# Optimal Packet Length for Free-Space Optical Communications with Average SNR Feedback Channel

Authors: Ghassan Alnwaimi; Hatem Boujemaa; Kamran Arshad
Journal: Journal of Computer Networks and Communications
 (2019)

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## Abstract

In this article, a method to enhance data rates of free-space optical (FSO) systems using packet length optimization is proposed. The average signal-to-noise ratio (ASNR) is measured at the receiver and sent back to the transmitter to optimize packet length. In addition, the length of packet is optimized to enhance the average throughput. We concluded that packet length can be reduced at low ASNR. However, packet length should be increased at higher values of received ASNR. For each ASNR, we also choose the optimal modulation and coding scheme (MCS) and optimal packet length to maximize the throughput. Different MCSs are investigated such as 4-pulse amplitude modulation (PAM) with and without channel coding, 8-PAM, 16-PAM, and 32-PAM. The proposed method gives 0.8–1.9 dB gain with respect to conventional FSO with adaptive modulation and coding (AMC) and fixed packet length. This is the first paper to deal with packet length optimization for FSO systems.


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## Body

## 1. Introduction

FSO communications allow high data rates as compared to radio frequency (RF) communications [1–5]. It is easy to install and does not require uncovering of walkways to introduce fiber joins. Moreover, FSO spectrum is license free. Therefore, FSO communications become viable due to its low implementation cost from one side and to overcome RF spectrum scarcity on the other side. However, the performance of FSO systems can be degraded due to rain, mist, tide, warm, or pointing errors [1–10]. To circumvent this drawback, we can use cooperative or spatial diversity combined with error control coding [11–17]. In cooperative FSO, relays amplify or decode the source message before transmitting it to the destination. Cooperative protocols allow us to benefit from cooperative diversity as the same information is transmitted over independent relayed channels [11–17]. Cooperative communications and channel coding techniques allow us to combat the effects of atmospheric turbulence and improve the performance of FSO systems.Adaptive modulation and coding (AMC) allows us to increase data rates in free-space optical (FSO) communications. In fact, the best modulation and coding scheme (MCS) is selected for each instantaneous or average SNR. At high SNRs, we can use 32-pulse amplitude modulation (32-PAM) to increase data rates since its spectral efficiency is 5 bit/s/Hz. At low SNR, robust on-off keying (OOK) modulation with channel coding can be used. It is well known that channel coding is required to overcome channel impairments and noise. Different MCSs can be deployed for each range of SNR: i.e.,MCSi is used if Di≤SNR<Di+1. In this paper, we will provide a methodology to derive the values of thresholds Di for fixed and optimal packet length. The thresholds for optimal packet length are denoted by Di, and thresholds for fixed packet length are denoted by Si.In all previous studies, packet length is fixed and only the MCS is varied where different-PAM can be used with different channel encoders [18–26]. The main contribution of this paper is to optimize packet length in order to maximize the system throughput. For each average signal-to-noise ratio (ASNR), we choose the optimal packet length and MCS to have the largest throughput. The proposed solution allows 0.8–1.9 dB gain with respect to conventional FSO systems with AMC and constant packet length. This is the first paper to deal with packet length optimization for FSO systems.The system model is presented in Section2. The packet error probability is analyzed in Section 3 in the absence and presence of channel coding. The optimal packet length is derived in Section 4. Adaptive packet length and MCS using ASNR is described in Section 5. Numerical results are given in Section 6. Section 7 concludes the paper.


## 2. Signal Model

In FSO communications, the received signal is written as follows:(1)yl=Γ¯gIpxl+nl,where xl is the transmitted signal, l is the symbol index, g is the atmospheric fading coefficient, Ip is the effect of pointing errors, nl is an AWGN with normalized variance, and Γ¯ is the average SNR.The average SNR can be expressed as follows:(2)Γ¯=EbN0,where Eb is the transmitted energy per bit and N0 is the power spectral density of the noise.The instantaneous photo SNR is expressed as follows:(3)γ=Γ¯g2Ip2.We denote(4)h=g2,as the atmospheric turbulence.Using a Gamma-Gamma distribution for atmospheric turbulence and in the presence of pointing errors, the probability density function (PDF) is equal to [27](5)fγx=b2xΓζΓεG1,33,0ζεxΓ¯b2,ζ,εb2+1,where G is Meijer’s G function and b is the ratio between the equivalent beam radius at the receiver and pointing error displacement standard deviation. For small pointing errors, i.e., b⟶+∞, the above distribution converges to the nonpointing error case, where ζ=2.064 and ε=1.342 correspond to strong turbulence, ζ=2.296 and ε=1.822 correspond to moderate turbulence, and ζ=2.902 and ε=2.51 correspond to weak turbulence [28, 29].The cumulative distribution (CDF) of SNR is given by the following equation [27]:(6)Fγx=b2xΓζΓεG2,43,1ζεxΓ¯b2,ζ,ε1,b2+1.


