A photon-counting spatial-diversity-and-multiplexing MIMO scheme for Poisson atmospheric channels relying on Q-ary PPM

Xiaolin Zhou; Dingchen Zhang; Rong Zhang; Lajos Hanzo

doi:10.1364/OE.20.026379

A photon-counting spatial-diversity-and-multiplexing MIMO scheme for Poisson atmospheric channels relying on Q-ary PPM

Xiaolin Zhou, Dingchen Zhang, Rong Zhang, and Lajos Hanzo

Optics Express
Vol. 20,
Issue 24,
pp. 26379-26393
(2012)
•https://doi.org/10.1364/OE.20.026379

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Xiaolin Zhou, Dingchen Zhang, Rong Zhang, and Lajos Hanzo, "A photon-counting spatial-diversity-and-multiplexing MIMO scheme for Poisson atmospheric channels relying on Q-ary PPM," Opt. Express 20, 26379-26393 (2012)

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Related Topics
Table of Contents Category
- Atmospheric and Oceanic Optics
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Atmospheric turbulence (010.1330)
- Optical communications (060.4510)
- Free-space optical communication (060.2605)

About this Article
History
- Original Manuscript: September 17, 2012
- Revised Manuscript: October 26, 2012
- Manuscript Accepted: October 28, 2012
- Published: November 8, 2012

Accessible

Open Access

Abstract

A novel Photon-Counting Spatial-Diversity-and-Multiplexing (PC-SDM) scheme is proposed for high-speed Free-Space Optical (FSO) transmission over shot-noise limited Poisson channels experiencing turbulence-induced fading. In particular, Iterative Parallel Interference Cancellation (Iter-PIC) aided Q-ary Pulse Position Modulation (Q-PPM) is employed. Simulation results demonstrate that our proposed scheme exhibits a high integrity and a high throughput, while mitigating the effects of multi-stream interference and background radiation noise.

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References

View by:
Article Order
|
Year
|
Author
|
Publication

V. W. S. Chan, “Free-space optical communications,” J. Lightwave Technol. 24(12), 4750–4762 (2006).
[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. IEEE 100(13), 1853–1888, (2012).
[Crossref]
M. Niu, J. Cheng, and J. F. Holzman, “Exact error rate analysis of equal gain and selection diversity for coherent free-space optical systems on strong turbulence channels,” Opt. Express 18(13), 13915–13926 (2010).
[Crossref] [PubMed]
A. A. Farid and S. Hranilovic, “Diversity gain and outage probability for MIMO free-space optical links with misalignment,” IEEE Trans. Commun. 60(2), 479–487 (2012).
[Crossref]
S. G. Wilson, M. Brandt-Pearce, Q. Cao, and J. H. Leveque, “Free space optical MIMO transmission with Q-ary PPM,” IEEE Trans. Commun. 53(8), 1402–1412 (2005).
[Crossref]
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]
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]
A. Paraskevopoulos, J. Vucic, S. H. Voss, R. Swoboda, and K. D. Langer, “Optical wireless communication systems in the Mb/s to Gb/s range, suitable for industrial applications,” IEEE/ASME Trans. Mech. 15(4), 541–547 (2010).
[Crossref]
J. Wells, “Faster than fiber: the future of multi-Gb/s wireless,” IEEE Microw. Mag. 10(3), 104–112 (2009).
[Crossref]
S. M. Aghajanzadeh and M. Uysal, “Diversity-multiplexing trade-off in coherent free-space optical systems with multiple receivers,” J. Opt. Commun. Netw. 2(12), 1087–1094 (2010).
[Crossref]
E. Bayaki, R. Schober, and R. Mallik, “Performance analysis of MIMO free-space optical systems in Gamma-Gamma fading,” IEEE Trans. Commun. 57(11), 3415–3424 (2009).
[Crossref]
I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “Multilevel coding in free-space optical MIMO transmission with Q-ary PPM over the atmospheric turbulence channel,” IEEE Photon. Technol. Lett. 18(14), 1491–1493 (2006).
[Crossref]
S. G. Wilson, M. Brandt-Pearce, Q. Cao, and M. Baedke, “Optical repetition MIMO transmission with multipulse PPM,” IEEE J. Select. Areas Commun. 23(9), 1901–1910 (2005).
[Crossref]
L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE Press, 2005).
[Crossref]
M. Uysal, J. Li, and M. Yu, “Error rate performance analysis of coded free-space optical links over Gamma-Gamma atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 5(6), 1229–1233 (2006).
[Crossref]
C. Berrou and A. Glavieux, “Near optimum error correcting coding and decoding: turbo-codes,” IEEE Trans. Commun. 44(10), 1261–1271 (1996).
[Crossref]
C. Georghiades, “Modulation and coding for throughput-efficient optical systems,” IEEE Trans. Inform. Theory 40(5), 1313–1326 (1994).
[Crossref]
R. Zhang and L. Hanzo, “Space-time coding for high-throughput interleave division multiplexing aided multi-source cooperation,” Electron. Lett. 44(5), 367–368 (2008).
[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]

