## Performance and capacity analysis of Poisson photon-counting based Iter-PIC OCDMA systems |

Optics Express, Vol. 21, Issue 22, pp. 25954-25967 (2013)

http://dx.doi.org/10.1364/OE.21.025954

Acrobat PDF (1048 KB)

### 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

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. 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]

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]

*analyse the Poisson counting based Iter-PIC OCDMA systems’s convergence behaviour with the aid of extrinsic information transfer (EXIT) charts [16].*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.

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

## 2. System model

### 2.1. Optical transmitter

*K*simultaneous users,

*k*∈ {1, 2,...,

*K*} is the user index. The information bit sequence

**= {**

*d*_{k}*d*(

_{k}*n*

_{d}),

*n*

_{d}= 1, 2,...,

*N*

_{info}} of user

*k*is encoded by a length-

*N*

_{c}repetition code,

**= [+1, −1, +1, −1...], where**

*v**R*

_{c}= 1/

*N*

_{c}is the coding rate. The coded sequence

**= {**

*c*_{k}*c*(

_{k}*l*),

*l*= 1, 2,...,

*L*} (

*L*=

*N*

_{info}×

*N*

_{c}) is then interleaved by the interleaver

*π*, producing the interleaved version of

_{k}**, namely**

*c*_{k}**= {**

*x*_{k}*x*(

_{k}*l*),

*l*= 1, 2,...,

*L*}, whose elements are referred to as “chips”. The chip sequence

**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**

*x*_{k}*n*

_{s}. These photons are then transmitted for each chip “1”, while no photons are transmitted for each chip “0”. Hence,

*n*

_{s}should be multiplied by the coderate, namely by 2/

*N*

_{c}, for each chip “1”. Given the coding rate of

*R*

_{c}= 1/

*N*

_{c}, the number of transmitted signal photons for each chip “1” is 2

*n*

_{s}

*R*

_{c}, while the number of transmitted signal photons for each chip “0” is zero.

### 2.2. Photon-counting receiver

*n*

_{b}=

*ηE*

_{b}/(

*hf*) is the number of equivalent photons per bit imposed by the background radiation, where

*E*

_{b}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,

*n*

_{b}should be multiplied by the code-rate, namely by 1/

*N*, for each chip. Given the coding rate of

_{c}*R*

_{c}= 1/

*N*

_{c}, the number of equivalent background radiation photons for each chip “1” and chip “0” is

*n*

_{b}

*R*

_{c}.

*r*(

*l*) received for the

*l*th 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]

*h*is the channel fading factor of user

_{k}*k*. For simplicity, in this paper we assume

*h*= 1, ∀

_{k}*k*∈ {1, 2,...,

*K*}.

## 3. Iterative algorithm

*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. 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

*a priori*LLR sequence {

*λ*

_{ese,in}[

*x*(

_{k}*l*)],

*l*= 1, 2,...,

*L*}, while its output is represented by the extrinsic LLR sequence {

*λ*

_{ese,out}[

*x*(

_{k}*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]

*k*.

### 3.2. The DEC function

*a priori*LLRs {

*λ*

_{dec,in}[

*c*(

_{k}*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]

*d*(

_{k}*n*

_{d}) are carried out as follows where sgn{.} is the sign function. The output of the DEC block of Fig. 1 is defined by the extrinsic LLRs as The output sequence of DEC will then be interleaved by

*π*

_{k}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.

## 4. BER performance analysis

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

*μm*wavelength, we assume

*η*= 0.5 and

*E*

_{b}= −170dBJ, which results in

*n*

_{b}= 39.

### 4.1. Introduction of EXIT charts

*I*

_{in}=

*I*(

*X*;

*λ*

_{in}) and

*I*

_{out}=

*I*(

*X*;

*λ*

_{out}) [17

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

*X*∈ {0, 1} being the chip value,

*λ*

_{in}the

*a priori*LLR value and

*λ*

_{out}is the extrinsic LLR value. More explicitly, we have and

*i*th iteration, we have

*R*

_{c}= 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.

