ( I ese , in , n s , n b ) ,
I dec , out = T dec ( I dec , in ) .
I ese , out new = T ese [ T dec ( I ese , out old ) , n s , n b ] .
I ese , out = T ( n s , n b ) ,
I ese , out = T ( n s ) .
I ese , out = f ( n ˜ s ) ,
P E = g ( n ˜ s ) .
n ˜ s = f 1 ( I ese , out ) .
P E = g [ f 1 ( I ese , out ) ] .
λ ese , out [ x k ( l ) ] = r ( l ) ln [ 1 + 2 n ˜ s n b ] 2 n ˜ s R c .
P r [ λ ese , out [ x k ( l ) ] | x k ( l ) ] = P r [ r ( l ) | x k ( l ) ] .
I ese , out = 1 2 x = 0 , 1 r = 0 P r [ r | X = x ] × log 2 [ 2 × P r [ r | X = x ] P r [ r | X = 0 ] + P r [ r | X = 1 ] ] .
I ese , out = 1 2 r = 0 + ( 2 n ˜ s R c + n b R c ) r r ! e ( 2 n ˜ s R c + n b R c ) log 2 [ 2 1 + ( n b 2 n ˜ s + n b ) r e 2 n ˜ s R c ] + 1 2 r = 0 + ( n b R c ) r r ! e ( n b R c ) log 2 [ 2 1 + ( 2 n ˜ s + n b n b ) r e 2 n ˜ s R c ] ,
i = 1 N c ( 1 ) i 1 λ dec , in [ c k ( n d N c N c + i ) ] = i = 1 N c [ r ( n d N c N c + i ) ln ( 1 + 2 n ˜ s n b ) 2 n ˜ s R c ] ( 1 ) i 1 = ln ( 1 + 2 n ˜ s n b ) [ m = 1 , 3 , 5 , N c 1 r ( n d N c N c + m ) n = 2 , 4 , 6 , N c r ( n d N c N c + n ) ] .
M X ( t ) = e μ X ( e t 1 ) ,
M S n ( t ) = i = 1 n M X i ( t ) .
M r ( n d N c N c + m ) ( t ) = e ( 2 n ˜ s R c + n b R c ) ( e t 1 ) , m = 1 , 3 , , N c 1 ,
M r ( n d N c N c + n ) ( t ) = e ( n b R c ) ( e t 1 ) , n = 2 , 4 , , N c .
M R 1 ( n d ) ( t ) = m = 1 , 3 , 5 , N c 1 M r ( n d N c N c + m ) ( t ) = m = 1 , 3 , 5 , N c 1 e ( 2 n ˜ s R c + n b R c ) ( e t 1 ) = e N c 2 ( 2 n ˜ s R c + n b R c ) ( e t 1 ) = e ( n ˜ s + n b 2 ) ( e t 1 ) ,
M R 2 ( n d ) ( t ) = n = 2 , 4 , 6 , N c M r ( n d N c N c + n ) ( t ) = n = 2 , 4 , 6 , N c e ( n b R c ) ( e t 1 ) = e N c 2 ( n b R c ) ( e t 1 ) = e ( n b 2 ) ( e t 1 ) .
P r 1 [ Δ R ( n d ) ] = e ( n ˜ s + n b 2 + n b 2 ) ( n ˜ s + n b 2 n b 2 ) Δ R ( n d ) / 2 I | Δ R ( n d ) | [ 2 ( n ˜ s + n b 2 ) ( n b 2 ) ] ,
P E 1 = Δ R ( n d ) = 1 P r 1 [ Δ R ( n d ) ] .
P r 0 [ Δ R ( n d ) ] = e ( n b 2 + n ˜ s + n b 2 ) ( n ˜ s + n b 2 n b 2 ) Δ R ( n d ) / 2 I | Δ R ( n d ) | [ 2 ( n b 2 ) ( n ˜ s + n b 2 ) ] .
P E 0 = Δ R ( n d ) = 1 P r 0 [ Δ R ( n d ) ] .
P E = 1 2 ( P E 1 + P E 0 ) = P E 1 = P E 0 .
C = ln 2 H ( P E ) ,
H ( p ) = p ln p ( 1 p ) ln ( 1 p ) .
C ph = C n s .

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