Blind Detection for Serial Relays in Free Space Optical Communication Systems

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Introduction
Optical wireless communication refers to the transmission of data in an unguided propagation media through the use of optical carriers [1][2][3][4][5].Optical wireless communications (OWCs) can be divided into five categories; ultra-short range OWC, short-range OWC, medium range OWC, long-range OWC and ultra-long range OWC [1,2,4].Long range OWC systems are referred to as Free Space Optical (FSO), which is the OWC used between buildings [1,2,4].
Free-space optical (FSO) communications have recently been adopted due to its high data rate (up to 10 Gbps), large bandwidth, transmission distance up to several kilometers, frequency above 300 GHz, which is a license-free spectrum, easily and quickly deployed and finally, reinstallation capability of the system [1][2][3][4]6].
To evaluate the atmospheric turbulence effect, two methods were used.The first method is the Channel State Information (CSI) method, where pilot signals were sent to detect the channel condition [7,13].For the mentioned systems, the receiver would require knowledge of the CSI to adjust the detection threshold [17].To increase the bandwidth efficiency of FSO channels and to decrease the complexity, the second method was introduced in [13,[17][18][19].The second method is the blind detection method.The blind detection does not use pilot signals.It decreases the complexity of the receiver and increases the throughput [7,13].
A blind practical method for detecting wireless optical communication was introduced in [18].In [20], the authors proposed an iterative sequence detector based on [18].Due to the Bit Error Rate (BER) in [18,20] being floored for a small size observation window, the authors of [7] introduced an improved data detection methodology.This methodology, introduced in [7], used the specification of channel probability density function (PDF) and proposed a new algorithm that decreased the error floor.The authors of [13] have also derived an analytical expression of a conditional BER of the system proposed on [18]; however, the average BER was not derived.
One of the mitigation techniques used for the atmospheric turbulence effect and path loss attenuation is the assisted relay technique [2,6].The assisted relay technique helps in shortening the hops, and thus, improving the performance significantly [2,6].
Table 1 accommodates some of the key pros and cons of the previous literature that used the blind detection method with FSO communication systems.

Number
Pros Cons [7] Novel data detection methodology No mathematical model [13] Two decision steps Monte Carlo simulations is missing [15] Gamma-gamma channel which covers moderate to strong range of atmospheric turbulence Missing of weak turbulence regime [16] Proposed data efficient channel estimation method Single iteration method [18] Two decision steps No closed form equation In this paper, we employed a blind detection method for a serial decode and forward relay of the FSO system over a log-normal channel.The novelty of this work is as follows:

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The combined advantages of blind detection and assisted relays motivated the research group to investigate the combined system in more details.

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Approximated average BER, which was not derived earlier in [7,13], is derived within the paper.
Although the derivation of the analytical model is more difficult than the Monte Carlo simulation, it allows a better understanding of the proposed system and an easier comparison between the proposed model and other ones.

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The Monte Carlo simulation verifies the correctness of the derived expression.

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The obtained results show the closeness of the employed method using small windows compared with channel state information.

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The new results would promote the use of blind detection for more applications.
The remainder of this paper is organized as follows.Section 2 describes the system model.Section 3 presents our derived closed-form error probability for blind OOK detection.Our numerical results and discussions are presented in Section 4. Section 5 is devoted to the main conclusions.

System Model
An FSO system with IM/DD and OOK modulation is considered.The received signal of hop i, where s i [k] ∈ {0, 1}, is the transmitted OOK symbol and n i [k] is the signal-independent additive white Gaussian noise with zero mean and variance σ 2 n = N 0 /2.The channel coefficient can be formulated as h = h l h a , where fading coefficient h a , is the time-varying channel state due to atmospheric turbulence and is considered to be constant over a large number of transmitted bits.h l is the channel attenuation.
The average electrical signal to noise ratio (SNR) is defined as denotes the statistical expectation.The proposed system uses a single decode and a forward relay.The relay would help to shorten the communication link as shown in Figure 1.The impact of the decrease of the link length would significantly reduce the turbulence effect and path loss.

