Abstract

In this paper, we design a novel Poisson photon-counting based iterative successive interference cancellation (SIC) scheme for transmission over free-space optical (FSO) channels in the presence of both multiple access interference (MAI) as well as Gamma-Gamma atmospheric turbulence fading, shot-noise and background light. Our simulation results demonstrate that the proposed scheme exhibits a strong MAI suppression capability. Importantly, an order of magnitude of BER improvements may be achieved compared to the conventional chip-level optical code-division multiple-access (OCDMA) photon-counting detector.

© 2013 OSA

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References

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  1. V. W. S. Chan, “Free-space optical communications,” J. Lightwave Technol.24(12), 4750–4762 (2006).
    [CrossRef]
  2. 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. IEEE100(special centennial issue), 1853–1888 (2012).
    [CrossRef]
  3. 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. Express20(8), 9284–9295 (2012).
    [CrossRef] [PubMed]
  4. 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. Express18(10), 9879–9890 (2010).
    [CrossRef] [PubMed]
  5. J. Jiang, D. Wu, and P. Fan, “General constructions of optimal variable-weight optical orthogonal codes,” IEEE Trans. Inf. Theory57(7), 4488–4496 (2011).
    [CrossRef]
  6. G. C. Yang, C. H. Chen, and W. C. Kwong, “Accurate analysis of double-weight optical CDMA with power control,” IEEE Trans. Commun.60(2), 322–327 (2012).
    [CrossRef]
  7. Y. Du, S. J. B. Yoo, and Z. Ding, “Nonuniform spectral phase encoding in optical CDMA networks,” IEEE Photon. Technol. Lett.18(23), 2505–2507 (2006).
    [CrossRef]
  8. M. Jazayerifar and J. A. Salehi, “Atmospheric optical CDMA communication systems via optical orthogonal codes,” IEEE Trans. Commun.54(9), 1614–1623 (2006).
    [CrossRef]
  9. C. Goursaud, A. Vergojanne, C. Berthelemot, J. Cances, and J. Dumasl, “DS-OCDMA receiver based on parallel interference cancellation and hard limiter,” IEEE Trans. Commun.54(9), 1663–1671 (2006).
    [CrossRef]
  10. A. O. M’foubat, I. Dayoub, J. M. Rouvaen, W. Hamouda, and A. Mazen, “Approach to interference cancellation in DS-CDMA optical networks,” J. Opt. Commun. Netw.1(3), 204–212 (2009).
    [CrossRef]
  11. H. Mrabet, I. Dayoub, R. Attia, and S. Haxha, “Performance improving of OCDMA system using 2-d optical codes with optical SIC receiver,” J. Lightwave Technol.27(21), 4744–4753 (2009).
    [CrossRef]
  12. X. Yuan, Q. Guo, and L. Ping, “Low-complexity iterative detection in multi-user MIMO ISI channels,” IEEE Signal Processing Lett.15(1), 25–28 (2008).
    [CrossRef]
  13. H. V. Poor, “Iterative multiuser detection,” IEEE Sig. Processing Mag.21(1), 81–88 (2004).
    [CrossRef]
  14. 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 (2012).
    [CrossRef]
  15. X. Wang and H. V. Poor, “Iterative (turbo) soft interference cancellation and decoding for coded CDMA,” IEEE Trans. Commun.467, 1046–1061 (1999).
    [CrossRef]
  16. H. Shalaby, “Complexities, error probabilities, and capacities of optical OOK-CDMA communication systems,” IEEE Trans. Commun.50(12), 2009–2017 (2002).
    [CrossRef]
  17. L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE Press, 2005).
    [CrossRef]
  18. W. Gappmair and S. S. Muhammad, “Error performance of PPM/Poisson channels in turbulent athmosphere with Gamma-Gamma distribution,” Electron. Lett.43(16), 880–882 (2007).
    [CrossRef]
  19. C. Berrou and A. Glavieux, “Near optimum error correcting coding and decoding: turbo-codes,” IEEE Trans. Commun.44(10), 1261–1271 (1996).
    [CrossRef]
  20. A. C. Reid, T. A. Gulliver, and D. P. Taylor, “Convergence and errors in turbo-decoding,” IEEE Trans. Commun.49(12), 2045–2051 (2001).
    [CrossRef]

