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基于孔径接收的UWOC系统模型如图 1所示。发光光源和光电探测器(photodetector, PD)分别位于发送和接收平面。发送端信号被PPM调制后驱动激光二极管(laser diode, LD)发光,高斯光束的光源尺寸定义为αs,2αs2=w02,w0是由场振幅的1/e点定义的光斑半径。光束经过各向异性海洋湍流信道到达接收端,接收端高斯透镜有效投射半径为WG,焦距为FG,高斯透镜孔径直径为D。光电检测器位于高斯透镜的焦距处,激光光源和高斯透镜之间的距离为L。假设各向异性海洋湍流仅存在发送平面(接收平面)和高斯透镜之间,以保证高斯透镜前面的闪烁指数等于在透镜的光瞳平面上的闪烁指数。发送平面高斯光束的光场分布可表示为[16]:
$ {U_0}\left( {r, L = 0} \right) = {\rm{exp}}\left[ { - {r^2}/\left( {2{\alpha _s}^2} \right) - {\rm{j}}k{r^2}/\left( {2{F_0}} \right)} \right] $
(1) 式中,r为横向方向距光束中心线的距离,${\rm{j}} = \sqrt { - 1}$,波数k=2π/λ,λ为波长,F0为曲率半径。
在水下传输介质中,海洋湍流主要由温度、盐度和密度波动引起。湍流产生的波动通过一个由湍流涡旋组成的频谱模型研究,湍流涡旋的内尺度湍流到外尺度湍流会引起透镜尺寸的连续随机变化,从而导致图 1中透镜随机影响传播光束,引起波前畸变,进而使接收强度发生波动,亦称闪烁[14]。
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海水为复杂传输介质,含有大量的盐、浮游生物、悬浮体以及溶解物质,是一个复杂的物理、化学、生物系统。因此光信号在海水中传播时会与各种成分相互作用,导致光信号的传播状态发生改变,这些海水成分最终会对光波造成吸收和散射。吸收和散射造成的能量总损失为:
$ J\left( \lambda \right) = a\left( \lambda \right) + b\left( \lambda \right) $
(2) 式中,a(λ)为吸收系数,b(λ)为散射系数。光波能量损失会随光波长λ和海水类型变化而变化[17],当波长在450nm~550nm之间时,吸收和散射效果最弱。因此,UWOC系统采用蓝绿光发送和接收数据,本文中选取波长λ=532nm的绿光,以尽量降低海水环境对光信号的衰减。由于海水环境复杂多变,并且伴有水深和季节因素的影响,目前,一般用指数分布近似表达光波在海水中的能量衰减规律[18],假设I0为初始光强,那么接收光强为:
$ I = {I_0}{\rm{exp}}\left[ { - J\left( \lambda \right)L} \right] $
(3) -
根据修正的Rytov理论,光束经过海洋湍流后的闪烁系数为[19]:
$ {\sigma _0}^2 = {\rm{exp}}\left[ {{\sigma _{{\rm{LS}}}}^2\left( D \right) + {\sigma _{{\rm{ss}}}}^2\left( D \right)} \right] - 1 $
(4) 式中,σLS2(D)和σSS2(D)分别是大尺度和小尺度对数强度方差,可由(5)式、(6)式给出[20]:
$ \begin{array}{c} {\sigma _{{\rm{LS}}}}^2\left( D \right) = 0.49{\left( {\frac{{{\mathit{\Omega }_{\rm{G}}} - {\mathit{\Lambda }_1}}}{{{\mathit{\Omega }_{\rm{G}}} + {\mathit{\Lambda }_1}}}} \right)^2}{\sigma _1}^2 \times \\ \left[ {1 + \frac{{0.4\left( {2 - {{\mathit{\bar \Theta }}_1}} \right){{\left( {{\sigma _1}/{\sigma _{\rm{R}}}} \right)}^{12/7}}}}{{\left( {{\mathit{\Omega }_{\rm{G}}} + {\mathit{\Lambda }_1}} \right){{\left( {\frac{1}{3} - \frac{1}{2}{{\mathit{\bar \Theta }}_1} + \frac{1}{5}{{\mathit{\bar \Theta }}_1}^2} \right)}^{6/7}}}}} \right.\\ {\left. {0.56\left( {1 + {{\mathit{\bar \Theta }}_1}} \right){\sigma _1}^{12/5}} \right]^{ - 7/6}} \end{array} $
(5) $ \begin{array}{C} {\sigma _{{\rm{ss}}}}^2\left( D \right) = \\ \underline {\left( {0.51{\sigma _1}^2} \right)/{{\left( {1 + 0.69{\sigma _1}^{12/5}} \right)}^{5/6}}} \\ 1 + \frac{{1.20{{\left( {{\sigma _{\rm{R}}}/{\sigma _1}} \right)}^{12/5}} + 0.83{\sigma _{\rm{R}}}^{12/5}}}{{{\mathit{\Omega }_{\rm{G}}} + {\mathit{\Lambda }_1}}} \end{array} $
