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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.
© 2012 Optical Society of America
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, 2]. 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, 4]. 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]. Recently, the Poisson Counting Process (PCP) established itself as a good model of shot-noise limited FSO systems [6–8]. In [6], Gappmair and Muhammad analyzed the error performance of Pulse Position Modulation (PPM) aided Poisson channels subject to Gamma-Gamma turbulence fading. In [7], 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 [9,10]. As a benefit of introducing the spatial domain for increasing the degree of freedom, MIMO FSO systems constitute promising solutions [11–14]. In [11], 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], 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, 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.
2.1. Optical transmitter
At the transmitter, the information data package d={d1, ⋯,dm, ⋯,dM} 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 is encoded by a Forward Error Correcting (FEC) code, generating a binary chip sequence , where Lb is the information bit frame length and Lc is the FEC-encoded chip frame length. After a stream-specific interleaver Πm, the permuted chip sequence becomes . Then, each set of log2 Q FEC-encoded chips is grouped together and mapped to a Q-PPM symbol, yielding the modulated sequence , where Ls 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 Im,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]
Furthermore, the scintillation parameters α and β are respectively defined as [16] where the Rytov variance is given by 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, 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 per slot follows the Poisson distribution given by [5]
where ns = ηPrTslot/(Mhf) is the average number of signal-related photon-electrons per slot at the PD, while nb = ηPbTslot/(hf) is the average number of photon-electrons per slot due to background radiation. More explicitly, Pr represents the received signal’s optical power at a single PD and Pb is the incident background power, while Tslot 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 are calculated for Q-PPM based Iter-PIC detection. For simplicity, the 2-PPM case is treated first, which is formulated as , where we have and , as illustrated in Fig. 2. The signals , represent 2-PPM slots modulated by the chip of the mth data stream. Furthermore, the slot interval is Tslot = Tsym/2, where Tsym is the duration of a modulated 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 = {Im,n, ∀m,n}, the a posteriori LLR of the chip can be expressed as
where and denote the extrinsic LLR and the a priori LLR of the chip , respectively. Furthermore, the extrinsic LLR can be calculated as where and 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
where the mean value of the slot is and where .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 are summed as . Note that since , n ∈ [1, N], are N LLRs of the same chip transmitted through different independent turbulence channels, diversity gain can be achieved by combining the components of
3.1.4. DEC stage
After diversity combining, 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]. If repetition coding is adopted, the decoded LLRs can be generated by linearly combining the chip-level LLRs as follows:
where sz ∈ s. s = [1, −1, 1, −1,⋯, 1, −1] represents the repetition code’s spreading scheme having a length of . The corresponding repetition-based FEC code is . Indeed, we can treat the decoding of a repetition code as a linear despreading process formulated as: After hard decision, the variable 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
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 , where is the kth Q-PPM symbol, while is the encoded chip-word corresponding to the symbol , k ∈ [0, (Ls/Q − 1)]. Furthermore, the pair of notations, and 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 is determined by .
Table 2. Natural Mapping for Q-PPM Symbols
The calculation of the extrinsic LLR 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
where the estimation of the equivalent noise and . Moreover, . Based on the calculated extrinsic LLRs , 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 , where Tchip and Tbit 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 bits/slot [18, 19], where Rc 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], 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 PbTbit= − 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 Eb ≜ PrTslot/(log2 Q · Rc)=PrTbit, 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 Rc = 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 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.
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 Rc = 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], 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 Rc = 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 Rc = 1/8 and Eb = −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.
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 used in Section 3.
Based on the Q-PPM mapping method in Table 2, the a posteriori LLR of is given by
where denotes the received slot sequence on the nth PD from the mth source-laser.According to Eq. (13), the extrinsic LLR can be derived as
where denotes the encoded chip-word with . Likewise, denotes the chip-word with . Moreover, and denote the set of chip-words having , , respectively.Then, we let the slot-word and , which corresponds to the Q-PPM mapping method of Table 2. Thus, the further derivation of the extrinsic LLR can be
where without . without . In addition, , .After that, we let For Q-PPM, only one slot is “1” while other Q−1 slots are “0”, as shown in Table 2. For , indicates that . For , implies that . Thus,
where with . Furthermore, , and . Meanwhile, and denote the index set {i} where in the chip-word , , respectively.Then, the equivalent noise can be estimated by
where and . As shown in Table 1, in the first iteration, we have . Then, is updated by the LLR fed back from the DEC block of Fig. 1. Based on Eq. (17), the extrinsic LLR can also be calculated.Finally, with Eqs. (16)–(18), the Q-PPM based extrinsic LLR can be achieved as
Based on the calculated extrinsic LLRs , 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.
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