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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 22 — Nov. 4, 2013
  • pp: 25954–25967
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Performance and capacity analysis of Poisson photon-counting based Iter-PIC OCDMA systems

Lingbin Li, Xiaolin Zhou, Rong Zhang, Dingchen Zhang, and Lajos Hanzo  »View Author Affiliations


Optics Express, Vol. 21, Issue 22, pp. 25954-25967 (2013)
http://dx.doi.org/10.1364/OE.21.025954


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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

1. Introduction

Free space optical (FSO) communications may be regarded as an attractive alternative to radio frequency (RF) communications due to its wider bandwidth and higher information security [1

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]

4

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

]. In high-rate FSO systems, optical code-division multiple-access (OCDMA) has been proposed for multiuser situations [5

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]

7

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]

]. However, the strong multiple access interference (MAI) erodes the performance of conventional chip-level OCDMA systems.

Hence, the novelty of this paper is that we analyse the Poisson counting based Iter-PIC OCDMA systems’s convergence behaviour with the aid of extrinsic information transfer (EXIT) charts [16

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

].
We will demonstrate the validity of our EXIT charts analysis with the aid of Monte Carlo simulations, showing that the Iter-PIC OCDMA systems outperform the conventional chip-level OCDMA systems in terms of both their BER performance as well as their capacity and spectral efficiency.

The remainder of this paper is organized as follows. Section 2 highlights the FSO system considered, which relies on an OOK transmitter and Poisson counting receiver. Section 3 develops the Iter-PIC aided APP detection algorithm proposed for multiuser-access. Section 4 and Section 5 analyse the BER and the capacity of this system, respectively. Section 6 compares the Iter-PIC OCDMA and the conventional chip-level OCDMA system with the aid of both simulations and theoretical results. We conclude in Section 7.

2. System model

Figure 1 shows a typical narrow-beam line-of-sight atmospheric optical communications system relying on intensity modulation/direct detection (IM/DD), which consists of an optical transmitter and a photon-counting receiver.

Fig. 1 Model of the Iter-PIC OCDMA system based on Poisson photon-counting reception and iterative detection.

2.1. Optical transmitter

In our Iter-PIC OCDMA system supporting K simultaneous users, k ∈ {1, 2,...,K} is the user index. The information bit sequence dk = {dk (nd), nd = 1, 2,...,Ninfo} of user k is encoded by a length-Nc repetition code, v = [+1, −1, +1, −1...], where Rc = 1/Nc is the coding rate. The coded sequence ck = {ck (l), l = 1, 2,...,L} (L = Ninfo × Nc) is then interleaved by the interleaver πk, producing the interleaved version of ck, namely xk = {xk (l), l = 1, 2,...,L}, whose elements are referred to as “chips”. The chip sequence xk is then modulated using OOK, for driving the laser source, which then transmits the appropriate number of photons for each chip duration. In this paper, we assume that the average number of signal photons transmitted for each information bit equals to ns. These photons are then transmitted for each chip “1”, while no photons are transmitted for each chip “0”. Hence, ns should be multiplied by the coderate, namely by 2/Nc, for each chip “1”. Given the coding rate of Rc = 1/Nc, the number of transmitted signal photons for each chip “1” is 2nsRc, while the number of transmitted signal photons for each chip “0” is zero.

2.2. Photon-counting receiver

At the receiver, the effect of background radiation has to be carefully considered. Explicitly, nb = ηEb/(hf) is the number of equivalent photons per bit imposed by the background radiation, where Eb denotes the incident background energy per slot, η is the quantum efficiency, h is the Planck constant and f is the central optical frequency. The background radiation is imposed on in all chips of a bit. Thus, nb should be multiplied by the code-rate, namely by 1/Nc, for each chip. Given the coding rate of Rc = 1/Nc, the number of equivalent background radiation photons for each chip “1” and chip “0” is nbRc.

The Poisson photon-counting model is adopted here. The electron-count r (l) received for the lth chip obeys the Poisson distribution given by [13

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]

]
Pr[r(l)]={k=1K[hk×xk(l)×(2nsRc)]+(nbRc)}r(l)r(l)!e{k=1K[hk×xk(l)×(2nsRc)]+(nbRc)},
(1)
where hk is the channel fading factor of user k. For simplicity, in this paper we assume hk = 1, ∀k ∈ {1, 2,...,K}.