2012 (4)

G. C. Yang, C. H. Chen, and W. C. Kwong, “Accurate analysis of double-weight optical CDMA with power control,” IEEE Trans. Commun.60(2), 322–327 (2012).
[CrossRef]

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. IEEE100(special centennial issue), 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. Express20(8), 9284–9295 (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 (2012).
[CrossRef]

2011 (1)

J. Jiang, D. Wu, and P. Fan, “General constructions of optimal variable-weight optical orthogonal codes,” IEEE Trans. Inf. Theory57(7), 4488–4496 (2011).
[CrossRef]

2010 (1)

2009 (2)

2008 (1)

X. Yuan, Q. Guo, and L. Ping, “Low-complexity iterative detection in multi-user MIMO ISI channels,” IEEE Signal Processing Lett.15(1), 25–28 (2008).
[CrossRef]

2007 (1)

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

2006 (4)

Y. Du, S. J. B. Yoo, and Z. Ding, “Nonuniform spectral phase encoding in optical CDMA networks,” IEEE Photon. Technol. Lett.18(23), 2505–2507 (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]

C. Goursaud, A. Vergojanne, C. Berthelemot, J. Cances, and J. Dumasl, “DS-OCDMA receiver based on parallel interference cancellation and hard limiter,” IEEE Trans. Commun.54(9), 1663–1671 (2006).
[CrossRef]

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

2004 (1)

H. V. Poor, “Iterative multiuser detection,” IEEE Sig. Processing Mag.21(1), 81–88 (2004).
[CrossRef]

2002 (1)

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

2001 (1)

A. C. Reid, T. A. Gulliver, and D. P. Taylor, “Convergence and errors in turbo-decoding,” IEEE Trans. Commun.49(12), 2045–2051 (2001).
[CrossRef]

1999 (1)

X. Wang and H. V. Poor, “Iterative (turbo) soft interference cancellation and decoding for coded CDMA,” IEEE Trans. Commun.467, 1046–1061 (1999).
[CrossRef]

1996 (1)

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

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE Press, 2005).
[CrossRef]

Attia, R.

Berrou, C.

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

Berthelemot, C.

C. Goursaud, A. Vergojanne, C. Berthelemot, J. Cances, and J. Dumasl, “DS-OCDMA receiver based on parallel interference cancellation and hard limiter,” IEEE Trans. Commun.54(9), 1663–1671 (2006).
[CrossRef]

Cances, J.

C. Goursaud, A. Vergojanne, C. Berthelemot, J. Cances, and J. Dumasl, “DS-OCDMA receiver based on parallel interference cancellation and hard limiter,” IEEE Trans. Commun.54(9), 1663–1671 (2006).
[CrossRef]

Chan, V. W. S.

Chen, C. H.

G. C. Yang, C. H. Chen, and W. C. Kwong, “Accurate analysis of double-weight optical CDMA with power control,” IEEE Trans. Commun.60(2), 322–327 (2012).
[CrossRef]

Dayoub, I.

Ding, Z.

Y. Du, S. J. B. Yoo, and Z. Ding, “Nonuniform spectral phase encoding in optical CDMA networks,” IEEE Photon. Technol. Lett.18(23), 2505–2507 (2006).
[CrossRef]

Du, Y.

Y. Du, S. J. B. Yoo, and Z. Ding, “Nonuniform spectral phase encoding in optical CDMA networks,” IEEE Photon. Technol. Lett.18(23), 2505–2507 (2006).
[CrossRef]

Dumasl, J.

C. Goursaud, A. Vergojanne, C. Berthelemot, J. Cances, and J. Dumasl, “DS-OCDMA receiver based on parallel interference cancellation and hard limiter,” IEEE Trans. Commun.54(9), 1663–1671 (2006).
[CrossRef]

Fan, P.

J. Jiang, D. Wu, and P. Fan, “General constructions of optimal variable-weight optical orthogonal codes,” IEEE Trans. Inf. Theory57(7), 4488–4496 (2011).
[CrossRef]

Fu, S.

Gao, Z.

Gappmair, W.

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

Glavieux, A.

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

Goursaud, C.

