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不同于大气环境,水下无线光通信的信道干扰因素更为复杂,例如叶绿体、悬浮体所带来的干扰,而水介质对光的吸收及散射也会给光信号带来一定程度的衰减[7],结合水下环境给光信号带来的各类影响因素,建立信道激光光源模型。UOWC信道激光光源模型如图 1所示。
信道函数如下式所示:
$ \begin{gathered} P_{\mathrm{r}}=P_{\mathrm{t}} \times \eta_{\mathrm{t}} \times \eta_{\mathrm{r}} \times \frac{a_{\mathrm{r}}{}^{2}}{\left(d \tan \theta+a_{\mathrm{t}}\right)^{2}} \times \\ \exp [-c(\lambda, h, D) \times d] \end{gathered} $
(1) 式中,Pt和Pr分别为信号的传输光功率和接收光功率,ηt和ηr分别表示信号传输及接收的效率,at与ar分别为传输和接收天线的孔径,d是传输距离,c(λ, h, D)是水下衰减系数,λ为所选用激光波长, h为叶绿素的密度,D为悬浮粒子浓度,θ为激光光源发散角。
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正交频分复用调制是一种多载波调制技术,首先将串行的数据流进行相移键控(phase shift keying, PSK)或者正交振幅调制(quadrature amplitude modulation, QAM)映射,接着对映射后数据进行并行处理,之后对其进行逆傅里叶变换(inverse Fourier transform, IFFT)运算,即得到调制后的OFDM信号[8]。
设Xi[k]是位于第k个子载波的第i个符号(i=0, 1, …, ∞; k=0, 1, …,N-1)。连续时域OFDM信号如下式所示:
$ x_{i}(t)=\sum\limits_{i=0}^{\infty} \sum\limits_{k=0}^{N-1} X_{i}[k] \exp \left[{\rm{j}} 2 {\rm{ \mathsf{ π} }} f_{k}(t-i T)\right] $
(2) 式中, T表示每个OFDM符号的传输周期,fk表示第k个子载波的频率,与(2)式对应的离散时域OFDM信号如下式所示:
$ x_{i}[n]=\sum\limits_{k=0}^{N-1} X_{i}[k] \exp \left(\frac{\mathrm{j} 2 {\rm{ \mathsf{ π} }} k n}{N}\right) $
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在OFDM系统中,多载波调制后各个子载波发生叠加,会产生较大的峰值,高峰值信号的输入会导致功率放大器的非线性所引起的带内失真以及带外非线性辐射[9],从而致使系统无法进行远距离低误比特率的传输,因此PAPR抑制是OFDM系统的重要技术,PAPR表达式如下所示:
$ r_{\mathrm{PAPR}}(x(t))=\frac{\max |x(t)|^{2}}{E|x(t)|^{2}} $
(4) 式中,E为平均功率。
通常采用互补累积分布函数(complementary cumulative distribution function, CCDF)衡量PAPR,表示如下:
$ P_{\mathrm{b}}=\left[P_{\mathrm{b}}\left(Z_{\max }>Z\right)\right]>1-\left(1-\mathrm{e}^{-Z^{2}}\right)^{N} $
(5) 式中,Pb表示Zmax>Z的概率; Z表示复数采样信号的幅值; Zmax表示信号通频带的波峰因数(crest factor, CF),其数值为PAPR的平方根,即(5)式表示了CF超过Z的概率。
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子载波预留法将载波划分为数据载波X和峰值抑制载波S两个部分[10],在此设B为峰值抑制信号所保留的子载波数的集合,Bc表示其补集,分配给数据信号作为载波。其思路是运用S[k]将高峰值信号抵消掉,从而达到抑制PAPR的目的。该方法的频域表达如下式所示:
$ X[k]+S[k]=\left\{\begin{array}{l} X[k],(k \in B) \\ S[k],\left(k \in B^{c}\right) \end{array}\right. $
(6) 时域上,常规TR法的PAPR为:
$ r_{\text {PAPR }}\left(x_{n}\right)=\frac{\max \limits_{0 \leqslant n \leqslant N-1}\left|x_{n}+s_{n}\right|^{2}}{E\left(\left|x_{n}\right|^{2}\right)} $
(7) 式中,N为子载波数,xn为时域数据信号,sn为时域峰值抵消信号。
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TR法需要找到削峰因子p,使峰值抑制信号sn与p相乘后尽可能逼近信号限幅噪声f,从而降低发送信号的峰值,达到峰均比抑制的目的。TR-LSA的时域迭代公式如下式所示:
$ x_{n}{}^{i+1}=x_{n}{}^{i}-p \cdot s_{n} $
(8) 首先,通过下式软限幅找到所需要的第n个限幅噪声:
$ f_{n}=\left\{\begin{array}{l} 0,\left(\left|x_{n}\right| \leqslant R\right) \\ x_{n}-R\mathrm{e}^{\mathrm{j} \theta_{n}},\left(\left|x_{n}\right|>R\right) \end{array}\right. $