3. Iterative algorithm

The receiver employs the soft output parallel interference cancellation structure for its detection and decision making, which consists of an elementary signal estimator (ESE) and K single-user a posteriori probability decoders (APP-DECs), as illustrated in Fig. 1. The ESE and DEC blocks of Fig. 1 operate in an iterative manner [11

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

, 12

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]

].

3.1. The ESE function

The input of ESE is defined by the a priori LLR sequence {λese,in [xk (l)], l = 1, 2,...,L}, while its output is represented by the extrinsic LLR sequence {λese,out [xk (l)], l = 1, 2,...,L}. According to [9

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]

], we have:
λese,out[xk(l)]=r(l)ln[1+2nsni,kest(l)+nb]2nsRc,
(2)
where
ni,kest(l)=2nskkKeλese,in[xk(l)]1+eλese,in[xk(l)],
(3)
is the estimated value of the number of photons imposing interference on user k.

3.2. The DEC function

The output sequence of ESE is then de-interleaved by the de-interleaver πk1 of Fig. 1 for generating the input sequence of DEC, namely the a priori LLRs {λdec,in [ck (l)], l = 1, 2,...,L}. According to the standard APP decoding mechanism [12

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]

], when using a repetition code, the hard decisions concerning a bit dk (nd) are carried out as follows
dk^(nd)=sgn{i=1Nc(1)i1λdec,in[ck(ndNcNc+i)]}+12,
(4)
where sgn{.} is the sign function. The output of the DEC block of Fig. 1 is defined by the extrinsic LLRs as
λdec,out[ck(ndNcNc+i)]={j=1Nc(1)j1λdec,in[ck(ndNcNc+j)]}(1)i1λdec,in[ck(ndNcNc+i)].
(5)
The output sequence of DEC will then be interleaved by πk for producing the input sequence of ESE in Fig. 1. The iterative detection algorithm is summarized in Table 1 for the sake of explicit clarity.

Table 1. OOK based Parallel Iterative Detection Algorithm

table-icon
View This Table

4. BER performance analysis

In this section, the BER performance of OCDMA systems relying on our Iter-PIC scheme is analysed with the aid of the EXIT charts developed in [16

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

]. For the optical systems operating at the 1.55μm wavelength, we assume η = 0.5 and Eb = −170dBJ, which results in nb = 39.

4.1. Introduction of EXIT charts

We assume that for each user the ESE and the DEC blocks of Fig. 1 constitute two individual soft-in-soft-out (SISO) decoders, where the mutual information (MI) between the soft-values and their hard-decision counterparts at the input and output are Iin = I (X; λin) and Iout = I (X; λout) [17

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

, 18

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

], with X ∈ {0, 1} being the chip value, λin the a priori LLR value and λout is the extrinsic LLR value. More explicitly, we have
Iin=12x=0,1Pr[λin|X=x]log2[2Pr[λin|X=x]Pr[λin|X=0]+Pr[λin|X=1]]dλin,0Iin1,
(6)
and
Iout=12x=0,1Pr[λout|X=x]log2[2Pr[λout|X=x]Pr[λout|X=0]+Pr[λout|X=1]]dλout,0Iout1.
(7)

Each decoder’s output sequence is interleaved/de-interleaved to be presented in the right order as the other decoder’s input sequence, but this process does not change the MI values. Thus, at the ith iteration, we have
{Idec,ini=Iese,outiIese,ini=Idec,outi1.
(8)

The exchange of extrinsic information can be visualized as a stair-case-shaped decoding trajectory as seen in Fig. 2 for Rc = 1/4, 1/8 and 1/16. To elaborate a little further, every time the ESE and DEC blocks of Fig. 1 provide each other with some additional extrinsic information, the trajectory moves a little step closer to the top-right corner of Fig. 2. The (1, 1) point in this diagram would correspond to perfect decoding convergence to a vanishingly low BER. Observe in Fig. 2 that this trajectory fails to reach the (1, 1) point and hence only a gradually decaying BER curve is expected. It has to be noted that the accuracy of EXIT-chart based performance evaluation relies on the assumption that the LLR values obey the Gaussian distribution, which may only be approximated in case of infinitely long interleavers. Since this is impractical, the LLR histogram was evaluated using Monte-Carlo simulations relying on a“free-running” iterative decoder. More explicitly, Fig. 2 plots the EXIT trajectories for a K = 4-user system, where every “step” in the trajectories suggests the start of a new iteration. Our inherent assumption is that the K = 4 users have the same power. It is observed that after a certain number of iterations, the MI improvements become marginal. Furthermore, the number of iterations required for convergence is reduced and so is the complexity, as the coding rate becomes lower.