## 3. Preliminary Results

The average packet error probability (PEP) can be tightly upper bounded by the following equation [30]:(7)PEP≤∫0w0fγxdx=Fγw0,where fΓγ is the PDF of SNR Γ and w0 is a waterfall threshold.Equation (7) shows that the PEP for a given instantaneous SNR γ≤w0 is approximated by 1. However, the PEP for a given instantaneous SNR γ>w0 is approximated by 0.The waterfall threshold can be written as follows [30]:(8)w0=∫0+∞gγdγ,where gγ is the PEP for a given instantaneous SNR γ.


### 3.1. Without Channel Coding

For uncodedM-PAM, we have(9)gγ=1−1−PesγN+nd/log2M,where N is the number of data bits per packet, nd is the number of parity bits per packet, and Pes is the symbol error probability (SEP) of M-PAM [31].(10)Pesγ=M−1Merfc3γlog2M2M−12M−1,using(11)erfcx≤e−x2.Using (10) and (11), we have(12)Pes≃a1e−c1γ,where(13)a1=M−1M,c1=3log2M2M−12M−1.


### 3.2. With Channel Coding

For convolutionally coded communications, we have(14)gγ=1−1−PEγN+nd/log2M,where(15)PEγ≤∑d=df+∞adPdγ,in which df and ad are, respectively, the free distance and distance spectrum.(16)Pdγ≃M−1MerfcRcd3γlog2M2M−12M−1,where Rc is the rate of convolutional encoding.Using equation (11), we have(17)PEγ≃a2e−c2γ,where(18)a2=adfM−1M,c2=Rcdf3log2M2M−12M−1.We have(19)gγ≃1−1−aie−ciγN+nd/log2M,where i=1 for coded communications and i=2 for uncoded communications.


### 3.3. Waterfall Threshold

Appendix shows that waterfall threshold is written as(20)w0≃k1lnN+ndlog2M+k2,where(21)k1=1ci,k2=E+lnaici,where E≃0.577 is the Euler constant.


## 3.1. Without Channel Coding

For uncodedM-PAM, we have(9)gγ=1−1−PesγN+nd/log2M,where N is the number of data bits per packet, nd is the number of parity bits per packet, and Pes is the symbol error probability (SEP) of M-PAM [31].(10)Pesγ=M−1Merfc3γlog2M2M−12M−1,using(11)erfcx≤e−x2.Using (10) and (11), we have(12)Pes≃a1e−c1γ,where(13)a1=M−1M,c1=3log2M2M−12M−1.


## 3.2. With Channel Coding

For convolutionally coded communications, we have(14)gγ=1−1−PEγN+nd/log2M,where(15)PEγ≤∑d=df+∞adPdγ,in which df and ad are, respectively, the free distance and distance spectrum.(16)Pdγ≃M−1MerfcRcd3γlog2M2M−12M−1,where Rc is the rate of convolutional encoding.Using equation (11), we have(17)PEγ≃a2e−c2γ,where(18)a2=adfM−1M,c2=Rcdf3log2M2M−12M−1.We have(19)gγ≃1−1−aie−ciγN+nd/log2M,where i=1 for coded communications and i=2 for uncoded communications.


## 3.3. Waterfall Threshold

Appendix shows that waterfall threshold is written as(20)w0≃k1lnN+ndlog2M+k2,where(21)k1=1ci,k2=E+lnaici,where E≃0.577 is the Euler constant.