System Model
An FSO system with IM/DD and OOK modulation is considered.The received signal of hop i,  The average electrical signal to noise ratio (SNR) is defined as , where

[.] E
denotes the statistical expectation.The proposed system uses a single decode and a forward relay.The relay would help to shorten the communication link as shown in Figure 1.The impact of the decrease of the link length would significantly reduce the turbulence effect and path loss.

Channel Model
A log-normal channel for both weak and moderate turbulence is used.The log-normal distribution for the channel coefficient is defined as [14] ( ) , where x is normally distributed random variable with mean and variance  .The channel attenuation can be modeled as [13] ( ) where r A is the aperture area, θ is divergence angle, 0 d is the distance between the transmitter and the receiver and 0  is extinction coefficient.

For Perfect CSI
For a receiver with perfect CSI, the maximum likelihood (ML) decision rule for the kth bit and the optimum threshold are defined, respectively, as in [13]:

Channel Model
A log-normal channel for both weak and moderate turbulence is used.The log-normal distribution for the channel coefficient is defined as [14] f LN (h a ) = 1 with h a = exp(2x), where x is normally distributed random variable with mean µ x and variance σ 2 x .The channel attenuation can be modeled as [13] where A r is the aperture area, θ is divergence angle, d 0 is the distance between the transmitter and the receiver and β 0 is extinction coefficient.

For Perfect CSI
For a receiver with perfect CSI, the maximum likelihood (ML) decision rule for the kth bit and the optimum threshold are defined, respectively, as in [13]: where ŝ[k] is the detected bit.

For Blind Data Detection
To estimate the channel, the two-stage block by block receiver is performed: where τ 1 is the first threshold used for the first step, and L is the observation window.Using the threshold, the first step of blind detection is: where s [k] is the first step of decision for s[k].
The second stage is performed as in [13] with the simplified form introduced in [7] where which would result in where s [k] is the second step of decision for s[k].In the proposed system, using the relay technique, stages one and two are performed at the relay and at the destination.

Derivation of Closed Form Average BER Expression
In this section, the equations used for calculating the average BER of the proposed scheme over the log-normal channel are discussed.The derivation of the average BER allowed us a better understanding of the proposed system, reduced long simulation time and an eased the comparison between the proposed model and other ones.

Average BER for Log-Normal Channel
The average BER of the proposed scheme is defined as in [13]: where the P(e/L, h) is defined as in [13]: where L M is the number of combinations of M items out of L.
P(e/L, M, h) is defined as in [13]: where is the number of ones in the observation window with length L.
The Gaussian Q function is [21]: The PDF of the fading coefficient, h, which is given as h = h l h a : Substituting ( 14) and ( 17) in ( 13), we get: Making the change of variables: This results in: This integration in (20) could be solved using Hermite polynomial approximation [22]: where w i and x i are the weights and the roots of the Hermite polynomial of order n, respectively.Applying Equation ( 21) on Equation ( 20) yields: For the dual hop system proposed, an approximated average bit error rate (ABER) is calculated [23]: where ABER 1 and ABER 2 are the ABER for the first and second node, respectively.

Numerical Results and Discussion
In the obtained simulation results, 10 8 bits were transmitted for each depicted SNR value and the Hermite polynomial order was n ≤ 50.Now, we evaluate the ABER performance of the dual hop blind detection FSO system for various locations of the relay.The ABER performance was calculated over weak and moderate turbulence in order to investigate the degradation effects.The direct link simulation results, investigated in [13], were used as a benchmark to show the performance difference between the direct link and the proposed model.For the direct link, the power value was multiplied by two to achieve a fair comparison [6].Table 2 shows the system parameters under investigation.Table 3 shows the standard deviation values with the multiple scenarios under weak and moderate atmospheric turbulence.