(6) 式中,ΩG=16L/(kD2)为一个表征聚光镜光斑半径的无量纲参量;Λ1=Λ0/(Θ02+Λ02),Θ1=Θ0/(Θ02+Λ02),Θ1和Λ1为高斯光束在自由空间的输出参量;Λ0=L/(kαs2),Θ0=1-L/F0,Θ0和Λ0为高斯光束在自由空间的输入参量;Θ1=1-Θ1,方差σ12可以表示为[20]:
$ \begin{array}{l} {\sigma _1}^2 \approx 3.86{\sigma _{\rm{R}}}^2\left\{ {0.40{{\left[ {{{\left( {1 + 2{\mathit{\Theta }_1}} \right)}^2} + 4{\mathit{\Lambda }_1}^2} \right]}^{12/5}} \times } \right.\\ \left. {{\rm{cos}}\left[ {\frac{5}{6}{\rm{arctan}}\left( {\frac{{1 + 2{\mathit{\Theta }_1}}}{{2{\mathit{\Lambda }_1}}}} \right) - \frac{{11}}{{16}}{\mathit{\Lambda }_1}^{5/6}} \right]} \right\} \end{array} $
(7) 式中,σR2为Rytoy方差,σR2=1.23C02k7/6L11/6。其中各向异性海洋湍流下的等效结构常数C02可以表示为[21]:
$ \begin{array}{l} {C_0}^2 = \frac{{3.044{\rm{ \mathsf{ π} }} \times {{10}^{ - 8}}{\mu _x}{\mu _y}X}}{{{\omega ^2}{\varepsilon ^{1/3}}}}{k^{ - 7/6}}{L^{ - 11/6}} \times \\ {\rm{Re}}\left\{ {\int_0^L {{\rm{d}}z} \int_{ - \infty }^\infty {{\rm{d}}{k_x}} \int_{ - \infty }^\infty {{\rm{d}}{k_y}\left[ {P\left( {z, {k_x}, {k_y}} \right) \times } \right.} } \right.\\ \left. {P\left( {z, - {k_x}, - {k_y}} \right) + {{\left| {P\left( {z, {k_x}, {k_y}} \right)} \right|}^2}} \right] \times \\ {\left( {{\mu _x}^2{k_x}^2 + {\mu _y}^2{k_y}^2} \right)^{ - 11/6}}\left[ {1 + 2.35{v^{1/2}}{\varepsilon ^{ - 11/6}}} \right. \times \\ \left. {{{\left( {{\mu _x}^2{k_x}^2 + {\mu _y}^2{k_y}^2} \right)}^{1/3}}} \right]\left[ {{\omega ^2}{\rm{exp}}\left( { - {A_0}\delta } \right) + } \right.\\ \left. {\left. {{\rm{exp}}\left( { - {A_1}\delta } \right) - 2\omega {\rm{exp}}\left( { - {A_2}\delta } \right)} \right]} \right\} \end{array} $
(8) 式中,参量P(z, κx, κy)表示为:
$ \begin{array}{c} P\left( {z, {k_x}, {k_y}} \right) = \\ {\rm{j}}k{\rm{exp}}\left[ { - 0.5{{\left( {kL} \right)}^{ - 1}}{\rm{j}}z\left( {L - z} \right)\left( {{k_x}^2 + {k_y}^2} \right)} \right] \end{array} $
(9) 式中,z表示光束传播方向;κx,κy是空间频率在图 1中任意一点H处的x,y方向的分量;μx,μy是海洋湍流分别在x,y方向上的各向异性因子;Χ为均方温度耗散率,取值范围为10-10K2/s~10-4K2/s; ε为湍流动能耗散率,数值范围为10-10m2/s3~10-1m2/s3;v为动力粘度,范围为0m2/s~10-5m2/s;ω为温度和盐度对海洋湍流功率谱变化贡献的比值,无量纲,ω=0表示盐度诱导湍流,ω=-5表示温度诱导湍流;参量A0=1.863×10-2,A1=1.9×10-4,A2=9.41×10-3。(8)式中的参量δ可以表示为:
$ \begin{array}{c} \delta = 8.284v{\varepsilon ^{ - 1/3}}{\left[ {{{\left( {{\mu _x}{k_x}} \right)}^2} + {{\left( {{\mu _y}{k_y}} \right)}^2}} \right]^{2/3}} + \\ 12.978{v^{3/2}}{\varepsilon ^{ - 1/2}}\left[ {{{\left( {{\mu _x}{k_x}} \right)}^2} + {{\left( {{\mu _y}{k_y}} \right)}^2}} \right] \end{array} $
(10) -
光束在水下传播时,海洋湍流会造成光信号强度波动,影响系统性能。由实验研究表明,光信号在海洋湍流中传输时强度服从gamma-gamma统计分布模型[20],表示为:
$ \begin{array}{*{20}{l}} {f\left( {{K_{\rm{s}}}} \right) = \frac{{2{{\left( {\alpha \beta } \right)}^{\left( {\alpha + \beta } \right)/2}}}}{{\Gamma \left( \alpha \right)\Gamma \left( \beta \right){K_{\rm{s}}}}}{{\left( {\frac{{{K_{\rm{s}}}}}{{{{\bar K}_{\rm{s}}}}}} \right)}^{\left( {\alpha + \beta } \right)/2}} \times }\\ {{{\rm{K}}_{{\rm{ }}\alpha - \beta }}\left( {\sqrt[2]{{\frac{{\alpha \beta {K_{\rm{s}}}}}{{{{\bar K}_{\rm{s}}}}}}}} \right),\left( {{K_{\rm{s}}} > 0} \right)} \end{array} $
(11) 式中,α={exp[σLS2(D)]-1}-1,β={exp[σSS2(D)]-1}-1分别为大尺度和小尺度散射系数;Kα-β(·)为(α-β)阶第2类修正贝塞尔函数;Γ(·)为伽马函数;Ks为每个PPM时隙的光子计数,Ks=ηλ〈Pave〉T1/(hc),其中η为探测器的量子效率, h为普朗克常数, c为真空光速, T1为时隙持续时间,T1=(Tblog2M)/M,Tb为位持续时间,Tb=1/Rb,Rb为光信号传输速率。
图 2为M进制PPM示意图,表示具有2位符号的数据序列时域波形以及M=4时对应的PPM时域波形。PPM作为一种正交调制技术,有利于功率高效传输,在PPM中,log2M数据位的每个块被映射到M个可能的符号中的1个,每个符号由1个时隙中出现的脉冲以及(M-1)个空时隙组成,脉冲的位置表示log2M数据位的十进制值,通过出现在相应时隙中的脉冲位置对信息进行编码。〈Pave〉为光信号通过海洋湍流时,时隙持续时间内检测到的平均光功率[20-22]为:
$ \left\langle {{P_{{\rm{ave}}}}} \right\rangle = {{\rm{ \mathsf{ π} }}^3}{\left( {\lambda L} \right)^{ - 2}}/\left( {{t_1}^2{t_2}^2{t_3}^2} \right) $
(12) 式中,参量定义为: t12=0.5αs-2-0.5jkL-1+ρ0-2,t22=0.5αs-2+0.5jkL-1+ρ0-2-t1-2ρ0-4和t32=8D-2+0.25kt1-2L-2+0.25kt2-2L-2(1-t1-2ρ0-2)2;ρ0为海洋湍流中光波相干长度,ρ0=(0.546C02k2L)-3/5。
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当UWOC系统采用M进制PPM调制,经过湍流信道传输,系统BER可以表示为[20]:
$ {P_{\rm{e}}} = \int_0^\infty Q \left( {\sqrt {\mathit{\Gamma }\left( {{K_{\rm{s}}}} \right)} } \right)f\left( {{K_{\rm{s}}}} \right){\rm{d}}{K_{\rm{s}}} $
(13) 式中,Q(·)为高斯Q函数,其中$Q\left( x \right) = \left( {1/\sqrt {2{\rm{ \mathsf{ π} }}} } \right) \times \int_x^\infty {{\rm{exp}}\left( { - 0.5{y^2}} \right){\rm{d}}y = 0.5{\rm{erfc}}\left( {x/\sqrt 2 } \right), {\rm{\Gamma }}\left( {{k_{\rm{s}}}} \right)} $定义为[23]:
$ \begin{array}{l} {\rm{\Gamma }}\left( {{K_{\rm{s}}}} \right) = {\left( {Gq} \right)^2}{K_{\rm{s}}}^2\left\{ {{{\left( {Gq} \right)}^2}\left( {2 + \zeta G} \right)} \right.\left[ {{K_{\rm{s}}} + } \right.\\ {\left. {\left. {2\eta \lambda {P_{{\rm{bg}}}}{T_1}/\left( {hc} \right)} \right] + 4{k_{\rm{B}}}{T_0}{T_1}/{R_0}} \right\}^{ - 1}} \end{array} $
(14) 式中,G为APD平均增益;q为电子电荷量;ζ为APD电离因子;Pbg背景辐射功率;kB为玻尔兹曼常数;T0为接收器开尔文温度;R0为等效负载电阻。
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基于第1节中的理论分析,在本节中,将观察孔径接收时的系统性能,分析讨论海洋湍流参量,传输距离等对系统误比特率BER的影响,初始仿真参量如表 1所示,调制初始阶数M=8,背景辐射功率Pbg设置为〈Pr〉的1%。
Table 1. Numerical simulation parameters
parameters value parameters value λ 532nm v 10-4m2/s L 70m ω -1 αs 5mm η 0.4 F0 ∞ ζ 0.028 ε 10-2m2/s3 T0 300K Χ 10-6K2/s R0 50Ω Rb 2.4Gbit/s 图 3~图 6中给出了系统在不同接收孔径D和各向异性因子下BER随不同海洋湍流参量,即湍流动能耗散率ε、均方温度耗散率X、温度和盐度对海洋湍流功率谱变化贡献的比值ω和动力粘度v的变化曲线。每幅图除改变相应参量外,其它参量如表 1所示。从图 3~图 6可以看出,固定湍流参量和各向异性因子,增大孔径直径D,系统BER性能明显改善;孔径相同D相同时,随着各向异性因子增大,系统BER性能也随之改善。