C. Goursaud, A. Vergojanne, C. Berthelemot, J. Cances, and J. Dumasl, “DS-OCDMA receiver based on parallel interference cancellation and hard limiter,” IEEE Trans. Commun.54(9), 1663–1671 (2006).
[CrossRef]

Gulliver, T. A.

A. C. Reid, T. A. Gulliver, and D. P. Taylor, “Convergence and errors in turbo-decoding,” IEEE Trans. Commun.49(12), 2045–2051 (2001).
[CrossRef]

Guo, Q.

X. Yuan, Q. Guo, and L. Ping, “Low-complexity iterative detection in multi-user MIMO ISI channels,” IEEE Signal Processing Lett.15(1), 25–28 (2008).
[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. IEEE100(special centennial issue), 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. IEEE100(special centennial issue), 1853–1888 (2012).
[CrossRef]

Hamouda, W.

Hanzo, 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. IEEE100(special centennial issue), 1853–1888 (2012).
[CrossRef]

Haxha, S.

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. IEEE100(special centennial issue), 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]

Jiang, J.

J. Jiang, D. Wu, and P. Fan, “General constructions of optimal variable-weight optical orthogonal codes,” IEEE Trans. Inf. Theory57(7), 4488–4496 (2011).
[CrossRef]

Kataoka, N.

Kwong, W. C.

G. C. Yang, C. H. Chen, and W. C. Kwong, “Accurate analysis of double-weight optical CDMA with power control,” IEEE Trans. Commun.60(2), 322–327 (2012).
[CrossRef]

Liu, J.

M’foubat, A. O.

Mazen, A.

Mrabet, H.

Muhammad, S. S.

W. Gappmair and S. S. Muhammad, “Error performance of PPM/Poisson channels in turbulent athmosphere with Gamma-Gamma distribution,” Electron. Lett.43(16), 880–882 (2007).
[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. IEEE100(special centennial issue), 1853–1888 (2012).
[CrossRef]

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE Press, 2005).
[CrossRef]

Ping, L.

X. Yuan, Q. Guo, and L. Ping, “Low-complexity iterative detection in multi-user MIMO ISI channels,” IEEE Signal Processing Lett.15(1), 25–28 (2008).
[CrossRef]

Poor, H. V.

H. V. Poor, “Iterative multiuser detection,” IEEE Sig. Processing Mag.21(1), 81–88 (2004).
[CrossRef]

X. Wang and H. V. Poor, “Iterative (turbo) soft interference cancellation and decoding for coded CDMA,” IEEE Trans. Commun.467, 1046–1061 (1999).
[CrossRef]

Reid, A. C.

A. C. Reid, T. A. Gulliver, and D. P. Taylor, “Convergence and errors in turbo-decoding,” IEEE Trans. Commun.49(12), 2045–2051 (2001).
[CrossRef]

Rouvaen, J. M.

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. IEEE100(special centennial issue), 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]

Shalaby, H.

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

Shao, Y.

Taylor, D. P.

A. C. Reid, T. A. Gulliver, and D. P. Taylor, “Convergence and errors in turbo-decoding,” IEEE Trans. Commun.49(12), 2045–2051 (2001).
[CrossRef]

Vergojanne, A.

C. Goursaud, A. Vergojanne, C. Berthelemot, J. Cances, and J. Dumasl, “DS-OCDMA receiver based on parallel interference cancellation and hard limiter,” IEEE Trans. Commun.54(9), 1663–1671 (2006).
[CrossRef]

Wada, N.

Wang, X.

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. Express18(10), 9879–9890 (2010).
[CrossRef] [PubMed]

X. Wang and H. V. Poor, “Iterative (turbo) soft interference cancellation and decoding for coded CDMA,” IEEE Trans. Commun.467, 1046–1061 (1999).
[CrossRef]

Wang, Z.

Wu, D.

J. Jiang, D. Wu, and P. Fan, “General constructions of optimal variable-weight optical orthogonal codes,” IEEE Trans. Inf. Theory57(7), 4488–4496 (2011).
[CrossRef]

Yang, G. C.