### 4.2. EXIT chart aided analysis

*I*

_{ese,in},

*I*

_{ese,out}) or (

*I*

_{dec,out},

*I*

_{dec,in}) of the the last iteration. We may then view

*I*

_{ese,out}as a function of

*I*

_{ese,in},

*n*

_{s}and

*n*

_{b}, which is formulated as and

*I*

_{dec,out}as a function of

*I*

_{dec,in}, namely as According to Eq. (8), we have Given the initialization of our PIC algorithm in Table 1, the

*n*

_{s}and

*n*

_{b}, which is formulated as: or, for a fixed background radiation

*n*

_{b}, as:

### 4.3. Relationship between I_{ese,out} and ñ_{s}

*K*= 1 and that DEC provides no extrinsic information for ESE of Fig. 1, Eq. (2) may be rewritten as By exploiting that

*λ*

_{ese,out}[

*x*(

_{k}*l*)] is a linear function of

*r*(

*l*), we have Therefore, we rewrite Eq. (7) for the ESE block of Fig. 1 as Upon combining Eqs. (1) and (20), we arrive at:

*f*(

*ñ*

_{s}) in Eq. (14).

*I*

_{ese,out}and

*ñ*

_{s}, 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.

### 4.4. Relationship between the BER and ñ_{s}

*ñ*

_{s}. In a single-user, non-iterative system, we have

*d*(

_{k}*n*

_{d}) = 1, {

*r*(

*n*

_{d}

*N*

_{c}−

*N*

_{c}+

*m*),

*m*= 1, 3,...,

*N*

_{c}− 1} is given by random variables obeying the Poisson distribution associated with the parameter (2

*ñ*

_{s}

*R*

_{c}+

*n*

_{b}

*R*

_{c}), while {

*r*(

*n*

_{d}

*N*

_{c}−

*N*

_{c}+

*n*),

*n*= 2, 4,...,

*N*} is represented by random variables obeying the Poisson distribution associated with the parameter (

_{c}*n*

_{b}

*R*

_{c}).

*X*is given by where

*μ*is the mean value of

_{X}*X*. For a sequence of independent random variables, namely for

*X*

_{1},

*X*

_{2},...,

*X*, the sum

_{n}*r*(

*n*

_{d}

*N*

_{c}−

*N*

_{c}+

*m*) is given by and the MGF of

*r*(

*n*

_{d}

*N*

_{c}−

*N*

_{c}+

*n*) is given by

*r*(

*n*

_{d}

*N*

_{c}−

*N*

_{c}+

*m*) and

*r*(

*n*

_{d}

*N*

_{c}−

*N*

_{c}+

*n*), their MGFs satisfy

*R*

_{1}(

*n*

_{d}) follows the Poisson distribution associated with the parameter

*R*

_{2}(

*n*

_{d}) obeys the Poisson distribution associated with the parameter

*R*(

*n*

_{d}) =

*R*

_{1}(

*n*

_{d}) −

*R*

_{2}(

*n*

_{d}), Δ

*R*(

*n*

_{d}) 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]

*I*[.] is the modified Bessel function of the first kind with order

_{k}*k*. Bit errors occur, when Δ

*R*(

*n*

_{d}) < 0. The error probability for bit “1” is given by Similarly, when

*d*(

_{k}*n*

_{d}) = 0,

*R*

_{1}(

*n*

_{d}) obeys the Poisson distribution associated with the parameter

*R*

_{2}(

*n*

_{d}) follows the Poisson distribution having the parameter of

*R*(

*n*

_{d}) is given by In this case, the bit errors occur, when Δ

*R*(

*n*

_{d}) > 0. Hence the error probability for a bit “0” is given by

_{E1}= P

_{E0}. 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

*R*

_{c}, which is also demonstrated in Fig. 4 by the simulated curves of

*R*

_{c}= 1/2 and

*R*

_{c}= 1/4.

