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

doi:10.1364/OE.20.026379

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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 »View Author Affiliations

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

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▸ Article Outline
– Abstract
1. Introduction
2. System model
3. Iterative receiver
4. Numerical results
5. Conclusions
– References and links

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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.

1. Introduction

Free-Space Optical (FSO) systems constitute an attractive design alternative to Radio-Frequency (RF) systems in the context of ultra-fast data transmission [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. IEEE 100(13), 1853–1888, (2012). [CrossRef]

]. However, the FSO links are typically vulnerable to the effect of absorption (e.g. fog), scattering and scintillation, which may reduce the received power [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]

, 4

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]

]. In low received signal FSO scenarios, the signal-dependent shot-noise becomes a dominant performance limiting factor, which is quite different from the conventional Additive White Gaussian Noise (AWGN) model of RF systems [5

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]

]. Recently, the Poisson Counting Process (PCP) established itself as a good model of shot-noise limited FSO systems [6

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]

–8

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]

]. In [6

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]

], Gappmair and Muhammad analyzed the error performance of Pulse Position Modulation (PPM) aided Poisson channels subject to Gamma-Gamma turbulence fading. In [7

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]

], Chakraborty et al. derived the exact expression for the outage capacity of Multiple-Input Multiple-Output (MIMO) FSO systems over Poisson channels.

In addition to shot-noise limitation and atmospheric turbulence fading, the rapid development of multi-Gb/s FSO systems is hampered by the limited availability of sufficient wavelength/time resources [9

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]

,10

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

]. As a benefit of introducing the spatial domain for increasing the degree of freedom, MIMO FSO systems constitute promising solutions [11

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]

–14

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]

]. In [11

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 coherent MIMO FSO scheme was developed for striking a diversity-multiplexing trade-off. Although coherent transmission is capable of achieving a better outage probability performance, the complexity of coherent optical receivers is much higher than that of the Intensity Modulation/Direct Detection (IM/DD) receiver proposed in this paper. In [12

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]

], uncoded MIMO FSO transmissions were investigated in Gamma-Gamma fading channels, where the performance analysis of the MIMO FSO scheme was based on the conventional AWGN model, rather than on the PCP model used in this study for low optical signal power scenarios. Furthermore, in [5

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]

, 13

], a widely used repetition-code aided MIMO FSO system was analyzed, where all source-lasers transmit the same symbol, hence a beneficial transmit diversity is achieved at the cost of sacrificing the achievable throughput. Against this background, we set out to beneficially exploit the available spatial resources for the sake of increasing both the system’s throughput as well as integrity.

More explicitly, the novel contribution of this paper is that we propose a Photon-Counting Spatial-Diversity-and-Multiplexing (PC-SDM) scheme for achieving a high-throughput, high-integrity as well as energy efficient, fading-resilient FSO transmission under the shot-noise limited model, where multiple transmitters are employed for parallel stream multiplexing and multiple receivers are invoked for diversity reception. This receiver was explicitly designed for combating the atmospheric turbulence fading, which relies on an Iterative Parallel Interference Cancellation (Iter-PIC) algorithm conceived for mitigating the effect of multi-stream interference. Moreover, a general Q-PPM based iterative Log-Likelihood Ratio (LLR) calculation algorithm is designed.

The remainder of this paper is organized as follows. Section 2 describes the proposed PC-SDM FSO system. The Q-PPM based Iter-PIC algorithm is derived in Section 3. Section 4 presents our simulation results for transmission over Gamma-Gamma fading channels. Finally, we conclude in Section 5.

2. System model

As shown in Fig. 1, the PC-SDM MIMO array has M source-lasers and N Photo-Detectors (PDs). For practicality, the IM/DD technique is employed in this paper.

Fig. 1 Model of the Q-PPM based PC-SDM FSO system over Poisson atmospheric channels.