2012 (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. IEEE 100(13), 1853–1888, (2012).
[Crossref]

A. A. Farid and S. Hranilovic, “Diversity gain and outage probability for MIMO free-space optical links with misalignment,” IEEE Trans. Commun. 60(2), 479–487 (2012).
[Crossref]

2010 (3)

M. Niu, J. Cheng, and J. F. Holzman, “Exact error rate analysis of equal gain and selection diversity for coherent free-space optical systems on strong turbulence channels,” Opt. Express 18(13), 13915–13926 (2010).
[Crossref] [PubMed]

S. M. Aghajanzadeh and M. Uysal, “Diversity-multiplexing trade-off in coherent free-space optical systems with multiple receivers,” J. Opt. Commun. Netw. 2(12), 1087–1094 (2010).
[Crossref]

A. Paraskevopoulos, J. Vucic, S. H. Voss, R. Swoboda, and K. D. Langer, “Optical wireless communication systems in the Mb/s to Gb/s range, suitable for industrial applications,” IEEE/ASME Trans. Mech. 15(4), 541–547 (2010).
[Crossref]

2009 (2)

J. Wells, “Faster than fiber: the future of multi-Gb/s wireless,” IEEE Microw. Mag. 10(3), 104–112 (2009).
[Crossref]

E. Bayaki, R. Schober, and R. Mallik, “Performance analysis of MIMO free-space optical systems in Gamma-Gamma fading,” IEEE Trans. Commun. 57(11), 3415–3424 (2009).
[Crossref]

2008 (3)

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]

R. Zhang and L. Hanzo, “Space-time coding for high-throughput interleave division multiplexing aided multi-source cooperation,” Electron. Lett. 44(5), 367–368 (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)

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

I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “Multilevel coding in free-space optical MIMO transmission with Q-ary PPM over the atmospheric turbulence channel,” IEEE Photon. Technol. Lett. 18(14), 1491–1493 (2006).
[Crossref]

M. Uysal, J. Li, and M. Yu, “Error rate performance analysis of coded free-space optical links over Gamma-Gamma atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 5(6), 1229–1233 (2006).
[Crossref]

2005 (2)

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and M. Baedke, “Optical repetition MIMO transmission with multipulse PPM,” IEEE J. Select. Areas Commun. 23(9), 1901–1910 (2005).
[Crossref]

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and J. H. Leveque, “Free space optical MIMO transmission with Q-ary PPM,” IEEE Trans. Commun. 53(8), 1402–1412 (2005).
[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]

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]

1994 (1)

C. Georghiades, “Modulation and coding for throughput-efficient optical systems,” IEEE Trans. Inform. Theory 40(5), 1313–1326 (1994).
[Crossref]

Aghajanzadeh, S. M.

S. M. Aghajanzadeh and M. Uysal, “Diversity-multiplexing trade-off in coherent free-space optical systems with multiple receivers,” J. Opt. Commun. Netw. 2(12), 1087–1094 (2010).
[Crossref]

Andrews, L. C.

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

Baedke, M.

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and M. Baedke, “Optical repetition MIMO transmission with multipulse PPM,” IEEE J. Select. Areas Commun. 23(9), 1901–1910 (2005).
[Crossref]

Bayaki, E.