G. C. Yang, C. H. Chen, and W. C. Kwong, “Accurate analysis of double-weight optical CDMA with power control,” IEEE Trans. Commun.60(2), 322–327 (2012).
[CrossRef]

Yang, Y.

Yoo, S. J. B.

Y. Du, S. J. B. Yoo, and Z. Ding, “Nonuniform spectral phase encoding in optical CDMA networks,” IEEE Photon. Technol. Lett.18(23), 2505–2507 (2006).
[CrossRef]

Yu, C.

Yuan, X.

X. Yuan, Q. Guo, and L. Ping, “Low-complexity iterative detection in multi-user MIMO ISI channels,” IEEE Signal Processing Lett.15(1), 25–28 (2008).
[CrossRef]

Zhong, W.

Zhou, X.

Chin. Opt. Lett. (1)

Electron. Lett. (1)

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

IEEE Photon. Technol. Lett. (1)

Y. Du, S. J. B. Yoo, and Z. Ding, “Nonuniform spectral phase encoding in optical CDMA networks,” IEEE Photon. Technol. Lett.18(23), 2505–2507 (2006).
[CrossRef]

IEEE Sig. Processing Mag. (1)

H. V. Poor, “Iterative multiuser detection,” IEEE Sig. Processing Mag.21(1), 81–88 (2004).
[CrossRef]

IEEE Signal Processing Lett. (1)

X. Yuan, Q. Guo, and L. Ping, “Low-complexity iterative detection in multi-user MIMO ISI channels,” IEEE Signal Processing Lett.15(1), 25–28 (2008).
[CrossRef]

IEEE Trans. Commun. (7)

G. C. Yang, C. H. Chen, and W. C. Kwong, “Accurate analysis of double-weight optical CDMA with power control,” IEEE Trans. Commun.60(2), 322–327 (2012).
[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]

C. Goursaud, A. Vergojanne, C. Berthelemot, J. Cances, and J. Dumasl, “DS-OCDMA receiver based on parallel interference cancellation and hard limiter,” IEEE Trans. Commun.54(9), 1663–1671 (2006).
[CrossRef]

X. Wang and H. V. Poor, “Iterative (turbo) soft interference cancellation and decoding for coded CDMA,” IEEE Trans. Commun.467, 1046–1061 (1999).
[CrossRef]

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

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

A. C. Reid, T. A. Gulliver, and D. P. Taylor, “Convergence and errors in turbo-decoding,” IEEE Trans. Commun.49(12), 2045–2051 (2001).
[CrossRef]

IEEE Trans. Inf. Theory (1)

J. Jiang, D. Wu, and P. Fan, “General constructions of optimal variable-weight optical orthogonal codes,” IEEE Trans. Inf. Theory57(7), 4488–4496 (2011).
[CrossRef]

J. Lightwave Technol. (2)

J. Opt. Commun. Netw. (1)

Opt. Express (2)

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. IEEE100(special centennial issue), 1853–1888 (2012).
[CrossRef]

Other (1)

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE Press, 2005).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

System model. (a) the proposed iterative SIC FSO system link with multiuser interference over atmospheric Poisson channels. (b) photon-counting based iterative SIC detector.

Fig. 2
Fig. 2

Illustration of the LLRs from interfering users at nth iteration.

Fig. 3
Fig. 3

Performance of the proposed Iter-SIC scheme in non-turbulent atmosphere. (a) Comparison between the conventional OCDMA CCR and chip-level receiver. (b) Comparison between the optimum OCDMA receiver. For CCR, chip-level detector of OCDMA, Rc is OOC spreading factor. For Iter-SIC receiver, Rc denotes the FEC coding rate. And, the background radiation photoelectrons per bit interval n b bit = n b / R c.

Fig. 4
Fig. 4

Performance of the proposed Iter-SIC scheme in non-turbulent atmosphere. (a) Comparison of two conventional OCDMA schemes. (b) Performance of the NE block. For single-user performance, Rc = 1/2(Turbo)*3/5(Rep.) for the same effective throughput.

Fig. 5
Fig. 5

(a) BER of the Iter-SIC scheme over Gamma-Gamma fading channels (σR = 0.25). (b) BER of the Iter-SIC scheme against the number of users for the same effective throughput RcK = 3/10.