*I*= 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.

_{t}## 5. Channel capacity analysis

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]

*H*(

*p*) is the binary entropy function, given by Let us furthermore define the normalised capacity as the capacity in nats per signal photon, namely as:

*n*

_{s}that maximizes the normalised capacity of the system. The optimum value of

*n*

_{s}varies for different values of

*K*and the maximum value of

*C*

_{ph}drops, as

*K*increases.

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

*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

*K*= 4 users at a coding rate of

*R*

_{c}= 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}.

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

*R*

_{c}= 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

*R*

_{c}= 1/10, while the conventional chip-level OCDMA scheme needs a coding rate of

*R*

_{c}= 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

*n*

_{s}is moderate (around the value of

*n*

_{b}). The channel capacity of Iter-PIC OCDMA is up to 0.1 nats higher than that of conventional chip-level OCDMA.

## 7. Conclusions

*R*

_{c}= 1/10.

## Acknowledgments

## References and links

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 |

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 |

3. | F. Yang and J. Cheng, “Coherent free-space optical communications in Lognormal-Rician turbulence,” IEEE Commun. Lett. |

4. | V. W. S. Chan, “Free-space optical communications,” J. Lightwave Technol. |

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 |

6. | H. M. H. Shalaby, “Complexities, error probabilities, and capacities of optical OOK-CDMA communication systems,” IEEE Trans. Commun. |

7. | M. Jazayerifar and J. A. Salehi, “Atmospheric optical CDMA communication systems via optical orthogonal codes,” IEEE Trans. Commun. |

8. | R. Zhang and L. Hanzo, “Three design aspects of multicarrier interleave division multiple access,” IEEE T. Veh. Technol. |

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 |

10. | X. Zhou, Y. Yang, Y. Shao, and J. Liu, “Photon-counting chip-interleaved iterative PIC detector over atmospheric turbulence channels,” Chin. Opt. Lett. |

11. | Li Ping, L. Liu, and W. K. Leung, “Interleave-division multiple-access,” IEEE Trans. Wireless Commun. |

12. | C. Berrou and A. Glavieux, “Near optimum limit error correcting coding and decoding: Turbo-codes,” IEEE Trans. Commun. |

13. | W. Gappmair and S. S. Muhammad, “Error performance of PPM/Poisson channels in turbulent atmosphere with Gamma-Gamma distribution,” Electron. Lett. |

14. | K. Chakraborty, S. Dey, and M. Franceschetti, “Outage capacity of the MIMO Poisson fading channels,” IEEE Trans. Infor. Theory |

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. |

16. | S. ten Brink, “Convergence behavior of iteratively decoded parallel concatenated codes,” IEEE Trans. Commun. |

17. | T. M. Cover and J. A. Thomas, |

18. | R. W. Hamming, |

19. | G. Casella and R. L. Berger, |

20. | D. Karlis and I. Ntzoufras, “Analysis of sports data by using bivariate Poisson models,” J. R. Statist. Soc. D |

**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

- 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, 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]
- F. Yang and J. Cheng, “Coherent free-space optical communications in Lognormal-Rician turbulence,” IEEE Commun. Lett.16(11), 1872–1875 (2012). [CrossRef]
- V. W. S. Chan, “Free-space optical communications,” J. Lightwave Technol.24(12), 4750–4762 (2006). [CrossRef]
- 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]
- 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]
- R. Zhang and L. Hanzo, “Three design aspects of multicarrier interleave division multiple access,” IEEE T. Veh. Technol.57(6), 3607–3617 (2008). [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. Express20(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).
- Li Ping, L. Liu, and W. K. Leung, “Interleave-division multiple-access,” IEEE Trans. Wireless Commun.5(4), 938–947 (2006). [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]
- 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. Theory54(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]
- S. ten Brink, “Convergence behavior of iteratively decoded parallel concatenated codes,” IEEE Trans. Commun.49(10), 1727–1737 (2001). [CrossRef]
- 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).
- 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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