E. Bayaki, R. Schober, and R. Mallik, “Performance analysis of MIMO free-space optical systems in Gamma-Gamma fading,” IEEE Trans. Commun. 57(11), 3415–3424 (2009).
[Crossref]

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]

Brandt-Pearce, M.

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and M. Baedke, “Optical repetition MIMO transmission with multipulse PPM,” IEEE J. Select. Areas Commun. 23(9), 1901–1910 (2005).
[Crossref]

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and J. H. Leveque, “Free space optical MIMO transmission with Q-ary PPM,” IEEE Trans. Commun. 53(8), 1402–1412 (2005).
[Crossref]

Cao, Q.

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and J. H. Leveque, “Free space optical MIMO transmission with Q-ary PPM,” IEEE Trans. Commun. 53(8), 1402–1412 (2005).
[Crossref]

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and M. Baedke, “Optical repetition MIMO transmission with multipulse PPM,” IEEE J. Select. Areas Commun. 23(9), 1901–1910 (2005).
[Crossref]

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.

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

Cheng, J.

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]

Djordjevic, I. B.

Farid, A. A.

A. A. Farid and S. Hranilovic, “Diversity gain and outage probability for MIMO free-space optical links with misalignment,” IEEE Trans. Commun. 60(2), 479–487 (2012).
[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]

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]

Georghiades, C.

C. Georghiades, “Modulation and coding for throughput-efficient optical systems,” IEEE Trans. Inform. Theory 40(5), 1313–1326 (1994).
[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]

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]

Gyongyosi, L.

Haas, H.

Hanzo, L.

R. Zhang and L. Hanzo, “Space-time coding for high-throughput interleave division multiplexing aided multi-source cooperation,” Electron. Lett. 44(5), 367–368 (2008).
[Crossref]

Holzman, J. F.

Hranilovic, S.

A. A. Farid and S. Hranilovic, “Diversity gain and outage probability for MIMO free-space optical links with misalignment,” IEEE Trans. Commun. 60(2), 479–487 (2012).
[Crossref]

Imre, S.

Lampe, L.

Langer, K. D.

Leveque, J. H.

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and J. H. Leveque, “Free space optical MIMO transmission with Q-ary PPM,” IEEE Trans. Commun. 53(8), 1402–1412 (2005).
[Crossref]

Li, J.

Mallik, R.

E. Bayaki, R. Schober, and R. Mallik, “Performance analysis of MIMO free-space optical systems in Gamma-Gamma fading,” IEEE Trans. Commun. 57(11), 3415–3424 (2009).
[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]

Neifeld, M. A.

Niu, M.

O’Brien, D.

Paraskevopoulos, A.

Phillips, R. L.

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

Riediger, M. L. B.

Rupp, M.

Schober, R.

E. Bayaki, R. Schober, and R. Mallik, “Performance analysis of MIMO free-space optical systems in Gamma-Gamma fading,” IEEE Trans. Commun. 57(11), 3415–3424 (2009).
[Crossref]

Swoboda, R.

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]

Uysal, M.

S. M. Aghajanzadeh and M. Uysal, “Diversity-multiplexing trade-off in coherent free-space optical systems with multiple receivers,” J. Opt. Commun. Netw. 2(12), 1087–1094 (2010).
[Crossref]

Vasic, B.

Voss, S. H.

Vucic, J.

Wells, J.

J. Wells, “Faster than fiber: the future of multi-Gb/s wireless,” IEEE Microw. Mag. 10(3), 104–112 (2009).
[Crossref]

Wilson, S. G.

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and J. H. Leveque, “Free space optical MIMO transmission with Q-ary PPM,” IEEE Trans. Commun. 53(8), 1402–1412 (2005).
[Crossref]

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and M. Baedke, “Optical repetition MIMO transmission with multipulse PPM,” IEEE J. Select. Areas Commun. 23(9), 1901–1910 (2005).
[Crossref]

Yu, M.

Zhang, R.