Symbol Value
A r 2π(0.1) 2 The derived analytical results were verified with Monte Carlo simulations.Figures 2-4 show the weak turbulence.Figure 5 shows the combined analytical values for moderate turbulence.Figures 2-5 show that the ABER decreased when a relay is added due to decreasing the distance.When comparing the figures with weak turbulence, Figure 2 with Figures 3 and 4, it can be seen that the best location for the relay is in the middle of the channel.We have tried all the ranges for the relay, which showed that the equidistant relay is the best location.Figure 5 shows that the best location for the relay for moderate turbulence is also in the middle of the channel.The derived analytical results were verified with Monte Carlo simulations.Figures 2-4 show the weak turbulence.Figure 5 shows the combined analytical values for moderate turbulence.Figures 2-5 show that the ABER decreased when a relay is added due to decreasing the distance.When comparing the figures with weak turbulence, Figure 2 with Figures 3 and 4, it can be seen that the best location for the relay is in the middle of the channel.We have tried all the ranges for the relay, which showed that the equidistant relay is the best location.Figure 5 shows that the best location for the relay for moderate turbulence is also in the middle of the channel.The generated analytical results are comparable to the CSI results as shown in Figures 2-4.The generated simulation results show the accuracy of the analytical results in Figures 2-4. Figure 2 shows an enhancement of around 10 dB between the direct link and using the relay.
Figures 3 and 4 also show an improvement of the proposed relay system by 6 dB. Figure 2 shows the equidistant relay outperforms its counterparts in Figures 3 and 4  As for the moderate turbulence, shown in Figure 5, an enhancement of 17 dB between the direct link and the equidistant relay can be seen.An enhancement of 12.5 dB was also obtained between the direct link and the other relays (600 m-400 m and 400 m-600 m).The generated analytical results are comparable to the CSI results as shown in Figures 2-4.The generated simulation results show the accuracy of the analytical results in Figures 2-4. Figure 2 shows an enhancement of around 10 dB between the direct link and using the relay.

Conclusions
Figures 3 and 4 also show an improvement of the proposed relay system by 6 dB. Figure 2 shows the equidistant relay outperforms its counterparts in Figures 3 and 4   As for the moderate turbulence, shown in Figure 5, an enhancement of 17 dB between the direct link and the equidistant relay can be seen.An enhancement of 12.5 dB was also obtained between the direct link and the other relays (600 m-400 m and 400 m-600 m).

Conclusions
A novel ABER performance analysis of a dual-hop blind detection FSO system was derived in this paper over a log-normal channel.Our Monte Carlo simulation results verified the validity of the theoretical predictions.Blind detection results were close to the CSI results while using a small window of length 32.Simulation results also showed that the proposed relay technique achieves better performance as compared to the results of a direct link, without a relay.It is worth to mention that as per the simulations, the equidistant relay is the optimum relay location.Additionally, the simulation results showed that even a non-equidistant relay would have a better performance than a direct link.
is the transmitted OOK symbol and the time-varying channel state due to atmospheric turbulence and is considered to be constant over a large number of transmitted bits.

Figure 1 .
Figure 1.Synoptic diagram of the proposed blind detection free-space optical (FSO) system.

Figure 1 .
Figure 1.Synoptic diagram of the proposed blind detection free-space optical (FSO) system.

Figure 2 .
Figure 2. ABER performance for the direct link and a single equidistant relay, over weak turbulence.Figure 2. ABER performance for the direct link and a single equidistant relay, over weak turbulence.

Figure 2 . 11 Figure 3 .
Figure 2. ABER performance for the direct link and a single equidistant relay, over weak turbulence.Figure 2. ABER performance for the direct link and a single equidistant relay, over weak turbulence.Appl.Sci.2018, 8, x 8 of 11

Figure 3 .
Figure 3. ABER performance for the direct link and a single relay after 400 m from the source, over weak turbulence.

Figure 3 .
Figure 3.ABER performance for the direct link and a single relay after 400 m from the source, over weak turbulence.

Figure 4 .
Figure 4. ABER performance for the direct link and a single relay after 600 m from the source, over weak turbulence.

Figure 4 .
Figure 4. ABER performance for the direct link and a single relay after 600 m from the source, over weak turbulence.Appl.Sci.2018, 8, x 9 of 11

Figure 5 .
Figure 5. ABER performance for the direct link and a single relay with different locations, over moderate turbulence.

Figure 5 .
Figure 5. ABER performance for the direct link and a single relay with different locations, over moderate turbulence.

Table 1 .
Pros and cons of Free-space optical (FSO) blind detection papers in the literature.