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.

4.2. EXIT chart aided analysis

With the aid of the EXIT trajectories of Fig. 2, we can now extract the coordinates (Iese,in, Iese,out) or (Idec,out, Idec,in) of the the last iteration. We may then view Iese,out as a function of Iese,in, ns and nb, which is formulated as
Iese,out=Tese(Iese,in,ns,nb),
(9)
and Idec,out as a function of Idec,in, namely as
Idec,out=Tdec(Idec,in).
(10)
According to Eq. (8), we have
Iese,outnew=Tese[Tdec(Iese,outold),ns,nb].
(11)
Given the initialization of our PIC algorithm in Table 1, the IEESE value of the last iteration can be consequently viewed as a function of ns and nb, which is formulated as:
Iese,out=T(ns,nb),
(12)
or, for a fixed background radiation nb, as:
Iese,out=T(ns).
(13)

4.3. Relationship between Iese,out and ñs

Assuming that the number of users is K = 1 and that DEC provides no extrinsic information for ESE of Fig. 1, Eq. (2) may be rewritten as
λese,out[xk(l)]=r(l)ln[1+2n˜snb]2n˜sRc.
(18)
By exploiting that λese,out [xk (l)] is a linear function of r (l), we have
Pr[λese,out[xk(l)]|xk(l)]=Pr[r(l)|xk(l)].
(19)
Therefore, we rewrite Eq. (7) for the ESE block of Fig. 1 as
Iese,out=12x=0,1r=0Pr[r|X=x]×log2[2×Pr[r|X=x]Pr[r|X=0]+Pr[r|X=1]].
(20)
Upon combining Eqs. (1) and (20), we arrive at:
Iese,out=12r=0+(2n˜sRc+nbRc)rr!e(2n˜sRc+nbRc)log2[21+(nb2n˜s+nb)re2n˜sRc]+12r=0+(nbRc)rr!e(nbRc)log2[21+(2n˜s+nbnb)re2n˜sRc],
(21)
which is the expression of f (ñs) in Eq. (14).

Figure 3 shows the relationship between Iese,out and ñs, as evaluated both from Eq. (21) and by simulations, whilst relying on the LLR histogram. The curves in Fig. 3 demonstrate the validity of our derivation of the function f (·). It is clearly seen in Fig. 3 that f (·) is monotonically increasing, thus f−1 (·) can be found with the aid of numerical analysis.

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.

4.4. Relationship between the BER and ñs

Let us now embark on quantifying the relationship between the BER and ñs. In a single-user, non-iterative system, we have
i=1Nc(1)i1λdec,in[ck(ndNcNc+i)]=i=1Nc[r(ndNcNc+i)ln(1+2n˜snb)2n˜sRc](1)i1=ln(1+2n˜snb)[m=1,3,5,Nc1r(ndNcNc+m)n=2,4,6,Ncr(ndNcNc+n)].
(22)

For dk (nd) = 1, {r (ndNcNc + m), m = 1, 3,...,Nc − 1} is given by random variables obeying the Poisson distribution associated with the parameter (2ñsRc + nbRc), while {r (ndNcNc + n), n = 2, 4,...,Nc} is represented by random variables obeying the Poisson distribution associated with the parameter (nbRc).

The moment generating function (MGF) [19

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

] for a Poisson random variable X is given by
MX(t)=eμX(et1),
(23)
where μX is the mean value of X. For a sequence of independent random variables, namely for X1, X2,...,Xn, the sum Sn=i=1nXi, has the MGF of
MSn(t)=i=1nMXi(t).
(24)

According to Eq. (23), the MGF of r (ndNcNc + m) is given by
Mr(ndNcNc+m)(t)=e(2n˜sRc+nbRc)(et1),m=1,3,,Nc1,
(25)
and the MGF of r (ndNcNc + n) is given by
Mr(ndNcNc+n)(t)=e(nbRc)(et1),n=2,4,,Nc.
(26)