2.1. Optical transmitter

At the transmitter, the information data package d={d₁, ⋯,d_m, ⋯,d_M} is subdivided into M data streams after a Serial-to-Parallel (S/P) converter. Let m ∈ [1,M] be the stream index. The mth stream

d_{m} = {d_{m}^{1}, \dots, d_{m}^{i}, \dots d_{m}^{L_{b}}}

is encoded by a Forward Error Correcting (FEC) code, generating a binary chip sequence

c_{m} = {c_{m}^{1}, \dots, c_{m}^{l}, \dots c_{m}^{L_{c}}}

, where L_b is the information bit frame length and L_c is the FEC-encoded chip frame length. After a stream-specific interleaver Π_m, the permuted chip sequence becomes

x_{m} = {x_{m}^{1}, \dots, x_{m}^{v}, \dots, x_{m}^{L_{c}}}

. Then, each set of log₂ Q FEC-encoded chips is grouped together and mapped to a Q-PPM symbol, yielding the modulated sequence

s_{m} = {s_{m}^{1}, \dots, s_{m}^{j}, \dots, s_{m}^{L_{s}}}

, where L_s is the symbol frame length. It is then used for driving the optical modulator and transmitted to the receiver via FSO turbulence fading channels.

2.2. Poisson atmospheric channel model

For the atmospheric turbulence-induced fading, we let I_m,n denote the real-valued fading gain between the mth source-laser and the nth receiver PD, which obeys the Gamma-Gamma distribution having the PDF of [15

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

]

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 .

(1)

Furthermore, the scintillation parameters α and β are respectively defined as [16

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]

]

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

(2)

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},

(3)

where the Rytov variance is given by

χ^{2} = 0.5 C_{n}^{2} κ^{7 / 6} L^{11 / 6}

and the geometry factor is defined as d = (κD²/(4L))^1/2, with L representing the distance, D denoting the diameter of the receiver lens aperture,

C_{n}^{2}

indicating the refractive index and κ = 2π/λ representing the wave number associated with the wavelength of λ. Finally, the scintillation index is defined as S.I. = α⁻¹ + β⁻¹ + (αβ)⁻¹.

At the receiver, the PCP model is adopted. Consequently, the received electron-count

r_{n}^{j}

per slot follows the Poisson distribution given by [5

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]

]

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})],

(4)

where n_s = ηP_rT_slot/(Mhf) is the average number of signal-related photon-electrons per slot at the PD, while n_b = ηP_bT_slot/(hf) is the average number of photon-electrons per slot due to background radiation. More explicitly, P_r represents the received signal’s optical power at a single PD and P_b is the incident background power, while T_slot is the slot duration, f is the optical frequency and h is Plank’s constant and finally η is the quantum efficiency. The subscripts m, n denote the signal transmitted from the mth source-laser and received at the nth PD. The superscript j is the index of the jth time slot.

3. Iterative receiver

The proposed iterative receiver of Fig. 1 consists of N Multi-stream Interference Cancellers (MIC), N iterative Noise Estimators (NE) and M channel DECoders (DECs).

3.1. 2-PPM based Iter-PIC detection

In the MIC block of Fig. 1, the extrinsic LLRs

{L_{MIC}^{e} (x_{m})}

are calculated for Q-PPM based Iter-PIC detection. For simplicity, the 2-PPM case is treated first, which is formulated as

s_{m} = [s_{m}^{1}, s_{m}^{2}] = map ([x_{m}^{1}])

, where we have

[1, 0] = map ([x_{m}^{1} = 0])

and

[0, 1] = map ([x_{m}^{1} = 1])

, as illustrated in Fig. 2. The signals

s_{m}^{1}

s_{m}^{2}

represent 2-PPM slots modulated by the chip

x_{m}^{1}

of the mth data stream. Furthermore, the slot interval is T_slot = T_sym/2, where T_sym is the duration of a modulated 2-PPM symbol.

Fig. 2 Mapping example of the 2-PPM symbol.

3.1.1. MIC stage

As shown in Fig. 1, in the MIC block, given the turbulence channel’s output observation I = {I_m,n, ∀m,n}, the a posteriori LLR of the chip

x_{m, n}^{1}

can be expressed as

\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}

(5)

where

L_{MIC}^{e} (x_{m, n}^{1})

and

L_{MIC}^{a} (x_{m, n}^{1})

denote the extrinsic LLR and the a priori LLR of the chip

x_{m, n}^{1}

, respectively. Furthermore, the extrinsic LLR

L_{MIC}^{e} (x_{m, n}^{1})

can be calculated as

\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}

(6)

where

ξ_{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}

and

ξ_{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}

are defined as the equivalent noise imposed by the multi-stream interference and background radiation.