## 4. Optimal Packet Length Using ASNR

The number of trials of HARQ is [32–34] calculated as follows:(22)Tr=∑i=1+∞iPEPi−11−PEP=11−PEP.The throughput in bit/s/Hz is expressed as follows:(23)Thr=log2MNN+ndTsBTr=log2MNN+nd1−PEP>log2MNN+nd1−Fγw0,where B=1/Ts is the used bandwidth and Ts is the symbol period. In equation (23), we used the expression of PEP provided in equation (7).The optimal packet length maximizing the throughput can be obtained using the gradient algorithm:(24)Ni+1=Ni+μ∂ThrN=Ni∂N.Usually, the gradient search is used to minimize a functionf, and we write Ni+1=Ni−μ∂fN=Ni/∂N. Here, we aim to maximize the throughput which is equivalent to minimize f=−Thr.The derivative of throughput with respect to packet length is equal to(25)∂Thr∂N=log2MndN+nd21−Fγw0−log2MNN+ndfγw0×k1N+nd.This expression is valid for Gamma-Gamma atmospheric turbulence with pointing errors.The used packet length is the closed solution to that obtained by gradient search such thatN+nd is a multiple of log2M so that bits can be converted to N+nd/log2M M-PAM symbols.


## 5. Optimal Packet Length with AMC

Figure1 shows the average throughput of FSO communications with respect to ASNR using optimal N obtained with gradient search equation (24). From Figure 1, we can deduce the following strategy:(i)
Γ¯<D1=14.5dB: use convolutionally coded 4-PAM Rc=1/2(ii)
D1<Γ¯<D2=22.2dB: use uncoded 4-PAM(iii)
D2<Γ¯<D3=29.2dB: use uncoded 8-PAM(iv)
D3<Γ¯<D4=35.9: use uncoded 16-PAM(v)
D4<Γ¯: use uncoded 32-PAMFigure 1
Average throughput of FSO systems for optimal packet length and different modulations and coding schemes with moderate turbulence.ThresholdsDi correspond to intersection of different curves in Figure 1.As shown in Figure2, a similar approach is used when packet length is fixed, N=410 and nd=10 with thresholds S1=16.1 dB, S2=23.3dB, and S3=30dB.Figure 2
Average throughput of FSO systems for fixed packet length and different modulations and coding schemes with moderate turbulence.Figure3 shows the proposed adaptive packet length and adaptive MCS using ASNR Γ¯. The ASNR is measured at the receiver to select the optimal packet length and MCS. The ASNR is compared to thresholds D1, D2, D3, and D4 to determine the optimal MCS. Packet length is computed using the gradient algorithm as explained in equations (24) and (25) .Figure 3
Packet length optimization using ASNR.