Figure 3. BER vs. the rate of dissipation of kinetic energy per unit mass of fluid ε for different receive aperture diameters D and anisotropy factor
Figure 4. BER vs. the rate of dissipation of mean-squared temperature X for different receive aperture diameters D and anisotropy factor
Figure 5. BER vs. the rate of temperature to salinity contributions to the refractive index spectrum ω for different receive aperture diameters D and anisotropy factor
观察图 3可以看到,ε增大,系统误比特率BER减小。湍流参量和各向异性因子相同时,大孔径3mm接收比小孔径1mm接收BER性能更好。其次,当各向异性因子μx和μy都增大到2时,系统误比特率BER变化趋势为先上升后下降,这是因为长期的光束扩展造成的。同时,从线距可以看出,相比于其它两种各向异性因子情况,μx和μy都为2时误比特率的改善情况更加明显。
从图 4可知,随着X增大,误比特率随之增大。X较小时,即X范围在10-7K2/s~5×10-7 K2/s时,大孔径接收系统对系统BER改善非常明显。随后X增大,大孔径接收系统对系统BER改善能力逐渐减弱。各向异性因子都增大到2时,误比特率只在5×10-7K2/s~5×10-5K2/s之间存在有效值,其余两种各向异性因子条件下,在5×10-5K2/s之后无法找到有效值。
图 5中,随着ω增大,系统误比特率随之增大。ω较小时,大孔径接收系统能更好地降低系统BER。各向异性因子越大,大孔径接收系统越能降低系统BER,改善系统性能。ω增大到-0.5附近时,大孔径接收系统和更大的各向异性因子改善系统性能的能力都逐渐变弱,表明海洋湍流在由盐度波动起主导作用时,系统性能会变差,即使采用大孔径接收系统,系统BER改变不大。
图 6说明,相同各向异性因子和孔径直径D,随着v增加,系统BER减小。相同v和孔径直径D,随着μx增大,系统BER减小,但μy增大到一定程度后,系统BER不再变化。当v分别为5×10-5m2/s和1×10-4m2/s时,误比特率的差距非常微小,几乎重合,以μx=6为例,差距仅为1.375×10-4。当v=5×10-4m2/s时,大孔径接收系统改善系统BER的能力非常明显,但在μx=5以后,开始趋于平缓。
Figure 6. BER vs. the rate of dissipation of μx for different receive aperture diameters D and the kinetic viscosity v
图 7显示了系统误比特率随传输距离L变化的曲线,可以看到,传输距离越远,系统误比特率越大,因为随着距离增大,湍流对传输链路的干扰越强。相同各向异性因子下,大孔径接收系统的系统BER性能更好。孔径相同,各向异性因子越大,系统性能越好。但是,随着传输距离进一步增加,即L>70m后,系统BER持续增大,孔径和各向异性因子的变化无法有效改善系统性能。
Figure 7. BER vs. the rate of transmission distance L for different receive aperture diameters D and anisotropy factor
图 8中给出了不同孔径直径D和各向异性因子下BER随APD增益G的变化曲线。可以看出,随着APD增益增大,系统误比特率先下降,之后基本趋于平缓或者出现增加的趋势,大孔径接收系统较好的改善了系统性能。当各向异性因子都为2时,无论是大孔径(3mm)还是小孔径(1mm),系统误比特率都先减小后增大,小孔径和大孔径分别在增益为150和100时,误比特率最小。而其它两种各向异性因子条件下,当增益大于150时,系统BER没有明显的增减趋势。
Figure 8. BER vs. the rate of APD gain G for different receive aperture diameters D and anisotropy factor
图 9中仿真了不同孔径直径D和各向异性因子下BER随调制阶数M的变化曲线。可以看出,调制阶数越高,系统误比特率越大。从星座图角度理解,调制阶数越高,星座点越来越密,星座点的距离代表了译码的差错概率,判决时容易被判定为其它符号,导致系统BER增大。相同调制阶数和各向异性因子下,大孔径接收系统可以有效降低系统BER,调制阶数越小,BER改善越明显。相同孔径直径D和调制阶数下,各向异性因子越大,系统BER性能越好,大孔径接收系统改善系统BER的能力越强。但当调制阶数M > 64时,系统误比特率变化程度逐渐饱和,改变调制阶数无法降低系统BER。
孔径接收下各向异性海洋湍流UWOC系统误码分析
Bit error rate analysis of anisotropic ocean turbulence UWOC system with aperture reception