3.1.2. NE stage

In the iterative NE block of Fig. 1, the equivalent noise can be estimated as

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},

(7)

where the mean value of the slot

s_{m, n}^{1}

\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}

(8)

and

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},

(9)

where

E [s_{m, n}^{2}] = Pr (s_{m, n}^{2} = 1) = Pr (x_{m, n}^{1} = 1) = \frac{exp [L_{MIC}^{a} (x_{m, n}^{1})]}{1 + exp [L_{MIC}^{a} (x_{m, n}^{1})]}

3.1.3. Diversity combining stage

For combating the turbulence-induced fading, the diversity reception is invoked. As shown in Fig. 1, the deinterleaved LLRs

{L_{DEC}^{a} (c_{m, n}^{1})}

are summed as

L_{DEC}^{a (com)} (c_{m}^{1}) = \sum_{n = 1}^{N} L_{DEC}^{a} (c_{m, n}^{1})

. Note that since

{L_{DEC}^{a} (c_{m, n}^{1})}

, n ∈ [1, N], are N LLRs of the same chip

c_{m}^{1}

transmitted through different independent turbulence channels, diversity gain can be achieved by combining the components of

{L_{DEC}^{a} (c_{m, n}^{1})}

3.1.4. DEC stage

After diversity combining,

{L_{DEC}^{a (com)} (c_{m}^{l}), l \in [1, L_{c}]}

is then forwarded as a priori information to the soft DEC of Fig.1. In this paper, we assume an idealised random interleaver/deinterleaver, hence the encoded chips become entirely independent. In the DEC block, the a posteriori probability (APP) decoder obeys the standard approach of [17

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

]. If repetition coding is adopted, the decoded LLRs

{L_{DEC}^{Bit} (d_{m}^{i}), i \in [1, L_{b}]}

can be generated by linearly combining the chip-level LLRs

{L_{DEC}^{a (com)} (c_{m}^{l})}

as follows:

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},

(10)

where s_z ∈ s. s = [1, −1, 1, −1,⋯, 1, −1] represents the repetition code’s spreading scheme having a length of

L_{c}^{FEC}

. The corresponding repetition-based FEC code is

c_{m}^{i} = {\begin{matrix} [1, 0, 1, 0, \dots] & d_{m}^{i} = 1 \\ [0, 1, 0, 1, \dots] & d_{m}^{i} = 0 \end{matrix}

. Indeed, we can treat the decoding of a repetition code as a linear despreading process formulated as:

[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 .

(11)

After hard decision, the variable

d_{m}^{i} = {\begin{array}{l} 1 & if & L_{DEC}^{Bit} (d_{m}^{i}) \geq 0 \\ 0 & if & L_{DEC}^{Bit} (d_{m}^{i}) < 0 \end{array}}

is used for BER calculation. For unambiguous presentation, the pseudo-code of our Iter-PIC detection algorithm is recorded in Table 1, which is based on the system architecture of Fig. 1.

Table 1 2-PPM based Iter-PIC Algorithm

Step 1: Initialization. Set v = 0 and

L_{MIC}^{a (v)} (x_{m, n}^{1}) = 0, \forall m, n

Step 2:

(2.1) In the NE block of Fig. 1, calculate $ξ_{m, n}^{Est (1)}$ , $ξ_{m, n}^{Est (2)}$ by Eq. (7), Eq. (9).
(2.2) In the MIC block of Fig. 1, calculate $L_{MIC}^{e} (x_{m, n}^{1})$ by Eq. (6).
(2.3) After deinterleaving, diversity gain is achieved by combining the LLRs as $L_{DEC}^{a (com)} (c_{m}^{1}) = \sum_{n = 1}^{N} L_{DEC}^{a} (c_{m, n}^{1})$ .
(2.4) APP decoding is used for generating the reliable LLR $L_{DEC}^{LLR} (c_{m}^{1})$ .
(2.5) To extract the extrinsic LLRs, the following subtraction is performed, $L_{DEC}^{e} (c_{m, n}^{1}) = L_{DEC}^{LLR} (c_{m}^{1} - L_{DEC}^{a}) (c_{m, n}^{1})$ .
(2.6) After interleaving, update $L_{MIC}^{a (v + 1)} (x_{m, n}^{1})$ for noise estimation.
(2.7) v ← v + 1 and go back to step (2.1).

