## 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

## 1. Introduction

Free-space optical (FSO) systems have received considerable research attention due to their advantages of low power consumption, wide bandwidth and high information security [1–3]. In this context, the optical code-division multiple-access (OCDMA) scheme constitutes a promising multiple access scheme for high-rate multiuser systems [4–7].

However, in conventional non-coherent FSO OCDMA systems, strong multiple access interference (MAI) is imposed owing to the non-zero cross correlations of bandwidth inefficient long unipolar optical orthogonal code (OOC) sequences [8]. Therefore, numerous OCDMA receivers have been proposed for reducing the MAI, including the conventional correlator (CCR), the chip-level detector as well as the parallel and serial interference cancellation (PIC and SIC) schemes [9–11].

Iterative detection is widely used in radio frequency (RF) communications and thermal-noise-limited optical systems [12, 13], which typically exploit the *signal-independent* nature of *additive* Gaussian noise [14]. However, in FSO systems operating in weak received signal scenarios, the employment of a shot-noise-limited Poisson photon-counting based iterative detector encounters *signal-dependent* Poissonian noise.

Hence, the novelty of this paper is *the conception of an efficient chip-level iterative a posteriori probability (APP) MAI cancellation technique for the Poisson photon-counting process, namely that of the photon-counting iterative serial interference cancellation (Iter-SIC) scheme*. We will demonstrate that the Iter-SIC scheme is capable of exceeding the optimum performance of the conventional chip-level OCDMA scheme [16]. Equally importantly, the proposed Iter-SIC scheme significantly reduces the spreading-code length required for achieving the BER performance of the classic chip-level OCDMA scheme. Moreover, it is resilient against background light noise and its flexible system structure makes the Iter-SIC scheme suitable for employment in practical scenarios.

The remainder of the paper is organized as follows. Section 2 describes the proposed Iter-SIC aided FSO system and the APP based iterative detection algorithm. Section 3 presents our simulation results. Section 4 makes discussions of the application in practical scenarios, before we conclude in Section 5.

## 2. System description

#### 2.1. Optical transmitter

The transmitter structure of the Iter-SIC FSO system is shown in Fig. 1. Let *k* ∈ [1, *K*] be the user index. The information bit sequence **d*** _{k}* = {

*d*(

_{k}*i*),

*i*= 1, ⋯ ,

*L*} of the

_{d}*k*th user is encoded by a forward error correcting (FEC) encoder, generating the coded sequence

**c**

*= {*

_{k}*c*(

_{k}*j*),

*j*= 1, ⋯ ,

*L*}, where

_{c}*L*is the information frame length and

_{d}*L*is the encoded frame length. Then, the encoded data is interleaved by a random user-specific interleaver Π

_{c}*, producing the sequence*

_{k}**x**

*= {*

_{k}*x*(

_{k}*j*),

*j*= 1, ⋯ ,

*L*}.

_{c}For practicality, the intensity modulation/direct detection (IM/DD) technique relying on on-off keying (OOK) is employed. The modulated sequence **s*** _{k}* = {

*s*(

_{k}*j*),

*j*= 1, ⋯ ,

*L*} is then used for driving the optical modulator to generate the appropriate photon counts per chip

_{c}*m*

_{0}= 0 and

*m*

_{1}=

*PT*/

_{c}*hυ*representing “0” and “1”, where

*P,T*,

_{c}*υ*and

*h*denote the transmitted power, chip duration, optical frequency and Plank’s constant, respectively. The elements of {

*x*(

_{k}*j*)} and {

*s*(

_{k}*j*)} are referred to as “chips”.