Upon defining the random variables R1(nd)=m=1,3,5,Nc1 r (ndNcNc + m) and R2(nd)=n=2,4,6,Nc r (ndNcNc + n), their MGFs satisfy
MR1(nd)(t)=m=1,3,5,Nc1Mr(ndNcNc+m)(t)=m=1,3,5,Nc1e(2n˜sRc+nbRc)(et1)=eNc2(2n˜sRc+nbRc)(et1)=e(n˜s+nb2)(et1),
(27)
MR2(nd)(t)=n=2,4,6,NcMr(ndNcNc+n)(t)=n=2,4,6,Nce(nbRc)(et1)=eNc2(nbRc)(et1)=e(nb2)(et1).
(28)
Thus, R1 (nd) follows the Poisson distribution associated with the parameter (n˜s+nb2), while R2 (nd) obeys the Poisson distribution associated with the parameter (nb2).

Upon defining ΔR(nd) = R1 (nd) − R2 (nd), ΔR(nd) follows the Skellam distribution having the probability mass function (PMF) given by [20

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]

]
Pr1[ΔR(nd)]=e(n˜s+nb2+nb2)(n˜s+nb2nb2)ΔR(nd)/2I|ΔR(nd)|[2(n˜s+nb2)(nb2)],
(29)
where Ik [.] is the modified Bessel function of the first kind with order k. Bit errors occur, when ΔR(nd) < 0. The error probability for bit “1” is given by
PE1=ΔR(nd)=1Pr1[ΔR(nd)].
(30)
Similarly, when dk (nd) = 0, R1 (nd) obeys the Poisson distribution associated with the parameter (nb2) and R2 (nd) follows the Poisson distribution having the parameter of (n˜s+nb2), then the PMF of ΔR(nd) is given by
Pr0[ΔR(nd)]=e(nb2+n˜s+nb2)(n˜s+nb2nb2)ΔR(nd)/2I|ΔR(nd)|[2(nb2)(n˜s+nb2)].
(31)
In this case, the bit errors occur, when ΔR(nd) > 0. Hence the error probability for a bit “0” is given by
PE0=ΔR(nd)=1Pr0[ΔR(nd)].
(32)

Therefore we have PE1 = PE0. Assuming that bit “1” and bit “0” appear with equal probabilities, the channel we consider is a binary symmetric channel (BSC), hence the average BER is expressed as
PE=12(PE1+PE0)=PE1=PE0.
(33)

Figure 4 shows the BER curves acquired both from Eqs. (29)(33) and by Monte Carlo simulation, for a single-user, non-iterative system. The correspondence of these curves verifies the accuracy of our derivation. We observe from both Eqs. (29) and (31) that in a single-user system, the BER is independent of the coding rate Rc, which is also demonstrated in Fig. 4 by the simulated curves of Rc = 1/2 and Rc = 1/4.

Fig. 4 BER performance for the single-user, non-iterative system, with nb = 39.

Figure 5 shows the simulated BER performance (color markers) for the multi-user system. The performance curves corresponding to the EXIT chart analysis associated with It = 50 iterations are also included in Fig. 5 (black solid lines). The single-user performance is also plotted for reference (black dashed lines), which represents the best possible performance of the multi-user system. Observe in Fig. 5 that after a few iterations, the simulation results become reasonably similar to the results calculated by EXIT chart analysis. Again, this confirms the viability of the performance analysis method discussed above. Furthermore, as the number of users increases, the number of iterations required for accurate EXIT chart analysis is increased. Naturally, the performance discrepancy between the single-user and multi-user system is also increased upon increasing the number of users.

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.

5. Channel capacity analysis

The channel capacity expressed in nats per channel use for a BSC is defined as [6

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]

]
C=ln2H(PE),
(34)
where H (p) is the binary entropy function, given by
H(p)=plnp(1p)ln(1p).
(35)
Let us furthermore define the normalised capacity as the capacity in nats per signal photon, namely as:
Cph=Cns.
(36)

Figure 6 plots the capacity per signal photon for different number of users for the Iter-PIC OCDMA system of Fig. 1. As Fig. 6 shows, there exits an optimum value of ns that maximizes the normalised capacity of the system. The optimum value of ns varies for different values of K and the maximum value of Cph drops, as K increases.

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.