Step 3: After a number of iterations, the recovered sequence d̃_m is obtained subject to hard decisions in the DEC block of Fig. 1. Then, after the Parallel-to-Serial (P/S) converter of Fig. 1, the detected information d̃ becomes available.

3.2. Q-PPM based Iter-PIC detection

The algorithm of the 2-PPM Iter-PIC detection is now extended to the general Q-PPM case. The simple natural mapping adopted is shown in Table 2, which is denoted as

s_{m}^{k} = map (x_{m}^{k})

, where

s_{m}^{k} = [s_{m}^{k, 1}, \dots, s_{m}^{k, q}, \dots, s_{m}^{k, Q}]

is the kth Q-PPM symbol, while

x_{m}^{k} = [x_{m}^{k, 1}, \dots, x_{m}^{k, z}, \dots, x_{m}^{k, {log}_{2} Q}]

is the encoded chip-word corresponding to the symbol

s_{m}^{k}

, k ∈ [0, (L_s/Q − 1)]. Furthermore, the pair of notations,

s_{m}^{j}

and

s_{m}^{k, q}

represent the same Q-PPM slot, when j = kQ+q. For simplicity, these two terms will be used interchangeably in the following. The position of pulse “1” in

s_{m}^{k}

is determined by

q = \sum_{z = 1}^{{log}_{2} Q} x_{k, z} 2^{z - 1} + 1

Table 2 Natural Mapping for Q-PPM Symbols

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

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

The calculation of the extrinsic LLR

L_{MIC}^{e} (x_{m, n}^{k, z})

in the general Q-PPM case is more complicated than for the 2-PPM case. Hence the full mathematical derivation is relegated to the Appendix, yielding

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})},}

(12)

where the estimation of the equivalent noise

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}

and

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

. Moreover,

L^{a} (x_{m, n}^{k, i}) = log \frac{Pr [x_{m, n}^{k, i} = 1]}{Pr [x_{m, n}^{k, i} = 0]}

. Based on the calculated extrinsic LLRs

{L_{MIC}^{e} (x_{m, n}^{k, z})}

, the iterative soft-information exchange continues between the MIC and DEC of Fig. 1 in a similar manner as in the 2-PPM case, which is shown in Table 1. For the Q-PPM case, we have

T_{chip} = T_{bit} R_{c} = T_{slot} \frac{Q}{{log}_{2} Q}

, where T_chip and T_bit denote the chip and bit interval, respectively. Furthermore, since the independent multiple-streams are transmitted simultaneously from different transmitters, spatial multiplexing gain is achieved. The achievable throughput of the PC-SDM MIMO system is

M R_{c} \frac{{log}_{2} Q}{Q}

bits/slot [18

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

, 19

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]

], where R_c is the FEC coding rate.

4. Numerical results

In this section, simulations are conducted for evaluating the attainable performance of the proposed PC-SDM scheme over Gamma-Gamma turbulence fading channels. Using the previously reported simulation parameters [5

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]

], the quantum efficiency is set to η = 0.5, while the optical center wavelength to λ = 1.55μm. The energy of the background radiation power per bit duration is set to P_bT_bit= − 170dBJ. In the (M × N)-element MIMO FSO system, the adequate alignment between the transmitter as well as receiver is assumed along with having a sufficiently high spatial separation amongst the front end devices of different channels. For a fair comparison, the total power of the laser array is fixed, which is assumed to be equally shared among the M source-lasers. The received signal energy per bit becomes E_b ≜ P_rT_slot/(log₂ Q · R_c)=P_rT_bit, which is the total energy impinging on a single PD from all the M lasers.