## 6. Numerical Results

Numerical results were obtained in the presence of both atmospheric turbulence and pointing errors. Except in Figure4, the ratio b between the equivalent beam radius at receiver and pointing error displacement standard deviation is 12. Weak, moderate, and strong turbulence were considered. Figures 5 and 6 show, respectively, the average throughput of FSO communications for 4-PAM, ASNR = 15 dB (respectively, ASNR = 20 dB) with respect to packet length N. In all results, nd=10. We notice that throughput can be maximized by optimizing packet length. In fact, we notice that we can choose the value of packet length N to maximize the throughput. As shown in Figure 5, increasing packet length does not mean increasing the throughput all the time and there is a certain length that maximizes the throughput. The optimal packet length increases as the average SNR increases, as shown in Figures 5 and 6. The optimal packet length is 60 (respectively, 120) for Eb/N0=15dB (respectively, 20 dB). By comparing the results of Figures 5 and 6, we concluded that packet length should be increased as the ASNR increases.Figure 4
Effect of pointing errors.Figure 5
Average throughput of FSO systems with respect to packet length for an average SNR of 15 dB.Figure 6
Average throughput of FSO systems with respect to packet length for an average SNR of 20 dB.Figure7 compares the average throughput of 4-PAM when packet length is optimal with respect to the fixed packet length (FPL) N of 200, 400, and 1000. We notice that packet length optimization allows 0.8 dB gains with respect to N = 200 and throughput equal to 0.5 bit/s/Hz. Optimal packet length allows 1.3 dB gains with respect to N = 400 and throughput equal to 0.5 bit/s/Hz. Optimal packet length allows 1.9 dB gains with respect to N = 1000 and throughput equal to 0.5 bit/s/Hz. At high ASNR, N = 1000 allows higher throughput than N = 400 and N = 200. At low ASNR, N = 200 allows higher throughput than N = 400 and N = 1000.Figure 7
Average throughput of FSO systems for optimal and fixed packet length: 4-PAM modulation and moderate turbulence.Figure1 shows the average throughput of FSO communications with optimal packet length for different modulations and coding schemes (MCSs): 4-PAM with Rc=0.5 convolutional coding; 4-PAM, 8-PAM, 16-PAM, and 32-PAM without channel coding. The following AMC strategy is suggested:(i)
Γ¯<D1=14.5dB: use convolutionally coded 4-PAM with Rc=1/2(ii)
D1<Γ¯<D2=22.2dB: use uncoded 4-PAM(iii)
D2<Γ¯<D3=29.2dB: use uncoded 8-PAM(iv)
D3<Γ¯<D4=35.9dB: use uncoded 16-PAM(v)
D4<Γ¯: use uncoded 32-PAMThe thresholdsDi correspond to the intersection of different curves.Figure2 shows the average throughput of FSO communications with N=410 for different modulations and coding schemes (MCSs): 4-PAM with Rc=0.5 convolutional coding; 4-PAM, 8-PAM, 16-PAM, and 32-PAM without channel coding. If 4-PAM is used, the packet of length N+nd=420 is converted to 420/2 = 210 symbols. If 8-PAM is used, the packet is converted to 420/3 = 140 symbols. If 16-PAM is used, the packet is converted to 420/4 = 105 symbols. If 32-PAM is used, the packet is converted to 420/5 = 84 symbols. Therefore, packet length should be a multiple of 60.The following AMC approach is suggested:(i)
Γ¯<S1=16.1dB: use 4-PAM with convolutional coding Rc=1/2(ii)
S1<Γ¯<S2=23.3dB: use uncoded 4-PAM(iii)
S2<Γ¯<S3=30dB: use uncoded 8-PAM(iv)
S3<Γ¯<S4=36.4dB: use uncoded 16-PAM(v)
S4<Γ¯: use uncoded 32-PAM. Packet length is equal to N = 410ThresholdsSi correspond to intersection of different curves.4-PAM transmits 2 bits per symbol, whereas 5 bits are coded in one 32-PAM symbol. As the number of bits per symbol increases, the spectral efficiency increases, but the PEP also increases since the symbols are closer to each other for a fixed transmit energy per bit.Figure8 shows the throughput of FSO systems with adaptive modulation and coding (AMC) for fixed and adaptive packet length. Optimal packet length offers 1 dB gain with respect to N=410. The curve of optimal packet length with AMC is the upper bound of 5 curves of Figure 1. The curve of fixed packet length N=410 with AMC is the upper bound of 5 curves of Figure 2.Figure 8
Throughput of FSO systems with AMC using ASNR: fixed versus optimal packet length and moderate turbulence.The effects of pointing errors are investigated in Figure4 for b=3 and b=12. b is the ratio between the equivalent beam radius at the receiver and pointing error displacement standard deviation. We notice that the proposed optimal packet length allows us to increase the throughput even in the presence of pointing errors. When b is small, the pointing errors increase and the throughput decreases.We compare in Figure9 the throughput derived in our paper for N = 400 to that obtained from [27]. We used the PDF of SNR [27] to compute the PEP and throughput of 4-PAM as an integral as follows:(26)PEP=∫0+∞gxfΓxdx,where gx is the PEP for SNR equal to x, as given in equation (12).Figure 9
Throughput computation using an integral and our approach.Another contribution of the paper is to show that PEP can be deduced from CDF of SNR and yields close results to the PEP computed using the above integral. Figure9 shows that our results are very close to those of [27]. We have provided in equation (10) a tight upper bound of PEP. Therefore, it is a tight lower bound on throughput.


## 7. Conclusion

In this paper, we have optimized the throughput for FSO communications. For each average SNR (ASNR), we choose the optimal packet length and MCS to have the largest throughput. Optimal packet length was obtained using the gradient search algorithm and yields 0.8–1.9 dB gain with respect to conventional FSO. We have shown that the optimal packet length should be increased with average SNR to obtain higher throughput. We have also identified threshold to select the appropriate MCS for each average SNR and ensure higher data rates. The average SNR is compared to thresholds: when average SNR (ASNR) is less than14.5dB, 4-PAM with half-rate channel coding is used. When 14.5 dB < ASNR ≤ 22.2 dB, uncoded 4-PAM is the best choice. When 22.2 dB < ASNR ≤ 29.2 dB, uncoded 8-PAM should be used. When 29.2 dB < ASNR ≤ 35.9 dB, uncoded 16-PAM is the best choice. When 35.9 dB < ASNR, uncoded 32-PAM is the best choice. In all ranges of SNR, an optimal packet length has been used with each optimal MCS.



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