R. Zhang and L. Hanzo, “Space-time coding for high-throughput interleave division multiplexing aided multi-source cooperation,” Electron. Lett. 44(5), 367–368 (2008).
[Crossref]

Electron. Lett. (2)

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]

R. Zhang and L. Hanzo, “Space-time coding for high-throughput interleave division multiplexing aided multi-source cooperation,” Electron. Lett. 44(5), 367–368 (2008).
[Crossref]

IEEE J. Select. Areas Commun. (1)

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and M. Baedke, “Optical repetition MIMO transmission with multipulse PPM,” IEEE J. Select. Areas Commun. 23(9), 1901–1910 (2005).
[Crossref]

IEEE Microw. Mag. (1)

J. Wells, “Faster than fiber: the future of multi-Gb/s wireless,” IEEE Microw. Mag. 10(3), 104–112 (2009).
[Crossref]

IEEE Photon. Technol. Lett. (1)

IEEE Trans. Commun. (5)

E. Bayaki, R. Schober, and R. Mallik, “Performance analysis of MIMO free-space optical systems in Gamma-Gamma fading,” IEEE Trans. Commun. 57(11), 3415–3424 (2009).
[Crossref]

A. A. Farid and S. Hranilovic, “Diversity gain and outage probability for MIMO free-space optical links with misalignment,” IEEE Trans. Commun. 60(2), 479–487 (2012).
[Crossref]

S. G. Wilson, M. Brandt-Pearce, Q. Cao, and J. H. Leveque, “Free space optical MIMO transmission with Q-ary PPM,” IEEE Trans. Commun. 53(8), 1402–1412 (2005).
[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]

C. Berrou and A. Glavieux, “Near optimum 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. Inform. Theory (1)

C. Georghiades, “Modulation and coding for throughput-efficient optical systems,” IEEE Trans. Inform. Theory 40(5), 1313–1326 (1994).
[Crossref]

IEEE Trans. Wireless Commun. (2)

IEEE/ASME Trans. Mech. (1)

J. Lightwave Technol. (1)

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

J. Opt. Commun. Netw. (1)

S. M. Aghajanzadeh and M. Uysal, “Diversity-multiplexing trade-off in coherent free-space optical systems with multiple receivers,” J. Opt. Commun. Netw. 2(12), 1087–1094 (2010).
[Crossref]

Opt. Express (1)

Proc. IEEE (1)

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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Fig. 1

Model of the Q-PPM based PC-SDM FSO system over Poisson atmospheric channels.

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Fig. 2

Mapping example of the 2-PPM symbol.

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Fig. 3

BER for the Gamma-Gamma fading link (α = 4, β = 4) relying on 2-PPM and R_c = 1/8 upon varying N and M.

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Fig. 4

BER for different scintillation indeces, 2-PPM and R_c = 1/8.

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Fig. 5

BER of multi-stream transmissions for M = 4, 5, 6 over the Gamma-Gamma fading links (α = 4, β = 4) for 2-PPM and R_c = 1/8.

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Fig. 6

Comparison of various Q-PPM schemes for Gamma-Gamma fading links (α = 4, β = 4) with R_c = 1/8.

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Fig. 7

Performance of the NE block for the Gamma-Gamma fading link (α = 4, β = 1) with R_c = 1/8 and E_b = −170dBJ.

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Tables (2)

Table 1 2-PPM based Iter-PIC Algorithm

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Table 2 Natural Mapping for Q-PPM Symbols

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Equations (19)

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Pr (I_{m, n}) = \frac{2 {(α β)}^{(α + β) / 2}}{Γ (α) Γ (β)} I_{m, n}^{(α + β) / 2 - 1} K_{α - β} (2 \sqrt{α β I_{m, n}}), I_{m, n} > 0 .