4.1. Benefit 1: the BER is enhanced owing to the diversity gain

Figure 3 shows the substantial diversity gain of PC-SDM FSO transmission for N = 1, 3, 5 and M = 1, even in the multi-stream interference scenarios of M = 3 and M = 5. Figure 3 also demonstrates the flexibility of the PC-SDM scheme upon increasing the number of source-lasers (M) and receiver PDs (N).

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.

Figure 4(a) confirms that for a strong turbulence fading channel (α = 4, β = 1), a significant diversity gain can be achieved. Furthermore, as seen in Fig. 4(b), for a less hostile turbulence fading (α = 4, β = 4), the diversity gain becomes higher. The reason is that the lower the turbulence, the more reliable the LLRs

{L_{DEC}^{a} (c_{m, n}^{k, l})}

become, yielding an increased diversity gain. The performance of the non-fading scenario is also included as a benchmark. Thanks to the attainable diversity- and array-gain, the PC-SDM FSO scheme relying on a multiple-PD receiver communicating over turbulence fading channels is capable of exceeding the BER performance of single-PD receiver (N=1) operating in non-fading channels. More explicitly, as shown in Fig. 4(b), the receiver (M=2, N=5) communicating over turbulence fading channels has a better BER performance than the receiver (M=2, N=1) communicating in non-fading channels.

Fig. 4 BER for different scintillation indeces, 2-PPM and R_c = 1/8.

4.2. Benefit 2: the throughput is increased owing to the multiplexing gain

Figure 5 shows the impact of simultaneous multi-stream transmissions leading to multiplexing gain. Observe in Fig. 5(a), that upon increasing the number of streams (M), the throughput is increased from 4/16 to 5/16 and 6/16 bits/slot, respectively. As seen in Fig. 5(b), upon using N=2 PDs, the BER performance improves quite sharply, as a result of having both multiplexing and diversity gain.

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.

4.3. Benefit 3: energy efficiency and rapid convergence

Figure 6(a) shows the attainable BER performance of various Q-PPM schemes. Explicitly, upon increasing Q, the BER performance is improved due to the better energy efficiency of Q-PPM. Figure 6(b) indicates that the PC-SDM scheme is capable of rapid convergence in turbulence-fading channel scenarios, provided that a sufficiently low repetition coding rate is used. Our convergence-speed evaluation method is based on quantifying the mean of the LLRs [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]

], assuming that if the mean of the LLRs becomes a sufficiently high near-constant value after a number of iterations, the iterative algorithm is deemed to have converged. In Fig. 6(b), the mean of the chip LLRs is seen to converge after 4 iterations to a fixed value, while the variance of the chip LLRs tends to zero, hence the algorithm becomes convergent.

Fig. 6 Comparison of various Q-PPM schemes for Gamma-Gamma fading links (α = 4, β = 4) with R_c = 1/8.

4.4. Benefit 4: efficient noise estimation

When evaluating the performance of the iterative NE block of Fig. 1 for transmission over strong turbulence fading channels, Fig. 7 demonstrates that the estimation of noise power becomes increasingly more accurate, as the number of iterations increases. This also empirically reflects the rapid convergence of our algorithm.

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.

5. Conclusions

In this paper, a new PC-SDM FSO system was proposed. For the shot-noise-limited Poisson atmospheric channel, a Q-PPM based multi-stream Iter-PIC algorithm was derived. Our simulation results demonstrate that the PC-SDM scheme is capable of achieving error-resilient and high-throughput FSO transmissions with the aid of our flexible MIMO architecture.

Appendices

Appendix: the algorithm of the Q-PPM based extrinsic LLR calculation

In this appendix, we derive the algorithm of calculating the Q-PPM based extrinsic LLRs

L_{MIC}^{e} (x_{m, n}^{k, z})

used in Section 3.

Based on the Q-PPM mapping method in Table 2, the a posteriori LLR of

x_{m, n}^{k, z}

is given by

\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}

(13)

where

r_{m, n}^{k} = {r_{m, n}^{k, Q}, \dots, r_{m, n}^{k, 2}, r_{m, n}^{k, 1}}

denotes the received slot sequence on the nth PD from the mth source-laser.