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摘要: 为了研究孔径接收对各向异性海洋湍流条件下水下无线光通信(UWOC)系统误比特率的影响, 系统采用高斯光束传输, 接收端通过孔径接收, 在脉冲位置调制方式下通过各向异性海洋湍流信道。引入各向异性海洋湍流结构常数, 通过对闪烁的形成原理和各向异性海洋湍流条件下闪烁系数的分析, 数值模拟得到了在不同接收孔径和各向异性因子下, 海洋湍流参量、传输距离、雪崩光电二极管(APD)平均增益和调制阶数对系统误比特率的影响。结果表明, 相同各向异性因子和海洋湍流参量下, 大孔径接收能有效提升系统误比特率性能; 相同孔径直径和海洋湍流参量下, 各向异性因子越大, 系统通信性能越好; 均方温度耗散率、温度和盐度对海洋功率谱变化贡献的比值较小, 湍流动能耗散率、动力粘度较大以及传输距离越短, 系统误码性能越好; APD增益为100或150时, 系统通信性能最佳; 调制阶数M=8时, 系统通信性能最佳, M>64时, 系统误比特率变化程度几乎饱和。该研究为UWOC系统平台搭建和性能估计提供了参考。Abstract: In order to study the effect of aperture reception on the bit error rate of underwater wireless optical communication (UWOC) system under the condition of anisotropic ocean turbulence, Gaussian beam transmission was adopted to pass through an anisotropic ocean turbulence channel under pulse position modulation.The signal was then received though the apertureby the acceptor, the effects of ocean turbulence parameters, transmission distance, average gain of avalanche photodiode (APD), and modulation order on the bit error rate of the system under different receiving aperture and anisotropy factor were numerically simulatedby introducing the anisotropic ocean turbulence structure constant and analyzing the formation principle of scintillation and scintillation index under the condition of anisotropic ocean turbulence. The results show that under the same anisotropy factor and ocean turbulence parameters, the bit error rate performance of the system can be effectively improved by large aperture receiver. The larger the anisotropy factor, the better the system communication performance for the same aperture diameter and ocean turbulence parameters. The bit error rate performance of the system becomes better when the ratios of mean square temperature dissipation rate, temperature, and salinity contribution to the variation of ocean power spectrum are smaller, while the turbulent kinetic energy dissipation rate and kinetic viscosity are larger and the transmission distance is shorter. The performance of system communication is optimal when the APD gain is 100 or 150. The optimal system communication performance is achieved when the modulation order M=8, and the degree of system bit error rate variation is almost saturated when M>64. This study provides a reference for UWOC system platform construction and performance estimation.
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Table 1. Numerical simulation parameters
parameters value parameters value λ 532nm v 10-4m2/s L 70m ω -1 αs 5mm η 0.4 F0 ∞ ζ 0.028 ε 10-2m2/s3 T0 300K Χ 10-6K2/s R0 50Ω Rb 2.4Gbit/s -
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