α = {[exp (\frac{0.49 χ^{2}}{{(1 + 0.18 d^{2} + 0.56 χ^{12 / 5})}^{7 / 6}}) - 1]}^{- 1},

β = {[exp (\frac{0.51 χ^{2} {(1 + 0.69 χ^{12 / 5})}^{- 5 / 6}}{{(1 + 0.9 d^{2} + 0.62 d^{2} χ^{12 / 5})}^{5 / 6}}) - 1]}^{- 1},

Pr [r_{n}^{j}] = \frac{{[n_{s} \sum_{m = 1}^{M} I_{m, n} s_{m, n}^{j} + n_{b}]}^{r_{n}^{j}}}{r_{n}^{j}!} exp [- (n_{s} \sum_{m = 1}^{M} I_{m, n} s_{m, n}^{j} + n_{b})],

\begin{array}{l} L_{MIC}^{apost} (x_{m, n}^{1}) & = log \frac{Pr [x_{m, n}^{1} = 1 | (r_{m, n}^{2}, r_{m, n}^{1})]}{Pr [x_{m, n}^{1} = 0 | (r_{m, n}^{2}, r_{m, n}^{1})]} \\ = log \frac{Pr [(r_{m, n}^{2}, r_{m, n}^{1}) | x_{m, n}^{1} = 1] Pr [x_{m, n}^{1} = 1]}{Pr [(r_{m, n}^{2}, r_{m, n}^{1}) | x_{m, n}^{1} = 0] Pr [x_{m, n}^{1} = 0]} \\ = \underset{L_{MIC}^{e} (x_{m, n}^{1})}{\underset{︸}{log \frac{Pr [(r_{m, n}^{2}, r_{m, n}^{1}) | x_{m, n}^{1} = 1]}{Pr [(r_{m, n}^{2}, r_{m, n}^{1}) | x_{m, n}^{1} = 0]}}} + \underset{L_{MIC}^{a} (x_{m, n}^{1})}{\underset{︸}{log \frac{Pr [x_{m, n}^{1} = 1]}{Pr [x_{m, n}^{1} = 0]}}}, \end{array}

\begin{array}{l} L_{MIC}^{e} (x_{m, n}^{1}) & = log \frac{Pr [(r_{n}^{2}, r_{n}^{1}) + (s_{m, n}^{2} = 1, s_{m, n}^{1} = 0)]}{Pr [(r_{n}^{2}, r_{n}^{1}) | (s_{m, n}^{2} = 0, s_{m, n}^{1} = 1)]} \\ = log \frac{Pr [r_{n}^{2} | s_{m, n}^{2} = 1] Pr [r_{n}^{1} | s_{m, n}^{1} = 0]}{Pr [r_{n}^{2} | s_{m, n}^{2} = 0] Pr [r_{n}^{1} | s_{m, n}^{1} = 1]} \\ = log \frac{exp {r_{n}^{2} log [1 + \frac{n_{s} I_{m, n}^{2}}{n_{s} \sum_{\begin{array}{l} \tilde{m} = 1 \\ \tilde{m} \neq m \end{array}}^{M} I_{\tilde{m}, n}^{2} s_{\tilde{m}, n}^{2} + n_{b}}] - n_{s} I_{m, n}^{2}}}{exp {r_{n}^{1} log [1 + \frac{n_{s} I_{m, n}^{1}}{n_{s} \sum_{\begin{array}{l} \tilde{m} = 1 \\ \tilde{m} \neq m \end{array}}^{M} I_{\tilde{m}, n}^{1} s_{\tilde{m}, n}^{1} + n_{b}}] - n_{s} I_{m, n}^{1}}} \\ = {r_{n}^{2} log [1 + \frac{n_{s} I_{m, n}^{2}}{ξ_{m, n}^{2}}] - n_{s} I_{m, n}^{2}} - {r_{n}^{1} log [1 + \frac{n_{s} I_{m, n}^{1}}{ξ_{m, n}^{1}}] - n_{s} I_{m, n}^{1}} \\ = r_{n}^{2} log [1 + \frac{n_{s} I_{m, n}^{2}}{ξ_{m, n}^{2}}] - r_{n}^{1} log [1 + \frac{n_{s} I_{m, n}^{1}}{ξ_{m, n}^{1}}] - n_{s} I_{m, n}^{2} + n_{s} I_{m, n}^{1}, \end{array}