According to Eq. (13), the extrinsic LLR

L_{MIC}^{e} (x_{m, n}^{k, z})

can be derived as

\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}

(14)

where

{x_{m, n}^{k, {log}_{2} Q}, \dots, x_{m, n}^{k, z} = 1, \dots, x_{m, n}^{k, 1}} = x_{m, n}^{k, z, (1)}

denotes the encoded chip-word with

x_{m, n}^{k, z} = 1

. Likewise,

{x_{m, n}^{k, {log}_{2} Q}, \dots, x_{m, n}^{k, z} = 0, \dots, x_{m, n}^{k, 1}} = x_{m, n}^{k, z, (0)}

denotes the chip-word with

x_{m, n}^{k, z} = 0

. Moreover,

Ω (x_{m, n}^{k, z} = 1)

and

Ω (x_{m, n}^{k, z} = 0)

denote the set of chip-words

{x_{m, n}^{k, z}}

having

x_{m, n}^{k, z} = 1

x_{m, n}^{k, z} = 0

, respectively.

Then, we let the slot-word

s_{m, n}^{k, (1)} = map (x_{m, n}^{k, z, (1)})

and

s_{m, n}^{k, (0)} = map (x_{m, n}^{k, z, (0)})

, which corresponds to the Q-PPM mapping method of Table 2. Thus, the further derivation of the extrinsic LLR

L_{MIC}^{e} (x_{m, n}^{k, z})

can be

\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}

(15)

where

{\tilde{x}}_{m, n}^{k, z, (1)} \subset x_{m, n}^{k, z, (1)}

without

x_{m, n}^{k, z} = 1

{\tilde{x}}_{m, n}^{k, z, (0)} \subset x_{m, n}^{k, z, (0)}

without

x_{m, n}^{k, z} = 0

. In addition,

s_{m, n}^{k, q} \in [0, 1]

x_{m, n}^{k, i} \in [0, 1]

After that, we let

L^{a} (x_{m, n}^{k, i}) = log \frac{Pr [x_{m, n}^{k, i} = 1]}{Pr [x_{m, n}^{k, i} = 0]}

For Q-PPM, only one slot is “1” while other Q−1 slots are “0”, as shown in Table 2. For

x_{m, n}^{k, i} \in x_{m, n}^{k, z, (1)}

u = \sum_{i = 1}^{{log}_{2} Q} x_{m, n}^{k, i} 2^{i - 1} + 1

indicates that

s_{m, n}^{k, u} = 1

. For

x_{m, n}^{k, i} \in x_{m, n}^{k, z, (0)}

w = \sum_{i = 1}^{{log}_{2} Q} x_{m, n}^{k, i} 2^{i - 1} + 1

implies that

s_{m, n}^{k, w} = 1

. Thus,

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})}},

(16)

where

\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}

(17)

with

ξ_{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}

. Furthermore,

L^{e} (s_{m, n}^{k, w}) = r_{m, n}^{k, w} log [1 + \frac{n_{s} I_{m, n}^{k, w}}{ξ_{m, n}^{k, w}}] - n_{s} I_{m, n}^{k, w}

, and

ξ_{m, n}^{k, w} = n_{s} \sum_{\tilde{m} = 1, \tilde{m} \neq m}^{M} I_{\tilde{m}, n}^{k, w} s_{\tilde{m}, n}^{k, w} + n_{b}

. Meanwhile,

Ψ (x_{m, n}^{k, i} = 1)

and

Θ (x_{m, n}^{k, i} = 1)

denote the index set {i} where

x_{m, n}^{k, i} = 1

in the chip-word

x_{m, n}^{k, z, (1)}

x_{m, n}^{k, z, (0)}

, respectively.