#### 2.2. Poisson atmospheric channel model

As shown in Fig. 1, the photons generated are transmitted to the receiver via the FSO channel. The atmospheric turbulence-induced fading coefficient *I _{k}* ≥ 0 is modelled by a Gamma-Gamma distribution with the PDF given by [17]

*I*denotes the channel’s fading coefficient between the

_{k}*k*th user laser and the receiving photon detector (PD). The scintillation parameters

*α*> 0 and

*β*> 0 of Eq. (1) are linked to the Rytov variance ${\sigma}_{\text{R}}^{2}$, which corresponds to the strength of turbulence [17, 18], while K

*(·) is the modified Bessel function of the second kind of order*

_{l}*l*. Finally, the scintillation index is defined as

*S.I.*=

*α*

^{−1}+

*β*

^{−1}+ (

*αβ*)

^{−1}.

At the receiver, the Poisson photon-counting model is adopted. Consequently, the received electron counts per chip *r*(*j*) follow a Poisson distribution and is given by,

*λ*] denotes Poisson distribution associated with the parameter

*λ*and the received photoelectron counts are denoted by

*k*th user in the

*j*th chip.

*η*represents the PD efficiency.

*n*=

_{b}*ηP*/(

_{b}T_{c}*hυ*) stands for the background radiation photoelectrons per chip interval.

*P*is the power incident on the PD owing to the background noise.

_{b}#### 2.3. Iterative SIC algorithm

As shown in Fig. 1, for each user, the multiuser detector (MUD) consists of an iterative noise estimation (NE) block, an external log-likelihood-ratio calculation (ELC) block and an APP decoder (DEC).

In the MUD block, given the turbulence channel observation **I** = {*I _{k}*, ∀

*k*}, the

*a posteriori*log-likelihood ratios (LLRs) of the encoded sequence {

*x*(

_{k}*j*)} are defined as

*L*

_{MUD_e}[

*x*(

_{k}*j*)] denotes the extrinsic LLR about

*x*(

_{k}*j*), and

*L*

_{MUD_a}[

*x*(

_{k}*j*)] denotes the

*a priori*LLR about

*x*(

_{k}*j*).

The ELC block is used for generating the extrinsic LLRs by taking into account the noise estimates provided by the NE block of Fig. 1. By exploiting Eq. (2) and Eq. (4), we have

On the other hand, the NE processes the *a priori* LLRs from the DEC and generates the noise estimates of *ξ _{k}* (

*j*) for supporting the operation of the corresponding MUD block. More explicitly, the estimated noise is

*j*th chip for the

*k*th user, which is

*L*

_{MUD_a}[

*x*(

_{k̃}*j*)] ∈ Θ

_{MUD_a}[

*x*(

_{k}*j*)], Θ

_{MUD_a}[

*x*(

_{k}*j*)] = {

*L*

_{MUD_a}[

*x*

_{1}(

*j*)],...,

*L*

_{MUD_a}[

*x*

_{k}_{−1}(

*j*)],

*L*

_{MUD_a}[

*x*

_{k}_{+1}(

*j*)],...,

*L*

_{MUD_a}[

*x*(

_{k}*j*)]} stands for the set of LLRs from interfering users. As illustrated in Fig. 1 and 2, { ${\mathrm{\Theta}}_{\text{MUD}\_\text{a}}^{\left(n\right)}\left({x}_{k}\right)$, 1 ≤

*k*≤

*K*} are serially updated, where

*n*is the iteration index.

In the DEC block, the APP decoding is a standard function [19]. If a low-complexity repetition coding scheme is used as our FEC code, the *a posteriori* LLRs of the information bits {
${L}_{\text{DEC}}^{\text{Bit}}\left[{d}_{k}\left(i\right)\right]$, *i* = 1, 2,..., *L _{d}*} can be obtained by combining the

*a priori*LLRs of the encoded chips {

*L*

_{DEC_a}[

*c*(

_{k}*j*)]

*, j*= 1, 2,...,

*L*} as follows:

_{c}*s*∈

_{z}**s**, $\mathbf{s}=\underset{{N}_{c}}{\underbrace{\left[+1,-1,\cdots ,+1,-1\right]}}$ is the spreading vector of the repetition code employed and