6. Comparison of Iter-PIC OCDMA and of conventional chip-level OCDMA

This section compares the performance of the proposed Iter-PIC OCDMA scheme and of the conventional chip-level OCDMA scheme for a K = 4-user system. It also illustrates the advantages of the Iter-PIC OCDMA scheme compared to the conventional chip-level OCDMA scheme in terms of its BER, spectral-efficiency and capacity.

6.1. Advantage 1: In contrast to conventional chip-level OCDMA, Iter-PIC OCDMA does not have an error floor

In Fig. 7, the Iter-PIC OCDMA scheme is benchmarked against the conventional chip-level OCDMA scheme in terms of its BER. It is shown that the conventional chip-level OCDMA scheme exhibits an error floor, which limits the attainable BER improvement, as the signal power increases. By contrast, the Iter-PIC OCDMA scheme does not suffer from this limitation. For a system supporting K = 4 users at a coding rate of Rc = 1/10, using the Iter-PIC OCDMA scheme of Fig. 1 reaches a BER of 10−4, or even lower, as more signal photons are transmitted, while the conventional chip-level OCDMA scheme fails to achieve a BER lower than 10−2.

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.

6.2. Advantage 2: Iter-PIC OCDMA improves the spectral-efficiency

In Fig. 7, we also used coding rates lower than Rc = 1/10 for the conventional chip-level OCDMA scheme for comparison. Observe that the Iter-PIC OCDMA scheme is capable of reaching the desired BER at a higher coding rate than the conventional chip-level OCDMA scheme. For example, the Iter-PIC OCDMA arrangement attains a BER of 10−4 at Rc = 1/10, while the conventional chip-level OCDMA scheme needs a coding rate of Rc = 1/40 for obtaining the same BER. Hence the Iter-PIC OCDMA scheme improves the spectral-efficiency by a factor of four under these conditions.

6.3. Advantage 3: The capacity of Iter-PIC OCDMA exceeds that of conventional chip-level OCDMA at moderate number of signal photons

Figure 8 compares both the channel capacity and the capacity per signal photon between our Iter-PIC OCDMA and conventional chip-level OCDMA. Quantitatively, Fig. 8 shows that although the channel capacity of conventional chip-level OCDMA is higher than that of Iter-PIC OCDMA when the number of signal photons is low, it soon saturates at its upper limit, while the capacity of Iter-PIC OCDMA continues to increase, as the number of signal photons increases. Both the channel capacity and the normalised capacity of Iter-PIC OCDMA may surpass those of conventional chip-level OCDMA, when the number of signal photons ns is moderate (around the value of nb). The channel capacity of Iter-PIC OCDMA is up to 0.1 nats higher than that of conventional chip-level OCDMA.

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.

7. Conclusions

A novel photon-counting based Iter-PIC scheme was proposed for OCDMA systems subjected to background radiation. We analysed the system’s performance and verified the analytical results by Monte Carlo simulations. Both the analysis and the simulation results indicate the capability of the proposed Iter-PIC OCDMA scheme to mitigate the MAI, despite using a higher coding rate than the conventional chip-level OCDMA scheme. Two orders of magnitude BER improvements, 0.1 nats of capacity enhancement and a quadrupled spectral-efficiency can be achieved at a coding rate of Rc = 1/10.

Acknowledgments

This work was supported in part by the National High Technology Research and Development Program of China under Grant No. 2011AA100701, the National Natural Science Foundation of China under Grant No. 60802011, and the National Science and Technology Major Project of China No. 2012ZX03001013. The support of the European Research Council’s Advanced Fellow Scheme is also gratefully acknowledged. The authors would like to thank the anonymous reviewers for their constructive comments.

References and links

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

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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]

OCIS Codes
(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence
(060.4510) Fiber optics and optical communications : Optical communications
(060.2605) Fiber optics and optical communications : Free-space optical communication

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: August 8, 2013
Revised Manuscript: October 14, 2013
Manuscript Accepted: October 14, 2013
Published: October 23, 2013

Citation
Lingbin Li, Xiaolin Zhou, Rong Zhang, Dingchen Zhang, and Lajos Hanzo, "Performance and capacity analysis of Poisson photon-counting based Iter-PIC OCDMA systems," Opt. Express 21, 25954-25967 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-22-25954


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References

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  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).
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  14. K. Chakraborty, S. Dey, and M. Franceschetti, “Outage capacity of the MIMO Poisson fading channels,” IEEE Trans. Infor. Theory54(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. D52(3), 381–393 (2003). [CrossRef]

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