ξ_{m, n}^{Est (1)} = n_{s} \sum_{\begin{array}{l} \tilde{m} = 1 \\ \tilde{m} \neq m \end{array}}^{M} I_{\tilde{m}, n}^{1} E [s_{\tilde{m}, n}^{1}] + n_{b},

\begin{array}{l} E [s_{m, n}^{1}] & = 1 \times Pr (s_{m, n}^{1} = 1) + 0 \times Pr (s_{m, n}^{1} = 0) \\ = Pr (s_{m, n}^{1} = 1) = Pr (x_{m, n}^{1} = 0) \\ = \frac{1}{1 + exp [L_{MIC}^{a} (x_{m, n}^{1})]}, \end{array}

ξ_{m, n}^{Est (2)} = n_{s} \sum_{\begin{array}{l} \tilde{m} = 1 \\ \tilde{m} \neq m \end{array}}^{M} I_{\tilde{m}, n}^{2} E (s_{\tilde{m}, n}^{2}) + n_{b},

L_{DEC}^{Bit} (d_{m}^{i}) = \sum_{z = 1}^{L_{c}^{FEC}} L_{DEC}^{a (com)} (c_{m}^{(i - 1) \cdot L_{c}^{FEC} + z}) \cdot s_{z},

[L_{DEC}^{LLR} (c_{m}^{(i - 1) L_{c}^{FEC} + 1}), L_{DEC}^{LLR} (c_{m}^{(i - 1) L_{c}^{FEC} + 2}), \dots, L_{DEC}^{LLR} (c_{m}^{i \cdot L_{c}^{FEC}})] = L_{DEC}^{Bit} (d_{m}^{i}) \cdot s .

L_{MIC}^{e} (x_{m, n}^{k, z}) = \frac{\sum_{Ω (x_{m, n}^{k, z} = 1)} exp {r_{m, n}^{k, u} log [1 + \frac{n_{s} I_{m, n}^{k, u}}{ξ_{m, n}^{Est (k, u)}}] - n_{s} I_{m, n}^{k, u} + \sum_{i \in Ψ (x_{m, n}^{k, i} = 1)} L^{a} (x_{m, n}^{k, i})}}{\sum_{Ω (x_{m, n}^{k, z} = 0)} exp {r_{m, n}^{k, w} log [1 + \frac{n_{s} I_{m, n}^{k, w}}{ξ_{m, n}^{Est (k, w)}}] - n_{s} I_{m, n}^{k, w} + \sum_{i \in Θ (x_{m, n}^{k, i} = 1)} L^{a} (x_{m, n}^{k, i})},}

\begin{array}{l} L_{MIC}^{aposteriori} (x_{m, n}^{k, z}) & = log \frac{Pr [x_{m, n}^{k, z} = 1 | r_{m, n}^{k}]}{Pr [x_{m, n}^{k, z} = 0 | r_{m, n}^{k}]} \\ = log \frac{Pr [x_{m, n}^{k, z} = 1 | {r_{m, n}^{k, Q}, \dots, r_{m, n}^{k, 2}, r_{m, n}^{k, 1}}]}{Pr [x_{m, n}^{k, z} = 0 | {r_{m, n}^{k, Q}, \dots, r_{m, n}^{k, z}, r_{m, n}^{k, 1}}]} \\ = \underset{L_{MIC}^{e} (x_{m, n}^{k, z})}{\underset{︸}{log \frac{Pr [r_{m, n}^{k} | x_{m, n}^{k, z} = 1]}{Pr [r_{m, n}^{k} | x_{m, n}^{k, z} = 0]}}} + \underset{L_{MIC}^{a} (x_{m, n}^{k, z})}{\underset{︸}{log \frac{Pr [x_{m, n}^{k, z} = 1]}{Pr [x_{m, n}^{k, z} = 0]}}}, \end{array}