(9) 式中,R为限幅阈值,θn表示第n个子载波符号的相位,当|xn|>R时,可滤掉一定数量的高峰值,为了保证信号的峰值尽可能地减小,因此运用一定的限幅比对阈值R进行迭代限幅,令R=2.17x,其中x表示x的均值。
限幅后信号如下所示:
$ x_{n}{ }^{i+1}=x_{n}{ }^{i}-f_{n} $
(10) 为了使(8)式与(10)式进一步的逼近,因此选用LSA对其进行处理,所构造优化函数T(p)如下所示:
$ T(p)=\min \left(\sum\limits_{n \in P}\left[p \cdot\left|s_{n}\right|-\left|f_{n}\right|\right]^{2}\right) $
(11) 式中,P为第n个信号幅度大于R的集合。通过上式对p取偏导运算,并令其等于0,即:
$ \begin{gathered} \frac{\partial T(p)}{\partial p}= \\ 2\left(p \cdot \sum\limits_{n \in P}\left|s_{n}\right|^{2}-\sum\limits_{n \in P}\left|s_{n}{ }^{i}\right| \cdot\left|f_{n}\right|\right)=0 \end{gathered} $
(12) 所求p的结果如下式所示:
$ p=\frac{\sum\limits_{n \in P}\left|s_{n}\right| \cdot\left|f_{n}\right|}{\sum\limits_{n \in P}\left|s_{n}{}^{i}\right|^{2}} $
(13) 将(13)式代入(8)式,得到TR-LSA结果,时域表达式如下式所示:
$ \begin{gathered} x_{n}{ }^{i+1}=x_{n}{ }^{i}-\frac{\sum\limits_{n \in P}\left|s_{n}\right| \cdot\left|f_{n}\right|}{\sum\limits_{n \in P}\left|s_{n}\right|^{2}} \cdot s_{n}= \\ x_{n}{ }^{i}-\frac{\sum\limits_{n \in P}\left|s_{n}\right| \cdot\left(x_{n}{ }^{i}-R\mathrm{e}^{\mathrm{j} \theta_{n}}\right)}{\sum\limits_{n \in P}\left|s_{n}\right|^{2}} \cdot s_{n} \end{gathered} $
(14) 上述公式的频域表达式如下式所示:
$ \begin{gathered} X_{n}{ }^{i+1}=X_{n}{ }^{i}-\left(p \cdot s_{n}\right) \cdot \boldsymbol{G}(n)= \\ {\left[x_{n}{ }^{i}-\frac{\sum\limits_{n \in P}\left|s_{n}\right| \cdot\left(x_{n}{ }^{i}-R\mathrm{e}^{\mathrm{j} \theta_{n}{ }^{i}}\right)}{\sum\limits_{n \in P}\left|s_{n}\right|^{2}} \cdot s_{n}\right] \cdot \boldsymbol{G}(n)} \end{gathered} $
(15) 式中,X为时域信号x的频域形式,G(n)为傅里叶变换系数矩阵。
A律压缩可以有效降低OFDM信号的PAPR[5],压缩曲线如图 2所示。输入输出呈一一映射的关系,弱放大高幅值信号,强放大低幅值信号,使信号的峰值与均值差距缩小,从而使OFDM信号的PAPR进一步地降低。A律压缩函数如下式所示:
$ A\left(X_{n}\right)=\left\{\begin{array}{l} \operatorname{sgn}\left|X_{n}\right| \cdot \frac{1+\ln \left(A\left|X_{n}\right|\right)}{1+\ln A},\left(\frac{1}{A} \leqslant\left|X_{n}\right| \leqslant 1\right) \\ \operatorname{sgn}\left|X_{n}\right| \cdot \frac{A\left|X_{n}\right|}{1+\ln A},\left(0 \leqslant\left|X_{n}\right| \leqslant \frac{1}{A}\right) \end{array}\right. $
(16) 图 3所示为不同处理阶段的信号极坐标散点图。原始OFDM信号散点分布均匀程度较差,在边缘处散点分布稀疏,表现出其具有较多的高峰值信号,相比之下,经过TR-LSA处理之后,信号峰值得到了明显的抑制,而后进一步通过A律压缩,信号散点边缘密度稀疏程度变小,并且散点分布更为均匀,因此信号经过TR-LSA-A处理后表现出更加优异的PAPR性能。
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设置1024个子载波和1000个OFDM符号,其它参量设置如表 1所示。
Table 1. System simulation parameters
parameter value parameter value ηt 0.91 λ 514nm ηr 0.91 θ 0.33rad D 1 neural network learning rate 0.01 h 5mg·m-3 genetic iterations 20 ar 0.003m nodes in single hidden layer 6 at 0.003m the population size 100 c 0.151 图 7显示了SLM-C、μ律压缩、A律压缩以及TR-LSA-A法的PAPR性能。通过仿真结果可以看出:TR-LSA-A比参量μ=2的SML-C、A律压缩和μ律压缩具有更加优异的PAPR抑制性能,能够将PAPR分别降低2.5dB, 4.2dB和4.9dB,其互补累计分布函数取10-3时,PAPR为2dB。