*N*denotes the repetition code length. The corresponding repetition code is ${c}_{k}\left(j\right)=\{\begin{array}{ll}\underset{{N}_{c}}{\underbrace{[1,0,1,0,\cdots ,1,0]}}\hfill & {d}_{k}\left(i\right)=1,\hfill \\ \left[0,1,0,1,\cdots ,0,1\right]\hfill & {d}_{k}\left(i\right)=0.\hfill \end{array}$

_{c}As shown in Fig. 1, the decoded LLRs of the feedback link can be generated as,

**d̃**

*} can be recovered by using hard decisions as*

_{k}#### 2.4. Summary of the iterative SIC scheme

For the sake of explicit clarity, the pseudo-code of our Iter-SIC scheme is provided in Algorithm I.

Below we analyze the computational complexity of Algorithm I. Firstly, as for the MUD block, according to Eq. (8) and Fig. 2, the noise estimates
$\left\{{\xi}_{k}^{\text{Est}\left(n\right)}\left(j\right)\right\}$ of the *k*th user can be invoked for the (*k* + 1)th user during the *n*th iteration as

*Nc*− 1) additions, 2

*Nc*multiplications per bit, namely (2 − 1/

*Nc*) addition, 2 multiplications per user per chip per iteration, if repetition coding is adopted.

Thus, the overall computational cost of Algorithm I is 1 exponentiation, 1 logarithm, (6−1/*Nc*) additions and 7 multiplications per user per chip per iteration. *The Iter-SIC scheme’s computational complexity of O*(*K*) *per chip per iteration is modest*. By contrast, some typical CDMA MUD algorithms have a substantially higher complexity of *O*(*K*^{2}) per chip per iteration, such as that of the well-known SIC-MMSE detector [15].

## 3. Simulation results

In this section, the BER performance of the proposed Iter-SIC FSO scheme is evaluated for transmission over Poisson atmospheric channels.

#### 3.1. Comparison with conventional OCDMA schemes

Firstly, we compare the performance of the Iter-SIC scheme of 5 iterations to that of the conventional OCDMA schemes [16] in Fig. 3 and 4, where we support *K* = 9 users.

In Fig. 3(a), the performance of OCDMA employing a CCR and a chip-level receiver is illustrated, showing that the proposed Iter-SIC scheme is capable of efficiently mitigating the MAI for Poisson channels and hence achieves significant BER performance improvements compared to the conventional OCDMA schemes. Furthermore, the performance of the optimum OCDMA receiver is illustrated in Fig. 3(b), demonstrating that upon increasing the average photon counts per bit, the proposed Iter-SIC scheme is capable of exceeding the best possible performance of the chip-level conventional OCDMA scheme without MAI cancellation [16].

For ease of comparison, we compare the results of two OCDMA detectors and of the Iter-SIC detector in Fig. 4(a). The attainable performance of turbo code aided Iter-SIC is also portrayed, exhibiting a powerful MAI mitigation capability. On the other hand, importantly, the repetition coding rate of the Iter-SIC scheme is *R _{c}* = 1/30, which is higher than that of the spreading code rate of the conventional OCDMA schemes, such as

*R*= 1/150, hence potentially improving the bandwidth-efficiency by a factor of 5.

_{c}More insightfully, the power of the NE block is illustrated in Fig. 4(b), where the estimation of the equivalent noise becomes increasingly more accurate as the number of iterations increases.

#### 3.2. Performance over atmospheric Poisson channels

Figure 5(a) investigates the performance of the Iter-SIC scheme for transmission over atmospheric Poisson channels associated with *σ*_{R} = 0.25. The number of users supported is set to *K* = 9 and the repetition coding rate is set to *R _{c}* = 1/30. Our simulation results show that the proposed Iter-SIC scheme exhibits a rapid convergence after 5 iterations.

Figure 5(b) demonstrates the effect of the number of users associated with different photon counts per bit by maintaining the same effective throughput of *R _{c}K* = 3/10 over turbulent fading channels. As expected, the BER performance degrades upon increasing the number of users and it improves upon increasing the photon counts per bit.