\begin{array}{l} L_{MIC}^{e} (x_{m, n}^{k, z}) = log \frac{Pr [r_{m, n}^{k} | x_{m, n}^{k, z} = 1]}{Pr [r_{m, n}^{k} | x_{m, n}^{k, z} = 0]} \\ = log \frac{\sum_{Ω (x_{m, n}^{k, z} = 1)} Pr [r_{m, n}^{k} | {x_{m, n}^{k, {log}_{2} Q}, \dots, x_{m, n}^{k, z} = 1, \dots, x_{m, n}^{k, 1}}] Pr [x_{m, n}^{k, {log}_{2} Q}, \dots, x_{m, n}^{k, z + 1}, x_{m, n}^{k, z - 1}, \dots, x_{m, n}^{k, 1}]}{\sum_{Ω (x_{m, n}^{k, z} = 0)} Pr [r_{m, n}^{k} | {x_{m, n}^{k, {log}_{2} Q}, \dots, x_{m, n}^{k, z} = 0, \dots, x_{m, n}^{k, 1}}] Pr [x_{m, n}^{k, {log}_{2} Q}, \dots, x_{m, n}^{k, z + 1}, x_{m, n}^{k, z - 1}, \dots, x_{m, n}^{k, 0}]}, \end{array}

\begin{array}{l} L_{MIC}^{e} (x_{m, n}^{k, z}) & = log \frac{\sum_{Ω (x_{m, n}^{k, z} = 1)} Pr [{r_{m, n}^{k, Q}, \dots, r_{m, n}^{k, 2}, r_{m, n}^{k, 1}} | s_{m, n}^{k, (1)}] Pr [{\tilde{x}}_{m, n}^{k, z, (1)}]}{\sum_{Ω (x_{m, n}^{k, z} = 0)} Pr [{r_{m, n}^{k, Q}, \dots, r_{m, n}^{k, 2}, r_{m, n}^{k, 1}} | s_{m, n}^{k, (0)}] Pr [{\tilde{x}}_{m, n}^{k, z, (0)}]} \\ = log \frac{\sum_{Ω (x_{m, n}^{k, z} = 1)} {\prod_{q = 1}^{Q} Pr [r_{m, n}^{k, q} | s_{m, n}^{k, q}] \times \prod_{i = 1}^{{log}_{2} Q - 1} Pr [x_{m, n}^{k, i}]}}{\sum_{Ω (x_{m, n}^{k, z} = 0)} {\prod_{q = 1}^{Q} Pr [r_{m, n}^{k, q} | s_{m, n}^{k, q}] \times \prod_{i = 1}^{{log}_{2} Q - 1} Pr [x_{m, n}^{k, i}]}} \\ = log \frac{\sum_{Ω (x_{m, n}^{k, z} = 1)} \frac{\prod_{q = 1}^{Q} Pr [r_{m, n}^{k, q} | s_{m, n}^{k, q}] \times \prod_{i = 1}^{{log}_{2} Q - 1} Pr [x_{m, n}^{k, i}]}{\prod_{q = 1}^{Q} Pr [r_{m, n}^{k, q} | s_{m, n}^{k, q} = 0] \times \prod_{i = 1}^{{log}_{2} Q - 1} Pr [x_{m, n}^{k, i} = 0]}}{\sum_{Ω (x_{m, n}^{k, z} = 0)} \frac{\prod_{q = 1}^{Q} Pr [r_{m, n}^{k, q} | s_{m, n}^{k, q}] \times \prod_{i = 1}^{{log}_{2} Q - 1} Pr [x_{m,}^{k, i}]}{\prod_{q = 1}^{Q} Pr [r_{m, n}^{k, q} | s_{m, n}^{k, q} = 0] \times \prod_{i = 1}^{{log}_{2} Q - 1} Pr [x_{m, n}^{k, i} = 0]}}, \end{array}

L_{MIC}^{e} (x_{m, n}^{k, z}) = log \frac{\sum_{Ω (x_{m, n}^{k, z} = 1)} exp {L^{e} (s_{m, n}^{k, u}) + \sum_{i \in Ψ (x_{m, n}^{k, i} = 1)} L^{a} (x_{m, n}^{k, i})}}{\sum_{Ω (x_{m, n}^{k, z} = 0)} exp {L^{e} (s_{m, n}^{k, w}) + \sum_{i \in Θ (x_{m, n}^{k, i} = 1)} L^{a} (x_{m, n}^{k, i})}},