Then, the equivalent noise

ξ_{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}

can be estimated by

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},

(18)

where

E [s_{m, n}^{k, u}] = Pr (s_{m, n}^{k, u} = 1) = Pr (x_{m, n}^{k}) = \sum_{j = 1}^{{log}_{2} Q} Pr (x_{m, n}^{k, j}) = \sum_{j = 1}^{{log}_{2} Q} Φ_{m, n}^{j}

and

Φ_{m, n}^{j} = {\begin{array}{l} \frac{exp [L^{a} (x_{m, n}^{k, j})]}{1 + exp [L^{a} (x_{m, n}^{k, j})]}, & if & Pr (x_{m, n}^{k, j}) = 1, \\ \frac{1}{1 + exp [L^{a} (x_{m, n}^{k, j})]}, & if & Pr (x_{m, n}^{k, j}) = 0 \end{array}

. As shown in Table 1, in the first iteration, we have

L^{a} (x_{m, n}^{k, j}) = 0

. Then,

L^{a} (x_{m, n}^{k, j})

is updated by the LLR

L_{DEC}^{e} (c_{m, n}^{k, l})

fed back from the DEC block of Fig. 1. Based on Eq. (17), the extrinsic LLR

L^{e} (s_{m, n}^{k, w})

can also be calculated.

Finally, with Eqs. (16)–(18), the Q-PPM based extrinsic LLR

L_{MIC}^{e} (x_{m, n}^{k, z})

can be achieved as

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})}} .

(19)

Based on the calculated extrinsic LLRs

{L_{MIC}^{e} (x_{m, n}^{k, z})}

, the Q-PPM based Iter-PIC algorithm can be performed in a similar manner as in the 2-PPM case, shown in Table 1.

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.

References and links

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. IEEE 100(13), 1853–1888, (2012). [CrossRef]
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]
4.	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]
5.	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]
6.	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]
7.	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]
8.	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]
9.	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]
10.	J. Wells, “Faster than fiber: the future of multi-Gb/s wireless,” IEEE Microw. Mag. 10(3), 104–112 (2009). [CrossRef]
11.	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]
12.	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]
13.	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]
14.	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]
15.	L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media , 2nd ed. (SPIE Press, 2005). [CrossRef]
16.	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]
17.	C. Berrou and A. Glavieux, “Near optimum error correcting coding and decoding: turbo-codes,” IEEE Trans. Commun. 44(10), 1261–1271 (1996). [CrossRef]
18.	C. Georghiades, “Modulation and coding for throughput-efficient optical systems,” IEEE Trans. Inform. Theory 40(5), 1313–1326 (1994). [CrossRef]
19.	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]
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]

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:
Atmospheric and Oceanic Optics

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

Citation
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)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-24-26379

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References

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. IEEE100(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. Express18(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. 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]
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. Theory40(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]

Aghajanzadeh, S. M.
Andrews, L. C.
Baedke, M.
Bayaki, E.
Berrou, C.
Brandt-Pearce, M.
Cao, Q.
Chakraborty, K.
Chan, V. W. S.
Cheng, J.
Dey, S.
Djordjevic, I. B.
Farid, A. A.
Franceschetti, M.
Gappmair, W.
Georghiades, C.
Glavieux, A.
Gulliver, T. A.
Gyongyosi, L.
Haas, H.
Hanzo, L.
Holzman, J. F.
Hranilovic, S.
Imre, S.
Lampe, L.
Langer, K. D.
Leveque, J. H.
Li, J.
Mallik, R.
Muhammad, S. S.
Neifeld, M. A.
Niu, M.
O’Brien, D.
Paraskevopoulos, A.
Phillips, R. L.
Reid, A. C.
Riediger, M. L. B.
Rupp, M.
Schober, R.
Swoboda, R.
Taylor, D. P.
Uysal, M.
Vasic, B.
Voss, S. H.
Vucic, J.
Wells, J.
Wilson, S. G.
Yu, M.
Zhang, R.

Electron. Lett.

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.

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.

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.

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]

IEEE Trans. Commun.

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

K. Chakraborty, S. Dey, and M. Franceschetti, “Outage capacity of the MIMO Poisson fading channels,” IEEE Trans. Infor. Theory54(11), 4887–4907 (2008). [CrossRef]

IEEE Trans. Inform. Theory

C. Georghiades, “Modulation and coding for throughput-efficient optical systems,” IEEE Trans. Inform. Theory40(5), 1313–1326 (1994). [CrossRef]

IEEE Trans. Wireless Commun.

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]
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]

IEEE/ASME Trans. Mech.

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]

Proc. IEEE

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(13), 1853–1888, (2012). [CrossRef]

Other

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

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