Fig. 6
Fig. 6

Analysis of convergence properties. (a) non-turbulence with user number K = 9. (b) turbulence-fading with user number K = 9. (c) turbulence-fading with user number K = 12. Repetition coding Rc = 1/30. Gamma-Gamma turbulence fading associated with σR = 0.25. Average transmit photon-counts are 200.

Fig. 7
Fig. 7

Impact of background light noise ( n b bit) on the photon-counting Iter-SIC system. (a) non-turbulent channels. (b) turbulent fading channels.

Fig. 8
Fig. 8

BER performance associated with different number of user numbers (K), and with different repetition coding rates (Rc), namely with K = 3, 6, 9, and with the repetition coding rate of R c = 1 10, 1 20, 1 30, respectively. For K = 1, we use R c = 1 4. For the chip-level OOC, we have K = 9 with R c = 1 120. (a) non-turbulent channels. (b) turbulent fading channels.

Tables (1)

Tables Icon

Algorithm I: Poisson photon-counting Iter-SIC

Equations (12)

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

f I k ( I k ) = 2 ( α β ) ( α + β ) / 2 Γ ( α ) Γ ( β ) I k ( α + β ) / 2 1 K α β ( 2 α β I k ) ,
Pr [ r ( j ) ] = [ n s R ( j ) + n b ] = [ n s R ( j ) + n b ] r ( j ) r ( j ) ! e [ n s R ( j ) + n b ] ,
n s R ( j ) = η k = 1 K I k m k Opt ( j ) ,
L MUD [ x k ( j ) ] = log Pr [ x k ( j ) = 1 | r ( j ) , I ] Pr [ x k ( j ) = 0 | r ( j ) , I ] = log Pr [ r ( j ) | x k ( j ) = 1 , I ] Pr [ r ( j ) | x k ( j ) = 0 , I ] L MUD _ e [ x k ( j ) ] + log Pr [ x k ( j ) = 1 ] Pr [ x k ( j ) = 0 ] L MUD _ a [ x k ( j ) ] ,
L MUD _ e [ x k ( j ) ] = r ( j ) log [ η I k m 1 η k ˜ = 1 , k ˜ k K I k ˜ m k ˜ Opt ( j ) + n b + 1 ] η I k m 1 = r ( j ) log [ η I k m 1 ξ k ( j ) + 1 ] η I k m 1 ,
ξ k Est ( j ) = η k ˜ = 1 , k ˜ k K I k ˜ E [ m k ˜ Opt ( j ) ] + n b ,
E [ m k Opt ( j ) ] = m 1 Pr [ m k Opt ( j ) = m 1 ] + m 0 Pr [ m k Opt ( j ) = m 0 ] = m 1 exp { L MUD _ a [ x k ( j ) ] } 1 + exp { L MUD _ a [ x k ( j ) ] } .
ξ k Est ( j ) = η k ˜ = 1 , k ˜ k K { I k ˜ m 1 exp { L MUD _ a [ x k ˜ ( j ) ] } 1 + exp { L MUD _ a [ x k ˜ ( j ) ] } } + n b ,
L DEC Bit [ d k ( i ) ] = z = 1 N c L DEC _ a [ c k ( ( i 1 ) N c + z ) ] s z ,
{ L DEC LLR [ c k ( j ) ] , j [ ( i 1 ) N c + 1 , , i N c ] } = L DEC Bit [ d k ( i ) ] s .
d ˜ k ( i ) = { 1 L DEC Bit [ d k ( i ) ] 0 , 0 L DEC Bit [ d k ( i ) ] < 0 .
ξ k + 1 Est ( n ) ( j ) = ξ k Est ( n ) ( j ) + η { I k E [ m k Opt ( n ) ( j ) ] I k + 1 E [ m k + 1 Opt ( n 1 ) ( j ) ] } = ξ k Est ( n ) ( j ) + η { I k m 1 exp { L MUD _ a ( n ) [ x k ( j ) ] } 1 + exp { L MUD _ a ( n ) [ x k ( j ) ] } I k + 1 m 1 exp { L MUD _ a ( n 1 ) [ x k + 1 ( j ) ] } 1 + exp { L MUD _ a ( n 1 ) [ x k + 1 ( j ) ] } } .

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