\begin{array}{l} L^{e} (s_{m, n}^{k, u}) = log \frac{Pr (r_{m, n}^{k, u} | s_{m, n}^{k, u} + 1)}{Pr (r_{m, n}^{k, u} | s_{m, n}^{k, u} = 0)} \\ = log \frac{\frac{{[n_{s} I_{m, n}^{k, u} + n_{s} \sum_{\tilde{m} = 1, \tilde{m} \neq m}^{M} I_{\tilde{m}, n}^{k, u} s_{\tilde{m}, n}^{k, u} + n_{b}]}^{r_{m, n}^{k, u}}}{r_{m, n}^{k, u}!} exp [- (n_{s} I_{m, n}^{k, u} + n_{s} \sum_{\tilde{m} = 1, \tilde{m} \neq m}^{M} I_{\tilde{m}, n}^{k, u} s_{\tilde{m}, n}^{k, u} + n_{b})]}{\frac{{[n_{s} \sum_{\tilde{m} = 1, \tilde{m} \neq m}^{M} I_{\tilde{m}, n}^{k, u} s_{\tilde{m}, n}^{k, u} + n_{b}]}^{r_{m, n}^{k, u}}}{r_{m, n}^{k, u}!} exp [- (n_{s} \sum_{\tilde{m} = 1, \tilde{m} \neq m}^{M} I_{\tilde{m}, n}^{k, u} s_{\tilde{m}, n}^{k, u} + n_{b})]} \\ = r_{m, n}^{k, u} log [1 + \frac{n_{s} I_{m, n}^{k, u}}{ξ_{m, n}^{k, u}}] - n_{s} I_{m, n}^{k, u}, \end{array}

ξ_{m, n}^{Est (k, u)} = n_{s} \sum_{\tilde{m} = 1, \tilde{m} \neq m}^{M} I_{\tilde{m}, n}^{k, u} E [s_{\tilde{m}, n}^{k, u}] + n_{b},

L_{MIC}^{e} (x_{m, n}^{k, z}) = \frac{\sum_{Ω (x_{m, n}^{k, z} = 1)} exp {r_{m, n}^{k, u} log [1 + \frac{n_{s} I_{m, n}^{k, u}}{ξ_{m, n}^{Est (k, u)}}] - n_{s} I_{m, n}^{k, u} + \sum_{i \in Ψ (x_{m, n}^{k, i} = 1)} L^{a} (x_{m, n}^{k, i})}}{\sum_{Ω (x_{m, n}^{k, z} = 0)} exp {r_{m, n}^{k, w} log [1 + \frac{n_{s} I_{m, n}^{k, w}}{ξ_{m, n}^{Est (k, w)}}] - n_{s} I_{m, n}^{k, w} + \sum_{i \in Θ (x_{m, n}^{k, i} = 1)} L^{a} (x_{m, n}^{k, i})}} .

$s_{m}^{k, Q}, \dots, s_{m}^{k, 4}, s_{m}^{k, 3}, s_{m}^{k, 2}, s_{m}^{k, 1}$	$x_{m}^{k, {log}_{2} Q}, \dots, x_{m}^{k, 2}, x_{m}^{k, 1}$

0,⋯, 0, 0, 0, 1	0,⋯, 0, 0
0,⋯, 0, 0, 1, 0	0,⋯, 0, 1
0,⋯, 0, 1, 0, 0	0,⋯, 1, 0
0,⋯, 1, 0, 0, 0	0,⋯, 1, 1
⋯	⋯
1,⋯, 0, 0, 0, 0	1,⋯, 1, 1

$s_{m}^{k, Q}, \dots, s_{m}^{k, 4}, s_{m}^{k, 3}, s_{m}^{k, 2}, s_{m}^{k, 1}$	$x_{m}^{k, {log}_{2} Q}, \dots, x_{m}^{k, 2}, x_{m}^{k, 1}$

0,⋯, 0, 0, 0, 1	0,⋯, 0, 0
0,⋯, 0, 0, 1, 0	0,⋯, 0, 1
0,⋯, 0, 1, 0, 0	0,⋯, 1, 0
0,⋯, 1, 0, 0, 0	0,⋯, 1, 1
⋯	⋯
1,⋯, 0, 0, 0, 0	1,⋯, 1, 1

Abstract

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