如图 8所示,TR-LSA-A算法在比特信噪比(即比特能量Eb与噪声功率密度N0之比)等于10dB时误比特率(bit error rate, BER)低于10-3, 即满足UOWC系统的误比特率要求。TR-LSA-A算法的误比特率性能优于对比算法。
如图 9所示,在水下强衰减信道中,未加均衡器的DCO-OFDM系统的采用TR-LSA-A算法抑制PAPR时,误比特率保持在10-1~10-2的范围之间;当信噪比大于7.4dB时,其性能劣势开始显现,无法达到水下无线光通信的误比特率要求。由此可见,在接收端加入均衡器对于UOWC系统通信性能的提升是十分关键的。
图 10a为发送端经QPSK映射后的原始数据星座图。该发送信号在UOWC信道中传输时,受水下信道严重衰减的影响,接收端星座图发生很大程度的扰乱。如图 10b所示,未加入均衡器的接收端信号星座图出现较强的噪声干扰,解调模块误判概率增加;加入均衡后的接收信号依然具有噪声干扰,如图 10c所示,但其干扰程度得到了较大程度的抑制,解调模块误比特率可被大幅降低。
原始OFDM信号的功率谱密度(power spectral density, PSD)主要集中在中央平台区。图 11为TR-LSA-A算法的PSD与原始OFDM信号、A律算法、μ律算法的对比图。与原始OFDM系统相比,TR-LSA-A算法在大幅降低PAPR的前提下,与其它对比算法带来的带外频谱扩散接近。
一种改进的OFDM水下可见光无线通信系统
An improved scheme of OFDM underwater visible wireless optical communication system
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摘要: 在水下直流偏置光无线通信正交频分复用(UOWC-DCO-OFDM)系统中, 为了保证光信号在发射端具有较低峰均比(PAPR)并在水下可进行远距离低误比特率传输,采用了子载波预留、最小二乘算法(TR-LSA)与压扩变换相结合的方法,同时运用优化的神经网络对水下环境进行信道估计,并基于该方法在接收端设计信道均衡器,以应对水下环境对光信号的强衰减。结果表明,UOWC-DCO-OFDM系统的PAPR降低9dB,且在信噪比为10dB时误比特率低于10-3,达到水下无线光通信的误比特率标准。该系统可实现光信号水下远距离、低误比特率传输。Abstract: In the underwater optical wireless communication-direct current bias-optical-orthogonal frequency division multiplexing (UOWC-DCO-OFDM) system, in order to ensure that the optical signal has a low peak-to-average power ratio (PAPR) at the transmitting end and can carry out long-distance low underwater, the method combining the tone reservation-least squares algorithm (TR-LSA) and companding transformation was adopted. At the same time, the optimized neural network was used to estimate the channel of the underwater environment. Based on this method, the channel equalizer was designed at the receiving end to deal with the strong attenuation of the optical signal in the underwater environment. The results show that the PAPR of the UOWC-DCO-OFDM system is reduced by 9dB, and the bit error rate is 10-3 lower when the signal-to-noise ratio is 10dB, which is under the bit error rate standard of wireless optical communication. The system can realize long-distance underwater transmission of optical signals with low bit error rate.
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Table 1. System simulation parameters
parameter value parameter value ηt 0.91 λ 514nm ηr 0.91 θ 0.33rad D 1 neural network learning rate 0.01 h 5mg·m-3 genetic iterations 20 ar 0.003m nodes in single hidden layer 6 at 0.003m the population size 100 c 0.151 -
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