#### 3.3. Convergence analysis

The achievable convergence speed is an important characteristic of iterative signal processing algorithms. Our convergence-speed evaluation method is based on quantifying both the minimum and the mean of LLRs [20], because if the minimum and the mean of the LLRs become large and reach their steady-state value after a number of iterations, then the algorithm is deemed to have converged.

In Fig. 6, both the minimum and the mean of the chip-LLRs is seen to converge after 7 iterations, while the variance of the chip-LLRs tends to zero. Hence the algorithm becomes convergent and therefore the BER also reaches its best possible value.

Thus, Fig. 6 reveals that the proposed Iter-SIC scheme is capable of rapid convergence both in the non-turbulence and turbulence-fading channel scenarios, provided that a sufficiently low FEC coding rate is adopted.

#### 3.4. Impact of background light noise

Figure 7 shows that the BER performance associated with
${n}_{b}^{\text{bit}}=30$ is close to that of
${n}_{b}^{\text{bit}}=60$, 90, 120, respectively, for both non-turbulent and turbulent fading (*σ*_{R} = 0.25) channels, which makes our proposed scheme suitable for employment in practical scenarios as it is resilient against the background-induced light-noise.

#### 3.5. Impact of the number of users with different repetition coding rates

Figure 8 depicts the BER curves associated with a different user numbers, and with different repetition coding rates. For a single-user operating in the absence of multiuser interference, Fig. 8(a) shows that the BER curve is reminiscent of a straight line, especially in the high average photon-count region. For *K* = 3, 6, 9 users, Fig. 8(a) also shows an approximately linear BER vs photon counts relationship. The reason is likely to because the multiuser interference is essentially eliminated and the multiuser scenario becomes similar to the single-user case. Thus, the BER curve exhibits an approximately linear trend. Moreover, the shot-noise is signal-dependent, which constitutes the reason for the performance gaps among the *K* = 3, 6, 9 cases, although the multiuser interference was substantially mitigated. By contrast, in the presence of multiuser interference, the BER curve of conventional chip-level OOC exhibits an error-floor, as shown in the dashed line marked by cross of Fig. 8(a). In turbulence fading channels associated with *α* = 4.1, *β* = 2.0, the error-floor occurs at a high BER level for both single-user and multiuser scenarios, as seen in Fig. 8(b).

The simulation results of Fig. 8 show that, for repetition coding, the relationship between the BER and the photon counts is nearly linear in non-turbulent channels. At the time of writing, the theoretical analysis of this phenomenon is an open problem, since the Poisson distributed iterative structure imposes a challenge.

## 4. Discussions on the employment in practical scenarios

The photon-counting Iter-SIC scheme has numerous advantages in practical situations. Firstly, our results demonstrated that the Iter-SIC scheme is resilient against the background-induced light-noise. Naturally, the additional employment of optical filters may also be necessary in practical scenarios for employment during the day time. Secondly, the Iter-SIC scheme may also be further developed to a multiple-lasers based multiple-PDs aided structure for the sake of mitigating the effects of turbulence-fading. Thus, a beneficial spatial diversity gain can be achieved for mitigating the effects of the turbulence fading. Apart from the above-mentioned factors, the absorption by water molecules also constitutes a critical impairment in FSO communications. The hybrid FSO/RF Iter-SIC scheme is an attractive candidate solution for employment in both foggy and rainy weather conditions.

## 5. Conclusions

A photon-counting Iter-SIC scheme is proposed for transmission over Poisson atmospheric channels. An efficient iterative MUD algorithm was conceived for mitigating the MAI in presence of shot noise, background radiation and turbulent fading. Our simulations demonstrated that the proposed scheme is capable of mitigating the effects of MAI at a significantly increased bandwidth-efficiency. Moreover, it is resilient against background-induced light-noise and its flexible system structure helps the Iter-SIC scheme to be employed in practical scenarios.

## Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grant No. 60802011 and the National High Technology Research and Development Program of China under Grant No. 2011AA100701. 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.

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